EPA 560/5-^5-015
                                               September 1990
     METHODS FOR ASSESSING EXPOSURE
          TO CHEMICAL SUBSTANCES
                  Volume 11
 Methodology for Estimating the Migration of Additives
       and Impurities from Polymeric Materials
                      by
               Arthur D. Schwope
                     and
               Rosemary Goydan
              Arthur D. Little, Inc.
             Cambridge, MA 02140

                     and

                 Robert C. Reid
        Massachusetts Institute of Technology
             Cambridge, MA 02139
          EPA Contract No. 68-D9-0166
                 Project Officer

                Thomas Murray
           Exposure Evaluation Division
            Office of Toxic Substances
             Washington, D.C. 20460
  U.S. ENVIRONMENTAL PROTECTION AGENCY
OFFICE OF PESTICIDES AND TOXIC SUBSTANCES
           WASHINGTON, D.C.  20460

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                                   DISCLAIMER
      This document has been reviewed and approved for publication by the Office of Toxic
Substances, Office of Pesticides and Toxic Substances, U.S. Environmental Protection
Agency.  The use of trade names or commercial products does not constitute Agency
endorsement or recommendation for use.
                                        11

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                                    FOREWORD

       This document is one of a series of volumes, developed for the U.S. Environmental
Protection Agency (EPA), Office of Toxic Substances (OTS), that provides methods and
information useful for assessing exposure to chemical  substances.  The methods described in
these volumes have been identified by EPA-OTS as having utility in exposure assessments on
existing and new chemicals in the OTS program. These methods are not necessarily the only
methods used by OTS, because the state-of-the-art  in exposure assessment is changing
rapidly, as is the availability of methods and tools.  There is no single correct approach to
performing an exposure assessment, and the methods in these volumes are accordingly
discussed only as options to be considered, rather than as rigid procedures.

       Perhaps more important than the optional methods presented in these volumes is the
general information catalogued. These documents  contain a great deal of non-chemical-
specific data which can be used for many types of exposure assessments. This information is
presented along with the methods in individual volumes and appendices.  As a set, these
volumes should be thought of as a catalog of information useful in exposure assessment, and
not as a "how-to" cookbook on the subject.

       The definition, background, and discussion  of planning exposure assessments are
discussed in the introductory volume of the series (Volume  1). Each subsequent volume
addresses only one general exposure setting.  Consult Volume 1  for guidance on the proper
use and interrelations of the various volumes and on the planning and integration of an  entire
assessment

       The tide of the nine basic volumes are as follows:

       Volume  1    Methods for Assessing Exposure to Chemical Substances
                   (EPA 560/5-85-001)

       Volume 2    Methods for Assessing Exposure to Chemical Substances in the
                   Ambient Environment (EPA  560/5-85-002)

       Volume 3    Methods for Assessing Exposure from Disposal of Chemical Substances
                   (EPA 560/5-85-003)

       Volume 4    Methods for Enumerating and Characterizing Populations Exposed to
                   Chemical Substances (EPA 560/5-85-004)

       Volume 5    Methods for Assessing Exposure to Chemical Substances in Drinking
                   Water (EPA 560/5-85-005)

       Volume 6    Methods for Assessing Occupational Exposure to Chemical Substances
                    (EPA 560/5-85-006)

       Volume 7    Methods for Assessing Consumer Exposure to Chemical Substances
                    (EPA 560/5-85-007)

                                          iii

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      Volume 8    Methods for Assessing Environmental Pathways of Food Contamination
                   (EPA 560/5-85-008)

      Volume 9    Methods for Assessing Exposure to Chemical Substances Resulting from
                   Transportation-Related Spills (EPA 560/5-85-009)

      Because exposure assessment is a rapidly developing field, its methods and analytical
tools are quite dynamic.  EPA-OTS intends to issue periodic supplements for Volumes 2
through 9 to describe significant improvements and updates for the existing information, as
well as adding short monographs to the series on specific areas of interest. The  first of these
monographs are as follows:

      Volume 11   Methodology for Estimating the Migration of Additives and Impurities
                   from Polymeric Materials (EPA 560/5-85-015)

      Volume 13   Methods for Estimating Retention of Liquids on Hands
                   (EPA 560/5-85-017)
                                             Thomas Murray, Chief
                                             Exposure Assessment Branch
                                             Exposure Evaluation Division (TS-798)
                                             Office of Toxic Substances
                                         IV

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                              TABLE OF CONTENTS

                                                                          Page No.

1.    SUMMARY 	      1

     1.1  Background	      1
     1.2  Scope  	      1
     1.3  Conclusions and Recommendations	      2

2.    FACTORS TFIAT AFFECT MIGRATION	      4

     2.1  Overview	      4
     2.2  Polymer	      4

         2.2.1   Glass Transition Temperature	      4
         2.2.2   Crystallinity 	      5
         2.2.3   Crosslinking	      5
         2.2.4   Branching	      5
         2.2.5   Molecular Weight	      5
         2.2.6   Plasticization	      8
         2.2.7   Degradation	      8
         2.2.8   Summary	......      8

     2.3  Migrant	8
     2.4  Migrant Diffusion Coefficient  	      9
     2.5  Migrant Concentration  Factors	 • • •     11
     2.6  External Phase	     11

         2.6.1   Physical State	     11
         2.6.2   Agitation  	11
         2.6.3   Partition Coefficient	     12
         2.6.4   Migrant Capacity in the External Phase	     12
         2.6.5   Diffusion Coefficient in the External Phase	     13
         2.6.6   Degradation 	     13
         2.6.7   Surfactants	     13
         2.6.8   Penetration	     14

     2.7  Temperature	     14
     2.8  Time	     14
     2.9  Overall Summary	     15

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                         TABLE OF CONTENTS (continued)

                                                                          Page No.

3.    MODELS FOR MIGRATION ESTIMATION ........................     16

     3.1  Introduction  ............................................     16
     3.2  Assumptions  ...........................................     16
     3.3  General Rate Concepts in Migration  ...........................     16

         3.3.1   The Polymer Film ..................................     16
         3.3.2   Migration into a Fluid External Phase ....................     18

                3.3.2.1   Partitioning  ................................     18
                3.3.2.2   Variation of the Additive Concentration in the
                          External Phase ....... ......................     18

         3.3.3   Migration into a Solid External Phase ....................     19

     3.4  General Mathematical Models  .............................. .     19

         3.4.1   Fluid External Phase ................................     19
         3.4.2   Fluid External Phase with a Large Mass
                Transfer Coefficient ........................... . .....    21
         3.4.3   Solid External Phase .................. ..... .........    21
         3.4.4   Summary  ....................... . ................    22

     3.5  Significance of the Dimensionless Groups .......... ........... . .    24

         3.5.1   -c  = Dpt/L2  ...................................... .    24
         3.5.2   a = KVe/A .......................................    24
         3.5.3   y  = kKL/D   ....... . ..............................    26
         3.5.4     = KOD1'2 . ..................................    27
    3.6  Illustrative Examples ......................................    27

         3.6.1   Fluid External Phase ................................    27
         3.6.2   Solid External Phase ............................ ....    27
                                        VI

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                         TABLE OF CONTENTS (continued)

                                                                            Page No.

4.    ESTIMATION OF VARIABLES	    31

     4.1   Introduction	    31
     4.2   Diffusion Coefficients of Additives in
          Polymers (Dp)  	    31
     4.3   Diffusion Coefficients of Additives in the
          External Phase (De)  	    38

          4.3.1    Air (Da)	'.	    38
          4.3.2    Water (Dw)	    41
          4.3.3    Other Materials	    45

     4.4   Partition Coefficient (K) 	    45

          4.4.1    Solubility of Additive in the Polymer (Cps) 	    45
          4.4.2    Solubility of Additives in the External
                 Phase (Ce s)  	    46

                 4.4.2.1   Air (Ca)  	    46
                 4.4.2.2   Water (Cw)  	    46

     4.5   Mass  Transfer Coefficients (k) for a Fluid External Phase	    47

          4.5.1    Migration to Water	    47

                 4.5.1.1   Hat Polymer Surfaces	    47
                 4.5.1.2   Polymer Pipes	    49

          4.5.2    Migration to Indoor Air	    51

                 4.5.2.1   Bulk Air Over Horizontal Polymer Surface	    51
                 4.5.2.2   Vertical Polymer Surfaces	  .    52
                 4.5.2.3   Thermally Driven Convection 	    54
                 4.5.2.4   Discussion	    56
                                        vn

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                        TABLE OF CONTENTS (continued)

                                                                       Page No.

5.   EXAMPLE MIGRATION CALCULATIONS  	    57

     5.1  Worst Case Examples	    57

         5.1.1   Background 	    57
         5.1.2   Initial Considerations	    57
         5.1.3   Estimations of Migration  	    58

                5.1.3.1   No Partitioning, No External" Mass
                        Transfer Resistance 	    58
                5.1.3.2   Two-Sided Loss	    58
                5.1.3.3   Air Row Through Room  	   59
                5.1.3.4   Diffusion Coefficient Variation  	    59

     5.2  Partition Limited Examples	    60

         5.2.1   Background	    60
         5.2.2   Initial Considerations	    60
         5.2.3   Estimation of Migration	    60

     5.3  Mass Transfer Examples  	    61

         5.3.1   Background 	    61
         5.3.2   Stagnant Water	    62
         5.3.3   Flowing Water	    64
         5.3.4   External Mass Transfer Resistance But
                No Partitioning	    64
         5.3.5   External Mass Transfer Resistance with Partitioning  	    66

6.   COMPUTER PROGRAM	    68

7.   NOMENCLATURE	    71

8.   REFERENCES  	    73

APPENDIX A - FORTRAN Code for Arthur D. Little Migration Estimation Model
         Computer Program	    76

APPENDIX B - AMEM Evaluation	   125
                                      vm

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Table 1.
                                LIST OF TABLES
Approximate Glass Transition Temperatures (T ) for Selected
                                        6
                                                                         Page No.

Table 2.
Table 3.

Table 4.
Table 5.
Table 6.
Table 7.

Table 8.
Table 9.
Table 10.
Table 11.
Table 12.
Polymers 	
Migration of BHT from LDPE and HDPE 	
Diffusion Coefficients in Natural Rubber as a Function of Molar
Volume at 40°C 	
Migration Model Equations 	 	
Variables for which Estimation Techniques are Required 	
Diffusion Coefficients for Selected Polymers at 25°C 	
Rank Ordering of Polymer Groups from High to Low Diffusion
Coefficients 	
Diffusion Coefficients for Selected Organic Chemicals in Air 	
Atomic Contributions to Estimate Vm In Eq. (4-1) 	
Diffusion Coefficients in Aqueous Solutions at Infinite Dilution ....
Atomic Contributions to Estimate Vm in Eq. (4-2) 	
Summary of Computer Program Input Requirements 	
6
7

10
23
32
36

37
39
40
42
43
69
                                        IX

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                                  LIST OF FIGURES

                                                                             Page No.

Figure 1.    Flow Chart for Migration Model Equation Selection  	    25

Figure 2.    Migration Estimates for the Fluid External Phase Cases Where
            the Mass Transfer Coefficient is Unlimited.  Eq. (3-23) Applies
            When a -» °o, All Other Curves from Eq. (3-26)	    28

Figure 3.    Migration Estimates for the Fluid External Phase Cases with a
            Finite Mass Transfer Coefficient, and a = 0.1	    29

Figure 4.    Migration Estimates for the Solid External Phase Cases	    30

Figure 5.    Diffusion Coefficients in Six Polymers as a Function of
            Molecular Weight of Diffusant, T = 25°C	    33

Figure 6.    Diffusion Coefficients for Four Polymer Groups Described
            in Table 7, T = 25°C	    35

Figure 7.    Diffusion Coefficients for Organic and Inorganic Chemicals in
            Water at 20°C	 44

Figure 8.    Estimation of Mass Transfer Coefficient for Water Flowing Over
            Flat Polymer Surfaces	    50

Figure 9.    Estimated Mass Transfer Coefficients for Migrants Into Air Due
            To Bulk How (300K, 1 Bar, Laminar Flow)	    53

Figure 10.   Estimated Mass Transfer Coefficients for Migrants In Air On
            Vertical Surfaces (300K, 1 Bar, Laminar Flow)	    55

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 1.     SUMMARY

                               — NOTE TO USERS —

       This report contains extensive documentation concerning the migration of additives
 from polymers to the environment.  However, for those users who only wish to use the
 computer to predict migration, it is recommended they proceed to Section 6 for instructions
 dealing with the computer program, as well as a list of the input data required for
 implementation.  Should one then need to estimate any of the input parameters, references are
 given to direct the user to the appropriate section.

 1.1    Background

       Plastic and elastomeric products are used in virtually all segments of the U.S.
 economy. These products are based on polymers  that are high molecular weight chains of
 low molecular weight monomers such as vinyl chloride, ethylene, styrene, and so forth.
 Although polymerization  and subsequent purification processes  have been greatly improved,
 in some cases the polymer product may contain impurities  such as catalyst residues, unreacted
 monomer, and relatively low molecular weight polymer molecules (i.e., oligomers).
 Furthermore, most polymers are compounded with a variety of relatively  low molecular
 weight chemicals to yield products with useful physical properties and service lives.
 Examples include plasticizers, UV and thermal stabilizers, antioxidants and biocides; other
 additives may be used to  facilitate processing the plastic compound into its final form.  New
 technology is under development to eliminate or minimize  additive migration, and processing
 stages can sometimes be added to remove impurities. Performance enhancing additives can
 often be covalently bonded to the polymer to provide a non-migrating performance additive.

       It is well-documented, however, that additives and monomer residues can migrate from
 the plastic or elastomer over time.  The rate  and extent of migration is dependent on many
 factors, such as temperature, the compatibility of the migrant with the polymer,  the molecular
 size of the migrant, the compatibility of the migrant with the phase external to the polymer,
 and the interactions that may occur between  the external phase and the polymer.

       The Exposure Evaluation Division (EED) of the Office of Toxic Substances is
 frequently required to assess the potential for exposure to chemicals that are used as additives
 in polymeric materials or are the monomers or low molecular weight oligomers contained in
polymers. Historically  this task has been difficult because, (1) the chemicals of concern are
 new  or complex  molecules for which there are no migration data in the literature, and (2) a
consistent model with which to make preliminary  estimates of migration has not been
 available.

 1.2    Scope

       This task was  undertaken with the objective of developing and documenting a
defensible approach to assess the potential for release of chemical additives and reaction
residues from polymeric materials.  (Throughout this report, additives and reaction residues

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such as monomers, oligomers, catalysts, etc., are collectively referred to as "additives" or
"migrants.")

       A family of mathematical models was developed and/or adapted for describing the
migration of additives to gases, liquids and solids in contact with polymeric materials.
Emphasis was placed on air and water as external phases.  The models are based on diffusion
and convection mass transfer theories  and have been organized to treat migration to fluids
(air, water) or to solids.  With the family of models, the user has an option to develop worst
case (total loss of additive) scenarios,  to allow partitioning to occur between the polymer and
the external phase, or to consider external mass  transfer resistances  and their effect on rates of
loss.  The more complex models require more input data.

       In all cases, the user must specify the physical situation (external phase, one- or two-
sided extraction, and polymer thickness), as well as an estimate of the diffusion coefficient of
the additive within the polymer.  Should partitioning effects be of interest, data or estimates
of the additive  solubilities in the polymer and external phase must be provided, as well as the
volume of the external phase and the polymer surface area for migration.  When external
mass transfer resistances are introduced, data must be given to define flow velocities and
physical properties of the external phase.

       Background material addressing these factors, as well as others  which may influence
migration, are presented in Section 2.  These discussions provide a basis for the models which
are developed in Section 3.  As noted above, the models require various data inputs in order
to yield migration estimations. It is highly unlikely that such data would be available at the
time of a premanufacture notification review.  For systems for which the necessary data are
incomplete, methods, figures and/or tabulations have been provided in Section 4 to
approximate missing values.

       Example calculations using  the various models are provided  in Section 5. Finally, the
computer program developed to implement these models is described in Section 6.  The
FORTRAN code and flow diagram for the AMEM computer model is provided in
Appendix A. Appendix B provides a  limited validation of the AMEM computer program by
comparing AMEM predictions with migration data from the literature for 13 example cases.

1.3    Conclusions and Recommendations

       Mathematical models have been developed to estimate the migration of additives and
impurities from polymeric materials to air, water, and solids. A thorough validation of the
models, however, has not been conducted. We recommend, therefore,  that the models be
tested against the considerable amounts and variety of data in the literature.  The validation
process would serve to establish practical limitations for the models that may not be
addressed by the model  assumptions.  Furthermore, the process is likely to identify key areas
for experimental study that might include frequently encountered migration scenarios for
which there are little or  no data.

       The models require the input of several physical properties of the migrant, the
polymer, and the external phase. Where values  for these inputs are not available, they must

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be estimated.  For those parameters that historically have been of interest to a large
community of scientists and engineers, techniques for estimating the inputs are fairly well
developed and substantiated.  These would include solubilities and diffusion coefficients in
water and air. Tabulations of, and methods for estimating, partition coefficients and
solubilities of additives in polymers are lacking.  Since values for these parameters are of
critical importance to several of the models presented herein, we recommend that further
investigation of these parameters be conducted.  The first step would be a focus sed review of
the literature.

       The methodology developed in this study  is based on two key assumptions:  (1) the
additive is initially distributed uniformly throughout the polymer and (2) penetration or
swelling of the polymer by the external phase does not occur to a significant extent.  That is,
absorption of an external phase by the polymer does not affect the diffusion coefficient or the
solubility of the migrant in the polymer. These assumptions greatly simplify the mathematics
but, in fact, may not always apply.  Furthermore, when these assumptions  are not valid, it is
likely that migration will occur at a rate higher than that predicted by the models. In the case
of a non-homogeneous additive distribution, the concentration at the surface may be higher
due to blooming or incompatibility. In the case of external phase penetration of the polymer,
the polymer may be swollen and the glass transition temperature lowered (essentially the
polymer is plasticized).  Thus, we recommend that the methodology be extended to cover
non-uniform additive concentrations (e.g., blooming) and external phase effects on migration.

       Finally, we suggest that the computer programs presented herein be further developed
and integrated with other exposure models, for example those for indoor air quality. Further
development of the programs could also be directed towards making them more adaptable to
the various forms of data commonly received in Premanufacture Notifications. An additional
possibility is to tailor the models for specific polymers or migrants that are of particular
interest to the EPA.

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 2.     FACTORS THAT AFFECT MIGRATION

 2.1    Overview

       The rate and extent of migration of an additive, monomer, or oligomer from a
 polymeric  material  depends  upon the properties of the polymer, the  migrant, and the
 external phase.  Interactions between these entities are also important.  System variables
 such as the geometry of the polymer-external phase contact, temperature,  and time must
 also be considered.

       The migration process involves the diffusion of the migrant through  the polymer to
 its surface, transfer of the migrant across the polymer/external phase interface and
 assimilation of the migrant into the bulk of the external  phase.  Movement  within the
 polymer depends upon the gradient in migrant concentration and upon an interaction
 parameter between the polymer and migrant termed the diffusion coefficient.

       Once the migrant reaches the polymer/external phase interface, several factors affect
 the rate of transfer  into the bulk of the  external phase.   These  include the relative
 solubilities of  the migrant in the polymer and external phase, the capacities  of both phases
 for the migrant, and the mechanism for  movement of the migrant in the external phase,
 e.g., by diffusion or by convection.

       In this  section, we discuss a variety of factors that influence  migration.  Most of
 these factors do not enter directly into any model calculation, but they are important in
 determining model parameters.  In many ways, this section provides  a background for  those
 interested in migration processes from polymer sheets. Migration model development and
 applications are described in Sections 3, 4, and 5.

 2.2    Polymer

       A single polymer molecule is composed of a repetition of simpler molecular units
 called monomers.  A polymeric material is typically envisioned  as a mass of intertwined
polymer molecules.   A key consideration in the migration process is the mobility of the
polymer molecules within the mass.  Depending upon the polymer, the  conditions under
which it was fabricated  and  the temperature, the polymeric material  contains varying
fractions of crystalline and amorphous regions.  In the amorphous regions, the molecules
 (or parts thereof) exhibit some degree of mobility. A consequence of their  combined
movements is  the formation, dissipation  and reformation  of "holes" through  which migrant
molecules  may move.  There are  several polymer properties and characteristics that relate
directly or  indirectly to the ease of migrant movement or loss during contact of the
polymer with the environment.  Some of the more important are discussed below.

2.2.1   Glass Transition Temperature

       The segmental mobility of a single polymer molecule in  a polymeric material is
dependent on temperature.  At a certain  temperature, the material changes from one in
which there is segmental motion and the polymeric material is  flexible  to one in which the
segments of the molecules become immobile and  the polymeric material is "stiff."  This
temperature is  called the glass transition temperature Tg (Rodriguez,  1982).  For any given
polymer, the potential for migration of an  additive is larger at temperatures  greater than Tg.

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This effect becomes more pronounced as the size of the migrant molecule increases (Crank
and Park, 1968), and there may be discontinuous increase in the diffusion coefficient as one
raises the temperature above T . Glass transition temperatures are shown for several
polymers in Table 1.

2.2.2  Crystallinity

       Polymers may exist in amorphous, partially crystalline, or crystalline states or regions.
Crystalline regions, as the name suggests, are highly ordered domains in which there is no
segmental mobility.  Migration, therefore, can occur only through the amorphous regions of
the polymer material. The tendency to crystallize is enhanced by regularity and polarity in
the polymer molecule (Rodriguez, 1982).  For example, trans- and cis-1,4-polybutadiene  have
crystallim'ties of 40% and 30% respectively.  However, a random mixture of the two isomers
has no crystallinity.  Similarly, a nonpolar polymer such as atactic polypropylene has no
crystallinity whereas  the polar polymer Nylon 6 is very crystalline.  The degree of
crystallinity can be influenced by the thermal and processing history of the polymer.

       It should be noted that crystallinity is different from glassiness although both are
indicative of loss  of chain mobility. Polymeric materials with crystalline regions exhibit
melting points (a  first-order transition) associated with the crystallites that are always higher
than the glass (second-order) transition temperature.

2.2.3   Crosslinking

       Crosslinks are covalent bonds between two or more polymer molecules.  In  the
vulcanization of rubber, sulfur is commonly used as  the crosslinking agent. Crosslinking
reduces polymer chain segmental motion and, therefore, decreases migration rates.

2.2.4   Branching

       Some polymer molecules are linear chains of monomers, while other polymer
molecules contain side chains branched from the main chain.  Branches hinder close ordering
of the polymer  molecules and result in a more open  network of polymer molecules. Higher
migration rates  would be expected from branched polymers (Flynn, 1982).  With reference to
Section 2.2.2 above, branched molecules are also less likely to form crystalline regions.  A
good example of  this is  a comparison between high- and low-density polyethylenes.  The
high-density material (HDPE) is composed of linear molecules that pack well and has a 70%
crystallinity whereas  low-density polyethylene (LDPE)  is branched and exhibits about 40%
crystallinity.  Data for the migration of the antioxidant 3,5-di-i-butyl-4-hydroxytoluene (BHT)
from LDPE and HDPE are compared in Table 2 (Till et al-> 1983).

2.2.5   Molecular Weight

       The average molecular weight of the molecules composing a polymeric material can
have a significant influence on many of its properties.  Molecular weight effects on migration,
however, are generally only a factor with relatively low-molecular weight polymers.  Such
polymers generally have properties that prevent their use as commercial

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     TABLE 1.  APPROXIMATE GLASS TRANSITION TEMPERATURES (Tg)
                          FOR SELECTED POLYMERS
                          (Seymour and Carraher,  1981)
                          Polymer                            (degrees kelvin)
               Cellulose acetate butyrate                             323
               Cellulose triacetate                                   430
               Polyethylene (LDPE)                                 148
               Polypropylene (atactic)             .                   253
               Polypropylene (isotactic)                              373
               Polytetrafluoroethylene                              160,400*
               Polyethyl aerylate                                    249
               Polymethyl acrylate                                   279
               Polybutyl methacrylate (atactic)                        339
               Polymethyl methacrylate (atactic)                       378
               Poly aery lonitrile                                      378
               Polypropylene (isotactic)                              263
               Poly vinyl acetate                                     301
               Poly vinyl alcohol                                     358
               Polyvinyl chloride                                    354
               Cis-poly-l,3-butadiene                                255
               Polyhexamethylene adipamide (nylon-66)                330
               Polyethylene adipate                                  223
               Polyethylene terephthalate                             342
               Polydimethylsiloxane (silicone rubber)                   150
               Polystyrene                                          373
Two major transitions observed.

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         TABLE 2. MIGRATION OF BHT FROM LDPE AND HDPE
                            (Till etal., 1983)
External Phase
Milk

Orange Juice

Margarine

Time
(days)
14
21
21
28
98
99
Temperature
(°C)
4
4
4
4
4
4
Migration*
(lie/dm2)
LDPE HDPE
65
1.5
96
- - 7.1
114
2.4
(--) Indicates no test data were reported for these conditions.

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 (solid) products.  Consequently, the molecular weight of the polymeric material is  not
 addressed further in this report.

       There are, in addition, polymeric products  that are available in the form of moisture
 activated or reactive, two-component systems.  These include various adhesives and sealants
 [e.g., epoxies  and room temperature vulcanizing silicones (RTV) and coatings (urethanes)].
 In these systems, a relatively low-molecular weight polymer reacts either with moisture in
 the air or with a second,  reactive  component  that  is mixed  with the low-molecular weight
 polymer at the time of use to form a high-molecular weight polymer.   During the  reaction
 (i.e., curing) period, carrier solvents and/or reaction products are released. They migrate
 through the polymer network that is increasing in  molecular weight with time.
 Furthermore, it is likely that a dynamic molecular weight gradient exists across the polymer
 during the  cure period. The modeling of migration in reacting  systems  is not considered in
 this report.

 2.2.6  Plasticization

       Plasticizers are additives, usually liquids, that  are used to make rigid  polymers more
 flexible.  As opposed  to solvents that also increase the flexibility of polymers, plasticizers
 have low volatility  and therefore remain in the polymer for longer periods so that  stable,
 plasticized  products are possible.  One example is the use of phthalate  esters to increase
 the flexibility  of poly vinyl chloride (PVC).

       Plasticizers lower the glass transition temperature of  the  polymer.  They  act similar
 to a solvent in that they increase the spacing  between polymer molecules and thereby
 increase  segmental chain mobility.  Diffusion  rates of additives, including the plasticizer,
 are very dependent upon the plasticizer concentration.

 2.2.7   Degradation

       Polymeric materials may be subject to  degradation by the external phase. In  such a
 process, polymer molecules are broken at points along their length (i.e., chain scission) or
 attacked  at side groups or at the end of the chain.  In these cases the molecular weight of
 the polymer may be decreased such that the physical integrity of the material is reduced;
 there may also be an increase in the diffusion rates of migrants in the polymer.

 2.2.8   Summary

       In this  brief discussion of polymers, we have  emphasized several key properties that
 affect migration.  None of these properties are employed directly in migration-rate  model or
 calculations but are only employed indirectly  as they affect  a key parameter  - the migrant
diffusion coefficient (see Section 2.4).

2.3    Migrant

       A migrant is, relative to  the polymer, a low molecular weight species contained or
dissolved within the network of polymer molecules.   The migrant may be:

       •       unreacted monomer,
       •       molecules composed of a few monomer units, i.e., oligomers,

                                            8

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        •      additives purposefully formulated with the polymer  to improve properties
              (e.g. stability during processing and use, flexibility,  or  appearance),
        •      degradation products  of certain additives, (e.g., antioxidants and thermal
              stabilizers), that are designed to react according to a mechanism that protects
              the polymer,
        «      impurities  present in  the monomers  as well as additives and by-products
              formed during polymer production, including residual catalysts, inhibitors, etc.

        The mobility  of a migrant in a polymer is  a function of the size and shape of the
 migrant molecule.  Both  linear and spherical low molecular weight (<100 g/mol) have
 mobilities that decrease as the size  of the molecule increases.  For molecular sizes larger
 than, for  example, butane, elongated or flattened molecules exhibit more mobility than
 spherical  molecules of approximately the same molecular weight (Barrens and Hopfenburg,
 1982).  Some of these effects  are demonstrated by the data in Table  3.

        Polymers are  typically formulated with several additives to yield a compound that
 has processing and ultimate properties appropriate  for the fabricated product.  Potentially,
 two or more of these additives could interact in such a way that their mobility
 characteristics in the polymer are affected.  Such interactions are not  considered in this
 report.

        Also, certain families of additives or migrants, such as antioxidants and thermal
 stabilizers, are formulated with the polymer for the purpose of protecting the polymer
 during  processing or prolonging the service life of the material.  These additives function
 by responding to external factors, e.g., heat, oxygen, sunlight, or by neutralizing reactive
 sites on the polymer chain.  In so doing, products of reaction are  generated.  These would
 be subject to the same considerations relative to migration as the initial additive.

 2.4    Migrant Diffusion Coefficient

       Depending upon their size, shape and  chemical properties, migrant molecules require
 different size holes in order to move through the polymer.  Consequently, each
 polymer-migrant pair is characterized by a  unique  property that describes the ability of the
 migrant to move through that particular polymer.  This property is termed the diffusion
 coefficient.

       The use of diffusion coefficients in  the mathematical models is covered in
 Section 3, but are considered here to note that they can vary by many  orders of magnitude
 between rubbery polymers and rigid, highly crystalline polymers for the same migrant at
 the same  temperature. Many of the factors noted in Section 2.2 have a large influence  on
 the magnitude of the diffusion coefficient.

       Diffusion coefficients can also be a function of the concentration of  constituents  or
penetrants in the polymer. Concentration-dependent diffusion coefficients can impede the
analysis of experimental data.  Exact expressions for concentration-dependent diffusion
coefficients are known for only a very limited number of polymer-migrant pairs.  This
problem is typically circumvented through the use  of  a so-called integral diffusion
coefficient that serves to  average the concentration effects.  It is this  "average" diffusion
coefficient which is employed  in this report.  Methods to estimate this  parameter are
discussed  in Section 4.2.

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TABLE 3.  DIFFUSION COEFFICIENTS IN NATURAL RUBBER
     AS A FUNCTION OF MOLAR VOLUME AT 40°C
                (Crank and Park, 1975)
Permeant
Methane
Ethane
Propane
Butane
n-Pentane
iso-Pentane
neo-Pentane
Molar
Volume
cmVmol
37.7
54.9
75.7
100.4
115
116
122
Diffusion Coefficient
x 108
Comment (cm2/s)
i 145
linear 54
increasing molecular size 34
4 34
linear 23
4 9
spherical 7
                       10

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 2.5    Migrant Concentration Factors

       Additives, monomers, and oligomers may be present in polymers at concentrations
 ranging from less than 0.1% to over 40%.   These  potential migrants may be present as
 discrete additive domains or, more usually, as dissolved material.  Indeed, many additives
 and polymerization residues form solid solutions with the polymer. In  general, the
 additives  are present in concentrations below their  solubility limit.  Because the driving
 force for  migration within the  polymer is the concentration gradient of  the additive
 (discussed later in Section 3.3.1), all other  conditions  being equal, higher migrations will be
 associated with higher concentrations of additives.  In fact, Pick's diffusion  laws predict
 that, for simple systems at  or below the solubility  limit, migration is directly proportional
 to concentration. Thus, for  example, doubling the concentration would double the rate of
 migration.

       In some cases where a  certain additive concentration is required  to achieve a desired
 effect but where the concentration is above the solubility limit, other additives may be
 formulated into the compound  to  increase the compatibility and, therefore, solubility of the
 additive.  In other cases,  additives may be  purposefully  added at concentrations above their
 solubility  limit in order to achieve a desired effect. For example, lubricants  which exude
 to the surface improve the processing of some polymers, and anti-fogging additives prevent
 condensation build-up on clear plastic films used to wrap meat and produce (Nah and
 Thomas,  1980).

 2.6    External Phase

       In  this report, the  external phase is defined  as  the material contacting the surface of
 the polymer.  It may be air, water,  soil, another polymer, body fluids, skin,  etc.
 Regardless of the nature of the external phase, its interactions with the  polymer
 (e.g., swelling) may have significant effects on migration.  Equally important from the
 point of view of predicting  migration is the degree of agitation of the external phase (Reid
 et aL, 1980).

 2.6.1  Physical State

       Simplistically, the  external phase may be considered vapor, liquid or  solid.   It is
 important  to note  that none of these external phases is more or less likely to be associated
 with high or low rates of migration.  Rather, the degree to which these different phases
 influence  migration is determined by:  the compatibilities and mobilities of the migrant in
 the polymer relative to those of the  external phase; the relative volumes and surface areas
of the polymer and external phase; and the degree  of  agitation of the external phase.
These factors  are addressed in  the remainder of this section.

2.6.2  Agitation

      As migrant moves to the surface of  the polymer, it must pass into the external
phase in order for migration to continue.  The overall rate at which the additive migrates is
defined by both the rate of migration through the polymer and the rate  of migration from
 the polymer surface into the external phase. In one extreme, when the  external phase is
 highly agitated and well mixed, the  rate  of migration from the surface will be rapid

                                           11

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 compared to the rate of diffusion through the polymer.  In this case, migration is controlled
 by the diffusion rate of the additive in the polymer.

        In the other extreme, when the external phase is totally stagnant, the migrant can
 move into the external phase only by a diffusional process.  The relative diffusivities of the
 migrant in the polymer and in the external phase determines whether the external phase
 controls the rate of migration.  The  external  phase  becomes controlling when the migrant
 diffusion rate within the external phase is much slower than that in the polymer. This may
 occur for certain combinations of diffusion coefficients and solubilities  in the two phases.

        An intermediate situation between the above extremes occurs when the external
 phase is only moderately agitated.  In this  case, a thin, stagnant layer (i.e., the boundary
 layer) exists  at  the polymer-external  phase  interface.  The  thickness of the boundary layer
 is a function of the degree of agitation of the external phase.   Depending upon the  rate of
 migration through the polymer relative to the rate of  transfer through the boundary  layer,
 the migration process can  be controlled by the external phase,  the polymer, or by a
 combination of both.  The  parameter that characterizes  the resistance to transfer through the
 boundary layer  is the mass transfer coefficient.  Methods to estimate this parameter are
 discussed in  Section 4.5.

 2.6.3  Partition Coefficient

       The partition coefficient is defined as  the ratio of the migrant concentration in the
 external phase to the migrant concentration in the polymer at equilibrium.  The partition
 coefficient may be calculated by dividing the saturation concentration of the additive in  the
 external phase by its saturation  concentration in the polymer.  Thus, the partition
 coefficient is an indicator of the relative  solubilities of the additive in the polymer and in
 the external phase.

       Knowledge  of the partition coefficient is necessary in order  to calculate migration
 rates  through a  stagnant or poorly mixed external phase  or to estimate the migration into
 an external phase that has a finite capacity for the  additive.  Techniques to calculate the
 partition coefficient are covered in Section  4.4.

 2.6.4  Migrant  Capacity in the External Phase

       For this  report, the external phase is always  considered  to be devoid of the migrant
of interest at the start of the migration process.  As migration occurs, the additive
concentration in the external phase increases.   As time passes,  one  of two situations will
develop  in the external phase that will depend on the volume of the external  phase, the
amount of additive available for migration, and the  partition coefficient.

       In one case,  the volume  of the external phase  is sufficiently large so that it  can
accept all the additive without surpassing about 10%  of the saturation concentration
possible if equilibrium existed between the external phase and polymer.  Below the  10%
level, the external phase appears to the polymer as  if its additive concentration is
essentially zero, and the presence of additive in the external phase  does not significantly
affect the concentration driving  force for migration.
                                           12

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       In the other case, at higher concentrations  of additive in the external phase, the
 driving force for additive migration into the external phase decreases.  Migration rates drop
 below those that would have occurred in the large external volume case and may even
 eventually cease.   At the point at which migration stops, a partition equilibrium exists
 between the additive in the polymer and the additive in the external phase.  Techniques for
 estimating the saturation concentration of a migrant in air and water external phases are
 provided in Section 4.4.2.

 2.6.5  Diffusion Coefficient in the External  Phase

       When the external phase is stagnant,  migrant molecules enter the external phase by
 diffusion.  In this case, similar to diffusion within the polymer, the migration  rate becomes
 a function of the concentration gradient  driving force and the diffusion  coefficient of the
 additive  in the external phase.  The diffusion coefficient is a property of the
 migrant-external phase pair.

       Diffusion coefficients of additives through  vapor are relatively high, generally in the
 10"1  to 10"2 cm2/s range, because the molecules of the vapor are widely  spaced and thus
 provide an open network for diffusion.   The molecules of a liquid are closer together and
 therefore holes for diffusion are reduced relative to a vapor external phase.  Typical
 diffusion coefficients for liquids are in the range of 10"5 to 10"* cm2/s.  Diffusion
 coefficients for polymeric solids are lower still and cover  a much broader range - from 10"7
 to 10"20 cm2/s.  As  noted earlier, the more  flexible  the polymer, the higher its  diffusion
 coefficient.  Estimation techniques for diffusion coefficients in air, water, and  polymers are
 covered in Sections 4.3.1, 4.3.2, and 4.2, respectively.

 2.6.6  Degradation

       In the  discussion of the migrant  capacity of the external phase (Section 2.6.4), the
 additive was considered inert.   The capacity  of the external phase was determined solely by
 its capacity for the additive.  The additive, however, may  react in  the external phase.  For
 example, an antioxidant that is designed to react with oxygen may do so in an external
 phase such as air or water containing dissolved oxygen.   In these cases, it is possible for
 higher levels of migration to occur than  would have been  expected on the basis of the
partition  coefficient of the unreacted antioxidant.  The kinetics of the reaction may even
control the migration process (Till et  al, 1983; Gandek, 1986).  Degradation of the  additive
 in the external phase is not considered further in this report.

 2.6.7  Surfactants

       Another factor that can  strongly  affect migration is the presence  of surface active
 agents (i.e., surfactants) in  the external phase.  Surfactants are molecules having both
hydrophilic and oleophilic segments.   As such, they promote  the solubilization of oleophilic
substances  in, for example, water. In such cases,  the effective capacity  of the external
phase can be  greatly increased giving a large "apparent" partition  coefficient.  Surfactants
can be effective at very low concentrations.
                                            13

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 2.6.8  Penetration

       Just as additives can migrate out of polymeric materials, the external phase may
 migrate into the polymer.  In some cases, the amount of external phase penetration into the
 polymer is  sufficient to influence migration  significantly. This effect occurs principally
 with liquid external phases. As the polymer absorbs liquid,  it may soften,  swell, become a
 gel  and, in the extreme, dissolve in the liquid.   As this process develops, the molecules of
 the  polymer become more  mobile.  This results in more and larger "holes" for diffusing
 molecules.  Thus the migrant diffusion coefficient increases.

       An interesting aspect of penetration is the case in which the external phase is a
 multicomponent liquid  containing penetrants at  relatively low concentrations.  Such
 penetrants may be preferentially absorbed by the polymer.  A possible  consequence is that
 the polymer exhibits migration  levels disproportionate with the nominal penetrant
 concentration in the external phase.  This  effect has  been observed in  the case of certain
 food packaging materials in contact with fruit juices that contain low concentrations of
 essential oils  (Till et. al., 1983).  Penetration of the  external  phase into the polymer is  not
 considered further in this report.

 2.7    Temperature

       Migration increases  strongly with temperature.  The principal reason is that the
 diffusion  coefficient follows an Arrhenius  relationship (i.e., it increases  exponentially with
 absolute  temperature).

       In addition, the  physical state of the polymer is a function of temperature.  As
 discussed earlier in Section 2.2.1, at temperatures below  the glass transition value, the
 polymer  molecules are  relatively immobile and  diffusion is slow.  Diffusion is more rapid
 above Tg.  As temperatures increase  further, crystallites,  if present, melt and the  polymer
 becomes amorphous - a condition also associated with more rapid diffusion.

       In the case of a liquid external phase, an increase in temperature also raises the
 diffusion coefficient of  any penetrating liquid into the polymer, thereby raising the rate and
 extent of swelling and extraction.  Temperature may also affect the solubilities of the
 additive in the polymer and external phase and, consequently, the partition coefficient.
Techniques  provided in Section 4 to estimate values  for  many of the required input
 parameters (e.g., the diffusion coefficient in  the polymer, the partition coefficient) apply
only at 25°C.  The model can,  however, be used to  estimate migration at higher
 temperatures but requires the user to input values for the required parameters at  the
 appropriate  temperature.

2.8    Time

       The  dependence of migration on time is addressed in  detail in Section 3 and is
discussed only briefly here.  In cases where migration is controlled by  a diffusive process,
initially,  migration is proportional to the square root  of time.  Accordingly, migration
occurs at  an ever-decreasing rate.  This applies to situations in which diffusion is the rate
limiting step.   In cases  where the migration is controlled by  mass transfer processes in the
external phase, the initial migration is proportional to linear time.  Thus the migration rate

                                           14

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will be constant until the external phase begins to exhibit finite characteristics or there is a
significant decrease in the additive concentration in the polymer.

2.9    Overall Summary

       While it is unlikely that the history of a polymer product will be known, it is
worthwhile to note that,  over time, a product will  be exposed to a variety of external
phases (Ayres et-al., 1983).  Some of these may be penetrating and  others inert; some may
be well-agitated and others stagnant; some may be essentially infinite in volume and others
finite.  Migration occurs to greater or  lesser extents to each of these external phases and,
in so doing, affects the additive concentration profile  within the polymer in  a variety  of
ways that may impact  subsequent migration.  To take all these potential exposures  into
account in a mathematical model would be a very complex task. Furthermore, to  attempt
to do so is likely to be impractical given that generally not much is known about  the basic
properties of the polymer, additive, and external phase or their interactions.  To circumvent
these problems, simplifying assumptions and approximations are necessary.  These
assumptions address, for example, the  concentration gradient across  the  polymer, the
transfer at the interface, and the conditions within  the external phase. Theoretical  and
experimental investigators have established a consistent set  of assumptions that  adequately
correlate migration data.   In Section 3 we present  models based on these assumptions that
provide procedures for estimating additive migration from polymeric  materials to a  variety
of external phases.
                                          15

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 3.     MODELS FOR MIGRATION ESTIMATION

 3.1    Introduction

       In the preceding section, many of the complex factors influencing migration were
 discussed.  An impractical level of testing would be required to examine all the parameters
 for the number of polymer-additive systems  of interest to the EPA.  Thus, there is a need
 for mathematical models to estimate the potential loss of additives to the environment.

       Such models should require minimum input data and produce an estimate of the
 fraction of additive initially present in the polymer that  might be  lost from the polymer as
 a function of time.  For  various scenarios, different inputs are required.   Several inputs are
 required  to physically characterize a system  (e.g., the polymer film thickness, the
 environment in contact with the polymer, etc.).  Other inputs are  required to define the
 basic migration parameters (e.g., the diffusion coefficient, the partition coefficient,
 solubilities of the additive, etc.).  Should values  for these inputs  not be available,  we
 present some techniques  in Section 4 to estimate them.

 3.2    Assumptions

       The mathematical models presented in the section are  based on the assumption of
 an ideal  polymer-additive system having the following properties:

       •      the polymer is  a flat sheet of  uniform thickness with no edges,
       •      initially, the additive is homogeneously distributed  throughout the polymer
             and there is no additive present in the external phase.  The additive of
             concern is  not affected by any other migration  mat  may be occurring,
             swelling or penetration does not occur, or, if it does, it does not affect the
             physical dimensions appreciably nor does  it affect the migration,
       •      the diffusion coefficient of the additive is a function only of temperature and
             not of position or time,
             the migration is isothermal, and
             Pick's laws apply.

3.3   General Rate Concepts in Migration

3.3.1  The Polymer Film

      As noted in the assumptions above, the additive is initially distributed uniformly
throughout the polymer film.   At time zero,  some change occurs at the film surface to
allow additive to leave the film.  This step establishes a lower additive concentration near
the film  surface and initiates  a diffusion process in  the polymer film that attempts to
reestablish a new uniform concentration.  Further movement of the additive to the external
phase necessitates additional diffusion to occur within the polymer.
                                          16

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       The movement of additive within the polymer towards the film surface and into the
 external phase is, therefore,  a function of gradients in additive concentration.  This
 generality is expressed in mathematical terms by Pick's first law.

       Diffusion flux = Dp (dCp/9x)                                               (3-1)

 The flux (mass/area-time) is, therefore, proportional to the gradient in concentration and the
 coefficient of proportionality is the diffusion coefficient of the migrant in the  polymer,  Dp.
 x is the distance measured into the polymer film and, because the concentration is always
 larger in the film interior during migration, 9Cp/dx > 0.  The units of Dp are  (length)2/time
 and are  usually expressed as cm2/s.

       To obtain the flux of additive leaving the film, Eq. (3-1) is employed  with the
 gradient (8Cp/3x) being evaluated at the polymer film interface (x=0).  Thus one must
 know  how Cp varies with x  at different times.  This is accomplished by  employing Pick's
 second law which is Eq. (3-1) applied  over a short distance dx in a short period of
 time dt.

       ac,/8t = D^cyax2)                                               .        (3-2)

 All models developed in this section utilize Eq. (3-2) to determine  Cp =  f(x,t).  Since
 Eq. (3-2) is a partial differential equation, the solution requires two initial or boundary
 conditions.  One of these conditions, common to all cases, is the assumption that the
 additive  is, initially, homogeneously distributed.  In mathematical terms,

       Cp(x)  = C^ for all x  at t = 0                                              (3-3)

 The remaining condition relates to events occurring at the film-external phase  boundary.
 Various physical situations lead to different boundary  conditions  and, therefore, to different
 Cp = f(x,t) relations.

       Before exploring  the various  types of boundary conditions, we note that there are
 two interpretations when defining the polymer thickness. The simplest case occurs for
 situations where the additive is  lost from only one side of the film (i.e.,  the other side is
 somehow prevented from losing additive). In this case, the  length variable x  begins at
 zero on the side where  loss takes place and increases to a value L  at the side where no
 loss occurs.  Thus, L is  the true film thickness.  The other case  is  when migration occurs
 from both faces of the polymer film.  In this situation there is no flux across  the mid-plane
of the  film and 3Cp/dx = 0 at this plane.  Thus, the system  behaves as though the film
 were only one-half its true thickness and, in the models, the characteristic film thickness is
only one-half the true thickness.

       Therefore, in employing  the models described later, it is necessary to provide the
 true film  thickness  and, also, to specify if migration occurs from only one or  from both
 surfaces.  When the variable L  is employed in equations, it  will  represent the  true film
thickness for one-sided  migration, but only one-half the film thickness for two-sided
migration.
                                           17

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3.3.2  Migration into a Fluid External Phase

       In cases where the additive leaves the polymer film and moves into a fluid external
phase (e.g., air and water), there are two important considerations that determine the
appropriate boundary condition that, with Eq.  (3-3), may be used to solve Eq. (3-2).

3.3.2.1  Partitioning

       It is generally assumed that during migration, the additive concentration within the
polymer is in equilibrium with the additive concentration in the fluid at the interface.  The
ratio of the equilibrium concentrations is the partition coefficient K.  (See Section 2.6.3.)
                                                    •                               (3-4)

The concept of equilibrium at the interface implies no rate restrictions (i.e., the additive can
move from the polymer into the external phase without hindrance).  Thus, any rate processes
relate only to movement of the additive into the bulk of the external phase.  The appropriate
boundary condition is then:

       Flux into the fluid external phase = k(Ce?interface-Ceibulk)                          (3-5)

where k is a mass  transfer coefficient. Denoting

       ^.interface = ^p,x=0                                                            @-6)

and using Eqs. (3-1) and (3-4), the boundary condition becomes

       Dp0C/)x)x=0 = k(KCpiX=0-Ce>bulk), t > 0                                      (3-7)

3.3.2.2         Variation of the Additive Concentration in the External Phase

       Next, one must examine whether Cebulk varies  appreciably during the migration
process.  First, we  note the limiting case where the volume of the external phase is very large
or where flow processes occur so that Ce bu]k is always essentially zero.  That is, there is no
measurable accumulation of migrant in the external phase, at least relative to KCp x=0.  Under
such circumstances, Cebu]k is set equal to zero in Eq. (3-7) to yield

       DpOCp/Sx)^ = kKCpiX=0    (Ce,bulk -» 0)                                     (3-8)

       Alternatively, for a smaller quantity of external fluid, one may have the situation
where Ce bu]k increases from an initial value of zero to approach Ce at the interface (i.e., equal
to KG  x=0).  In this case, the flux from the  polymer is hindered by the low driving force to
move additive away from the film surface.  This situation often occurs when the additive
solubility in the external phase is quite low  (small K)  so that the right-hand side of Eq. (3-7)
is initially small and decreases even further as Cebulk rises.
                                          18

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 3.3.3  Migration into a Solid External Phase

       If the external phase is a solid (or can be treated as  one because there are no
 convective currents), the physical processes in migration may be described as follows.
 Within the polymer film, the flux of additive to the  surface is identical to that occurring
 with  a fluid  external phase (Section 3.3.1).  At the film  surface, there  again  occurs a
 partitioning step as described in  Section 3.3.2.1.  However,  the mechanism to move the
 additive from the  interface  into the bulk solid phase  takes place by diffusion rather than by
 convection and  the boundary condition given by Eq.  (3-7) is replaced by

                                                                                 (3-9)

 Here,  y is the length variable in the solid external phase and is measured from the
 interface.  Ce is the additive concentration in the external phase.  De is the additive
 diffusion  coefficient in the  solid  external  phase.  Since 3Cy3y < 0, a negative sign is
 required in Eq.  (3-9).

       Within the solid external  phase, diffusion occurs and an equation comparable to
 Eq. (3-2) applies,  i.e.,

       acyat =  n^cjdf)                             -                         (3-io>
One has, in essence,  a dual diffusion problem with Eqs. (3-4) and (3-9) relating the
concentration in both solids.  The effective thickness of the  solid phase is normally set
equal to the volume of the external phase (VJ divided by the exposed surface area (A),
VyA.  For large values of VyA, the external solid phase is  essentially infinite in effective
thickness.

3.4    General Mathematical Models

3.4.1   Fluid External Phase

       For the  general case of migration into a fluid phase,  the rate model may be
formulated mathematically using Eqs. (3-2),  (3-3), and  (3-7)  with a variation  that Ceibulk may
be set equal to zero [as in Eq. (3-8)] provided the external phase volume  is sufficiently
large  or the additive solubility in this phase  is such that Q^ « KQ,^.  Thus, it is
necessary to specify Cpf, Dp, k, K, L, and A to compute Cp  = f(x,t) and, therefore, the
migrant flux from Eq. (3-1) at x = 0.

       Solutions have been obtained by Hatton and co-workers (1979,  1983)  and explored
by Gandek (1986) using the following dimensionless groups:

       T = Dpt/L2                                                                 (3-11)

       a = KVyAL                                                              (3-12)

       y = kKL/Dp                                                               (3-13)

where t is the time and, as noted  earlier, L is the film thickness for one-sided migration
and one-half the film thickness  for two-sided migration.

                                           19

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       We do not present the equations for Cp = f(x,t) (see Gandek, 1986).  Rather, we
 have utilized  them with the flux equation [Eq.  (3-1)] and, by integration,  have obtained the
 final relation  describing the total migration occurring at any time.  That is, with Cp =  f(x,t),

       Migrant flux = D^Cp/ax)^                                               (3-14)

 and

                                          t
        Total  mass migrated/area  =      f  (flux) dt                               (3-15)
                                      o

Further, we have non-dimensionalized the mass of additive migrated as

       M; = (total mass migrated/area)/LCpi0                                       (3-16)

so that Mj is the fraction of the original  additive  that has been lost from the polymer.

       For the  general case where  one must be concerned with the variation of the
concentration in the bulk of the external  phase, Cebuik,

                         oo
       M, = [o/(l+a)] + S (a2/a)exp(-G>n2T)                                    .   (3-17)
                        n=l

with

       Q, = a  + f[con + sin((Djcos(coJ]/{2coJ^os(G)>co>n(con)]2}                   (3-18)

and con are the  non-zero, positive roots of the characteristic equation

                                                                                 (3-19)
       If one is not concerned about variations in Ceibulk due to the large value of V/A  or
because C.^ « KC^^,  then the boundary condition given by Eq. (3-8) may be used.   In
this case,
       M, =  1  - 2Y2 E  [exp(-r02t)]/[rn2(rn2+f +y)l                                    (3-20)
                   n=l

where  rn are  the roots of. the equation

       rntan(rn)  = y                                                               (3-21)
                                           20

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 3.4.2  Fluid External Phase with a Large Mass Transfer Coefficient

       In some instances, one may desire to obtain an estimate of the migration in a
 worst-case analysis using a large value for the mass transfer coefficient, k.  Eqs. (3-17) and
 (3-20) could still be  employed by setting the dimensionless parameter y [Eq. (3-13)] to be
 large.  However, it is more expedient to resolve the basic  differential equations with k
 (or y)  -> oo.

       Two boundary conditions are possible. If the external fluid phase is large in extent
 so that the additive concentration remains very low or at least small compared to any
 saturation solubility,  then one can state:

               = *-e,buIk = " =

 For this simple case  (Crank, 1975),
       M, = 1 -  2 £ [expt-q/DJ/q/                                               (3-23)
                  n=0

where

       q, = (2n+l)7t/2                                                            (3-24)

       On the other hand, if C^^^ is sufficiently large  to affect the migration, then the
appropriate boundary condition is
       Ceju.rf.ce = C^fc = K(CpiX=0)                                                 (3-25)

where all the variables in Eq. (3-25) are functions of time except K.  In this case,

                            oo
       M, = [oc/(l+a)]  - 2a2 I  [expC-p^H/a-Kx+afo2)                             (3-26)
                           n=l

and pn are the non-zero, positive roots of

       tan(Pn) + ccpn =  0                                                          (3-27)

3.4.3   Solid External Phase

       The  appropriate boundary condition for this case is given by Eq.  (3-9) and the
solution to the coupled partial differential equations was only recently obtained by Gandek
(1986). Since no external mass-transfer coefficient is involved, the dimensionless group y
does not enter.  Instead, a new  dimensionless group  (5 is introduced,

       p =  K(De/Dp)1'2                                                            (3-28)
                                           21

-------
 Then,
               a     2  I         tan2(j)n exp(-0a2t)
           ~  I-KX

 with  n) + f3tan(on/p) = 0                                                   (3-30)

       If the  solid external phase volume is  very  large compared to the polymer film
 volume and K is  not exceedingly small, then the  dimensionless group a [Eq. (3-12)]
 becomes  a  large number.  In such cases, it is mathematically more convenient to employ
 Eq, (3-22) as the boundary condition i.e., a  — » °°.  This approach leads  to:
       M, = [2j3/(p+l)](t/n)1/2{l-[2p/(l+P)] S [(1-pya+p)]-1 x
                                         n=l

              [exp(-n2/t) - (n27t/t)1/2erfc(n/tI/2)]}            -                         (3-31)

Note that, in Eq. (3-31),  a does not enter as a variable.

3.4.4  Summary

       Six relations have been presented to compute the fraction of additive lost as  a
function of time.  Four of the equations apply for a fluid external phase and two for a
solid external phase.  The differences between the equations applicable to fluid or solid
external phases  relate to limiting cases when it may be assumed that some variable  is
sufficiently large such that one (or more) of the rate steps is of negligible importance.  We
summarize these equations, the input variables required, and the limiting constraints  in
Table 4.

       For an external fluid phase, Eq.  (3-17) represents the general case and, if desired,
could be  used for all external fluid phase situations.  Eq. (3-20) is a simplification for
those situations  where the external fluid phase is so large that the bulk  additive
concentration is essentially zero or for situations where the additive is very soluble in the
fluid.  Mass transfer effects  are considered though.  In Eq. (3-26), finite external phase
volumes or limited solubilities exist,  but there is no limit to the mass transfer coefficient.
Eq. (3-26), thus is  also a simplification but does  address solubility limitations.  Finally,
Eq. (3-23) allows one to  specify a worst case scenario with  a large external volume or no
solubility limitation in the fluid and with a large (—> °°) mass transfer coefficient.

       For a solid  external phase, Eq. (3-29) provides a solution for the general case while,
with Eq.  (3-31), one is not concerned with solubility limitations due either to a large value
of V/A.  or because the additive solubility in the solid is very high.
                                            22

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        TABLE 4.  MIGRATION MODEL EQUATIONS
                                 External Volume Large
                        Input      and/or No Solubility   Mass Transfer
External       Eq.     Variables      Limitations in the      Coefficient
 Phase        No.     Required*           Fluid              Large
 Fluid
3-17

3-20

3-23

3-26
K,k,Ve/A

  K,k



 K.V./A
No

Yes

Yes

No
No

No

Yes

Yes
 Solid
3-29    K,D.,V./A

3-31       K,De
                    No

                    Yes
                  NA"

                  NA
 In all cases L, t, and Dp are required.

 NA  = not applicable
                             23

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        The various models require different amounts of input data.  Eqs. (3-17) and (3-29)
 require the most information while the limiting cases require less due to the relaxation of
 some  constraints.

        In Figure 1  we present an  algorithm of the logic used in the computer program
 described in Section 6 to direct a  user to the various model equations.  At the start, one is
 required to define  the physical  scenario by specifying the time for migration, the film
 thickness (L),  and whether the  extraction  is one- or two-sided extraction.  Further, the
 diffusion coefficient of the additive in the polymer (Dp) must be given or estimated.

        The initial concentration  of the migrant is not required at this point since the
 equations yield the fraction of the additive that has migrated.  If one desires the absolute
 flux of migrant, Eq. (3-16) would be used and the total mass of migrant lost per unit  area
 would be the product of the fraction lost  (MJ the initial migrant concentration (CP_J and
 the polymer thickness (L).

       If the external phase is a fluid, the user must provide the mass  transfer
 coefficient (k)  or state that it is very large (e.g., to develop  a worst-case scenario).
 When k is large, one must then provide the solubility of the additive in the fluid phase or
 state that it is  unlimited.  If the latter, Eq. (3-23)  is employed to calculate the migration.
 If there is a finite  solubility, but a very large quantity of external phase, again Eq.  (3-23)
 is used. Both these situations will produce  a worst-case estimate of the migration.
 However, if there is a finite solubility  and a limited quantity of external phase, then the
 solubility of the additive in both the fluid and polymer must be specified to calculate the
 partition coefficient (K), and Eq. (3-26) is used to estimate  the fractional migration.

       Should  the user elect to  provide a  mass transfer coefficient, solubilities of the
 additive in the fluid and the polymer are also necessary.  Then, depending upon whether
 one states the external phase is unlimited  (or  the solubility of the additive in this phase is
 large)  or limited, Eq. (3-20) or  Eq. (3-17), respectively, is used to determine migration.

       In the case  of a solid external phase, specification of the solubilities of the additive
 in both phases must be given as well as the diffusion coefficient of the additive in  the
 solid external phase.  Depending upon  the choice of an unlimited or limited external phase,
 Eq. (3-31) or Eq.  (3-29) is employed in the computations.

 3.5    Significance  of the Dimensionless Groups

 3.5.1   t = Dpt/L2

       This group provides a convenient method to non-dimensionalize the time for a  given
physical situation.  For example, a thicker film would require longer times to attain the
 same value  of 1.

3.5.2   a = KV./AL

       This group relates primarily to situations where the external phase becomes saturated
with additive and migration ceases even though Mj, the fraction migrated, is still less than
unity.  To illustrate this  relationship, consider a case  in which equilibrium between  the two
phases is attained when the additive concentration in  the polymer has dropped from a

                                           24

-------
                                               Fluid
                                       f
Unlimited
                                    Specify k
(O
Unlimited
          imited I
             Specify Ce,sVe/A
                         1
Limited
            Eq. (3-23)
                              Specify Cp,s
                                       Eq. (3-26)
                                                                Start
                                                                               J


Identify Polymer, Migrant
                                                             Specify L, 1 or 2 sided, t,Dp
                                                                        T
                                                          Specify External Phase
                                                                      tLimi
                                                                               imited
                                                                          Specify
                                                           T
                                                            Unlimited
                                                                          Specify
                                                                                         Solid
                                                                                                    Specify
T
                                                                                             Unlimited
                                                                                                           Specify
                                                                                                                 Limited
                                                                               Limited
                                                                                                  Eq. (3-31)
                                                                                                           Eq. (3-29)
                                                                  Eq. (3-20)
                                                                           Eq. (3-17)
                               Figure 1.  Flow Chart for Migration Model Equation Selection

-------
 starting value of Cp,0 to Cp*.  Also,  the concentration of the additive in the external phase
 has  increased from zero to a  new value Ce*.  (Note that since an equilibrium state is
 assumed, neither Cp* nor Ce* are functions of position  or time.)  During  the actual
 migration,

       Mass lost from the polymer = AL(CP>0 -  Cp*)                                (3-32)

 This relationship would hold  for both one- or two-sided migration  because, in the former,
 L is the true film thickness and  A the one-sided area whereas in the latter, L is only one-
 half the  true film thickness but A now  represents a two-sided area.

       The mass lost from the polymer must  appear in the  external phase, i.e.,

       Mass in  the external phase = Ve(Ce* -  0)       '                            (3-33)

 Since, at equilibrium,

       K = C*/C*                                                              (3-34)

 then, equating Eqs. (3-32) and (3-33) and  using Eq. (3-34),

              - Cp*) = VeCe*                                                    (3-35)

              -  ALCP* = V.C.* = VCKCP*                                         (3-36)

       €,*/€„,„ = A1V(AL + KVJ = 1/[1 + (KVJAL)]  =  l/(l+cc)                    (3-37)

 Since M^*, the fraction migrated at this equilibrium, can be defined as

       M,* = (C^  - CV)/C^                                                      (3-38)

 then

       M,* = «/(!+cc)                                                            (3-39)

       Referring to the models in Section  3.4, we note  that, in the equations which assume
 limited external  volumes or solubility limitations [Eqs. (3-17), (3-26), and (3-29)], there is
 a leading term of o/(l+a) in  the expressions for M^  Therefore, as T becomes large  (e.g.,
 long times), M,.  is constant and equal to O/(!-KX).  The physical significance is that the
 external phase is saturated with additive and all migration ceases.

       If a » 1, migration can occur to deplete essentially all the additive from the
polymer  film, but for a « 1, the maximum fractional  migration is = a.  For a = 1, 50%
of the additive would migrate.

3.5.3 Y = kKL/Dp

      The dimensionless group Y provides some perspective about the magnitude of the
resistance to transfer across the external phase boundary layer relative to  the diffusional
resistance within the polymer.  Values of Y > 100 indicate that the boundary layer

                                          26

-------
 resistance is negligible and only the diffusional resistance within  the polymer film is
 important  In such a case, the use of Eq. (3-23) or Eq.  (3-26) would be acceptable.  (See
 Table 4.)  Values of y less than about  1  would indicate that the resistance to transfer
 across the boundary layer far exceeds the diffusional resistance in the polymer film and
 Eq. (3-17) or Eq. (3-20) must be used.
 3.5.4
                    1/2
       P  is a measure of the relative resistances to diffusion  in the solid external  phase and
 in the polymer.  If P > 10, diffusional resistances in the polymer greatly exceed those  in
 the solid  external phase.  Conversely, if p < 0.1, then the diffusional resistance has shifted
 to the solid external phase.  Note that, while De and Dp are important in defining p, the
 partition coefficient is  also a key parameter. Small partition  coefficients lead to a greater
 importance of external phase diffusion.

 3.6    Illustrative Examples

 3.6.1   Fluid External Phase

       In Figure 2, we have graphed the results  of applying Eq. (3-23) or Eq. (3-26).  For
 these two cases, the mass transfer coefficient is assumed to be very  large so that
 diffusional resistances within the  polymer film  greatly exceed any resistance  across the
 boundary layer.  When there is also no saturation limitation in the external fluid phase,
 a — »  oo and  Eq. (3-23) applies.   For a less than about 10, Eq.  (3-26) must be used. Here,
 at large values of I, the fractional migration levels out to a value of o/(l+a) as the
 polymer film and external fluid phase approach saturation equilibrium.  With large values
 of a, the fractional migration is  proportional to the square root of t  (or time) up to a point
 where about  50-60% of the additive has been lost.

       If  a finite mass transfer coefficient is employed, the fractional migration as  a
 function of t is shown in Figure  3 for different values of y but at a constant a = 0.1.  At
 small values  of y, essentially all resistance to migration resides in the boundary layer
 transfer.   When y >  100, the important resistance is the diffusion within the  polymer.  In
Figure 3,  the  curve f or y — » °o is identical to the a = 0.1 curve  in Figure 2  that was drawn
for an unlimited mass transfer coefficient  For a = 0.1, a partition equilibrium is
established at large values of t (long times) with M, — » o/(l+a) = 0.1/1.1  = 0.091.

3.6.2   Solid  External Phase

       We illustrate some results  for the solid external phase models in Figure 4.   The
maximum migration occurs when a. = p = oo which is  a special  case of Eq.  (3-31) with no
saturation limitations.   For  a values smaller than about 10, Eq. (3-29) applies and, again,
there  is a partition equilibrium of oc/(l+a).  Increasing  P leads to more migration;  this
increase can  result with systems  having a large  K,  a  large De, or when Dp  is small relative
to De.
                                            27

-------
                       Dimensionless  Time  (r)
FIGURE 2.   Migration Estimates for the Fluid External Phase Cases Where
            the Mass Transfer Coefficient is Unlimited. Eq. (3-23) Applies
            When a -> «>, All Other Curves from Eq. (3-26).  (Gandek, 1986.)
                                    28

-------
    G
    O


    
-------
       10C
  a
  o
        -i
 ^   iQ-1  -
  cO

  00
 •i—i
 2

 13
  c
  o
  o
  CO
io-2 F
      10
        -3
io-4
                   v-3
                                -i
io"a     10"*     io~J      10°

    Dimensionless Time  (r)
FIGURE 4.    Migration Estimates for the Solid External Phase Cases (Gandek, 1986).
                                 30

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 4.     ESTIMATION OF VARIABLES

 4.1    Introduction

       The models developed in Section 3 provide equations to predict additive migration
 from polymeric materials under a variety of conditions.  In order to use these models,
 input data are required.  In all cases, the initial concentration of the additive  in the
 polymer,  the polymer thickness, the time, and the diffusion coefficient of the  additive in
 the polymer are required.  Also, a specification of whether the additive is removed from
 one or both  faces of the polymer is necessary.  Values of the initial additive  concentration
 and polymer thickness are usually available or can be approximated within a  fairly narrow
 range based  on  the application of the polymeric material.  Values for the additive diffusion
 coefficient in the polymer, however, are generally not known nor readily  available.
 Consequently, they must be estimated.

       In some  instances, other input data are required.  These inputs include the partition
 coefficient (the ratio of the equilibrium solubility of the  additive in the external phase to
 that in the polymer), the external phase mass-transfer coefficient, or, if  the external phase
 is a solid or a  stagnant liquid, the diffusion coefficient of  the additive in  this  phase.

       Section 4 addresses the estimation of these input  data. Estimation techniques  are
 provided for the variables listed in Table  5.

 4.2    Diffusion Coefficients of Additives  in Polymers (Dr)

       The diffusion coefficient of an additive in a polymeric material is  a function of the
 segmental mobility of the polymer molecules and the size  and shape of the additive.  Over
 the past 30-40 years, researchers have  attempted to correlate the diffusion coefficient  with  a
 variety of properties of the polymer and additive.  These correlations range from simple to
 quite complex,  although a higher degree of correlation is not necessarily associated with the
 more complex methods.  Herein, we suggest a simple approach  which enables the
 estimation of Dp at 25°C knowing only the molecular  weight of the migrant and the
 general type  of polymer involved.

       In  Figure 5, experimentally determined diffusion coefficients for six common
polymers are plotted versus diffusant molecular weight on  log-log coordinates. Values  used
to generate Figure 5 were taken principally from Grun (1949), Flynn (1982),  and Park
 (1950,  1951). The degree of correlation is remarkably good.  As expected from theory, the
largest diffusion coefficients are associated with the most flexible  materials, silicone and
natural rubbers,  and the smallest diffusant molecules.   Diffusion coefficients for the more
flexible materials are less a function of molecular weight (or size) than those  for the stiffer
polymers [e.g.,  polyvinyl chloride  (PVC) and polystyrene (PS)].   Over a similar range of
molecular weights, Dp for the  flexible materials covers 3 to 5 orders of magnitude  while
for Dp  PVC ranges over 10 orders of magnitude.  Note,  however, that it appears that at
high molecular weights, Dp drops precipitously in even the flexible polymers.

       Correlation equations based on the curves in Figure 5 have been develoepd for each
of the six polymers and are used in the computer program described in Section 6.  Thus,
by identifying the polymer of interest and inputing a  value for the molecular  weight of the
                                          31

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          TABLE 5.  VARIABLES FOR WHICH ESTIMATION TECHNIQUES
                                 ARE PROVIDED
                      Variable                           Units          Section


Diffusion Coefficient in Polymer                           cm2/s          4.2

Diffusion Coefficient in External Phase

                 Air                                    cm2/s          4.3.1

                 Water                                  cm2/s          4.3.2

                 Other                                  cmVs          4.3.3

Partition Coefficient

            •    Additive Solubility in Polymer            g/cm3          4.4.1

                 Additive Solubility in External
                 Phase

                      Air                                g/cm3          4.4.2.1

                      Water                              g/cm3          4.4.2.2

Mass Transfer Coefficients

            •    Water                                  cm/s           4.5.1

            •    Air                                    cm/s           4.5.2
                                        32

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   10
                              1                       2
                              LOG MOLECULAR  WEIGHT
FIGURE 5.
Diffusion Coefficients in Six Polymers as a Function of Molecular Weight
of Diffusant, T = 25°C.
                                       33

-------
additive, a value for Dp can  be estimated.  Dp estimation equations are provided for silicone
rubber, natural rubber, LDPE, HDPE, polystyrene and unplasticized polyvinyl chloride.

       Most polymers do not have as extensive a compilation of Dp values as those shown
in Figure 5. Consequently, estimation of a diffusion coefficient for these other polymers is
more difficult and will be less exact.  One approach for approximating Dp in a polymer not
included in Figure 5 is to compare the diffusion coefficients of a common chemical
through the polymer of interest with  that for the polymers in Figure 5.  Nitrogen, oxygen,
and  carbon dioxide permeation have  been measured through a wide variety of polymers and
are useful  as "common chemicals."   A listing of diffusion coefficients for these gases in
polymers is given in Table 6.

       As  an example, suppose one was interested in estimating the diffusion coefficient  of
a chemical in Nylon 12 and  only polystyrene data were available for this chemical.  Using
the technique noted above with carbon dioxide as the common chemical, we find from
Table  6,

               diffusion coefficient (Nylon 12)         0.02
              diffusion coefficient (polystyrene)       0.06

This simple technique may lead to large errors, and somewhat different estimates can be
obtained by changing the common chemical.

       Another approach for estimating Dp for a polymer not addressed in Figure 5 is to
assign the polymer to a polymer group or class that represents a relatively narrow range of
diffusion coefficients.  As a first step  to such an approach, we reviewed the permeation
and migration literature and compiled  a list of diffusion  coefficients for a wide variety of
polymers and migrants.  From this compilation, we categorized the polymers into four
groups:

       •      rubbers,
       •      polyolefms I (mostly amorphous plastics above their glass transition
              temperatures),
       •      polyolefins H (mostly crystalline plastics above their glass transition
              temperature), and
              glassy polymers below their glass transition  temperature.

       Ranges were established for the diffusion coefficients in each polymer group.  These
ranges are  shown in Figure 6 which is identical to Figure  5 except that, for clarity, the
grid has been simplified and the datum points removed.  To facilitate the use  of Figure 6,
polymers representative of each group are identified in Table 7, in which they are arranged
in order of decreasing diffusion coefficient within their respective groups.  For example, if
one were interested in isotactic polypropylene with a high degree of crystallinity  ( = 60 to
65%),  then, from Table 7, this polymer is found near the  bottom of the polyolefin II
region.  If  the molecular weight of the migrant were, for example, 178 (log 178  = 2.25),
the estimated value of the diffusion coefficient of this migrant would be approximately
3xlO-u crnVs at  25°C.
                                           34

-------
    10- r

                               (unplasticized) V<&-?tV3fc
                               	'•-:':'J>'~:/
0.25
0.5    0.75
                               1     1.25    1.50   1.75     2
                               LOG  MOLECULAR WEIGHT
2-50   2.75
FIGURE 6.    Diffusion Coefficients for Four Polymer Groups Described in Table 7,
              T = 25°C.
                                         35

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   TABLE 6.  DIFFUSION COEFFICIENTS FOR SELECTED POLYMERS AT 25°C
D_ x 106 (craYs)
Polymer
Poly(l,3-butadiene)
Poly(butadiene-co-acrylonitrile)80/20
Poly (butadiene-co-acrylonitrile)6 1/39
Poly(butadiene-co-styrcne)92/8
Poly(carbonate) (Lexan)
Poly(chloroprene) (Neoprene)
Poly(dimethylbutadiene) (Methyl rubber)
Poly(dimethylsiloxane) (Silicone rubber)
Poly(ethylene) (Density 0.91)
Poly(ethylene) (Density 0.96)
Poly(ethylene-co-propylene)40/60
Poly (e thy lene terephthalate)
Poly(ethyl methacrylate)
Nitrocellulose
Nylon 66
Nylon 12
Poly(isoprene) (Natural rubber)
Poly(isoprene-co-acrylonitrile)74/26
Poly(isoprene-co-methylacrylonitrile)74/26
Poly(isobutene-co-isoprene)98/2 (Butyl rubber)
Poly (oxy-2,6-dimethyl- 1 ,4-phenylene)
Poly(oxymethylene)
Poly(propylene)
Poly(styrene)
Poly(tetrafluoroethylene) (Teflon)
Poly(tetrafluoroethylene-co-hexafluoroprene)
(Teflon FEP)
Poly(vinyl acetate)
Poly(vinyl butyral)
Poly(vinyl chloride) (Crystalline)
Poly(vinyldene fluoride-co-hexafluoropropylene)
(Viton A)
Nitrogen
1.1
0.5
0.06
1
. 0.2
0.3
0.08
16
0.3
0.09
0.7
0.001
0.02
0.01


1.2
0.05
0.1
0.05




0.09
0.09


0.6
0.004
0.03

Oxygen
1.5
0.8
1.4

0.02
0.4
0.14
21
0.4
0.17

0.003"
0.1
0.1


1.7
0.09
0.2
0.08



0.1
0.15
0.18

0.06

0.012
0.08

Carbon
Dioxide
1
0.42
0.04

0.005
0.3
0.06

0.4
0.12

0.0006
0.03
0.02
0.0008
0.02
1.3
0.03
0.09
0.06
0.06
0.01
0.08
0.06
0.09
0.1



0.002
0.03

(Yasuda and Stannett, 1975; Bixler and Sweeting, 1971)
                                   36

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            TABLE 7. RANK ORDERING OF POLYMER GROUPS FROM
                    HIGH TO LOW DIFFUSION COEFFICIENTS
                                 (SEE FIGURE 6)
I-     RUBBERS (AND SIMILAR MATERIALS)

      Polybutadiene
      Epichlorohydrin-ethylene oxide
      Natural
      Styrene-butadiene copolymer
      Neoprene
      Nitrile

n.    POLYOLEFINS-I (AND SIMILAR MATERIALS)

      Poly(ethylene-co-propylene)
      Polyurethane
      Hydrogenated butadiene
      Poly(4-methyl pentene-1)
      Ethylene-Vinyl Acetate (up to 40% VA)
      LDPE
      Polychlorotrifluoroethylene
      Isotactic polypropylene (30-40% crystallinity)

HI-    PQLYOLEFINS-II (AND SIMILAR MATERIALS)

      Butyl rubber
      HDPE
      Polyisobutylene
      Plasticized PVC*
      Teflon
      Isotactic polypropylene (60-65% crystallinity)
      Acrylonitrile-Butadiene-Styrene (ABS)

IV.    GLASSY POLYMERS

      Polymethyl methylacrylate (PMMA)
      Polystyrene
      Polyacrylonitrile
      Polyvinyl chloride (unplasticized)
      Polyamides
      Polyethylene terephthalate
      Very few data are available to indicate position.
                                       37

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       Figure 6 should only be used to determine order of magnitude values  of the
 diffusion coefficient.  Furthermore, the polymers listed in Table 7 form only  a small subset
 of the polymeric materials commercially available.   A polymer not addressed in Table  7
 may be  classified by comparing its properties with those of the polymers in each of the
 groups.  Key properties would be glass transition temperature, density, and coefficient of
 thermal  expansion.

       When using  the above approaches to estimate diffusion coefficients for use in the
 models developed in Section 3, one must note the limitations imposed by  two basic
 assumptions  of the models:

       •      Dp is  independent of migrant concentration

              Dp is  not influenced by the phase external to the polymer

 The first assumption is likely to hold for migrants initially present at less  than 1%
 concentration or for the initial  period of migration for concentrations above this level.  The
 second assumption will hold for virtually all cases with air as the external phase,  and for
 most cases with water as the external phase unless  the polymer  absorbs water.  When the
 external  phase readily  penetrates the polymer, however, it is likely to swell the polymer
 and increase  Dp.   Oils and solvents can penetrate many polymers. It is generally  true that
 the effect from any  given level of penetration will be relatively  greater for polymers having
 lower  initial  values of Dp.  For example, a  two-percent weight change due to solvent
 penetration is likely  to produce a much greater change in the  Dp of  a  glassy polymer than
 in the  Dp of  a rubber.

 4.3    Diffusion Coefficients of Additives in the External Phase (D.)

 4.3.1   Air (D.)

       At 25°C, diffusion coefficients of both inorganic and organic molecules in  air
 typically range from 0.04 to 0.23 cm2/s. A representative listing for organic molecules is
 shown in Table 8.  In general, these  diffusion coefficients  are only weakly dependent on
 temperature;  for example only a 6% reduction is expected  upon  lowering the  temperature
 from 25°C to 15°C.

       Several methods for estimating diffusion  coefficients are available and are reviewed
 in handbooks of chemical property estimation techniques.  See Chapter 11  of Reid,
 Prausnitz and Poling (1987) and Chapter 17 in Lyman, Reehl and Rosenblatt  (1982).  The
 former recommends  the method of Fuller and co-workers (1965,  1966,  1969).

       D. = 0.00143T175/{PMr1/2[V.!/3+VID1/3f}                                      (4-1)

V, and Vm are the characteristic or molar volumes of air and  the migrant,  respectively. For
 air, V. = 19.7 cmVmol, Vm is computed by an additive group procedure from the  migrant
chemical structure using the atomic values shown in Table 9.

       H =  2MJM.AM.+MJ                                                     (4-2)
                                           38

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   TABLE 8.  DIFFUSION COEFFICIENTS FOR SELECTED
              ORGANIC CHEMICALS IN AIR

         (Values of the diffuson coefficient, D., are for
                    25°C and 1 aim.)

Chemical
Hexane
Benzene
Toluene
Benzyl acohol
Chlorobenzene
Nitrobenzene
Benzyl chloride
o-Chlorotoluene
m- Chlorotoluene
p-Chlorotoluene
Diethyl phthalate
Dibutyl phthalate
Diisooctyl phthalate
Chloroform
Carbon tetrachloride
1 , 1 -Dichloroethane
1 ,2-Dichloroethane
1 , 1 -Dichloroethy lene
Vinyl chloride
1,1,1 -Trichloroe thane
1 , 1 ,2-Trichloroethane
1 , 1 ,2,2-Tetrachloroe thane
Trichloroe thylene
Tetrachloroethylene
Pentachloroethane
Hexachlorobenzene
1. Lugg, 1968.
2. Barr, Watts, 1972.
3. Farmer, Yang and Letey, 1980. (
Molecular
Weight
86.17
78.11
92,13
108.13
112.56
123.11
126.58
126.58
126.58
126.58
222.23
278.34
390.56
119.39
153.84
98.97
98.97
96.95
62.50
133.42
133.42
167.86
131.40
165.85
202.31
284.80


It should be noted
D.
(cm2/sec)
0.0732
0.0932
0.0849
0.0712
0.0747
0.0721
0.0713
0.0688
0.0645
0.0621
0.0497
0.0421
0.0377
0.0888
0.0828
0.0919
0.0907
0.1144
0.1225
0.0794
0.0792
0.0722
0.0875
0.0797
0.0673
0.12


that the value

Source
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
2
2
1
1
1
1
1
1
3


reported
by Farmer et al., was derived from data on soil volatilization rates of HCB.)
                          39

-------
TABLE 9.  ATOMIC CONTRIBUTIONS TO ESTIMATE Vm IN EQ. (4-1)
                  (Fuller et. al., 1965, 1966, 1969)
                Atom                              Contribution


                  C                                  15.9
                  H                                   2.31
                  O                                   6.11
                  N                                   4.54
                  F                                   14.7
                  Cl                    .              21.0
                  Br                                  29.8
                  I                                -  22.9

             Corrections

     Aromatic or heterocycylic ring                      -18.3
                               40

-------
where M, = 29 g/mol and M^ is the molecular weight of the migrant.  P is the pressure in
bars and equals unity at atmospheric pressure.  With the temperature in kelvins, D. is
calculated in units of cnf/s.

       As an  example, suppose one were interested in  estimating D, for allyl chloride
vapor in air at 25°C and atmospheric pressure.  From Table 9,
       Vm  = 3(C) + 5(H) +  1(C1) = (3)(15.9) + (5)(2.31) + (1)(21) = 80.25 cmVmol

With M,, = 76.5, then

       M,  = (2)(29)(76.5)/(29+76.5) = 42 g/mol

With P =  1 bar and T = 298 K, using Eq. (4-1),

       D. = (0.00143)(298)1J5/{(D(42)1/2[(19.7)1/3+(80.25)1/3]2} = 0.096 cm2/s

Lugg (1968) measured D. for allyl chloride to be 0.098 cm2/s.

       In general, the error in D. associated with  Fuller's method ranges from 5 to 15%.

4.3.2  Water (Dw)

       At 25°C, the diffusion coefficients for both inorganic and organic molecules in
water typically range from 0.4xlO~5 to 5xlO~5 cm2/s.  Similar to those for air, the diffusion
coefficients for water are not strong functions of temperature but, nevertheless, appear to
follow an Arrhenius relationship  (Reid, et  al. 1987).  A listing of diffusion coefficients for
solutes in water is presented in Table  10.

       For organic compounds diffusing in water  up to about 30°C, the Wilke-Chang
relation (Reid, et al. 1987) may be written  as

       Dw  = (5.1 x 10-7)T/Vm"                                                    (4-3)

with T in kelvins.  Vm is a characteristic volume  of the migrant which is  close to the
molar volume (cmVmol) at the boiling point at atmospheric pressure.  Vm may also be
estimated from the atomic contributions  in Table  11.  As an example, suppose one wants
to estimate Dw for ethylbenzene at 20°C (293 K).  Using Table  11,

       Vm  = 8(C) + 10(H) + 3(double bond) + ring =  (8)(7) + (10)(7) + (3)(7)  - 7
           = 140 cmVmol  ..

Then

       Dw  = (5.1xlO-7)(293)/(140)0j6 =  7.7X10"6 cm2/s

       It has also been suggested that, similar to  polymers, diffusion coefficients for  water
may be simply related to the molecular weight of the diffusant (Hober, 1945).  In
Figure 7, diffusion coefficient data given by Hober are plotted as a function of molecular
                                           41

-------
TABLE 10.  DIFFUSION COEFFICIENTS IN AQUEOUS
       SOLUTIONS AT INFINITE DILUTION
               (Reid et. al., 1978)
Chemical
Hydrogen
Oxygen

Nitrogen
Nitrous oxide
Carbon dioxide
Ammonia
Methane


n-Butane


Propylene
Methylcyclopentane



Benzene


Ethylbenzene



Methyl alcohol
T,
25
25
29.6
29.6
25
25
12
. 2
20
60
4
20
60
25
2
10
20
60
2
10
20
2
10
20
60
15
D^IO3
(cmVs)
4.8
2.41
3.49
3.47
2.67
2.00
1.64
0.85
1.49
3.55
0.50
0.89
2.51
1.44
0.48
0.59
0.85
1.92
0.58
0.75
1.02
0.44
0.61
0.81
1.95
1.26
Chemical
Ethyl alcohol


n-Propyl alcohol
Isoamyl alcohol
Allyl alcohol
Benzyl alcohol
Ethylene glycol




Glycerol
Acetic acid
Benzoic acid
Ethyl acetate
Urea

Diethylamine
Acetonitrile
Aniline
Pyridine
Vinyl chloride



T,
10
15
25
15
15
15
20
20
25
40
55
70
15
20
25
20
20
25
20
15
20
15
25
50
75

Dwxl05
(cmYs)
0.84
1.00
1.24
0.87
0.69
0.90
0.82
1.04
1.16
1.71
2.26
2.75
0.72
1.19
1.21
1.00
1.20
1.38
0.97
1.26
0.92
0.58
1.34
2.42
3.67

                     42

-------
    TABLE 11.  ATOMIC CONTRIBUTIONS TO ESTIMATE Vm IN EQ. (4-2)
                     Atom                              Contribution
c
H
O
N
Br
Cl
F
I
S
7
7
7
7
31.5
24.5
10.5
38.5
21
                     Ring*                                  -7

             Double bonds between
                 carbon atoms                               7

              Triple bonds between
                 carbon atoms                               14
3, 4, 5, or 6 membered; also include for naphthalene or anthracene.
                                   43

-------
 6
 o
W
H
U
I— I
fa
fa
W
O
o
o
M
M
Q
    10
      -4
    10
      -5
    10
      -6
    10
     -7

X

                                                                    \
                                                                             \
                                                                              \
         Log M.Wt   1
   FIGURE 7.    Diffusion Coefficients for Organic and Inorganic Chemicals in Water at
                 20°C (H6ber, 1945).
                                           44

-------
 weight on log-log coordinates.  The correlation is reasonable and may be adequate for an
 initial estimate of Dw.  The equation of the line in Figure 7 is
       Dw = 7.4xlO-3M*41                                                         (4-4)

 Eq. (4-4) is  the equation provided in the computer model described in Section 6 to
 estimate Dw  when necessary.

 4.3.3  Other Materials

       The range of external  phases obviously extends far beyond air and water.  Migrant
 diffusion coefficients in other materials can be approximated by logical processes in  which
 the external  phase in question is compared to air,  water, and solids  as represented by the
 polymers in  Figure  6.  For example, diffusion coefficients for skin would probably be less
 than those for  water but greater man those for low density polyethylene and perhaps
 greater than  those for natural rubber.  Recognizing that the result may be incorrect by as
 much  as a half order of magnitude, one could estimate De for skin by using the silicone
 rubber curve.

       Diffusion  coefficients for soils would be highly dependent on the moisture content
 of the soil.   For  a dry  sand, essentially all diffusion would occur through the air between
 the sand particles.  Thus De would be similar to that of air.  It may be also necessary to
 adjust migration to account for the polymer surface area occluded by the sand and the
 tortuous  path for diffusion within the sand.  For wet soils, De may be approximated using
 the values for water, again adjustments for occluded surface area and tortuosity may  be
 required  particularly for soils  such as clay.

 4.4    Partition Coefficient (K)

 4.4.1   Solubility  of Additive  in the Polymer (Cp,,)

       Cpit and C^ are, respectively,  the saturation  concentration of the migrant in the
polymer  and the  saturation concentration of the migrant in the external phase.  With  these
parameters, K,  the partition coefficient, can be calculated.

       K = Ce,/Cw                                                                (4-5)

K is often required in migration calculations. Unfortunately C^, is  known only for a very
few materials.  For many rubbery polymers or elastomers, Goydan  et al.  (1989) present an
estimation procedure for Cp,.  The technique, while complicated, has been coded for
computer application.  The FORTRAN program resides on personal computers within the
Chemical Engineering Branch of the EPA Office of Toxic Substances.  If this method is
inapplicable,  and no independent value of Cpi, is available, one may use  the initial
concentration of the additive in the polymer (C^)  as an initial guess for Cpf.   This
approach  usually  leads  to an estimated value for K that is greater  than the true K since
additives are usually present at  concentrations far below their solubility limits.  In this
sense,  such a choice  leads to  a "worst case"  situation since migration increases with  K.
Exceptions to this generalization are  additives that  are  present at concentrations above their
limit of saturation and  are actually designed to migrate to the surface (i.e., bloom).   In this
case, migration could be at rates greater than those predicted using the  assumption of C^

                                           45

-------
equal to C  0.  The migration of the bloom occurs by dissolution in the external phase rather
than by diffusion processes.  Dissolution is not addressed by the models presented herein.

4.4.2  Solubility of Additives in the External Phase (Ce s)

4.4.2.1        Air(Ca)

       The "solubility" of a chemical in air, Ca, is the same as its saturated vapor
       tration at a specified temperature.  The ideal gas law provides a sufficientl
       i ne  soiuDiiity  or a cnermcai in air, L,a, is tne same as its saturatea vapor
concentration at a specified temperature.  The ideal gas law provides a sufficiently accurate
basis for the calculation of this parameter if measured values are not available.

       Ca = (1.3 x 10-4)PVM/(RT)                                                   (4-6)

where

             Ca = "solubility" in air, g/cm3
             Pv = vapor pressure of chemical (torr)
             M = molecular weight of chemical, g/mol
             R = gas constant = 8.314 J/mol K
             T = temperature, kelvins

The  multiplier (1.3 x 10"4) in Eq. (4-6)  leads to the units of g/cm3 for Ca.

       For example, to estimate the value of Ca for trimethylphosphate (140.1 g/mol) at 25°C
where the vapor pressure is 3.80  mm Hg = 3.80 torr,

       Ca = (1.3 x 10'4)(3.80)(140.1)/[(8.314)(298)] = 2.79xlO'5 g/cm3

       The key chemical-specific input  for the calculation of Ca is the vapor pressure of the
chemical (Pv).  If a measured value of this property is not available, it may be estimated by
the methods described by Grain (1982).  The computerized chemical property estimation
system CHEMEST (Lyman, et al., 1982) contains estimation methods for vapor pressures that
are slightly more accurate than those covered by Grain.  The inputs  for the estimation
methods in the  cited works are a boiling point and, for solids only, a melting point.  If
necessary, both of these inputs can be estimated.

4.4.2.2         Water (Cw)

       Estimation methods for the solubility of chemicals in water, C^,, have been reviewed
by Lyman (1982), and the recommended methods have been incorporated into the
computerized chemical property estimation system CHEMEST (Lyman, et al., 1982).  In
general, the best approach is to estimate Cw from the octanol-water partition coefficient Kow,
which itself can be estimated from the chemical structure.  This approach involves relatively
simple calculations and can often provide estimates within a factor of 2 of the true value.
This technique provides estimates of Cw only at a temperature of 25°C.
                                           46

-------
       At present, it is recommended that the following regression equation be used to
predict Q,:
       log a = log M -  1.123 logK^- 0.0099Tm - 2.067                          (4-7)

where

       Q, =  water solubility at 25°C, g/cm3
       K,,w = octanol-water partition coefficient
       Tm =  melting point, °C, for  liquids use 25°C
       M = molecular weight of chemical, g/mol

       This equation is recommended by Yalkowsky (1982) and is based upon the studies
of Yalkowsky, et al. (1983) and Valvani, et al. (1981).  When used for neutral organic
chemicals, Eq. (4-7) is expected to  have an average method error of about a factor of 2;
this presumes that accurate values of K^, and Tm are available.

       When using Eq. (4-7), if the predicted value of Q, is greater than about 0.1 g/cm3,
the chemical should be assumed to  be  infinitely soluble in water.

4.5    Mass  Transfer Coefficients (k) for a Fluid External Phase

       In a migration process where the external phase is a  fluid (vapor or liquid),
consideration must be given to the  possibility that mass transfer resistances in the external
phase may influence the migration process.  Techniques for  estimating values of the  mass
transfer coefficient are empirical and highly dependent on the geometry of the physical
situation and the flow rate of the fluid. Mass transfer coefficients for migration to water
and air are addressed in this section.

4.5.1   Migration to Water

4.5.1.1   Rat Polymer Surfaces

       Polymers in contact with water  may be of various shapes and  sizes.  However,
some  geometry must be assumed to allow an estimation of the mass transfer coefficient.
We have chosen to express  the exposed area of the polymer in terms of an  equivalent flat
sheet  of area A with a characteristic length / in the direction of the bulk water flow. It
will also be assumed that, in the direction /,  there is an average  characteristic fluid
velocity v.

       In order to estimate the mass transfer coefficient, it is first necessary to determine
whether the water flow over the polymer surface is laminar  or turbulent.  This is typically
done by calculation of the length Reynolds number:

       Length Reynolds number = Re;  = Iv/v                                      (4-8)
                                          47

-------
where

       /  = the characteristic length in the direction of flow, cm
       v = characteristic water velocity, cnj/s
       v = kinematic viscosity of water, cm2/s, and is defined as the dynamic viscosity
         divided by the density of water. At room temperature v = 0.01 cm2/s.

Laminar and turbulent flow regimes  are delineated as

       Re, < 106 :  laminar
       Re; > 106 :  turbulent

The mass transfer coefficients in each regime may be estimated as proposed by Skelland
(1974):

Laminar

  k = CLOVOCRe/^CSc)1^                                              .          (4-9)

Turbulent

  k = Cr(Dw//)(Re/)a8(Sc)1/3                                                       (4-10)

where

       k = mass transfer coefficient, cm/s
       CL = constant ~ 0.664
       CT = constant « 0.037
                                                        n
       Dw = diffusion coefficient of the migrant in water, cm/s
       Sc.= Schmidt number = v/Dw

and /  and Re; are as defined in Eq. (4-8).

       Dw may be found in published tables such as Table  10 or estimated from Eq. (4-3) or
Eq. (4-4). In most cases Eq. (4-4) may be used as a good approximation for Dw when
calculating k. With it and setting v =0.01, then

Laminar

  k = l.SxlO'^v/O^/M0-27                                                        (4-11)

Turbulent

  k = S^xlO^v0-8//0-2)/^!0-27                                                      (4-12)


                                            48

-------
There is a weak dependence of k on the molecular weight M.  If M = 200, M0^7 = 4.2
while if M = 500, M0-27 = 5.4.  For simplicity,  M0-27 may be defined as a constant equal to
4.8. The final working equations then reduce to

Laminar

   k  = 5.2xlOJt(v//)1/2                                                             (4-13)

Turbulent

   k  = l^xlO^v0-8//0-2)                                                           (4-14)

with  the transition  from laminar to turbulent flow at Re, =  106.  Although this transition
point from the laminar to the turbulent regime is routinely  used, some  turbulence may be
present within the laminar domain.  We recommend, therefore, only the use of the
turbulent correlation for simple, approximate estimations of k for water flowing over flat
polymer surfaces.  Eq. (4-14) is plotted in Figure 8 to illustrate the dependence of k on
both  the water velocity and /.  Unless the water flow is very low (< 10 cm/s), this figure
yields reasonable values of k.

       For the special case where the water is essentially stagnant, there is no good
correlation for the estimation of the mass transfer coefficient.  Small eddies or instabilities
can lead to transient variations in  k.  If no such eddies are present, the stagnant water
might be modelled as  a "solid" external phase and any transfer of additive would be by
diffusion into the water.

4.5.1.2  Polymer Pipes

       If water is flowing  inside a polymer pipe and migrant from the  pipe wall is
transferred to the water, the mass  transfer coefficient may be estimated  by (Skelland,
1974):

       k = 0.023(DJd)(Rer(Sc)1/3                                                (4-15)

In this case

       Re = dv/v                                                                (4-16)

With  Sc defined  under Eq. (4-10), d is the pipe diameter (cm), v the water velocity (cm/s),
Dw the diffusion coefficient of the migrant in water (cm2/s), and v the  kinematic viscosity
of water (cm2/s).   Inserting these definitions,

       k = 0.023Dw2VJ/d°Va8-a33)                                                 (4-17)

With  v about 0.01  cm2/s and using Eq. (4-4) for Dw,

       k = S-SxlOV-V^M"7)                                                   (4-18)
                                           49

-------
   10'
 6
 o
 U
 O
 _l
 UJ
 cc
 UJ
    10
                                                                  i  11111
     10
       -4
    10"3                   10"2

MASS TRANSFER COEFFICIENT, cm/S
10"
FIGURE 8.    Estimation of Mass Transfer Coefficient for Water Rowing Over Hat
             Polymer Surfaces.
                                     50

-------
 Assuming, as before, Mo:n ~ 4.8

       k = 7.2xlOV-8/dM                                                        (4-19)

 with  k in cm/s.  The form of Eq. (4-19) is very similar to that of Eq. (4-14).  Eq. (4-19)
 should only be applied when Re > 2000 as, otherwise, laminar not turbulent flow would
 exist  In essentially all applications, water flow in pipes occurs in  the turbulent regime.

 4.5.2  Migration to Indoor Air

       The following discussion addresses polymers that are in some form of closed
 environment; not in the outside air.   We note, however, that the presentation in
 Section 4.5.2.1 is applicable  to polymers in both environments.  The shape and orientation
 of the polymer surfaces are important as is the motion of air within the environment.  In
 general, there are three different causes  for air circulation:

       1.      Bulk flow due to open windows, forced air circulation by fans, or wind.

       2.      Density driven convective flows on vertical  polymer surfaces due to the
              molecular weight difference between the migrant and air.

       3.      Thermally driven convection currents due to buoyancy effects.

       We consider each of these circulation flows separately although all three may be
 present simultaneously.  In general,  we conclude that, for indoor air, thermally  driven
 convection currents usually dominate and establish the degree of air turbulence  that
 determines the mass transfer coefficient.

 4.5.2.1  Bulk Air Over Horizontal Polymer Surface

       The polymer surface is modelled as a flat, horizontal plate of area A with a
 characteristic length / in the direction of the bulk air flow. In essentially all cases of
 interest, the flow is laminar and Eq. (4-9) is applicable.  The kinematic viscosity of  air is
 0.16 cm2/s at 25°C at  atmospheric pressure.  Thus, in a form comparable to Eq. (4-13),

       k = '0.90D.OT(v//)lfl                                                         (4-20)

 where D,  is the diffusion coefficient of the migrant in air  (cmz/s), v is the  air velocity
 (cm/s), and / is measured in  the direction of v (cm),  k is the mass transfer coefficient
 (cm/s). Values of D. may be estimated by the method suggested in Section 4.3.1.  For
 cases  where the migrant is primarily composed of carbon and  hydrogen and where the
migrant molecular weight, M, is much larger than the molecular weight of air,  the diffusion
coefficient at 20  to 30°C and atmospheric pressure can be approximated as

       D. - 3.3/(2.5 + M1/3)2 in crrrYs                                              (4-21)

For example, if the migrant were the antioxidant butylated hydroxytoluene  (BHT), M = 220
and, with  Eq.  (4-21), D, » 0.045 cm2/s.  For Irganox 1010, a proprietary antioxidant with
M = 1176, D. ~  0.02 cm2/s.  While such values are only approximate, they are satisfactory
                                           51

-------
 for use in determining k with Eq. (4-20), which becomes, with the substitution of D, from
 Eq.  (4-21),
       k  = 2/K2-.5  + M'TW*]                                                 (4-22)

 To employ Eq. (4-22), let us assume the migrant was BHT (M = 220) and the ratio (//v)
 was 3600 s.  Then

       k  » 2/{[2.5+(220)1/3]4/3(3600)1/2} -  1.9X10'3 cm/s

 To facilitate the estimation of k for horizontal surfaces, Eq. (4-22) is shown plotted in
 Figure 9  as a function of (//v) and migrant molecular weight.

 4.5.2.2   Vertical Polymer Surfaces

       In this case,  we assume a vertical polymer surface of average height H (cm) from
 which an additive is migrating.  Because the mass density of the migrant-air mixture near a
 vertical surface exceeds the average mass density of air in the bulk, a natural circulation
 flow is initiated down the surface.  For a laminar flow situation (Skelland, 1974),

       k = (4D./3H)Cvf(Sc)Gr1/4                                                  (4-23)

 where

       C¥ = constant ~ 0.50

       f(Sc) = Sc1/2/(0.952+Sc)1*                                                  (4-24)

       Sc = Schmidt number  defined under Eq. (4-10)

       Gr  = Grashof number  = (gHYv^Kp.-pJ/pJ                                (4-25)

       g = acceleration due to gravity = 980 cm/s2

       v = kinematic viscosity of air = 0.16 cm2/s at 25°C and 1 bar

       p.. = mass density of the bulk air

       p0  = mass density  of the migrant- air mixture at the surface

If we set

       p_ = PM./RT                                                            (4-26)

       p0  = (PVM/RT)  + (P-PJM./RT                                             (4-27)

where M  and  M, are the molecular weights of the migrant and air and Pv is the partial
pressure of the migrant in  the air-migrant mixture at the surface.  (P, would equal the
migrant vapor pressure if the  polymer were saturated with migrant.)  Then, with
Eqs. (4-26) and (4-27),

                                          52

-------

X.

£




fe  10
UJ

o

u.
u.
UJ
o
o

o:
LU
U,
CO



2  10"
        ,-1
      ID'3
                                                                     Molecular Weight

                                                                        of  Migrant
         tor'
                                                10
102
10-
                                                         //v , seconds
FIGURE 9.  Estimated Mass Transfer Coefficients for Migrants  Into Air Due To Bulk Flow (300K,  1  Bar, Laminar Flow).

-------
       (Po-P J/P~ = (IVP)[(M-Ma)/MJ                                           (4-28)

Substituting Eq. (4-28) into Eq. (4-23) with v = 0.16 cm2/s,

       k = 3.7(PV/P)1/4(Z/H1/4)                    •                               (4-29)

where

       Z = Da1/2{ [(Ma-M)/Ma][l/(0.952-f0.16/Da)]}1/4                               (4-30)

       From Eq. (4-21), we can approximate Da as a function of M so that Z then depends
only on the migrant molecular weight. Also, if one  computes Z for various values of M, Z is
found only to range from 0.23 for low values of M (=  100) to 0.20 for M ~  1000. Assuming
a mean value of Z  = 0.22, then Eq. (4-29) simplifies to

       k = [0.41 (PV/P)/H]1/4                                                      (4-31)

where H is in cm and  k in cm/s.

       As an example, suppose we were interested in the mass transfer of the antioxidant
BHT from a vertical polymer  surface  to air.  At 25°C,  Pv for BHT = 0.76 torn This vapor
pressure will be used since the actual  BHT partial pressure over the polymer is unknown.
(The use of this vapor pressure represents  the greatest driving force for migration, i.e., it is a
worst case.) The height of the vertical surface, H, is 3 m or 300 cm.  With P = atmospheric
pressure = 760 torr,

       k = [0.41(0.76/760)/300]1/4 = 0.034 cm/s

In Figure  10, the mass transfer coefficient  is shown for various values of (PV/P) and H.

4.5.2.3 Thermally  Driven Convection

       A large body of data has been developed by  the American Society of Heating,
Refrigeration, and Air Conditioning Engineers  (ASHRAE, 1980) to estimate  heat transfer
coefficients in closed environments where  there exist temperature gradients due to heating or
cooling surfaces. Typical natural convection heat transfer coefficients, h, range from about 1
to 8  W/m2K.

       Using the analogy between heat and mass transfer processes and  noting that for air,
the Lewis number (the ratio of the Prandtl to the Schmitt number) is essentially unity, then
the Sherwood number equals the Nusselt number:

       Sh = k//Da                                                               (4-32)
       Nu = h/A                                                               (4-33)

thus

       k = (Da/X)h                                                             (4-34)

                                          54

-------
                              10
                                 -6
                                                vp
10"5               10'
 (migcant) /P (atmospheric)
10
FIGURE 10.    Estimated Mass Transfer Coefficients for Migrants In Air On Vertical Surfaces (300K,  1 Bar, Laminar Flow).

-------
Here, Sh is the Sherwood number, Nu the Nusselt number, k the mass transfer coefficient,
h the convective heat-transfer coefficient, Dm the diffusion coefficient of the migrant in air,
X the thermal conductivity of air and / some characteristic length.  A, is about 0.026 W/mK
for air near ambient conditions.  Thus, using an average  value of h ~ 3.4 W/m2K,
Eq. (4-34)  becomes

       k = D.(3.4/0.026)( 1/100)  = 1.3 D.                                         (4-35)

with k in cm/s.  The factor  of 100 in Eq. (4-35) was inserted to convert meters to
centimeters.  D. may be  estimated from Eq. (4-21) in terms  of the migrant molecular
weight M or calculated as described  in Section 4.3.1. Then, for example, with M = 220,
D. - 0.045 cmVs and k= 0.059 cm/s.  Increasing M  to 1000, D. - 0.021 cnf/s and
k « 0.027 cm/s.

4.5.2.4   Discussion

       Of the three air movement mechanisms likely to be present in enclosed spaces,  the
largest mass-transfer coefficient is usually found when considering thermally-driven
convection.  Since the largest k  is the most important, Eq. (4-35) is usually only  necessary
to  estimate k. In cases where it is known that no  thermal convective currents are
important, however, k may be estimated from  the bulk-flow  model (Section 4.5.2.1) or the
vertical convective model (Section 4.5.2.2).  For example, the bulk-flow model would be
more appropriate outdoors.
                                          56

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 5.     EXAMPLE MIGRATION CALCULATIONS

       Examples are provided to illustrate the use of the migration equations described in
 Section 3 and the techniques to estimate the  input variables described in Section 4.  In
 most cases, these example calculations were performed using the computer program
 described in Section 6.

 5.1    Worst Case Examples

       Often one is interested in "worst case" scenarios to  initially estimate additive
 migration because,  if the results from such cases  are acceptable, more complicated
 estimations  are unnecessary.  Several "worst case" examples illustrate the procedures  and
 results.

 5.1.1  Background

       There is  a large TV console in a room of a house.  The dimensions of the room
 are 3 by 3 by 2.1 m (10 by  10 by 7 feet) and the volume  is 19 m3  (700 feet).  This
 console is made of a plastic that is believed to be ABS (acrylonitrile-butadiene-styrene
 copolymer).  The migrant is styrene (molecular weight = 104) at an  initial  concentration of
 450 ppm.  The cabinet is 2 mm thick and migration only occurs from the outside  surfaces.
 The exposed surface area (neglecting the base) is 9,000 cm2.  We are interested in finding
 the loss of the styrene  as a function of time to the room air.

       First obtain a value for the diffusion coefficient of styrene in  the ABS plastic. ABS
 is not one of the six polymers for which the  computer program includes a  subroutine for
 estimation of Dp.  Thus, Dp must be estimated by classifying ABS into one of the  four
 generic polymer classes and using Figure 6 to estimate a value for Dp.   From Table 7,
 ABS is ranked at the low end of the Polyolefins II class and with log(104) = 2.0,
 Dp » 3 x 10-'° cmVs from Figure 6.

       Thus, the required input data are:

             Dp = 3 x 10-10 cnrYs

             L  = 0.2  cm

       •      One-sided migration

5.1.2   Initial Considerations

       Neglecting any  consideration of rates,  one  can readily obtain the  total migration of
the styrene through a simple calculation given a value for the initial  concentration, C^.
                                          57

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       Total Migration  =  (area)(L)(Cpt0)

                        =  (9,000 cm2)(0.2 cm)(450 x 10"6 g/cm3)

                        =  0.81 g styrene

 This value could be employed to obtain a maximum possible concentration of styrene in
 the room air. At 25°C, there are (19)(10S)/(8.314)(298) = 77 moles of air in the room,
 calculated using the ideal gas law.   If 0.81 g (0.0081 moles) of styrene were added, the
 styrene concentration would be (0.0081/77) x  106 = 105 ppm (by volume).  If this
 maximum possible concentration  were high relative to exposure risk, one  would proceed to
 consider migration rates over time using the computer model for more realistic estimates of
 the styrene concentrations in the  air.

 5.1.3   Estimations of Migration

 5.1.3.1   No  Partitioning, No External Mass Transfer Resistance

       In this case, we assume that  there is no partitioning between the styrene  in the ABS
 and the room air and, in addition, there is no external mass transfer resistance for styrene
 leaving the TV console and entering the air.  Thus, the solubility of the styrene in air is
 unlimited as  is the external phase mass transfer coefficient.  With reference  to the
 computer program, partitioning and mass transfer options are not selected  and,
 consequently, the computer program proceeds to Eq. (3-23).

       Suppose we choose times  equal to one day, one week, one month, and one year
 after installation of the console at which to estimate the fraction migrated, M,.   Then with
 Dp = 3  x 10-10 cmVs and L = 0.2 cm,

                                                           Fraction Migrated,
              Time                 t = Drt/L2                   EQ. (3-23)

              1 day                 6.5 x 10"*                      0.03
              1 week                4.5 x 10'3                      0.08
              1 month                 0.020                        0.16
              1 year                   0.24                        0.55

This worst case situation indicates that about 8% of the styrene is lost within a  week  and
 some 50% within  a year.  We  next  illustrate some simple perturbations in the problem.

5.1.3.2   Two-Sided Loss

       Suppose one assumes  that styrene can be lost from both sides of the console (i.e.,
the  styrene leaving the  interior face  would exit to the room through vents in the console).
Then, in the  problem formulation, two-sided loss would be specified and  the "L" term in
Eq. (3-23) becomes one-half  the true thickness, 0.2/2  = 0.1 cm. After one day, then
T = (3  x  10-10)(8.64 x  lO4)/^.!)2 = 2.6 x  10'3, four times larger than in the one-sided case.
With Eq.  (3-23), the fraction migrated is then estimated as 0.06 after one day which, for
this particular example, is twice that for the one-sided situation.
                                          58

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       Should one be interested in converting the fraction migrated estimate to the mass of
styrene lost in the two cases, Eq.  (3-16) is used with the initial concentration, effective
thicknesses, and areas.

                                    Duration:   1 day

                                          One-Sided                 Two-sided

Fraction Migrated                           0.03                       0.06
Mass Lost/Area, (M.Cp^L)           (0.03)(450 x 10*)(0.2)      (0.06)(450 x  10^)(0.1)
  g/cm2                                  = 3.0 x 10"6                = 3.0 x  10*

Area for Loss, cm2                          9,000                    (2)(9,000)

Total Styrene Loss, g                        0.03                       0.06

Note that all migration estimates by the computer program  are provided as the fraction
migrated, M^.  These values can be readily converted to a mass/area or total mass
migrated using the initial concentration, the polymer thickness, and the polymer surface
area.

5.1.3.3   Air Row Through Room

       Instead of having a constant closed air reservoir in the room, suppose there was air
flow in and out equivalent to  5 room volume changes per day.  Then one could compute
the average styrene concentration for any given day by computing the total lost at  the end
of the day and subtracting the loss at the start of  the day and dividing by the total air
moved through the room during the day.   Obviously, the average concentration  would be
highest on the first day and decrease thereafter.  Such concentration calculations, however,
are not included as part of the computer model.  The user should refer to other volumes in
this  series for guidance.

5.1.3.4   Diffusion Coefficient Variation

       The estimate of 3 x 10"10 cm2/s as the diffusion coefficient for styrene in the ABS
TV cabinet was, at best, approximate.   Suppose one perturbed this estimate an order of
magnitude in either direction,  i.e., consider Dp = 3 x 10"9 cm2/s  and Dp = 3 x 10"n cm2/s.
Let the time  be 1 day and assume one-sided migration with L = 0.2 cm:

                                   Duration:  1 day
                                                             Fraction  Migrated
                  P^                    T = Drt/L2                Eq. (3-23)

             3 x 10'9 cm2/s              6.5  x  10'3                   0.09
             3 x 10'10  cnf/s              6.5  x  10^                   0.03
             3 x 10'11  cm2/s              6.5  x  10'5                   0.01

In the case for the highest estimate of Dp, 9% of  the styrene is lost in the first day
whereas for the lowest estimate only 1%  was lost.

                                           59

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5.2    Partition Limited Examples

       In some instances, the migration of an additive from a polymer decreases in rate
and eventually stops as the concentration of the additive increases in the external phase
until it is in equilibrium with the additive in the polymer.  Such partitioning effects are
normally important only when the volume of the external phase is small and the solubility
of the  additive in  it is low.  We consider such a case here but allow the external mass
transfer resistance  to be  unlimited.

5.2.1   Background

       A 5-liter polypropylene (isotactic) bag is to be produced that will contain potable
water.  The wall thickness is  about 10 mils (0.025 cm) and the area =  1,400 cm2.  The
polypropylene contains an  antioxidant (molecular weight is 250) that is  initially  present at a
level of 1,200 ppm. The density of the polypropylene is close to 1  g/cm3.

5.2.2   Initial Considerations

       A worst case analysis as illustrated in Section 5.1 would indicate that the total
antioxidant present is:
                  (area)(L)(Cp>0) = (1,400)(0.025)( 1,200 x 10^) = 0.042 g

If this amount of the antioxidant were to enter the 5 liters of water, a concentration of
about 8.4 ppm would result.  However, this value is above the saturation concentration
(6 ppm)  of the antioxidant in water.  Thus, partitioning must be considered.

       To use the computer program, an estimate of the diffusion coefficient of the
antioxidant in the isotactic polypropylene is required.   Again,  this polymer is not one of
the six for which an estimation equation is provided in the computer model.  From
Table 7,  we find that this polymer is listed near the bottom of the class Polyolefins II.
So, with  a molecular weight of the antioxidant of 250 from Figure 6, Dp ~ 5 x 10"u cm2/s.
The migration is one-sided with L = 0.025 cm.  We have no data concerning the solubility
of the antioxidant in the polymer.  As suggested  in Section 4, we could assume the value
was the same as C^ (1,200 ppm) although this value is probably low.
5.2.3   Estimation of Migration

       The necessary input data for the case of partitioning but no external mass transfer
resistance are shown below for the example under consideration.

             Dp = 5 x 10-11  cmVs

             L = 0.025 cm

       •      One-sided exposure

             Solubility in polypropylene, Clf - C^ = 1,200 ppm (1.2 x 10'3 g/cm3)
                                          60

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       •      Solubility in water, C,f = 6 ppm (6 x 10* g/cm3)

       •      Volume of external phase, Ve = 5,000 cm3

              Area for migration, A = 1,400 cm2

Eq. (3-26) is employed and, here,

       K = CM (water)/Cw (polypropylene) = 6/1,200 = 0.005

       a = Ve K/AL = (5,000)(0.005)/[(1,400)(0.025)] = 0.71

The value of a (= 0.7) is significant because, by Eq. (3-39), the maximum fraction
migrated is o/(l + a) = 0.42 before a partition equilibrium is achieved.  This migration
would correspond to that after long times and would give a concentration of about
                       = (0.42)(1200 x  10^)(1400)(0.025)(106)/(5000) = 3.5 ppm

To estimate the effect of time on the fraction migrated, with Eq.  (3-26) we find

                                                        - Fraction Migrated
              Time                T = Drt/L2                 Eg. (3-26)

              1 day                 6.9 x 10'3                   0.08
              1 week                4.8 x 10'2                   0.19
              1 month                 0.21                       0.32
              3 months                0.62                       0.40
              1 year                    2.5                       0.42
Therefore, if the water is held in the bag for only one day, approximately 8% of the
antioxidant would migrate and the expected concentration would be less than 1 ppm.  After
a week, the concentration would be ~ 2 ppm while if the water is stored for more than
three months, the antioxidant attains a partition equilibrium between the water and polymer
with a concentration of ~ 3.5 ppm in die water.  If the bag were used for multiple fillings
and drainings, however, the antioxidant level  in the polypropylene available for migration
would  soon drop to a low value.

5.3    Mass  Transfer Examples

5.3.1   Background

       This example concerns the loss of a plasticizer DEHP (di-2-ethylhexyl phthaiate)
from PVC  (polyvinyl chloride) tubing into  water.  The PVC is fabricated into a tube about
2 cm in inner diameter, d, with  a wall thickness of L = 0.1 cm.  The  tubing length is
about 23 m (75  feet).  The  initial loading of the DEHP is 50  phr (parts per hundred resin)
or 0.33 weight fraction.  Since the density  of PVC (without plasticizer) is approximately
1.4 g/cm3 and the density of DEHP is approximately 1  g/cm3,  the initial density of the
plasticized  PVC  tube is
                                          61

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            plasticized PVC density =  (50 +  100)/[50 + (100/1.4)] = 1.24 g/cm3

 Also, the initial concentration of the DEHP in the PVC tubing is

                              C^ =  phr/[phr + (100/1.4)]

                           = 50/[50 + (100/1.4)] = 0.41 g/cm3.

 The diffusion coefficient of DEHP within plasticized PVC is not well-documented.
 Quackenbos (1954) reported values obtained by a variety of techniques  over a temperature
 range from 20°C to 90°C.  We will consider two temperatures, 20°C and 80°C and, from
 his data  for a loading of 47 phr,

                           DM = 3.5 x 1Q-12 cmVs

                           DJO = 3 x  10-10 cm2/s

       The solubility of DEHP in water is not well  known.  A review of the literature
 indicates an average value of about 1 ppm or  1 x 10"6 g/cm3.  For this  example, this value
 of C^ is assumed to be temperature independent.  For Cw the solubility of DEHP in PVC,
 we know the value is greater than 50 phr or 0.41 g/cm3, but since we do not have  a
 definitive value, we will  use the initial concentration of 0.41  g/cm3.  Thus, the estimated
 partition  coefficient is

                              K = KW0.41 = 2.4 x 10-6

       As water is inside the tubing, migration is one-sided.

       With these estimates  of parameters, let  us next consider some examples.

 5.3.2    Stagnant Water

       The maximum transfer of DEHP from the PVC tube to stagnant water in the tube
 would occur with a fractional loss at the partition equilibrium given  by  Eq. (3-39)

                                   M, = o/(l + a)

 where  a  = KV./LA.  (V«/A) is the external phase volume per surface area of transfer.  For
 a length  of tubing = Z,

                                   Ve = (7cd2/4)(Z)
                                     A = (7td)(Z)
                             (VyA) = (d/4)  = 0.5  cmVcm2

 and, with K « 2.4 x 10* and L = 0.1 cm,

                         a = (2.4 x iO^XO.SyO.l = 1.22 x Ws.
With Eq. (3-39), the fraction of DEHP in the tubing which would migrate to stagnant
water at equilibrium is 1.22 x  10"s.

                                          62

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 Thus, the mass of DEHP lost from the tubing of length  Z = 2300 cm is

              HC^LA = (1.2 x 10-5)(0.41)(0.1)[(0.2)(TC x 2)(2300)J = 0.007 g

 The volume of water in length Z = 2300 cm is

                      Ve = (jcie  « 1 ppm (1 x 10"*  g/cm3)

             Ve = 7226 cm3

             A = 14451 cm2

For this example, K = 2.4  x 10/6 and a = 1.22 x 10"s and the computer program gives the
following estimates of  the fraction migrated over time using Eq. (3-26).
                                                          Fraction Migrated
              Time                T  = Drt/L2                 (Eq. 3-26)
              11	                  IM    t  "'"'            ^—^—i     ITL  ~~  ~~— —

            0.5 hours                6.3 x  10'7                  1.20 x 10"s
             1  hour                 1.3 x  10^                  1.21  x 10'5
              1 day                 3.0 x  10'3                  1.22 x 10'5

As the results show, the partition equilibrium is  reached in almost one hour.  Because of
the partitioning limitations only a very small fraction of the DEHP initially present in the
tubing is estimated to migrate.  The partition limitation is driven mainly by the low value
for K (i.e., the  solubility of DEHP in water relative to that in the PVC).
                                          63

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5.3.3  Flowing Water

       Next we consider the situation in which fresh water flows through the tubing at
such a high rate that the external mass  transfer coefficient is very large and there is no
partitioning of DEHP in the water.  In  this "worst case" (maximum DEHP loss),
Eq.  (3-23) is operative. The same input parameters are used except in this case we will
treat two  temperatures, 20°C and 80°C  with D^ =. 3.5  xlO'12 cnf/s and D^ = 3.0 x 10'10
cm2/s.  In these cases,

                                                 Fraction Migrated
                                        	Eq. (3-23)	
                 Time                   T_=_20^C                  T = 80°C

                 1 day                      0.006                     0.06
                 1 week                     0.016                     0.15
                 1 month                    0.034                     0.32
                 1 year                      0.12                       0.92

Since a very high flow rate of water was assumed,  the DEHP concentration in the outlet
flow would be negligible, but, clearly,  at 80°C, large losses  are estimated at long times.
Such losses actually  invalidate the procedure  since the diffusion coefficient  decreases as the
plasticizer concentration drops.  When  encountering situations where large losses are
predicted and Dp is a strong function of concentration, the simple equations given in this
report are not valid.

5.3.4   External Mass Transfer Resistance But No Partitioning

       In this case, we assume that there is a low external mass transfer coefficient but
that there is a sufficiently high volume to minimize partitioning.  That is, we still set
a —> oot but we must now consider the dimensionless group y = KkL/Dp.

       Mass transfer from walls of plastic tubes to water was considered in Section 4.5.1.2
and Eq. (4-19) developed.  However, to illustrate the procedure, let us first return to the
basic relation  [Eq. (4-15)] to estimate k,  and  compare results with values from the
simplified equation [Eq. (4-19)].

       To employ Eq. (4-15), we require some physical constants.

       «      kinematic viscosity of water
                    v = 0.01  cnf/s at 20°C
                    v = 0.0036 cnf/s at 80°C

       •      diffusion coefficient of DEHP in water
                    We employed Eq. (4-3)  with Vm determined from Table 11.  Since
                    there are 24 carbon atoms, 4 oxygen atoms,  38 hydrogen atoms, one
                    ring, and three double bonds in DEHP,

            Vm = (24)(7) + (4)(7) + (38)<7)  + (l)(-7) + (3)(7)  =  476 cm'/mole
                                          64

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              and

                               Dw = (5.1 xlO-7)(T)/(476)°-6

                               Dw (20°C) = 3.7 xlO^ cm2/s

                               Dw (80°C) = 4.4 xlO^ crnVs

       •      Schmidt number = v/Dw

              (Sck = 2700

              (ScXo = 820

Then, with d = 2.0 cm and the basic relation Eq. (4-15),

              kj,, = 4.1 x lO'5 v°-8  (cm/s)

              kgo = 7.4 x 10-5 v°-8  (cm/s)

with v,  the water velocity, in cm/s.

       If we had employed the  simplified Eq. (4-19), we would have obtained,  for both
temperatures,

       k = 6.3 x 10-5 v°-8 (cm/s)

Thus, one can see that temperature is not an important variable in estimating mass transfer
coefficients and the simplified Eq. (4-19) will provide reasonable accuracy.  For this
example, we will select a velocity of 10 cm/s and, thus, the estimated value for k is
1.2 x 10"5 cm/s at both temperatures.

       The dimensionless group y = KkL/Dp can then be estimated since K - 2.4 x 10"*,
k =  1.2 x lO'5 cm/s, L = 0.1 cm and Dp = 3.5 x 10'n cm2/s at 20°C and 3.0 x  10'10 cm2/s
at 80°C.

      720 = 0.82

      Ys, = 0.01

The  fraction migrated is then found from Eq. (3-20) for conditions with external phase
mass transfer resistances but no solubility limitations.
                                           65

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T = 20°C
2.5 x 10-5
1.7 x 10*
7.3 x 10-4
8.5 x 10-3
T = 80°C
2.5 x 10-5
1.7 x 10*
7.4 x 10^
9.0 x ID'3
                                                 Fraction Migrated
                                        	Eq. (3-20)
                 Time

                  1 day
                  1 week
                  1 month
                  1 year

       In this example, the mass transfer resistance in the external phase is controlling the
migration process.  Thus, there are only small differences in the amount migrated estimates
at the two temperatures even though Dp varied by two orders of magnitude over  this
temperature range.  These results, in comparison with the results reported for the no mass
transfer/no partitioning case  in Section 5.3.3, illustrate that the mass transfer resistance in
the PVC-water boundary layer reduces the  amount of DEHP estimated to migrate into the
water flowing through the PVC  tubing. However, these amounts migrated are higher than
those estimated when partitioning alone was considered.

5.3.5  External Mass Transfer Resistance With Partitioning

       In a real case, with a long plastic tube,  it is possible to attain a situation where
there are no partitioning effects  near the inlet, but, due  to accumulation of migrant during
flow, the plastic tube walls downstream "see" water with a  sufficiently high concentration
of additive to cause partitioning.  The models developed in  this report do not address such
a case with variable conditions in the external phase during the migration process.
However, to illustrate the applicability of Eq. (3-17) which allows for both external mass
transfer, as well as partitioning,  we assume that the water in the plastic tubing is
recirculated sufficiently fast  so that there is an  "average" additive concentration in the
water independent of position.

       For this example, we will use the conditions and parameters specified earlier for
DEHP migration from PVC  tubing into water at 20°C:

              Dp = 3.5  x 10-u cmYs at 20°C

       •   •    L = 0.1 cm

              d =  2.0 cm

       «       One-sided migration

              Solubility in PVC, C,^ - C^ = 0.41 g/cm3

              Solubility in water, CM « 1.0 x 10* g/cm3

              Ve = 7226 cm3

              A = 14451 cm2

       •       v  = 10 cm/s

                                          66

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             k =  1.2 x 10'5 cm/s

With these values,

       K = 2.4 x 10^

       a = 1.22 x 10'5

       y = 0.82

       The fraction migrated estimates as a function of time at 20°C are:

                                                         Fraction Migrated
              Time                 t = Drt/L2                 (Eg. 3-27)

             1 hour                1.3 x 10*                 1.0 x  10^
             1 day                 3.0 x 10'5                 1.1 x  10'5
             1  week                2.1 x 1Q-4                 1.22 x 10'5 -
             1 month               9.1 x 10^                 1.22 x 10'5

       In this example, a partition equilibrium is reached after approximately one week.  In
comparison, when partitioning but no mass transfer was considered in Section 5.3.2, the
equilibrium was reached in about one hour.  Thus, the combined  effects of mass transfer
and partitioning increase the time required to reach the partition equilibrium.
                                         67

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6.     COMPUTER PROGRAM

       AMEM (Arthur D. Little Migration Estimation Model) is a computer program that
facilitates the rapid estimation of the fraction of the additive originally in a polymer sheet that
will migrate under the various conditions defined in Section 3.  AMEM incorporates many of
the physical property estimation techniques presented in Section 4.  AMEM was coded in
FORTRAN for operation on ffiM-AT personal computers and compatibles.  The program
should be run  on a personal computer with a math coprocessor.   The FORTRAN code and
operational flowchart are provided in Appendix A. The program and floppy diskette on
which it was delivered should be considered integral parts of this report.

       With AMEM, one can estimate migration to both fluid and solid external phases.  The
program is designed to enable its user to first estimate the maximum rate of migration from
the polymer with the minimum set of inputs.  If this fraction migrated is below a "trigger"
level, then the user need proceed no further.  If the fraction migrated is above the trigger
level or if the  user wants to explore other scenarios, then additional inputs are required of the
user. Because of coprocessor limitations regarding the convergence  of some infinite series
terms near zero, the smallest fraction migrated value predicted by AMEM under worst case
conditions is 1 x 10~4 fraction migrated.  For fluid external phases, estimates are made for the
worst-case scenario in which migration is not hindered by partitioning or mass transfer
resistances in the external phase [Eq. (3-23)], for scenarios that consider partitioning effects
[Eq. (3-26)], and for scenarios that consider mass transfer [Eq. (3-20)], and for scenarios  that
consider both partitioning and mass transfer limitations [Eq. (3-17)].  For solid external
phases, the computer program estimates migration  for the worst  case scenario (Eq. (3-23)] and
for scenarios with and without partitioning limitations [Eq. (3-29) and (3-31)].

       AMEM contains subroutines to estimating values for the diffusion coefficient of the
additive in the polymer, air, and water, for the partition coefficient; and for the external phase
mass transfer coefficient.  The equations and techniques used in these subroutines are
designated in the appropriate paragraphs of Section 4. The FORTRAN code for these
routines is given in Appendix A.

       AMEM's user interface was developed to facilitate data input selections.  The  user is
queried for the required input data through a series of information screens and selection
menus.  Table  12 summarizes the input data requirements for the various migration scenarios
and equations.  The program calculates the fraction migrated for the conditions specified.
The output screen repeats the key inputs as well as presents the  results of the calculations.
The user may execute "Print Screen" or print the appropriate Eq*.LIS file to document the
results.  The user may then re-run the program changing either one or more of the data
inputs, or terminate the session.

       Appendix B provides a limited validation of AMEM in which the AMEM predictions
are compared with migration data from the technical literature for 13 example cases.
                                          68

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    TABLE 12.  SUMMARY OF COMPUTER PROGRAM INPUT REQUIREMENTS
               Input Variable
   Section
POLYMER

   Thickness, cm
   Exposure to External Phase, one or both sides

EXTERNAL PHASE

   Air, Water, or Solid

TIME

   Period over which migration occurs, hrs

DIFFUSION COEFFICIENT OF  MIGRANT IN POLYMER, cm2/s

   Estimated by program if unknown using:

     Migrant Molecular Weight, g/mole
     Polymer Type, select from menu:

       Silicone  Rubber
       Natural Rubber
       LDPE
       HDPE
       Polystyrene
       Polyvinyl Chloride (unplasticized)

PARTITIONING EFFECTS (Air  or Water External Phase)

   To consider, input:

   Volume of External Phase, m3
   Surface Area  of Polymer, cm2
   Partition Coefficient - to calculate input:

     Saturation Concentration of Migrant in Polymer, g/cm3

       If unknown use Initial Concentration, g/cm3
Scenario Specific
Scenario Specific
Scenario Specific
       4.2
       4.4
       4.4.1
                                                                     (continued)
                                      69

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                               TABLE 12.  (continued)
                Input Variable                                            Section*


PARTITIONING EFFECTS (Air or Water External Phase) (continued)

     Saturation Concentration of Migrant in External Phase, g/cm3               4.4.2

       To estimate input:

          for AIR:        Migrant Vapor Pressure, torr.                        4.4.2.1

          for WATER:    Migrant Melt Temp., °C                            4.4.2.2
                         Migrant Octanol/Water Partition Coefficient

MASS TRANSFER EFFECTS (Air or Water External Phase)

   To consider, input:

   Mass Transfer Coefficient, cm/s                                           4.5

      for AIR:        Air Flow Velocity, cm/s                                4.5.2
                     Polymer Position -  Horizontal or Vertical
                     Polymer Location - Indoors or Outdoors

      for WATER:    Water Flow Velocity, cm/s                             4.5.1
                     Polymer Plate Surface Length or Pipe
                     Diameter, cm

   Partition Coefficient - see above

SOLID EXTERNAL PHASE

   To consider, input:

   Diffusion Coefficient in External Phase, cm2/s                               4.3.3
                                                                          (4.2)
   Section of report that describes the input variable and documents the procedures used to
   estimate it in the computer program.
                                        70

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7.     NOMENCLATURE
This section provides definitions of the symbols and equation variables used throughout the
text. When applicable, the variable name used in the AMEM program code provided in
Appendix A is listed in parenthesis.
A           Surface area of polymer exposed to external phase, cm2; (SAP)
Ca           Solubility of migrant in air, g/cm3; (CSATA)
Ce           Concentration of migrant in external phase, g/cm3
                                                    f*
Ce s          Solubility of migrant in external phase, g/cm ; (CSATE)
                                                   *-t
Cp           Concentration of additive in polymer, g/cm
Cp 0          Initial concentration of additive in polymer, g/cm3;  (CINTT)
Cp s          Solubility of aditive in polymer, g/cm3; (CSATP)
Cw          Solubility of migrant in water, g/cm3; (CSATW)
d            Diameter of pipe, cm; (DPEPE)
Da           Diffusion coefficient of migrant in air, cm2/s; (DAIR)
De           Diffusion coefficient of migrant in external phase, cm2/s; (DEXT)
D           Diffusion coefficient of migrant in polymer, crr^/s;  (DP)
                                                    f\
Dw          Diffusion coefficient of migrant in water, cm /s; (DMIG)
                                              SJ
g            Acceleration due to gravity, 980 cm/s
h            Convective  heat transfer coefficient
H           Height of vertical polymer surface over which air is flowing, cm; (HEIGHT)
k            Mass transfer coefficient, cm/s; (RK)
K           Partition coefficient; ratio of additive concentration in external phase to
             additive concentration in polymer at equilibrium;  (PC)
Kow          Octanol-water partition coefficient; (XKOWM)
/             Characteristic length of polymer in direction of external phase flow, cm;
              (RSURFA or RSURFW)
L            Thickness of polymer in cases of one-sided exposure to external phase;
              half-thickness of polymer in cases of two-sided exposure to external phase,
              cm; (XLEN)
M           Molecular weight, g/mol
Ma          Molecular weight of air, 29 g/mol
             Molecular weight of migrant,  g/mol; (XMW)
             Fraction migrated; (FRMIG)
                                         71

-------
P           Pressure, atm
Pv          Vapor pressure of migrant, ton; (VPMT)
R           Gas constant
t           Time, seconds; (TSEC)
T           Temperature, degrees kelvin
Tm         Melting temperature of migrant, °C; (TMM)
v           Bulk velocity of external phase, cm/s; (AIRVEL or WATVEL)
                            o
V           Molar volume, cm /mol
                                     «3
Va         Molar volume of air, 19.7 cm /mol
Ve         External phase volume; (VHP)
Vm         Molar volume of migrant, cm3/mol
Vp         Polymer volume
5           Boundary layer thickness, cm
v           Kinematic viscosity, cm2/s
p           Density, g/cm
X,           Thermal conductivity of air
                           DIMENSIONLESS GROUPS
a           (Ve/AXK/L); (ALPHA)
P           KOVDp)0"5; (BETA)
T           Dpt/L2; (TAU)
7           kKL/Dp; (GAMMA)
Gr          Grashof number, (gH3/v2)[(p0-p J/p J
Nu          Nusselt number, J\lfk
Re          Reynolds number, dv/v
Re,          Length Reynolds number, v//v
Sc          Schmidt number, v/De
Sh          Sherwood number, k//De
                                      72

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8.     REFERENCES
Arthur D. Little, Inc. 1983. "Migration of BHT and Irganox  1010 from Low Density
Polyethylene".  Summary Report on FDA Contract 223-77-2360 (1983).

ASHRAE Handbook and Product Directory. 1980.  American Society of Heating,
Refrigeration and Air-Conditioning Engineers, Inc., New York. Chapter 21.

Ayres, J.L.,  J.C. Osborne, H.B. Hopfenberg, and WJ. Koros.  1983. "Effect of Variable
Storage Times on the Calculation of Diffusion Coefficients Characterizing  Small Molecule
Migration in Polymers".  Ind. Eng.  Chem. Prod. Res. Dev. Vol. (22):86-89.

Barr,  R.F. and H. Watts. 1972. "Diffusion of Some Organic and Inorganic Compounds in
Air".  J. Chem. Eng. Data. Vol. (17):45-46.

Barrens, A.R. and H.B. Hopfenberg. 1982.  "Diffusion of Organic Vapors at Low
Concentrations in Glassy PVC, Polystyrene, and PMMA". J.  Membrane Sci. Vol. (10):283-
30.

Bixler, HJ.  and O.J. Sweeting. 1971. "Barrier Properties of Polymer Films". The Science
and Technology of Polymer Films, Vol. II". Sweeting (ed), Wiley & Sons, NY.

Crank, J.. 1975. The Mathematics of Diffusion.  Clarendon Press,  Oxford.

Crank, J. and G.S. Park.  1975. Diffusion in Polymers. Academic  Press, NY.

Eckert, E.R.G. and R.M. Drake. 1972. Analysis of Heat and  Mass Transfer. McGraw-Hill,
New York.

Farmer, W.J., M.S. Yang and J. Letey. 1980. "Land Disposal of Hexachlorobenzene
Wastes—Controlling Vapor Movement in  Soil". U.S. Environmental Protection Agency
Report No. EPA-600/2-80-119.

Flynn, J.H..  1982. "A Collection of Kinetic Data for the Diffusion of Organic  Compounds
in Polyolefms". Polymer. Vol. (23):1325-1344.

Fuller, E.N.  and J.C. Giddings. 1965. "A Comparison of Methods for Predicting Gaseous
Diffusion Coefficients". J. Gas Chromatogr. Vol. (3):222-7.

Fuller, E.N., K. Ensley, and J.C.  Giddings. 1969. "Diffusion  of Halogenated Hydrocarbons
in Helium".  J. Phys. Chem. Vol.  (73):3679-3685.

Fuller, E.N., P.D. Schettler, and J.C. Giddings.  1966. "A New Method for Prediction of
Binary Gas-Phase Diffusion Coefficients". Ind. Eng. Chem. Vol. (58)(5): 18-27.

Gandek, T.P. 1986.  "Migration of Phenolic Antioxidants From Polyolefms  to Aqueous
Media with Application to Indirect Food Additive Migration". Ph.D.  Thesis. Mass.  Inst.  of
Tech.  Cambridge, MA.  .
                                         73

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Goydan, R., R.C. Reid, and H.-S. Tseng. 1989. "Estimation of the Solubilities of Organic
Compounds in Polymers by Group Contribution Methods". Ind. Eng. Chem. Res. Vol.
(28):445-454.

Grain, C.F.  1982. "Vapor Pressure". Handbook of Chemical Property Estimation Methods.
W. Lyman,  W. Reehl, and D. Rosenblatt (eds), McGraw-Hill Book Co., NY.

Grun, F. 1949.  "Measurements of Diffusion in Rubber". Rubber Chem. and Technol. Vol.
(22):316-319.

Hatton, T.A., A.S. Chiang, P.T. Noble, and E.N. Lightfoot.  1979. "Transient Diffusional
Interactions  Between Solid Bodies  and Isolated Fluids". Chem. Eng. Sci. Vol. (34): 1339-
1344.

Hatton, T.A. 1985. "On the Calorimeter Problem for Finite Cylinders and Rectangular
Prisms". Chem. Eng. Sci. Vol. (40): 167-170.

Hayduk, W. and H. Laude.  1974. "Prediction of Diffusion Coefficients  for Nonelectrolytes
in Dilute Aqueous Solutions". AICHE J. Vol. (20):611-615.

Hober, R. 1945. Physical Chemistry of Cells and Tissues. Blakiston Company, PA (1945).

Lugg, G.A.  1968. "Diffusion Coefficients of Some Organic and Other Vapors in Air".
Anal. Chem. Vol. (40): 1072-1077.

Lyman, W.J., R.G. Potts and G.C.  Magil. 1982. User's Guide  [tol  CHEMEST. prepared for
the U.S. Environmental Protection Agency, Office of Toxic  Substances, Washington, DC,
and the U.S. Army Medical Bioengineering Research and Development Laboratory, Fort
Dedrick, Frederick, MD.

Lyman, W.J. 1982. "Solubility in Water" in Handbook of Chemical Property Estimation
Methods. W. Lyman, W. Reehl and D. Rosenblatt (eds), McGraw-Hill Book Co., NY.

Lyman, W.J., W. Reehl, and D. Rosenblatt. 1982. Handbook of Chemical Property
Estimation Methods. McGraw-Hill Book Co.,  NY.

Nan, S.H. and A.G. Thomas. 1980.  "Migration and Blooming of Waxes to the Surface of
Rubber Vulcanizates". J. Poly. Sci.: Poly Phys. EA Vol. (18):511-521.

Park, G.S. 1950. "The Diffusion of Some Halo-methanes in  Polystyrene". Trans. Faraday
Soc. Vol. (46):684-697.

Park G.S. 1951.  "The Diffusion of Some Organic Substances in Polystyrene". Trans.
Faraday Soc. Vol. (47):1007-1013.

Quackenbos, H.M. 1954. "Plasticizers  in Vinyl Chloride Resins, Migration of Plasticizer".
Ind. Eng. Chem. Vol.  (46):1335-1344.

Reid, R.C., J.M. Prausnitz, and B.E. Poling.  1987. The Properties of Gases and Liquids.
McGraw-Hill Book Co., NY. 4th edition.

                                         74

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 Reid, R.C., J.M. Prausnitz, and T.K. Sherwood. 1977. The Properties of Gases and Liquids.
 McGraw-Hill Book Co., NY.

 Reid, R.C., K.R. Sidman, A.D. Schwope, and D.E. Till. 1980. "Loss of Adjuvants from
 Polymer Films to Foods or Food Simulants.  Effect of the External Phase". I&EC Prod.
 Res. and Dev. Vol. (19):580-587.

 Rodriquez, F. 1982. Principles of Polymer Systems. McGraw-Hill Book Co., NY. 2nd
 edition.

 Seymour, R.B. and C.E. Carraher. 1981. Polymer Chemistry. Marcel Dekker,  NY.

 Skelland, A.H.P. 1974. Diffusional Mass Transfer. Wiley and Sons, NY.

 Smith, L.E., I.C. Sanchez, S.S. Chang, and F.L. McCrackin. 1979. "Models for the
 Migration of Paraffinic Additives in Polyethylene". National Bureau of Standards, NBSIR
 79-1598, pp 4-11.

 Till, D.E., R.C. Reid,  P.S. Schwartz, K.R. Sidman, J.R. Valentine, and R.H. Whelan. 1982.
 "Plasticizer Migration  from Polyvinyl  Chloride Film to Solvents  and Foods". Fd.  Chem.
 Toxic. Vol. (20):95-104.

 Till, D.E., D.J. Ehntholt, A.D. Schwope, K.R. Sidman, R.H. Whelan, and R.C. Reid.  1983.
 "A  Study of Indirect Food Additive Migration".  Final Report on FDA Contract Number
 223-77-2360.

 Valvani, S.C., S.H. Yalkowsky and T.J. Roseman. 1981. "Solubility and Partitioning.  IV.
 Aqueous Solubility and Octanol-Water Partition Coefficients of Liquid Non-electrolytes". J.
 Pharm. Sci. Vol. (70):502-507.

 Yalkowsky, S.H. 1982. (University of Arizona, Tuscon, AZ).  Personal communication to
 W.  Lyman of Arthur D. Little, Inc.

 Yalkowsky, S.H., S.C. Valvani, and D. Mackay.  1983. "Estimation of the Aqueous
 Solubility of Some Aromatic Compounds". Residue Rev. Vol.  (85):43-45.

Yasuda, H. and V. Stannett. 1975. "Permeability Coefficients". Chap. HI.9 in  Polymer
Handbook. 2nd Ed. Bandrup and Immergut (eds), Wiley and Sons, NY.
                                         75

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                                  APPENDIX A
                            FORTRAN Code for AMEM
             Arthur D. Little Migration Estimation Model Computer Program
   This appendix provides a listing of the FORTRAN code for the AMEM computer
program, a brief description of the functions and subroutines used, a definition list for the
program variables, and a program flowchart with example AMEM input/output screens.
                                       76

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                                    c
   Start
                                   Select External Phase
                                  Specify 1 or 2 sided, L,t,Dp
                                    Worst Case Migration
                                         Eq (3-23)
                                       Module Eq 23
                      Modify Worst
                    Case Input Values
  Consider
 Partitioning
Mass Transfer/
                                                               Quit
                                            Yes


Confirm Initial Input Value
Specify Cp,s

Air or Water

f
Consider
Mass Transfer/
Partitioning
I



Phase

iMass Transfer 1 Partitioning 1 Mass Transfer
f Only * Only * & Partitioning
Specify k, K or Ce,s
Specify K or Ce,s
Ve,A
Specify k, K or
Ve,A
Ce,s
1 t I
Eq (3-20)
Module Eq 20
Eq (3-26)
Module Eq 26
Eq (3-17)
Module Eq 1 7
s,

Solid
i
f
Conside
Partition!
I
iNo
"Partitioning
Specify Da,
Ce,s or K
T
Eq(3-31)
Module Eq 31

r
ig

f Partitioning
Specify De, Ce,s or K,
Ve, A
1
Eq (3-29)
Module Eq 29
Figure A-1.  Flowchart for AMEM Migration Estimation Computer Program

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      DEFINITION OF VARIABLES USED IN PROGRAM AMEM
                                                      0990
Variable
            Description
                                                      Units
AIRVEL  Velocity of air as the external phase, as it moves     cm/s
        across the polymer surface.
ALPHA
BETA
CINIT

CSATA

CSATE

CSATP

CSATW

DAIR


DEXT

DMIG


DP



DPIPE

FRMIG


GAMMA
Non-dimensional variable calculated as the product of
the partition coefficient and the external phase volume,
divided by the exposed surface area and polymer length.

Non-dimensional variable defined as the product of the
partition coefficient and the square root of the
ratio of the migrant-external phase diffusion coefficient
and the migrant-polymer diffusion coefficient.

Initial concentration of migrant in polymer            g/cm3

Saturation concentration of migrant in air             g/cm3

Saturation concentration of migrant in external phase  g/cm3

Saturation concentration of migrant in polymer         g/cm3

Saturation concentration of migrant in water           g/cm3

                                                       cm2/s
Diffusion coefficient of migrant in air,  calculated
as a function of the migrant molecular weight.

Diffusion coefficient of migrant in external phase
                                                       cm2/s
Diffusion coefficient of migrant in water, calculated  cm2/s
as a function of the migrant molecular weight.
Diffusion coefficient of migrant in polymer,
which can be user-specified or calculated as a
function of the migrant molecular weight.

Diameter of polymer pipe through which water flows

Fraction of migrant that diffuses into external
phase

Non-dimensional variable that is calculated as the
product of the mass transfer coefficient, partition
coefficient, and polymer length, all divided by the
polymer diffusion coefficient.
                                                       cm2/s
                                                       cm
                                     78

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HEIGHT  Average height  of vertical polymer surface from which   cm
        a migrant  is migrating.

IPOLY   Index  to specify one of six polymer types, as follows:

        01- Silicone rubber
        02- Natural rubber
        03= Low-density polyethylene
        04— High-density polyethylene
        05= Polystyrene
        06- Unplasticized Polyvinylchloride

IR      Integer variable that routes program to read from input file

IRWAT   Integer variable that is set equal to 1 for water
        flowing through polymer pipe, and 2 for water flowing
        over a polymer  plate

IW      Integer variable that routes program to write to output file

JEXT    Variable used in the solid-solid diffusion case that
        defines the type of external polymer phase. See variable
        IPOLY  for 01-06 polymer type definitions.

KHORZ   Integer variable to define polymer position in air:
        l=Horizontal, 2-Vertical

NEXT    Integer variable to define external phase type:  01-Air, 02-Water

NMAX      : Maximum Iteration Counter (1000) in series convergence routines
        within Equation 23 and Equation 31 modules.

NPLC    Polymer location in air (1-Indoors; 2-Outdoors)

NRTBFFMAX : Maximum Iteration Counter (1000) in series convergence routine
        within Equation 17 module.

NRTBFIMAX : Maximum Iteration Counter (1000) in series convergence routine
        within Equation 20 module.

NRTDFFMAX : Maximum Iteration Counter (1000) in series convergence routine
        within Equation 29 module.

NRTMFFMAX : Maximum Iteration Counter (1000) in series convergence routine
        within Equation 26 module.

NSIDE   Integer variable to define number of polymer sides
        involved in diffusion:   01—One-sided,  02—Two-sided

PC      Partition coefficient,a non-dimensional variable defined
        as the ratio of the saturation concentration of the
        migrant in the external phase to the saturation concen-
        tration of the migrant in the polymer.  This variable can
        be user-specified or estimated by the program.
                                      79

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RK
RSURFA

RSURFW

SAP

TAU



THRS

TITLE

TMM


TSEC

VEP

VEPCC

VPM

VPMT


WATVEL


XKOWM


XL

XLEN


XMW
Mass  transfer coefficient can be specified by the      cm/s
user  or computed within the program as a function
of the polymer position(horizontal,vertical),location
(indoors.outdoors), characteristic length(or diameter),
the type of external phase(water,air), and the
external phase flow velocity.

Surface length of polymer exposed to air               cm

Plate length of polymer exposed to water               cm

Surface area of polymer exposed to external phase      cm2

Non-dimensional variable defined as the product of
the diffusion coefficient of the migrant in the polymer
and time divided by the polymer thickness squared.

Exposure time input by user                            hrs

One-line alphanumeric variable to define scenario

Migrant melt temperature for calculation of            deg C
saturation concentration of migrant in water

Exposure time (units converted hrs->sec)               sec

Volume of external phase as input by user              m3

Volume of external phase (units converted m3->cm3)     cm3

Migrant vapor pressure (units converted torr->atm)     atm

Migrant vapor pressure as input by user to estimate    torr
saturation concentration in air

Velocity of water as the external phase, as it moves   cm/s
across the polymer surface (min.  value lOcm/s)

Migrant octanol-water partition coefficient for
calculation of saturation concentration in water
Total polymer film thickness

Polymer film thickness redefined, based on number
of sides exposed to external phase for diffusion

Molecular weight of migrant
cm
cm
g/g-mol
                                   80

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        FUNCTION SUBROUTINES IN PROGRAM AMEM            0990
     REF: GANDEK.T.P., PH.D. THESIS, M.I.T., CAMBRIDGE, MA 1986

1. DOMBFF:
       Domain BFF:  Finite polymer in contact with a finite fluid external
       phase. Boundary layer resistance.  Returns fractional migration assuming
       the migrant is initially uniformly distributed in the polymer and
       the initial external phase concentration is zero.
       Requires subprogram ROOTBFF

       Note that TAU   = Dpt/L2
                 R     - kKL/Dp
                 ALPHA - KV/AL

2. ROOTBFF:
       Finds roots of the equation
         Tan(BN) - R*BN/(BN2-N)
       or
         Cot(BN) - (BN2-N)/(R*BN)  .
       where
         I  - number of desired root
         R  - kKL/Dp
         ALPHA = KV/AL
         N  - R/ALPHA

3. DOMBFI
       Domain BFI:  Finite polymer in contact with an infinite fluid
       external phase. Boundary layer resistance.   Returns fractional
       migration assuming the fluid phase concentration is initially zero
       and the migrant is uniformly distributed in the polymer.
       Requires subprogram ROOTBFI
       Note that R   - kKL/Dp
                 Tau - Dpt/L2

4. ROOTBFI
       Returns roots of the equation
         tan(GN) - R/GN      or       cot(GN) - GN/R
       where R is a constant

5. DOMMFI
       Domain MFI: Finite polymer in contact with an infinite, well-stirred
       fluid external phase.  Returns fractional migration assuming the
       initial migrant concentration in the fluid external phase is zero and
       the polymer distribution is uniform.

       Note that TAU - Dpt/L2

6. DOMMFF
       Domain MFF:  Finite polymer in contact with a finite, well-stirred


                                  81

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       fluid external phase.  Returns fractional migration, assuming that
       the migrant concentration in the fluid external phase is initially
       zero, and the distribution in the polymer is uniform.
       Requires the subprogram ROOTMFF.

       Note that TAU   - Dpt/L2
                 ALPHA - KV/AL

7. ROOTMFF
       Finds roots of the equation Tan (BN) + ALPHA*BN - 0
       Requires argument ALPHA — KV/AL and N — root #

8. DOMDFF
       Domain DFF:  Finite polymer in contact with a finite solid
       external phase. Returns fractional migration assuming the migrant
       concentration in the solid external phase is initially zero and
       initial concentration in the polymer is.uniform.
       Requires subprogram ROOTDFF

       Note that TAU   - Dpt/L2
                 BETA  = Ksqrt(De/Dp)
                 DOMDFF(TAU,ALPHA,BETA)
       In order to avoid problems with the rootfinding routine,  ROOTDFF,
       ALPHA is perturbed slightly.   The potential problems occur when round
       figures are used for BETA and ALPHA, when either are simply related
       multiples or are common factors of a larger number,  because then
       asymptotes of the two tangent functions (in the characteristic eqn.)
       can coincide.
       If alpha > beta it is faster to calculate migration by letting
       alpha and beta equal their reciprocals and tau - tau times the
       square of beta/alpha.  The value of migration thus calculated
       is multiplied by the true value of alpha to give migration for
       the original case.

9. ROOTDFF
       Finds Nth root of the equation
          tan (LN) + BETA*tan(ALPHA*LN/BETA) - 0
       given the parameters ALPHA and BETA.  The root for N-0 is 0.
         N    — root number; root 0-0
         Nl   — N + 1,  used for calcaulation of asymptotes
         EPSl - desired accuracy of root
         EPS2 — part of interval away from asymptote where  the search begins
         EPS3 - EPS2*BETA/ALPHA  if BETA < ALPHA
              - EPS2             .     if BETA > ALPHA
         ETA  - 1/(1 + BETA/ALPHA)
         BOA  - BETA/ALPHA
         BAB  - BOA     if BETA < ALPHA
              - I/BOA   if BETA > ALPHA
         ETA  - ETA              if ALPHA < BETA
              - 1/(1+ALPHA/BETA) if ALPHA > BETA

       To trace program execution for diagnostic purposes,  set IDIAG — 1
       by passing a negative value of N to ROOTDFF

       When roots cannot be determined (message indicates such), the returned


                                     82

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       root will be negative.

       The two asymptotes bound the desired root.  To find the root, we
       start at the right and use Newton's method to 'walk down the
       curve' to the root.  If an inflection point is encountered, we
       start from the other direction.  If the root is within EPS3 of
       an asymptote, an alternate form of the residual is used.
       Note that TAU   - Dpt/L2
                 ALPHA - KV/AL
                 BETA  = Ksqrt(De/Dp)

10. DOMDFI
       Domain DFI:   Finite polymer in contact with a semi-infinite solid
       external phase.  Returns fractional migration assuming the external
       phase concentration is initially zero and the migrant is uniformly
       distributed in the polymer.
       Requires the subprograms ERFCA, SHANK

       Note that TAU   - Dpt/L2
                 ALPHA - KV/AL
                 BETA  = Ksqrt(De/Dp)

11. ERFCA
       Function ERFCA(Z) calculates the complementary error
       function of the real,  positive argument Z.  The result
       is accurate to about 8 significant figures at worst.
       Requires function SHANK.

12. SHANK
       Function SHANK uses Shanks transform to accelerate
       convergence of a series.
       S is a vector (length 25)  containing N consecutive
       terms of the series.   S is destroyed during the calculation.
                                83

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      PROGRAM NAME:  EQ17
KM DILWALI 0990
 C  THIS  PROGRAM MODULE  SERVES AS THE  I/O SHELL TO
 C  COMPUTE THE FRACTION OF ADDITIVE MIGRATION
 C  ASSUMING THE FOLLOWING CONDITIONS:   (BFF)
 C  - BOUNDARY LAYER RESISTANCE  EFFECTS  CONSIDERED
 C  - POLYMER PHASE THICKNESS MUST BE  CONSIDERED FINITE
 C  - EXTRACTANT PHASE FINITE;I.E. PARTITIONING EFFECTS CONSIDERED
 C  INPUT REQUIREMENTS:
 C        IPOLY : 01-06  POLYMER  TYPE
 C          01- SILICONS RUBBER   04-HDPE
 C          02= NATURAL  RUBBER   05=POLYSTYRENE
 C          03- LDPE              06-PVC(UNPLAS'TICIZED)
 C        XMW     MOLECULAR WEIGHT OF MIGRANT              G/MOL
 C        DP      DIFFUSION COEFF. OF ADDITIVE IN POLYMER  CM2/S
 C        THRS    TIME                                     HRS
 C        XL      POLYMER FILM THICKNESS                   CM
 C        NSIDE   01-ONE-SIDED , 02-TWO-SIDED DIFFUSION
 C        CINIT   INIT MIGRANT CONCN IN  POLYMER           G/CM3  .
 C                (ONLY  REQUIRED IF CSATP IS UNKNOWN)
 C      > NEXT    EXT PHASE TYPE 01-AIR, 02-H20 ,  03-SOLID
 C        VEP     VOL OF EXT PHASE      ,M3
 C        SAP     EXPOSED SURFACE AREA  ,CM2
 C        PC      PARTITION COEFF; 0 IF UNKNOWN
 C        CSATP   SAT CONCN OF MIGR IN POLYMER OR 0.,G/CM3
 C        CSATA   SAT CONCN OF MIGR IN AIR     OR 0.,G/CM3
 C        CSATW   SAT CONCN OF MIGR IN H20     OR 0.,G/CM3
 C        CSATE   SAT CONCN OF MIGR IN SOLID (N/A)   G/CM3
 C      > VPMT    MIGR VAPOR PRESSURE,TORR FOR CSATA,RK CALCS
 C        TMM     MIGR MELT TEMP DEG C    FOR CSATW CALC
 C        XKOWM   MIGR OCT-H20 PART COEFF FOR CSATW CALC
C        RK      MASS TRANSFER COEFF CM/S;  0.  IF UNKNOWN
C        KHORZ   AIR:  POLYMER POSITION 1-HORIZONTAL, 2-VERTICAL
C        NPLC    AIR:  POLYMER LOCN 1-INDOORS,  2-OUTDOORS
C        RSURFA  AIR:  SURFACE LENGTH  ,CM
C        AIRVEL  AIR:  AIRFLOW VELOCITY, CM/S
C        HEIGHT  AIR:  SURFACE HEIGHT FOR VERTICAL POSITION, CM
C        WATVEL  H20:  WATER FLOW VELOCITY ,  CM/S
C        IRWAT   H20:  01-FLOW THRU PIPE; 02-FLOW OVER PLATE
C        RSURFW  H20:  FOR IRWAT-2,PLATE LENGTH,  CM
C        DPIPE   H20:  FOR IRWAT-l.PIPE  DIAMETER, CM
C********************
      IMPLICIT REAL*8 (A-H.O-Z)
      COMMON/PARAM3/B3(1000),NRTBFF,A3LAST,R3IAST,NRTBFFMAX
      TAU - 0.0
      ALPHA - 0.0
      GAMMA - 0.0
      NRTBFFMAX - 1000
C********************
      COMMON/IO/IR.IW
                                 84

-------
   CHARACTER*2
   IR - 5
   IW - 6
   OPEN(IR,
   OPEN(IW,
READ INPUT
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
INITIALIZE
   IF(RK.NE
              TITLE(36)
               •'EQ17.INP'
               •'EQ17.LIS'
FILE
FILE
DATA
'(36A2)')TITLE
'(I2)')IPOLY
'(F10.2)')XMW
'(F10.2)')DP
'(F10.2)')THRS
'(F10.2)')XL
'(I2)')NSIDE
'(F10.2)')CINIT
'(12)')NEXT
'(F10.2)')VEP
'(F10
'(F10
'(F10
'(F10
               STATUS-'UNKNOWN')
                .2)
                .2)
                .2)
                .2)'
        )SAP
        )PC
        )CSATP
        )CSATA
     2)')CSATW
     2)')VPMT
     2)')TMM
     2)')XKOWM
     2)')RK
     )KHORZ
     )NPLC
     2)')RSURFA
     2)')AIRVEL
     2)')HEIGHT
(F10.2)')WATVEL
(I2)')IRWAT
(F10.2)')RSURFW
(F10.2)')DPIPE
           (F10,
           (F10.
           (F10.
           (F10.
           (F10.
           (12)'
           (12)'
           (F10.
           (F10.
           (F10.
           0..AND.WATVEL.EQ.O.)WATVEL-10.
  XLEN - XL
  IF(NSIDE.EQ.2)XLEN-XL/2.
  TSEC - THRS*3600.
ECHO INPUT
           201)TITLE
           210)
           EQ.0)WRITE(IW,209)
              1)WRITE(IW,211)
              2)WRITE(IW,212)
              3)WRITE(IW,213)
              4)WRITE(IW,214)
              5)WRITE(IW,215)
              6)WRITE(IW,216)
              )WRITE(IW,220)XMW
           221)THRS,XL
           0.)WRITE(IW,225)DP
           EQ.1)WRITE(IW,230)
           EQ.2)WRITE(IW,235)
   WRITE(IW
   WRITE(IW
   IF(IPOLY
   IF(IPOLY
   IF(IPOLY
   IF(IPOLY
   IF(IPOLY
   IF(IPOLY
   IF(IPOLY
   IF(XMW.NE.O
   WRITE(IW
   IF(DP.NE
   IF(NSIDE
   IF(NSIDE
 EQ.
 EQ.
 EQ.
 EQ.
 EQ.
 EQ.
   WRITE(IW,280)VEP
                                 85

-------
      WRITE(IW,290)SAP
      IF(NEXT.EQ.1)WRITE(IW,260)
      IF(NEXT.EQ.2)WRITE(IW,265)
      IF(PC.NE.O.)WRITE(IW,331)PC
        IF(PC.NE.O.)GO TO 15
      IF(CSATP.NE.O.)WRITE(IW,266)CSATP
      IF(CSATP.EQ.O.)WRITE(IW,267)CINIT
      IF(NEXT.EQ.1.AND.CSATA.NE.O.)WRITE(IW,268)CSATA
      IF(NEXT.EQ.2.AND.CSATW.NE.O.)WRITE(IW,269)CSATW
      IF(NEXT.EQ.1.AND.CSATA.EQ.O.)WRITE(IW,270)VPMT
      IF(NEXT.EQ.2.AND.CSATW.EQ.O.)WRITE(IW,275)TMM,XKOWM
15    IF(RK.NE.O.)WRITE(IW,330)RK
        IF(RK.NE.O.)GO TO 30
        IF(NEXT.EQ.2)GO TO 20
      IF(KHORZ.EQ.1)WRITE(IW,335)
      IF(KHORZ.EQ.2)WRITE(IW,340)
      IF(NPLC.EQ.1)WRITE(IW,345)
      IF(NPLC.EQ.2)WRITE(IW,350)
      WRITE(IW,355)AIRVEL
      IF(KHORZ.EQ.1)WRITE(IW,360)RSURFA
      IF(KHORZ.EQ.2)WRITE'(IW,365)HEIGHT
      GO TO 30
20    WRITE(IW,370)WATVEL
      IF(IRWAT.EQ.1)WRITE(IW,375)
      IF(IRWAT.EQ.2)WRITE(IW,380)
      IF(IRWAT.EQ.1)WRITE(IW,390)DPIPE
      IF(IRWAT.EQ.2)WRITE(IW,360)RSURFW
30    CONTINUE
C   LIST OUTPUT
      WRITE(IW,310)
C   CALC DIFFUSION COEFF IN POLYMER (DP) IF NOT USER-SPECIFIED
      IF(IPOLY.EQ.O.AND.DP.EQ.O.)GO TO 900
      IF(DP.EQ.0.)CALL DPCALC(IPOLY,XMW,DP)
C   CALC PARTITION COEFF IF NOT USER-SPECIFIED
      VPM - VPMT/760.
      IF(PC.EQ.0.)CALL KCALC(NEXT,CSATP,CSATA,CSATW,CINIT,CSATE,
     1                       XMW,VPM,TMM,XKOWM,PC)
C   CALC MASS TRANSFER COEFF IF NOT USER-SPECIFIED
      IF((NEXT.EQ.2).AND.(WATVEL.LT.10.))GO TO 850
      IF(RK.EQ.O.)CALL RKCALC(NEXT,KHORZ,NPLC,AIRVEL,RSURFA,VPM,
     1 HEIGHT,WATVEL,IRWAT,RSURFW,DPIPE,XMW,RKO)
      IF(RK.EQ.O.)WRITE(IW,395)RKO
      IF(RK.EQ.O.)RK - RKO
C CALC NONDIMENSIONAL PARAMETERS
      TAU - DP*TSEC/XLEN/XLEN
      VEPCC - VEP*1.0E6
      ALPHA - PC*VEPCC/SAP/XLEN
      GAMMA - RK*PC*XLEN/DP
      WRITE(IW,320)TAU,ALPHA,GAMMA
      FRMIG - DOMBFF(TAU,ALPHA,GAMMA)
 201  FORMAT(6X,36A2//)
 210  FORMAT(6X,'** INPUT PARAMETERS **'/)
 209  FORMAT(6X,'POLYMER CATEGORY:',T60,'  UNDEFINED        ')
 211  FORMAT(6X,'POLYMER CATEGORY:',T60,'  SILICONE RUBBER  ')
 212  FORMAT(6X,'POLYMER CATEGORY:',T60,'  NATURAL  RUBBER  ')
                                    86

-------
213
214
215
216
220
221

225

230
235
C
260
265
266
FORMAT (6X,
FORMAT (6X,
FORMAT (6X,
FORMAT (6X,
FORMAT (6X,
FORMAT (6X,
1 6X,
FORMAT (6X,
1 T60
FORMAT (6X,
FORMAT (6X,

FORMAT (6X,
FORMAT (6X,
FORMAT (6X,
267  FORMAT(6X,
268
269
270
275

280
290

310
320
 FORMAT(6X,
 FORMAT(6X,
 FORMAT(6X,
 FORMAT(6X,
1       6X,
 FORMAT(6X,
 FORMAT(6X,
'POLYMER CATEGORY:',T60,'   LDPE            ')
'POLYMER CATEGORY:',T60,'   HOPE            ')
'POLYMER CATEGORY:',T60,'   POLYSTYRENE     ')
'POLYMER CATEGORY:',T60,'   PVC(UNPLASTICIZED)')
'MOLECULAR WEIGHT OF ADDITIVE   ',T60,1PE10.2)
'TIME  (MRS)       ',T60,1PE10.2/
'TOTAL POLYMER SHEET THICKNESS  (CM)',T60,1PE10.2)
'USER-SPECIFIED DIFFUSION  COEFFICIENT(CM2/S)',
.1PE10.2)
'DIFFUSION SPECIFIED AS',T60,'   ONE-SIDED')
'DIFFUSION SPECIFIED AS',T60,'   TWO-SIDED')

 EXTERNAL PHASE IS         ',T60,'   AIR')
 EXTERNAL PHASE IS         ',T60,'   WATER')
 SATURATION CONC.  OF MIGRANT IN POLYMER (G/CM3)',
 T60,1PE10.2)
 LET SATUR. CONC.  IN POLYMER - -INIT CONC.  (G/CM3)',
 T60,1PE10.2)
 SATURATION CONC.  IN  AIR (G/CM3)',T60,1PE10.2)
 SATURATION CONC.  IN  WATER (G/CM3)',T60,1PE10.2)
 MIGRANT VAPOR PRESSURE (TORR)   ',T60,1PE10.2)
 MIGRANT MELT  TEMP  (DEC C)',T60,1PE10.2/
 MIGRANT OCTANOL-WATER PART.  COEF.',T60,1PE10.2)
VOLUME OF EXTERNAL  PHASE  (M3)',T60,1PE10.2)
 SURFACE AREA  OF POLYMER (CM2)',T60,1PE10.2)
 FORMAT(//6X,'** OUTPUT VALUES (MODULE: EQUATION 17) **'/)
 FORMAT(6X,'TAU                          ',T60,1PE10.2,/
1       6X,'ALPHA                        ',T60,1PE10.2,/
1       6X,(GAMMA                        ',T60,1PE10.2)
                USER-SPECIFIED MASS TRANSFER COEFFICIENT  (CM/S)',
                1PE10.2)
330

331
335
340
345
350
355
360
365
370
375
380
390
395
C

850
702
FORMAT (6X,
1 T60
FORMAT (6X,
FORMAT (6X,
FORMAT (6X,
FORMAT (6X,
FORMAT (6X,
FORMAT (6X,
FORMAT (6X,
FORMAT (6X,
FORMAT (6X,
FORMAT (6X,
FORMAT (6X,
FORMAT (6X,
FORMAT (6X,

GO TO 999
WRITE (IW, 7i
FORMAT (/6X
                                                       ,T60,1PE10.2)
                                         VERTICAL
                                         INDOORS
                                         OUTDOORS
900
            'USER-SPECIFIED PARTITION COEFFICIENT
            1 POLYMER POSITION',T60,'   HORIZONTAL')
            'POLYMER POSITION',T60,
            POLYMER LOCATION',T60,
            POLYMER LOCATION',T60,
            AIRFLOW VELOCITY(CM/S)',T60,1PE10.2)
            POLYMER PLATE SURFACE LENGTH (CM)',T60,1PE10.2)
            POLYMER PLATE HEIGHT (CM)',T60,1PE10.2)
            WATER FLOW VELOCITY (CM/S)',T60,1PE10.2)
            WATER FLOW IS    ',T60,'   THROUGH PIPE')
            WATER FLOW IS    ',T60,'   OVER PLATE   ')
            POLYMER PIPE DIAMETER (CM)',T60,1PE10.2)
            ESTD. MASS TRANSFER COEFFICIENT (CM/S)',T60,1PE10.2)
 FORMAT(/6X,'THE WATER FLOW VELOCITY GIVEN IS  '.1PE10.2,' CM/S',
1 /6X,'WATER CAN BE CONSIDERED ESSENTIALLY STAGNANT AT VELOCITIES',
2 /6X,'LESS THAN 10 CM/S. MODIFY INPUT TO EXECUTE PROGRAM.'
3 /6X,'RECOMMENDATION:  RE-DEFINE EXTERNAL PHASE AS "SOLID".')
 GO TO 999
 WRITE(IW,801)
                                    87

-------
 801  FORMAT(/6X,'PROGRAM STOP. INPUT ERROR. FOR A USER-SPECIFIED'
     1       /6X,'POLYMER CLASS, THE DIFFUSION COEFFICIENT OF
     2       /6X,'ADDITIVE IN POLYMER MUST BE A NON-ZERO VALUE.   ')
 999  STOP
      END
C       File DOMBFF
C     REF: GANDEK.T.P., PH.D. THESIS, M.I.T., CAMBRIDGE, MA 1986
C       Domain BFF:  finite slab in contact with a finite bath.
C        Boundary layer resistance.  Returns fractional migration assuming
C        the solute is initially uniformly distributed in the solid and
C        the initial fluid concentration is zero.
C       Requires subprogram ROOTBFF
C       Note that TAU   - Dpt/L2
C                 R       kKL/Dp
C                 ALPHA - KV/AL
        DOUBLE PRECISION FUNCTION DOMBFF(TAU,ALPHA,R)
     IMPLICIT REAL*8 (A-H.O-Z)
     COMMON/PARAM3/B(1000),NROOT,ALAST,RIAST,NRTBFFMAX
      COMMON/IO/IR.IW
     IF ((ALAST.NE.ALPHA).OR.(RLAST.NE.R)) NROOT-0
        EPS - l.d-8
        IEPS - 0
        SUM - l.dO
        I - -1
10      I - I + 1
        IF (I.EQ.NRTBFFMAX)  GOTO 30
        IF (I.LT.NROOT) GOTO 20
          J - 1+1
          B(J)  - ROOTBFF(I,ALPHA,R)
          NROOT - J
20      BN - B(J)
        SIN1 - SIN(BN)
        COS1 - COS(BN)
        EN   - (R*COS1 - BN*SIN1)
        Dl   - EN*EN*ALPHA/(R*R)
        D2   - 0.5dO
        IF (I.NE.O) D2 - SINl*COSl/(2.dO*BN)
        DEN  - 0.5dO + Dl + D2
        Tl - l.dO
        IF (I.NE.O) Tl - SIN1/BN
        T2 - O.dO
        T3 - -BN*BN*TAU
        IF (T3.GT.-120.dO) T2 - EXP(T3)
        TERM - T1*T1*T2/DEN
        SUM - SUM - TERM
        RAT - ABS(TERM/SUM)
        IF (RAT.LT.EPS) IEPS - IEPS + 1
        IF (IEPS.LT.3) GOTO 10
        GOTO 40
30      write(iw.lOl)
101     FORMAT(/6X,'NOTE:  GIVEN THE ABOVE INPUT AND THE ESTIMATED'
     1  /6X,'TAU VALUE, MIGRATION IS VERY CLOSE TO ZERO {<1.0E-04}.'
     2  //6X,'RECOMMENDATION: INCREASE EXPOSURE TIME, INCREASE
     3  /6X,'MIGRANT-POLYMER DIFFUSION COEFFICIENT, OR REDUCE POLYMER'
     4  /6X,'THICKNESS, AND RE-RUN SCENARIO.'/)

-------
40      DOMBFF  -  SUM
      WRITE (IW ,.100) DOMBFF
100   FORMAT(6X/FRACTION MIGRATED             ',T60,1PE10.2)
        RETURN
        END
C       File  ROOTBFF
C     REF: GANDEK.T.P.,  PH.D.  THESIS, M.I.T.,  CAMBRIDGE, MA  1986
C       Finds roots  of  the equation
C         Tan(BN) -  R*BN/(BN2-N)
C       or
C         Cot(BN) -  (BN2-N)/(R*BN)
C       where
C         I   -  number of desired  root
C         R   -  kKL/Dp
C         ALPHA - KV/AL
C         N   -  R/ALPHA
        DOUBLE  PRECISION FUNCTION ROOTBFF(I,ALPHA,R)
     IMPLICIT REAL*8 (A-H.O-Z)
        REAL  N
        IF (I.LE.O)  GOTO 39
        EPS1  -  l.d-3
        EPS2  -  l.d-12
        RI    -  FLOAT(I)
        PI    -  2.dO*ACOS(0.dO)
        RIPI  -  RI*PI
        N     -  R/ALPHA
        SRN   -  SQRT(N)
        REPS1-  (PI/2.dO)-EPSl
        ISTEP-  0
        ITAN  -  0
        FLAG  -  l.dO
        CTR   -  (RI-0.5dO)*PI
        IF(RIPI.GT.SRN)  FLAG - -l.dO
        BN    -  CTR + FLAG*REPS1
10      BN2   -  BN*BN
        IF(ITAN.EQ.l) GOTO 15
        COT1  -  l.dO/TAN(BN)
        CSC1  -  l.dO/SIN(BN)
        CSC2  -  CSC1*CSC1
        RES  -  COT1 + (N-BN2)/(R*BN)
        DR   — CSC2  - (BN24-N)/(R*BN2)
        DDR  -  2.dO*CSC2*COTl 4- 2.dO*N/(R*BN2*BN)
        GOTO  18
15      TAN1 -  TAN(BN)
        SEC1 -  l.dO/COS(BN)
        SEC2 -  SEC1*SEC1
        BN2N -  BN2-N
        RES  -  TAN1 - R*BN/BN2N
        DR   -  SEC2 + R*(BN2+N)/(BN2N*BN2N)
        DDR  -  2.dO*SEC2*TANl - 2.dO*R*BN*(BN2+3.dO*N)/(BN2N*BN2N*BN2N)
18      BNN  - BN - (RES/DR)
        ISTEP-  ISTEP + 1
        IF ((RES*FLAG.GT.O.).AND.(ISTEP.EQ.l)) GOTO 30
        IF (ABS(BNN-BN).LT.EPS2) GOTO 40
        IF ((FLAG*DDR).GT.O.dO) GOTO 20
                                     89

-------
        IF (ABS(BNN-CTR).GT.PI/2.dO)  GOTO 20
        BN   - BNN
        GOTO 10
C       Inflection point, start search from other asymptote
20      FLAG - -l.dO*FLAG
        BN   - CTR + FIAG*REPS1
        IF (ITAN.EQ.l) BN - CTR - FLAG*REPSl
        ISTEP - 0
        GOTO 10
C       Root within EPS1 of asymptote;  solve reciprocal (tan) equation
30      CTR  - RIPI
        IF (FLAG.LT.O.dO) CTR - CTR-PI
        REPS1- EPS1
        ITAN - 1
        BN   - CTR - FLAG*REPS1
        GOTO 10
39      BNN  - O.dO
40      ROOTBFF - BNN
        RETURN
        END
                                    90

-------
      PROGRAM NAME:  EQ20
                                    KM DILWALI 0990
 C
 C
 C
 C
 C
 C
 C
 C
 C
 C
 C
 C
 C
 C
 C
 C
 C
 C
 C
 C
 C
 C
 C
 C
 C
 C
 C
 C
 C
 C
 C
 C
C
C
C
C
C
C
 THIS PROGRAM MODULE SERVES AS THE I/O SHELL TO
 COMPUTE THE FRACTION OF ADDITIVE MIGRATION
 ASSUMING THE FOLLOWING CONDITIONS:   (BFI)
 -  BOUNDARY LAYER RESISTANCE EFFECTS  CONSIDERED
 -  POLYMER PHASE THICKNESS  MUST BE CONSIDERED FINITE
 -  EXTRACTANT PHASE INFINITE;I.E.  LIMITED PARTITIONING  EFFECTS*
* NOTE:  PART COEFF.  STILL REQD FOR CALCN OF NON-DIMNAL  MASS
        TRANSFER PARAMETER  GAMMA.  CAN BE USER-SPECD OR  CALCD.
 INPUT REQUIREMENTS:
       IPOLY :  01-06  POLYMER TYPE
         01- SILICONS RUBBER   04-HDPE
         02- NATURAL   RUBBER   05-POLYSTYRENE
         03- LDPE              06-PVC(UNPLASTICIZED)
XMW
DP
THRS
XL
NSIDE
CINIT
NEXT
PC
CSATP
CSATA
CSATW
CSATE
VPMT
TMM
XKOWM
RK
KHORZ
NPLC
RSURFA
AIRVEL
HEIGHT
WATVEL
IRWAT
RSURFW
DPIPE
              MOLECULAR WEIGHT OF MIGRANT              G/MOL
              DIFFUSION COEFF. OF ADDITIVE IN POLYMER  CM2/S
              TIME                                     HRS
              POLYMER FILM THICKNESS                   CM
              01=ONE-SIDED , 02=TWO-SIDED DIFFUSION
              INIT MIGRANT CONG. IN POLYMER G/CM3
              EXT PHASE TYPE 01-AIR, 02=H20
              PARTITION COEFF; 0 IF UNKNOWN
              SAT CONG. OF MIGR IN POLYMER OR 0.,G/CM3
              SAT CONG. OF MIGR IN AIR     OR 0.,G/CM3
              SAT CONG. OF MIGR IN WATER   OR 0.,G/CM3
              SAT CONG. OF MIGR IN SOLID(N/A)    G/CM3
              MIGR VAPOR PRESSURE, TORR FOR CSATA  , RK CALCS
              MIGR MELT TEMP DEC C    FOR CSATW CALC
              MIGR OCT-H20 PART COEFF FOR CSATW CALC
              MASS TRANSFER COEFF CM/S; 0. IF UNKNOWN
              AIR: POLYMER POSITION 1-HORIZONTAL, 2-VERTICAL
              AIR: POLYMER LOCN 1-INDOORS, 2-OUTDOORS
              AIR: SURFACE LENGTH ,CM
              AIR: AIRFLOW VELOCITY, CM/S
              AIR: SURFACE HEIGHT FOR VERTICAL POSITION, CM
              H20: WATER FLOW VELOCITY ,  CM/S
              H20: 01-FLOW THRU PIPE; 02-FLOW OVER PLATE
              H20: FOR IRWAT-2,PLATE LENGTH,  CM
              H20: FOR IRWAT-l.PIPE  DIAMETER, CM
                      FOR IRWAT-1
C********************
      IMPLICIT REAL*8 (A-H,0-Z)
      TAU - 0.0
      GAMMA - 0.0
      NRTBFIMAX - 1000
C********************
      COMMON/PARAM2/B(1000),NROOT,RLAST,NRTBFIMAX
      COMMON/IO/IR.IW
      CHARACTER*2 TITLE(36)
      IR - 5
                                      91

-------
         FILE-'EQ20.INP')
         FILE='EQ20.LIS',STATUS-'UNKNOWN')
         DATA
         '(36A2)')TITLE
         '(12)'
         '(F10.
         '(F10.
         '(F10.
         '(F10.
         '(12)'
         '(F10.
         '(12)'
         '(F10.
         '(F10.
         '(F10.
         '(F10.
         '(F10
         r(F10
         '(F10
          (F10
          (F10
          (F10
          (12)
          (12)
          (F10
          (F10
          (F10
          (F10
          (12)
          (F10
          (F10
     2)
     2)
     2)
     .2)
     .2)
     .2)'
     2)'
   IW -  6
   OPEN(IR
   OPEN(IW
READ INPUT
   READ(IR
   READ(IR
   READ(IR
   READ(IR
   READ(IR
   READ(IR
   READ(IR
   READ(IR
   READ(IR
   READ(IR
   READ(IR
   READ(IR
   READ(IR
   READ(IR
   READ(IR
   READ(IR:
   READ(IR;
   READ(IR;
   READ(IR;
   READ(IR1
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
INITIALIZE
   IF(RK.NE.O..AND.WATVEL.EQ.O.)WATVEL-10.
   XLEN - XL
   IF(NSIDE.EQ.2)XLEN-XL/2.
   TSEC - THRS*3600.
 ECHO INPUT
            201)TITLE
            210)
            EQ.1)WRITE(IW,
               2)WRITE(IW,
               3)WRITE(IW,
               4)WRITE(IW,
               5)WRITE(IW,
               6)WRITE(IW,216)
               )WRITE(IW,220)XMW
            221)THRS,XL
            0.)WRITE(IW,225)DP
            EQ.1)WRITE(IW,230)
            EQ.2)WRITE(IW,235)
   IF(NEXT.EQ.1)WRITE(IW,260)
   IF(NEXT.EQ.2)WRITE(IW,265)
   IF(PC.NE.O.)WRITE(IW,331)PC
     IF(PC.NE.O.)GO TO 15
')IPOLY
,2)')XMW
,2)')DP
 2)')THRS
 2)')XL
 )NSIDE
 2)')CSATP
 )NEXT
 2)')VEP
    )SAP
    )PC
    )CINIT
    )CSATA
    )CSATW
    )VPMT
    )TMM
    )XKOWM
    )RK
' )KHORZ
')NPLC
.2)')RSURFA
 2)')AIRVEL
 2)')HEIGHT
 2)')WATVEL
 )IRWAT
 2)')RSURFW
 2)')DPIPE
WRITE(IW
WRITE(IW
IF'CIPOLY
IF(IPOLY
IF(IPOLY
IF(IPOLY
IF(IPOLY
IF(IPOLY
IF(XMW.NE.O
WRITE(IW
IF(DP.NE
IF(NSIDE
IF(NSIDE
EQ.
EQ.
EQ.
EQ.
EQ.
         ,211)
         ,212)
         ,213)
         ,214)
         ,215)
                               92

-------
15
20
30
C
    IF(CSATP.NE.O.)WRITE(IW,266)CSATP
    IF(CSATP..EQ.O.)WRITE(IW,267)CINIT
    IF(NEXT.EQ.1.AND.CSATA.NE.O.)WRITE(IW,268)CSATA
    IF(NEXT.EQ.2.AND.CSATW.NE.O.)WRITE(IW,269)CSATW
    IF(NEXT.EQ.1.AND.CSATA.EQ.O.)WRITE(IW,270)VPMT
    IF(NEXT.EQ.2.AND.CSATW.EQ.0.)WRITE(IW,275)TMM,XKOWM
    IF(RK.NE.O.)WRITE(IW,330)RK
      IF(RK.NE.O.)GO TO 30
      IF(NEXT.EQ.2)GO TO 20
    IF(KHORZ.EQ.1)WRITE(IW,335)
    IF(KHORZ.EQ.2)WRITE(IW,340)
    IF(NPLC.EQ.1)WRITE(IW,345)
    IF(NPLC.EQ.2)WRITE(IW,350)
    WRITE(IW,355)AIRVEL
    IF(KHORZ.EQ.1)WRITE(IW,360)RSURFA
    IF(KHORZ.EQ.2)WRITE(IW,365)HEIGHT
    GO TO 30
    WRITE(IW,370)WATVEL
    IF(IRWAT.EQ.1)WRITE(IW,375)
    IF(IRWAT.EQ.2)WRITE(IW,380)
    IF(IRWAT.EQ.1)WRITE(IW,390)DPIPE
    IF(IRWAT.EQ.2)WRITE(IW,360)RSURFW
    CONTINUE
  LIST OUTPUT
    WRITE(IW,310)
  CALC DIFFUSION COEFF IN POLYMER (DP) IF NOT USER-SPECIFIED
    IF(IPOLY.EQ.O.AND.DP.EQ.O.)GO TO 900
    IF(DP.EQ.O.)CALL DPCALC(IPOLY,XMW,DP)
  CALC PARTITION COEFF IF NOT USER-SPECIFIED
    VPM - VPMT/760.
    IF(PC.EQ.O.)CALL KCALC(NEXT,CSATP,CSATA,CSATW.CINIT.CSATE,
   1                       XMW,VPM,TMM,XKOWM,PC)
  CALC MASS TRANSFER COEFF IF NOT USER-SPECIFIED
    IF((NEXT.EQ.2).AND.(WATVEL.LT.10.))GO TO 850
    IF(RK.EQ.O.)CALL RKCALC(NEXT,KHORZ,NPLC,AIRVEL,RSURFA,VPM,
   1 HEIGHT,WATVEL,IRWAT,RSURFW,DPIPE,XMW,RKO)
    IF(RK.EQ.O.)WRITE(IW,395)RKO
    IF(RK.EQ.O.)RK - RKO
CALC NONDIMENSIONAL PARAMETERS
    TAU - DP*TSEC/XLEN/XLEN
    GAMMA - RK*PC*XLEN/DP
    WRITE(IW,3 20)TAU,GAMMA
    FRMIG - DOMBFI(TAU,GAMMA)
               6A2//)
               ** INPUT PARAMETERS **
               POLYMER CATEGORY:',160
               POLYMER CATEGORY:
               POLYMER CATEGORY:
               POLYMER CATEGORY:
               POLYMER CATEGORY:
               POLYMER CATEGORY:
201
210
211
212
213
214
215
216
220
221

FORMAT (6X
FORMAT (6X
FORMAT (6X
FORMAT (6X
FORMAT (6X
FORMAT (6X
FORMAT (6X
FORMAT (6X
FORMAT (6X
FORMAT (6X
1 6X
                                           SILICONE RUBBER
                                           NATURAL  RUBBER
                                           LDPE
                                           HOPE
                                           POLYSTYRENE
                   T60,
                   T60,
                   T60,
                   T60,
                   T60,
MOLECULAR WEIGHT OF ADDITIVE   ',T60,1PE10.2)
TIME  (HRS)      ',T60,1PE10.2/
TOTAL POLYMER SHEET THICKNESS  (CM)',T60,1PE10.2)
                                           PVC(UNPLASTICIZED)')
 225   FORMAT(6X,'USER-SPECIFIED DIFFUSION COEFFICIENT(CM2/S)',
                                     93

-------
     1       T60.1PE10.2)
 230  FORMAT(6X,'DIFFUSION SPECIFIED AS',T60,'  ONE-SIDED')
 235  FORMAT(6X,'DIFFUSION SPECIFIED AS',T60,'  TWO-SIDED')
C
 260  FORMAT(6X,'EXTERNAL PHASE  IS         ',T60,'  AIR')
 265  FORMAT(6X,'EXTERNAL PHASE  IS         ',T60,'  WATER')
 266  FORMAT(6X,'SATURATION CONG. IN POLYMER  (G/CM3)',T60,1PE10.2)
 267  FORMAT(6X,'LET SATUR. CONG. IN POLY  - INIT CONG.  (G/CM3)',
     1       T60.1PE10.2)
 268  FORMAT(6X,'SATURATION CONG. IN AIR •     (G/CM3)',T60,1PE10.2)
 269  FORMAT(6X,'SATURATION CONG. IN WATER   (G/CM3)',T60,1PE10.2)
 270  FORMAT(6X,'MIGRANT VAPOR PRESSURE  (TORR)  ',T60,1PE10.2)
 275  FORMAT(6X,'MIGRANT MELT TEMP  (DEC C)',T60,1PE10.2/
     2       6X,'MIGRANT OCTANOL-WATER PART.  COEFF.',T60,1PE10.2)
C
 310  FORMAT(//6X,'** OUTPUT VALUES (MODULE:  EQUATION 20) **'/)
 320  FORMAT(6X,'TAU                           .' ,T60 , 1PE10. 2 ,/
     1       6X,'GAMMA                         ',T60,1PE10.2)
 330  FORMAT(6X,'USER-SPECIFIED MASS TRANSFER COEFFICIENT (CM/S)',
     1       T60.1PE10.2)
 331  FORMAT(6X,'USER-SPECIFIED PARTITION  COEFFICIENT ',T60,1PE10.2)
 335  FORMAT(6X,'POLYMER POSITION',T60,'   HORIZONTAL')
 340  FORMAT(6X,'POLYMER POSITION',T60,'   VERTICAL   ')
 345  FORMAT(6X,'POLYMER LOCATION',T60,'   INDOORS    ')
 350  FORMAT(6X,'POLYMER LOCATION',T60,'   OUTDOORS   ')
 355  FORMAT(6X,'AIRFLOW VELOCITY(CM/S)',T60,1PE10.2)
 360  FORMAT(6X,'POLYMER PLATE SURFACE LENGTH  (CM)',T60,1PE10.2)
 365  FORMAT(6X,'POLYMER PLATE HEIGHT (CM)',T60,1PE10.2)
 370  FORMAT(6X,'WATER FLOW VELOCITY (CM/S)',T60,1PE10.2)
 375  FORMAT(6X,'WATER FLOW IS   ',T60,'   THROUGH PIPE')
 380  FORMAT(6X,'WATER FLOW IS   ',T60,'   OVER PLATE  ')
 390  FORMAT(6X,'POLYMER PIPE DIAMETER (CM)',T60,1PE10.2)
 395  FORMAT(6X,'ESTD.  MASS TRANSFER COEFFICIENT (CM/S)',T60,1PE10.2)
C
      GO TO 999
850   WRITE(IW,702)WATVEL
702   FORMAT(/6X, "THE WATER FLOW VELOCITY  GIVEN IS '.1PE10.2,' CM/S',
     1 /6X, 'WATER CAN BE CONSIDERED ESSENTIALLY STAGNANT AT VELOCITIES',
     2 /6X,'LESS THAN 10 CM/S. MODIFY INPUT TO EXECUTE  PROGRAM.'
     3 /6X,'RECOMMENDATION: RE-DEFINE EXTERNAL PHASE AS "SOLID".')
      GO TO 999
900   WRITE(IW,801)
801   FORMAT(/6X,'PROGRAM STOP. INPUT ERROR.   FOR A USER-SPECIFIED'
     1       /6X,'POLYMER CLASS,  THE DIFFUSION COEFFICIENT OF
     2       /6X,'ADDITIVE IN POLYMER MUST BE A NON-ZERO VALUE.   ')
 999  STOP
      END
C       File DOMBFI                                       0990
C     REF: GANDEK,T.P., PH.D. THESIS,  M.I.T.,  CAMBRIDGE, MA 1986
C       Domain BFI: finite slab in contact with an infinite bath.
C        Boundary layer resistance.   Returns  fractional migration assuming
C        the fluid phase concentration is  initially  zero and the solute is
C        uniformly distributed in the slab.
C       Requires subprogram ROOTBFI
C       Note that R   - kKL/Dp


                                     94

-------
C                 Tau - Dpt/L2
        DOUBLE PRECISION FUNCTION DOMBFI  (TAU.R)
     IMPLICIT REAL*8 (A-H.O-Z)
        COMMON/PARAM2/B(1000),NROOT,RLAST,NRTBFIMAX
      COMMON/IO/IR.IW
     IF (RLAST.NE.R) NROOT-0
        R2 - R*R
        EPS1 - l.d-8
        SUM - l.dO
        N - 0
10      N - N + 1
        IF (N.EQ.NRTBFIMAX) GOTO 30
        IF (N.LE.NROOT) GOTO  20
          B(N) - ROOTBFI(N,R)
          NROOT - N
20      GN - B(N)
        GN2 - GN*GN
        El - O.dO
        IF ((GN2*TAU).LT.120.) El - EXP(-GN2*TAU)
        TERM - 2.dO*R2*El/(GN2*(GN2+R2-fR))
        SUM - SUM - TERM
        IF (ABS(TERM/SUM).GT.EPS1) GOTO 10
        GOTO 40
30      write(iw.lOl)
101     FORMAT(/6X,'NOTE:  GIVEN THE ABOVE INPUT AND THE ESTIMATED'
     1  /6X/TAU VALUE, MIGRATION IS VERY CLOSE TO ZERO (<1.0E-04).'
     2  //6X,'RECOMMENDATION: INCREASE EXPOSURE TIME, INCREASE
     3  /6X,'MIGRANT-POLYMER DIFFUSION COEFFICIENT, OR REDUCE POLYMER'
     4  /6X,'THICKNESS, AND RE-RUN SCENARIO.'/)
40      DOMBFI - SUM
      WRITE(IW,100)DOMBFI
100   FORMAT(6X,'FRACTION MIGRATED            ',T60,1PE10.2)
        RETURN
        END
C       File ROOTBFI                                  0990
C     REF:  GANDEK.T.P., PH.D. THESIS,  M.I.T., CAMBRIDGE, MA 1986
C       Returns roots of the  equation
C         tan(GN) - R/GN      or       cot(GN) - GN/R
C       where R is a constant
        DOUBLE PRECISION FUNCTION ROOTBFI (N,R)
     IMPLICIT REAL*8 (A-H.O-Z)
        IF (N.LE.O) GOTO 29
        EPS1 - l.d-3
        EPS2 - l.d-12
        RN   - FLOAT (N)
        PI   - 2.dO*ACOS(0.dO)
        RNPI1- (RN-l.dO)*PI
        ISTEP- 0
        GN   - RNPI1 -I- EPS1
10      COT1 - l.dO/TAN(GN)
        CSC1 - l.dO/SIN(GN)
        CSC2 - CSC1*CSC1
        RES  - COT1 - GN/R
        ORES - -CSC2 -  l.dO/R
        GNN  - GN - (RES/DRES)


                                   95

-------
        ISTEP- ISTEP + 1
        IF ((GNN.LT.GN).AND.(ISTEP.EQ.l)) GOTO 20
        IF (ABS(GNN-GN).LT.EPS2) GOTO 30
        GN   - GNN
        GOTO 10
C       Root close to asymptote of cot; use tan form.
20      TAN1 - TAN(GN)
        SEC1 - l,dO/COS(GN)
        SEC2 - SEC1*SEC1
        RES  - TAN1 - R/GN
        DRES - SEC2 + R/(GN*GN)
        GNN  - GN - (RES/DRES)
        IF (ABS(GNN-GN).LT.EPS2) GOTO 30
        GN   - GNN
        GOTO 20
29      GNN - O.dO
30      ROOTBFI - GNN
        RETURN
        END
                                   96

-------
     PROGRAM NAME: EQ23
                                   KM DILWALI 0990
C  THIS PROGRAM MODULE SERVES AS THE I/O SHELL TO
C  COMPUTE THE FRACTION OF ADDITIVE MIGRATION
C  ASSUMING THE FOLLOWING CONDITIONS:  (MFI)
C  - NO BOUNDARY LAYER RESISTANCE;I.E INFINITE WELL-STIRRED BATH
C  - POLYMER PHASE THICKNESS MUST BE CONSIDERED FINITE
C  - EXTRACTANT PHASE INFINITE;I.E. PARTITIONING EFFECTS NEGL.
C  INPUT REQUIREMENTS:
C        IPOLY : 00-06 POLYMER TYPE
C          01- SILICONE RUBBER   04-HDPE
C          02-= NATURAL  RUBBER   05-POLYSTYRENE
C          03- LDPE              06-PVC(UNPLAS'TICIZED)
C        XMW     MOLECULAR WEIGHT OF MIGRANT              G/MOL
C        DP      DIFFUSION COEFF.  OF ADDITIVE IN POLYMER  CM2/S
C        THRS    TIME                                     HRS
C        XL      POLYMER FILM THICKNESS                   CM
C        NSIDE   01-ONE-SIDED , 02-TWO-SIDED DIFFUSION
      COMMON/IO/IR,IW
      IMPLICIT REAL*8 (A-H.O-Z)
      CHARACTER*2 TITLE(36)
    '  IR - 5
      IW - 6
      OPEN(IR,FILE-'EQ23.INP')
      OPEN(IW,FILE-'EQ23.LIS',STATUS-'UNKNOWN')
C  READ INPUT DATA
      READ(IR,'(36A2)')TITLE
               (12)')IPOLY
               (F10.2)')XMW
               (F10.2)')DP
               (F10.2)')THRS
               (F10.2)')XL
               (I2)')NSIDE
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
INITIALIZE
   XLEN - XL
   IF(NSIDE.EQ.2)XLEN-XL/2.
   TSEC - THRS*3600.
 ECHO INPUT
   WRITE(IW,201)TITLE
   WRITE(IW,210)
   IF(IPOLY.EQ.1)WRITE(IW,211)
   IF(IPOLY.EQ.2)WRITE(IW,212)
   IF(IPOLY.EQ.3)WRITE(IW,213)
   IF(IPOLY.EQ.4)WRITE(IW,214)
   IF(IPOLY.EQ.5)WRITE(IW,215)
   IF(IPOLY.EQ.6)WRITE(IW,216)
   IF(XMW.NE.O.)WRITE(IW,220)XMW
   WRITE(IW,221)THRS,XL
   IF(DP.NE.O.)WRITE(IW,225)DP
   IF(NSIDE.EQ.1)WRITE(IW,230)
                                     97

-------
       IF(NSIDE.EQ.2)WRITE(IW,235)
 C    LIST OUTPUT
       WRITE(IW,310)
 C  CALC NONDIMENSIONAL PARAMETERS
       IF(IPOLY.EQ.O.AND.DP.EQ.O.)GO TO 900
       IF(DP.EQ.O.)CALL DPCALC(IPOLY,XMW,DP)
       TAU - DP*TSEC/XLEN/XLEN
       WRITE(IW,320)TAU
       FRMIG - DOMMFI(TAU)
201
210
211
212
213
214
215
216
220
221

FORMAT (6X
FORMAT (6X
FORMAT (6X
FORMAT (6X
FORMAT (6X
FORMAT (6X
FORMAT (6X
FORMAT (6X
FORMAT (6X
FORMAT (6X,
1 6X.
                 '** INPUT PARAMETERS  **'/)
                 'POLYMER CATEGORY:',T60,'
                 'POLYMER CATEGORY:',T60,'
                 'POLYMER CATEGORY:',T60,'
                 'POLYMER CATEGORY:',T60,'
                 'POLYMER CATEGORY:',T60,'
                 'POLYMER CATEGORY:',T60,'
SILICONS RUBBER ')
NATURAL  RUBBER ')
LDPE            ')
HOPE            ')
POLYSTYRENE     ')
PVC.(UNPLASTICIZED) ' )
                 'MOLECULAR WEIGHT OF ADDITIVE    ',T60,1PE10.2)
                 'TIME   (HRS)      ',T60,1PE10.2/
                 'TOTAL  POLYMER  SHEET THICKNESS   (CM)',T60,1PE10.2)
 225  FORMAT(6X,'USER-SPECIFIED DIFFUSION COEFFICIENT  (CM2/S)',
     1       T60,1PE10.2)
 230  FORMAT(6X,'DIFFUSION SPECIFIED AS',T60,'  ONE-SIDED')
 235  FORMAT(6X,'DIFFUSION SPECIFIED AS',T60,'  TWO-SIDED')
 310  FORMAT(//6X,'** OUTPUT VALUES (MODULE: EQUATION  23)  **'/)
 320  FORMAT(6X,'TAU                    ',T60,1PE10.2)
      GO TO 999
 900  WRITE(IW,801)
 801  FORMAT(/6X,'PROGRAM STOP. INPUT ERROR. FOR A USER-SPECIFIED'
     1       /6X,'POLYMER CLASS, THE DIFFUSION COEFFICIENT OF
     2       /6X,'ADDITIVE IN POLYMER MUST BE A NON-ZERO VALUE.   ')
 999  STOP
      END
C       File DOMMFI                                     0990
C     REF: GANDEK.T.P., PH.D. THESIS, M.I.T., CAMBRIDGE, MA 1986
C       Domain MFI: Finite slab in contact with an infinite, well-stirred
C        bath.  Returns fractional migration assuming  the  initial solute
C        concentration  in the bath is zero and the slab distribution is
C        uniform.
C       Note that TAU - Dpt/L2
        DOUBLE PRECISION FUNCTION DOMMFI(TAU)
     IMPLICIT REAL*8 (A-H,0-Z)
        DOUBLE PRECISION QN
      COMMON/IO/IR,IW
        EPS - l.OD-8
        PI - 3.1415926535898dO
        N - 0
        NMAX - 1000
        SUM - l.dO
10      N - N + 1
        IF(N.GT.NMAX)GO TO 99
        RN - FLOAT(N)
C       QN - (2.dO*RN + 1.dO)*PI/2.dO
        QN - (2.dO*RN - l.dO)*PI/2.dO
        QN2 - QN*QN
                                      98

-------
        TERM - 2.dO*EXP(-QN2*TAU)/QN2
        SUM - SUM - TERM
        RAT - ABS(TERM/SUM)
        IF (RAT.GT.EPS) GOTO 10
        DOMMFI - SUM
        WRITE(IW,100)DOMMFI
100     FORMAT(6X,'FRACTION MIGRATED     ',T60,1PE10.2)
        RETURN
 99     WRITE(IW,101)
        DOMMFI - SUM
        RETURN
101     FORMAT(/6X,'NOTE:  GIVEN THE ABOVE INPUT AND THE ESTIMATED'
     1  /6X,'TAU VALUE, MIGRATION IS VERY CLOSE TO ZERO {<1.0E-04}.'
     2  //6X,'RECOMMENDATION:  INCREASE EXPOSURE TIME, INCREASE
     3  /6X,'MIGRANT-POLYMER DIFFUSION COEFFICIENT, OR REDUCE POLYMER'
     4  /6X,'THICKNESS, AND RE-RUN SCENARIO.'/)
        END
                                      99

-------
     PROGRAM NAME: EQ26
KM DILWALI 0990
   THIS PROGRAM MODULE  SERVES AS THE I/O SHELL TO
   COMPUTE THE FRACTION OF ADDITIVE MIGRATION
   ASSUMING THE FOLLOWING CONDITIONS:   (MFF)
   - NO BOUNDARY LAYER  RESISTANCE;I.E INFINITE WELL-STIRRED BATH
   - POLYMER PHASE THICKNESS MUST BE CONSIDERED FINITE
   - EXTRACTANT PHASE FINITE;I.E. PARTITIONING EFFECTS CONSIDERED
   INPUT REQUIREMENTS:
         IPOLY : 01-06  POLYMER TYPE
           01- SILICONS RUBBER   04-HDPE
           02= NATURAL  RUBBER   05-POLYSTYRENE
           03- LDPE              06-PVC(UNPLASTICIZED)
         XMW     MOLECULAR WEIGHT OF MIGRANT              G/MOL
         DP      DIFFUSION COEFF. OF ADDITIVE IN POLYMER  CM2/S
         THRS    TIME                                     HRS
         XL      POLYMER SHEET THICKNESS                  CM
         NSIDE   01-ONE-SIDED , 02-TWO-SIDED DIFFUSION
         CINIT   INIT MIGRANT CONCN IN POLYMER            G/CM3
                 (ONLY REQUIRED IF CSATP NOT AVAILABLE)
         NEXT    EXT PHASE TYPE 01-AIR, 02-H20 ,  03=SOLID(N/A)
         VEP     VOL OF EXT PHASE      ,M3
         SAP     EXPOSED SURFACE AREA  ,CM2
         PC      PARTITION COEFF; 0 IF UNKNOWN
         CSATP   SAT CONCN OF MIGR IN POLYMER OR 0.,G/CM3
         CSATA   SAT CONCN OF MIGR IN AIR     OR 0.,G/CM3
         CSATW   SAT CONCN OF MIGR IN H20     OR 0.,G/CM3
         CSATE   SAT CONCN OF MIGR IN SOLID EXT PHASE (N/A),G/CM3
         VPMT    MIGR VAPOR PRESSURE,TORR FOR CSATA CALC
         TMM     MIGR MELT TEMP DEC C    FOR CSATW CALC
         XKOWM   MIGR OCT-H20 PART COEFF FOR CSATW CALC
C********************
         IMPLICIT REAL*8 (A-H.O-Z)
         COMMON/PARAM4/B4(1000),NRTMFF,A4LAST,NRTMFFMAX
         TAU - 0.0
         ALPHA - 0.0
         NRTMFFMAX - 1000
C********************
      COMMON/IO/IR,IW
      CHARACTER*2 TITLE(36)
      IR - 5
      IW - 6
      OPEN(IR,FILE-'EQ26.INP')
      OPEN(IW,FILE-'EQ26.LIS',STATUS-'UNKNOWN')
C  READ INPUT DATA
      READ(IR,'(36A2)')TITLE
      READ(IR,'(12)')IPOLY
      READ(IR,'(F10.2)')XMW
      READ(IR,'(F10.2)')DP
      READ(IR,'(F10.2)')THRS
                                    100

-------
READ(IR,'(F10.2)')XL
READ(IR,'(I2)')NSIDE
         (F10.2)')CINIT
         (12)'
         (F10.
')NEXT
,2)')VEP
         (F10.2)')SAP
         (F10.2)')PC
         (F10.2)
             .2)
      READ(IR,
      READ(IR,
      READ(IR,
      READ(IR,
      READ(IR,
      READ(IR,'(F10.2)')CSATP
      READ(IR,'(F10.2)')CSATA
      READ(IR,'(F10.2)')CSATW
      READ(IR,'(F10.2)')VPMT
      READ(IR,'(F10.2)')TMM
      READ(IR,'(F10.2)')XKOWM
C  INITIALIZE
      XLEN - XL
      IF(NSIDE.EQ.2)XLEN-XL/2.
      TSEC - THRS*3600.
C   ECHO INPUT
      WRITE(IW,201)TITLE
      WRITE(IW,210)
      IF(IPOLY.EQ.1)WRITE(IW,211)
      IF(IPOLY.EQ.2)WRITE(IW,212)
      IF(IPOLY.EQ.3)WRITE(IW,213)
      IF(IPOLY.EQ.4)WRITE(IW,214)
      IF(IPOLY.EQ.5)WRITE(IW,215)
      IF(IPOLY.EQ.6)WRITE(IW,216)
      IF(XMW.NE.O.)WRITE(IW,220)XMW
      WRITE(IW,221)THRS,XL
      IF(DP.NE.O.)WRITE(IW,225)DP
      IF(NSIDE.EQ.1)WRITE(IW,230)
      IF(NSIDE.EQ.2)WRITE(IW,235)
      WRITE(IW,280)VEP
      WRITE(IW,290)SAP
      IF(NEXT.EQ.1)WRITE(IW,260)
      IF(NEXT.EQ.2)WRITE(IW,265)
      IF(PC.NE.O.)WRITE(IW,331)PC
        IF(PC.NE.O.)GO TO 15
      IF(CSATP.NE.O.)WRITE(IW,266)CSATP
      IF(CSATP.EQ.O.)WRITE(IW,267)CINIT
      IF(NEXT.EQ.1.AND.CSATA.NE.0.)WRITE(IW,268)CSATA
      IF(NEXT.EQ.2.AND.CSATW.NE.O.)WRITE(IW,269)CSATW
      IF(NEXT.EQ.l.AND.CSATA.EQ.O.)WRITE(IW,270)VPMT
      IF(NEXT.EQ.2.AND.CSATW.EQ.O.)WRITE(IW,275)TMM,XKOWM
C   LIST OUTPUT
15    WRITE(IW,310)
C   CALC DIFFUSION COEFF IN POLYMER (DP) IF NOT USER-SPECIFIED
      IF(IPOLY.EQ.O.AND.DP.EQ.O.)GO TO 900
      IF(DP.EQ.O.)CALL DPCALC(IPOLY,XMW,DP)
C   CALC PARTITION COEFF IF NOT USER-SPECIFIED
      VPM - VPMT/760.
      IF(PC.EQ.0.)CALL KCALC(NEXT,CSATP,CSATA,CSATW,CINIT,CSATE,
     1                       XMW,VPM,TMM,XKOWM,PC)
C CALC NONDIMENSIONAL PARAMETERS
      TAU   - DP*TSEC/XLEN/XLEN
      VEPCC - VEP*1.0E6
                               101

-------
 201
 210
 211
 212
 213
 214
 215
 216
 220
 221

 225

 230
 235

 260
 265
 266
                                           SILICONS RUBBER
                                           NATURAL  RUBBER
                                           LDPE
                                           HOPE
                                           POLYSTYRENE
                                           PVC(UNPLASTICIZED)')
 ALPHA - PC*VEPCC/SAP/XLEN
 WRITE (IW,. 3 2 0) TAU, ALPHA
 FRMIG - DOMMFF(TAU,ALPHA)
 FORMAT(6X,36A2//)
 FORMAT(6X,'** INPUT PARAMETERS **'/)
 FORMAT(6X,'POLYMER CATEGORY:',T60,'
 FORMAT(6X,'POLYMER CATEGORY:',T60,'
 FORMAT(6X,'POLYMER CATEGORY:',T60,'
 FORMAT(6X,'POLYMER CATEGORY:',T60,'
 FORMAT(6X,'POLYMER CATEGORY:',T60,'
 FORMAT(6X,'POLYMER CATEGORY:',T60,'
 FORMAT(6X,'MOLECULAR WEIGHT OF ADDITIVE   ',T60,1PE10.2)
 FORMAT(6X,'TIME  (HRS)      ',T60,1PE10.2/
1       6X,'TOTAL POLYMER SHEET THICKNESS  (CM)',T60,1PE10.2)
 FORMAT(6X,'USER-SPECIFIED DIFFUSION COEFFICIENT (CM2/S)',
1       T60.1PE10.2)
 FORMAT(6X,'DIFFUSION SPECIFIED AS',T60,'.  ONE-SIDED')
 FORMAT(6X,'DIFFUSION SPECIFIED AS',T60,'  TWO-SIDED')

 FORMAT(6X,'EXTERNAL PHASE IS        ',T60,'   AIR')
 FORMAT(6X,'EXTERNAL PHASE IS        ',T60,'   WATER')
 FORMAT(6X,'SATURATION CONG.  OF MIGRANT IN POLYMER (G/CM3)',
1           T60.1PE10.2)
 FORMAT(6X,'LET SATUR.  CONG.  IN POLYMER - INIT CONG.(G/CM3)*,
I           T60,1PE10.2)
 FORMAT(6X,'SATURATION CONG.  IN AIR (G/CM3)',T60,1PE10.2)
 FORMAT(6X,'SATURATION CONG.  IN WATER (G/CM3)',T60,1PE10.2)
 FORMAT(6X,'MIGRANT VAPOR PRESSURE (TORR)  ',T60,1PE10.2)
 FORMAT(6X,'MIGRANT MELT TEMP (DEC C)',T60,1PE10.2/
1       6X,'MIGRANT OCTANOL-WATER PART.  COEFF.',T60,1PE10.2)
 FORMAT(6X,'VOLUME  OF EXTERNAL PHASE  (M3)',T60,1PE10.2)
 FORMAT(6X/SURFACE AREA OF POLYMER (CM2)',T60,1PE10.2)
 268
 269
 270
 275

 280
 290
•^
 310
 320

 331
      FORMAT(//6X,'** OUTPUT VALUES (MODULE: EQUATION 26) **'/)
      FORMAT(6X,'TAU                          ',T60,1PE10.2,/
     1       6X,'ALPHA                        '.T60.1PE10.2)
      FORMAT(6X,'USER-SPECD. PARTITION COEFFICIENT  ',T60,1PE10.2)
C
      GO TO 999
 900  WRITE(IW,801)
 801  FORMAT(/6X,'PROGRAM STOP. INPUT ERROR. FOR A USER-SPECIFIED'
     1       /6X,'POLYMER CLASS, THE DIFFUSION COEFFICIENT OF
     2       /6X,'ADDITIVE IN POLYMER MUST BE A NON-ZERO VALUE.   ')
 999  STOP
      END
C       File DOMMFF                                            0990
C     REF: GANDEK.T.P., PH.D. THESIS, M.I.T., CAMBRIDGE, MA 1986
C       Domain MFF:  Finite slab in contact with a  finite, well-stirred bath.
C        Returns fractional migration, assuming that the solute concentration
C        in the bath is initially zero, and the distribution in the slab  is
C        uniform.
C       Requires the subprogram ROOTMFF.
C       Note that TAU   - Dpt/L2
C                 ALPHA - KV/AL
C
                                   102

-------
     DOUBLE  PRECISION  FUNCTION  DOMMFF(TAU,ALPHA)
     IMPLICIT REAL*8 (A-H,0-Z)
         COMMON/PARAM4/B(1000),NROOT,ALAST,NRTMFFMAX
         DOUBLE  PRECISION  PN
       COMMON/IO/IR.IW
     IF  (ALAST.NE.ALPHA)  NROOT-0
         EPS  - l.Od-8
         SUM  - ALPHA/(l.dO+ALPHA)
         DEN1 -  (l.dO/ALPHA)*(l.dO+l.dO/ALPHA)
         N -  0
 10       N -  N 4- 1
         IF (N.EQ.NRTMFFMAX) GOTO  30
         IF (N.LE.NROOT) GOTO  20
          B(N)  - ROOTMFF  (N,ALPHA)
          NROOT - N
 20       PN - B(N)
         PN2  - PN*PN
         TERM -  2.dO*EXP(-PN2*TAU)/(PN2+DENl)
         SUM  - SUM-TERM
         RAT  - ABS(TERM/SUM)
         IF (RAT.GT.EPS) GOTO  10
         GOTO 40
 30       WRITE(IW,101)
 101      FORMAT(/6X,'NOTE: GIVEN THE ABOVE INPUT AND THE ESTIMATED'
     1   /6X,'TAU VALUE, MIGRATION IS VERY CLOSE TO ZERO {<1.0E-04}.'
     2   //6X,'RECOMMENDATION: INCREASE EXPOSURE TIME, INCREASE
     3   /6X,'MIGRANT-POLYMER DIFFUSION COEFFICIENT, OR REDUCE POLYMER'
     4   /6X,'THICKNESS, AND RE-RUN SCENARIO.'/)
40       DOMMFF - SUM
         WRITE(IW,100)DOMMFF
100      FORMAT(6X,'FRACTION MIGRATED            ',T60,1PE10.2)
         RETURN
         END
C        ROOTMFF                                        0990
C     REF:  GANDEK.T.P., PH.D. THESIS, M.I.T., CAMBRIDGE, MA 1986
C        Finds roots of the equation Tan (BN) + ALPHA*BN - 0
C       Requires argument ALPHA - KV/AL and N - root #
         DOUBLE PRECISION FUNCTION ROOTMFF(N,ALPHA)
     IMPLICIT REAL*8 (A-H,0-Z)
      COMMON/I0/IR,IW
         ITR - 2
         IF (N.GT.O) GOTO 3
        BNN - O.dO
         IF (N.EQ.O) GOTO 45
        GOTO 30
3       EPS1 - O.ldO
        EPS2 - l.d-12
        RN - N
        PI - 3.141592653589793dO
        B2 - RN  * PI
         IF(ITR.EQ.1)WRITE(IW,*)'RUNTIME ERROR.Upper bound on BN -',B2
5       Bl - (RN-0.5dO)*PI -I- EPS1
        IF (ITR.EQ.1)WRITE(IW,*)'RUNTIME ERROR.Lower bound on BN ~',B1
        BN - Bl
10      TAN1 - TAN(BN)
                                  103

-------
        COS1 - COS(BN)
        SEC2 - l.dO/(COSl*COSl)
        R    - TAN1 + ALPHA*BN
        DR   - SEC2 + ALPHA
C       DDR  - 2.dO*TANl*SEC2
        BNN  - BN - (R/DR)
        IF (ITR.EQ.l) WRITE (IW,*) 'New BN -',BNN,'Residual -',R
        ETEST - ABS(BNN/BN - l.dO)
        IF (ETEST.LT.EPS2) GOTO 40
        IF (BNN.LT.BN) GOTO 20
        IF (BNN.GT.B2) GOTO 30
        BN - BNN
        GOTO 10
C       Need smaller EPS
20      EPS1 - EPS1 * O.ldO
        IF (EPSl.GE.(100.dO*EPS2)) GOTO 5
C       Failed
30      WRITE (IW,*) 'RUNTIME ERROR.Root search failed in ROOTMFF. N - 'N
        GOTO 45
C35      WRITE (IW,*) 'RUNTIME ERROR.Root search ends with large residual in ROOTMFF'
C     +, ' ,  N -' ,N
C        GOTO 45
C       Found root
40      IF (ABS(R).GT.l.dO) GOTO 30
45      ROOTMFF - BNN
        RETURN
        END
                                    104

-------
      PROGRAM NAME:  EQ29
                           KM DILWALI 0990
    THIS  PROGRAM MODULE  SERVES AS THE  I/O  SHELL TO
    COMPUTE  THE FRACTION OF ADDITIVE MIGRATION
    ASSUMING THE FOLLOWING CONDITIONS:   (OFF)
    -  FINITE SIAB(POLYMER) IN CONTACT  WITH ANOTHER FINITE SLAB(FOOD)
    -  FOOD PHASE CONCENTRATION INITIALLY IS  ZERO
    -  MIGRANT IS UNIFORMLY DISTRIBUTED IN  POLYMER
    INPUT REQUIREMENTS:
         IPOLY : 01-06  POLYMER TYPE (IF DIFF COEFF UNKNOWN)  -
            01- SILICONE RUBBER   04-HDPE
               NATURAL   RUBBER   05-POLYSTYRENE
02-
03- LDPE
                                 06-PVC(UNPLASTICIZED)
         XMW
         DP
         THRS
         XL
         NSIDE
         CINIT
         NEXT
         VEP
         SAP
         PC
         CSATP
         CSATA
         CSATW
         CSATE
         VPMT
         TMM
         XKOWM
         JEXT
                                               G/MOL
                                               CM2/S
                                               HRS
                                               CM
MOLECULAR WEIGHT OF MIGRANT
DIFFUSION COEFF. OF ADDITIVE IN POLYMER
TIME
POLYMER FILM THICKNESS
01-ONE-SIDED ,  02-TWO-SIDED DIFFUSION
INIT MIGRANT CONCN IN POLYMER G/CM3
EXT PHASE TYPE 01-AIR, 02-H20 ,  03-SOLID
VOL OF EXT PHASE      ,M3
EXPOSED SURFACE AREA  ,CM2
PARTITION COEFF; 0 IF UNKNOWN
SAT CONCN OF MIGR IN POLYMER OR 0.
SAT CONCN OF MIGR IN AIR     OR 0.
SAT CONCN OF MIGR IN H20     OR 0.
SAT CONCN OF MIGR IN EXT.PHASE
MIGR VAPOR PRESSURE,TORR FOR CSATA CALC
MIGR MELT TEMP DEC C    FOR CSATW CALC
MIGR OCT-H20 PART COEFF FOR CSATW CALC
TYPE OF EXTERNAL PHASE-0(OTHER);  01-06(POLYMER)
DIFFUSION COEFF. IN EXTERNAL PHASE, CM2/S
                                        ,G/CM3
                                        ,G/CM3
                                        , G/CM3
                                        , G/CM3
         DEXT
C********************
      IMPLICIT REAL*8 (A-H.O-Z)
      TAU - 0.
      ALPHA - 0.
      BETA  - 0.
      NRTDFFMAX - 1000
C********************
      COMMON/PARAM5/B(1000),NROOT,ALAST,BLAST,NRTDFFMAX
      COMMON/IO/IR,IW
      CHARACTER*2 TITLE(36)
      IR - 5
      IW - 6
      OPEN(IR,FILE-'EQ29.INP')
      OPEN(IW,FILE-'EQ29.LIS',STATUS-'UNKNOWN')
C  READ INPUT DATA
      READ(IR,'(36A2)')TITLE
      READ(IR,'(12)')IPOLY
      READ(IR,'(F10.2)')XMW
                                     105

-------
      READ(IR,'(F10
      READ(IR,'(F10
      READ(IR,'(F10
      READ(IR,'(12)
      READ(IR,'(F10
      READ(IR,'(I2)
      READ(IR,'(F10
      READ(IR,'(F10.2)
      READ(IR,'(F10.2)
      READ(IR,'(F10
      READ(IR,'(F10
      READ(IR,'(F10
      READ(IR,'(F10
      READ(IR,'(F10
      READ(IR,'(F10
      READ(IR,'(F10
      READ(IR,'(12)
      READ(IR,'(F10
   INITIALIZE
      XLEN - XL
      IF(NSIDE.EQ.2)XLEN=XL/2
      TSEC - THRS*3600.
    ECHO INPUT
               ,201)TITLE
               ,210)
               .EQ.1)WRITE(IW
               .EQ
               .EQ
               .EQ
               .EQ
              2)')DP
              2)')THRS
              2)')XL
              )NSIDE
              2)')CINIT
              )NEXT
              2)')VEP
                 )SAP
                 )PC
                 )CSATP
                 )CSATA
                 )CSATW
                 )CSATE
                 )VPMT
                 )TMM
                 )XKOWM
              )JEXT
              2)')DEXT
.2)
•2)
.2)
.2)
.2)
.2)
.2)
WRITE(IW,
WRITE(IW,
IF(IPOLY.
IF(IPOLY.
IF(IPOLY.
IF(IPOLY.
IF(IPOLY.
IF(IPOLY.EQ.
IF(XMW.NE.O.
          211)
          212)
          213)
          214)
          215)
            2)WRITE(IW,
            3)WRITE(IW,
            4)WRITE(IW,
            5)WRITE(IW,
            6)WRITE(IW,216)
            )WRITE(IW,220)XMW
WRITE(IW,221)THRS,XL
IF(DP.NE.O.)WRITE(IW,225)DP
IF(DEXT.NE.O.)WRITE(IW,226)DEXT
IF(NSIDE.EQ.1)WRITE(IW,230)
IF(NSIDE.EQ.2)WRITE(IW,235)
WRITE(IW,280)VEP
WRITE(IW,290)SAP
IF(NEXT.EQ.1)WRITE(IW,260)
IF(NEXT.EQ.2)WRITE(IW,265)
IF(NEXT.EQ.3)WRITE(IW,261)
IF(JEXT.EQ.O)WRITE(IW,410)
IF(PC.NE.O.)WRITE(IW,331)PC
  IF(PC.NE.O.)GO TO 15
IF(CSATP.NE.O.)WRITE(IW,266)CSATP
IF(CSATP.EQ.O.)WRITE(IW,267)CINIT
IF(NEXT.EQ.1.AND.CSATA.NE.O.)WRITE(IW,268)CSATA
IF(NEXT.EQ.2.AND.CSATW.NE.O.)WRITE(IW,269)CSATW
IF(NEXT.EQ.1.AND.CSATA.EQ.0.)WRITE(IW,270)VPMT
IF(NEXT.EQ.2.AND'.CSATW.EQ.O.)WRITE(IW,275)TMM,XKOWM
IF(NEXT.EQ.3)WRITE(IW,285)CSATE
C   LIST OUTPUT
15    WRITE (IW,. 310)
C   CALC DIFFUSION COEFF
                   IN POLYMER (DP) IF NOT USER-SPECIFIED
                                      106

-------
      IF(IPOLY.EQ.O.AND.DP.EQ.O.)GO TO 900
      IF(DP.EQ.O.)CALL DPCALC(IPOLY,XMW,DP)
C   CALC PARTITION COEFF IF NOT USER-SPECIFIED
      VPM - VPMT/760.
      IF(PC.EQ.0.)CALL KCALC(NEXT,CSATP,CSATA,CSATW,CINIT,CSATE,
     1                       XMW,VPM,TMM,XKOWM,PC)
C   CALC DIFFUSION COEFF IN EXTERNAL PHASE(DEXT) IF NOT USER-SPECIFIED
C     IF(JEXT.GT.6)GO TO 860
C     IF(JEXT.NE.O.AND.DEXT.EQ.O.)CALL DPCALC(JEXT,XMW,DEXT)
      IF(DEXT.NE.O.)GO TO 20
      CALL DPCALC(JEXT,XMW,DEXT)
      IF(JEXT.EQ.1)WRITE(IW,411)
      IF(JEXT.EQ.2)WRITE(IW,412)
      IF(JEXT.EQ.3)WRITE(IW,413)
      IF(JEXT.EQ.4)WRITE(IW,414)
      IF(JEXT.EQ.5)WRITE(IW,415)
      IF(JEXT.EQ.6)WRITE(IW,416)
C CALC NONDIMENSIONAL PARAMETERS
20    TAU - DP*TSEC/XLEN/XLEN
      VEPCC - VEP*1.0E6
      ALPHA - PC*VEPCC/SAP/XLEN
      BETA - PC*SQRT(DEXT/DP)
      WRITE(IW,320)TAU,ALPHA,BETA
      FRMIG - DOMDFF(TAU,ALPHA,BETA)
201
210
211
212
213
214
215
216
220
221

225

226

230
235
C
260
265
261
266

267

268
269
270
275

FORMAT (6X,
FORMAT (6X,
FORMAT (6X,
FORMAT (6X,
FORMAT (6X,
FORMAT (6X,
FORMAT (6X,
FORMAT (6X,
FORMAT (6X,
FORMAT (6X,
1 6X,
FORMAT (6X,
1 T60
FORMAT (6X,
1 T60
FORMAT (6X,
FORMAT (6X,

FORMAT (6X,
FORMAT (6X,
FORMAT (6X,
FORMAT (6X,
1
FORMAT (6X,
1
FORMAT (6X,
FORMAT (6X,
FORMAT (6X,
FORMAT (6X,
1 6X,
                'POLYMER CATEGORY:
                'POLYMER CATEGORY:
                'POLYMER CATEGORY:
                'POLYMER CATEGORY:
                'POLYMER CATEGORY:
                'POLYMER CATEGORY:
,T60,
,T60,
,T60,
,T60,
,T60,
,T60,
                           SILICONE  RUBBER ')
                           NATURAL  RUBBER ')
                           LDPE             ')
                           HOPE             ')
                           POLYSTYRENE      ')
                           PVC(UNPLASTICIZED)')
'MOLECULAR WEIGHT  OF ADDITIVE    ',T60,1PE10.2)
'TIME  (HRS)       ',T60,1PE10.2/
'TOTAL POLYMER SHEET THICKNESS   (CM)',T60,1PE10.
'USER-SPECIFIED DIFF.  COEFF. IN  POLYMER (CM2/S)1
.1PE10.2)
'USER-SPECIFIED DI3
,1PE10.2)
'DIFFUSION SPECIFIED AS',T60,
'DIFFUSION SPECIFIED AS',T60,
                                                                2)
                                      COEFF.  IN EXTERNAL PHASE (CM2/S)',
                                                ONE-SIDED')
                                                TWO-SIDED')
                'EXTERNAL PHASE IS        ',T60,'   AIR')
                'EXTERNAL PHASE IS        ',T60,'   WATER')
                'EXTERNAL PHASE IS        ',T60,'   SOLID')
                'SATURATION CONG.  OF MIGRANT IN POLYMER (G/CM3)',
                 T60,1PE10.2)
                'LET SATUR. CONC.  IN POLYMER - INIT CONG.(G/CM3)',
                 T60,1PE10.2)
                'SATURATION CONC.  IN AIR     (G/CM3)',T60,1PE10.2)
                'SATURATION CONC.  IN WATER   (G/CM3)',T60,1PE10.2)
                'MIGRANT VAPOR PRESSURE (TORR)  ',T60,1PE10.2)
                'MIGRANT MELT TEMP (DEC C)',T60,1PE10.2/
                'MIGRANT OCTANOL-WATER PART.  COEFF.',T60,1PE10.2)
 280  FORMAT(6X,'VOLUME OF EXTERNAL PHASE (M3)',T60,1PE10.2)
                                       107

-------
 285  FORMAT(6X,'SATUR. CONG. IN EXTERNAL PHASE  (G/CM3)',T60,1PE10.2)
 290  FORMAT(6X,'SURFACE AREA OF POLYMER (CM2)',T60,1PE10.2)
C
 310  FORMAT(//6X,'** OUTPUT VALUES  (MODULE: EQUATION 29) **'/)
 320  FORMAT(6X/TAU                           ' ,T60,1PE10. 2 ,/
     1       6X,'ALPHA                         ',T60,1PE10.2,/
     1       6X/BETA                          ' ,T60,1PE10.2)
 331  FORMAT(6X,'USER-SPECIFIED PARTITION COEFFICIENT',T60,1PE10.2)
 410  FORMAT(6X,'EXTERNAL PHASE POLYMER CATEGORY   :',
     1           T60,'  UNDEFINED')
 411  FORMAT(6X,'EXTERNAL PHASE POLYMER CATEGORY   :',
     1           T60,'  SILICONS RUBBER')
 412  FORMAT(6X,'EXTERNAL PHASE POLYMER CATEGORY   :',
     1           T60,'  NATURAL RUBBER')
 413  FORMAT(6X,'EXTERNAL PHASE POLYMER CATEGORY   :',
     1           T60,'  LDPE')
 414  FORMAT(6X,'EXTERNAL PHASE POLYMER CATEGORY   :',
     1           T60,'  HOPE')
 415  FORMAT(6X,'EXTERNAL PHASE POLYMER CATEGORY   :',
     1           T60,'  POLYSTYRENE')
 416  FORMAT(6X,'EXTERNAL PHASE POLYMER CATEGORY   :',
     1           T60,'  PVC(UNPLASTICIZED)')
C
      GO TO 999
 900  WRITE(IW,801)
 801  FORMAT(/6X,'PROGRAM STOP.  INPUT ERROR. FOR A USER-SPECIFIED'
     1       /6X,'POLYMER CLASS, THE DIFFUSION COEFFICIENT OF
     2       /6X,'ADDITIVE IN POLYMER MUST BE A NON-ZERO VALUE.  ')
 999  STOP
      END
C       File DOMDFF                                       0990
C     REF: GANDER,T.P., PH.D. THESIS, M.I.T., CAMBRIDGE, MA 1986
C       Domain OFF:  finite slab (polymer)  in contact with a finite slab (food)
C        Returns fractional migration assuming the solute concentration in
C        the food is initially zero and the concentration in the polymer is
C        initially uniform.
C       Requires subprogram ROOTDFF
C       Note that TAU   - Dpt/L2
C                 BETA  - Ksqrt(De/Dp)
        DOUBLE PRECISION FUNCTION DOMDFF(TAU,ALPHA,BETA)
     IMPLICIT REAL*8 (A-H,0-Z)
        COMMON/PARAM5/B(1000),NROOT,ALAST,BLAST,NRTDFFMAX
      COMMON/IO/IR.IW
        REAL LN
C       In order to avoid problems with the rootfinding routine, ROOTDFF,
C        ALPHA is perturbed slightly.  The potential problems occur when round
C        figures are used for BETA and ALPHA, when either are simply related
C        multiples or are common factors of a larger number, because then
C        asymptotes of the two tangent functions (in the characteristic eqn.)
C        can coincide.
C    If alpha > beta it is faster to calculate migration by letting
C     alpha and beta equal their reciprocals and tau - tau times the
C     square of beta/alpha.  The value of migration thus calculated
C     is multiplied by the true value of alpha to give migration for
C     the original case.


                                         108

-------
     IF  ((ALAST.NE.ALPHA).OR.(BLAST.NE.BETA)) NROOT-0
        ASAVE - ALPHA
        ALPHA - ALPHA * l.OOOOldO
        EPS1 - l.d-8
        SUM - l.dO - I.d0/(l.d0 + ALPHA)
        IFLAG - 0
        N - 0
10      N - N + 1
        IF (N.EQ.NRTDFFMAX) GOTO 40
        IF (N.LE.NROOT) GOTO  20
          B(N) = ROOTDFF(N,ALPHA,BETA)
          NROOT - N
20      LN - B(N)
        TAN1 - TAN(LN)
        DEN - l.dO + ALPHA +  (l.dO+ALPHA/(BETA*BETA))*TANl*TANl
        EN - 2.dO * TAN1*TAN1/(DEN*LN*LN)
        ARG1 - LN*LN*TAU
        ARG2 - LOG(EN)
        TERM - O.dO
        IF ((ARG2-ARGl).GT.-120.dO) TERM - EXP(ARG2-ARG1)
        SUM - SUM - TERM
        RAT - ABS(TERM/SUM)
        IF (RAT.LT.EPS1) GOTO 30
        IFLAG - 0     -
        GOTO 10
30      IFLAG - IFLAG + 1
        IF (IFLAG.LT.3) GOTO  10
        GOTO 50
 40     WRITE(IW.lOl)
101     FORMAT(/6X,'NOTE: GIVEN THE ABOVE INPUT AND THE ESTIMATED'
     1  /6X/TAU VALUE, MIGRATION IS VERY CLOSE TO ZERO {<1.0E-04}.'
     2  //6X,'RECOMMENDATION: INCREASE EXPOSURE TIME, INCREASE
     3  /6X,'MIGRANT-POLYMER DIFFUSION COEFFICIENT, OR REDUCE POLYMER'
     4  /6X,'THICKNESS, AND RE-RUN SCENARIO.'/)
        RETURN
50      ALPHA - ASAVE
        DOMDFF - SUM
      WRITE(IW,100)DOMDFF
100   FORMAT(6X,'FRACTION MIGRATED            ' ,T60,1PE10.2)
        RETURN
        END
C       File ROOTDFF, function ROOTDFF                0990
C     REF: GANDEK.T.P., PH.D. THESIS, M.I.T., CAMBRIDGE, MA 1986
C       Finds Nth root of the equation
C          tan (LN) + BETA*tan(ALPHA*LN/BETA) - 0
C       given the parameters ALPHA and BETA.  The root for N-0 is 0.
        DOUBLE PRECISION FUNCTION ROOTDFF(N,ALPHA,BETA)
     IMPLICIT REAL*8 (A-H.O-Z)
        DOUBLE PRECISION LN1,LN2,LNN
      COMMON/IO/IR.IW
C         N    - root number; root 0-0
C         Nl   - N + 1, used for calculation of asymptotes
C         EPS1 - desired accuracy of root
C         EPS2 - part of interval away from asymptote where the search begins
C         EPS3 - EPS2*BETA/ALPHA  if BETA < ALPHA
                                    109

-------
C              - EPS2                 if BETA > ALPHA
C         ETA  - 1/(1 + BETA/ALPHA)
C         BOA  - BETA/ALPHA
C         BAB  - BOA     if BETA < ALPHA
C              - I/BOA   if BETA > ALPHA
C         ETA  - ETA              if ALPHA < BETA
C              - 1/(1+ALPHA/BETA) if ALPHA > BETA
C
C       To trace program execution for diagnostic purposes, set IDIAG - 1
C        by passing a negative value of N to ROOTDFF
C
C       When roots cannot be determined, the returned root will be
C        negative. In order to trace the calculations via output
C        messages, asterisked comments C* can be activated. This module
C        can be recompiled and relinked accordingly.
        IDIAG - 0
        IF (N.LT.O) IDIAG - 1
C*      IF (IDIAG.EQ.l) WRITE (IW,*) 'N,ALPHA,BETA-',N,ALPHA,BETA
        IF (N.EQ.O) GOTO 160
        BOA - BETA/ALPHA
        BAB - BOA
        EPS1 - l.d-12
        EPS2 = O.ldO
        EPS3 - EPS2
C       if BETA.LT.ALPHA,  change scale to do calculations
        IF (BOA.LT.l.dO) BAB - l.dO/BOA
        IF (BAB.NE.BOA) EPS3 - EPS2*BOA
        PI1 - 2.dO*ACOS(0.dO)
        PI2 - PI1 * BAB
        Nl - N + 1
        IF (IDIAG.EQ.l) Nl - 1 - N
        RN - FLOAT(Nl)
        ETA - BAB/(l.dO + BAB)
C       Locate the two asymptotes
        DEC1 - RN*(l.dO -  ETA)
        DEC2 - RN*ETA
        TRU1 - AINT(DECl)
        TRU2 - AINT(DEC2)
        REM1 - DEC1 - TRU1
        REM2 - DEC2 - TRU2
        DIF - ABS(1.5dO-ETA-REMl)
C*      IF (IDIAG.EQ.l) WRITE (IW,*) DEC1,DEC2,TRU1,TRU2,REM1,REM2
        IF ((REMl.GE.O.SdO).AND.((REM1.LE.1.5dO-ETA)
     +.OR.(DIF.LT.l.d-13)))
     + GOTO 20
          AS2 - (TRU2 + 0.5dO) * PI1
          IF (REMl.GT.O.SdO) AS2 - (TRU2-0.5dO)*PIl
          AS1 - (TRU2 - 0.5dO) * PI1
          IF (REMl.GT.O.SdO) AS1 - (TRU2 -  1.5dO)*PIl
          AS1T - (TRU1 + O.SdO) * PI2
          IF ((AS1T.GT.AS1).AND.(AS1T.LT.AS2))  AS1 - AS1T
          GOTO 30
20      CONTINUE
          AS2 - (TRU1 + 0.5dO) * PI2
          AS1 - (TRU2 - O.SdO) * PI1
                                     110

-------
30      IF  (ABS(AS2-AS1).LT.(l.d-13*BOA)) GOTO 40
        IF  (ABS(AS2-ASl).LT.(100.dO*EPSl)) GOTO 50
C       Have two distinct asymptotes which bound the desired root
35      IF  (BAB.EQ.BOA)  GOTO 60
C         revert scale  (changed because  BETA.LT.ALPHA)
          AS1 - AS1*BOA
          AS2 - AS2*BOA
        GOTO 60
C       Asymptotes coincide, root — asymptote
40      ROOT - -AS2
C*      IF  (IDIAG.EQ.l)  WRITE  (IW,*) 'Asymptotes are ',AS1,AS2
C       Revert scale if  BETA.LT.ALPHA
C*      WRITE (IW,*) '---asymptotes coincide, N ~',N
        IF  (BAB.NE.BOA)  ROOT - ROOT*BOA
        GOTO 150
C       Two asymptotes within 10*EPSl of each other (but not identical)
C*50    WRITE (IW,*) 'Asymptotes too close.  Root is arithmetic mean.'
C*      WRITE (IW,*) 'Asymptotes are ',AS1,AS2
50      ROOT - -(AS2 + ASl)/2.dO
        GOTO 150
C       The two asymptotes bound the desired root.  To find the root, we
C         start at the right and use Newton's method to 'walk down the
C         curve' to the  root.  If an inflection point is encountered, we
C         start from the other direction.  If the root is within "EPS3 of
C         an asymptote,  an alternate form of the residual is used.
60      BMIN - l.dO
C*      IF  (IDIAG.EQ.l) WRITE  (IW,*) 'Asymptotes are ',AS1,AS2
        IF  (BETA.LT.l.dO) BMIN - BETA
        LEFT - 0
        LN1 - AS2 - EPS3*(AS2-AS1)
C*      IF  (IDIAG.EQ.l) WRITE (IW,*) '    START FROM RIGHT'
101     LN2 - LN1/BOA
        TAN1 - TAN(LNl)
        TAN2 - TAN(LN2)
        SEC1 - l.dO/COS(LNl)
        SEC2 - l.dO/COS(LN2)
        R    - TAN1 + BETA*TAN2
        DR   - SEC1*SEC1 + ALPHA*SEC2*SEC2
        DDR  - 2.dO*SECl*SECl*TANl + 2.dO*(ALPHA/BOA)*SEC2*SEC2*TAN2
        LNN  - LN1 - (R/DR)
        IF  (((LEFT.EQ.O).AND.(R.LT.O.dO)).OR.((LEFT.EQ.l)
     -KAND.(R.GT.O.dO)))
     + GOTO 130
        ICODE - 120
        IF  ((ABS(LNN-LNl).LT.EPSl).AND.(ABS(R).LT.BMIN)) GOTO 140
        IF  ((LEFT.EQ.O).AND.((DDR.LE.O).OR.(LNN.LE.AS1)))  GOTO 110
        ICODE - 121
        IF  ((LEFT.EQ.l).AND.((DDR.GE.O.dO).OR.(LNN.GE.AS2))) GOTO 120
C       Update guess (take one Newton step)
        LN1 - LNN
C*      IF  (IDIAG.EQ.l) WRITE (IW,*) '     Take one Newton step...'
        GOTO 101
C       Newton's search  stalled by inflection point.  Switch sides.
110     LEFT - 1
C*      IF  (IDIAG.EQ.l) WRITE (IW,*) '    Stalled--switch to left side'
                                       111

-------
         LN1  - AS1 + EPS3*(AS2-AS1)
         GOTO 101
 C        Search method unsuccessful
 120      WRITE (IW,*)  'Root search failed in routine ROOTDFF,  N -',N
 C*       WRITE (IW,*)  'ROOT,  CODE -',LNN,ICODE
         ROOT - -LNN
         GOTO 150
 C        Root is 'close'  to an asymptote;  use alternate  form for
 C          the residual to  minimize calculational  error
 130      IF (LEFT.EQ.O) AS1 - LN1
         IF (LEFT.EQ.l) AS2 - LNl
 C*       IF (IDIAG.EQ.l)  WRITE (IW,*)  '   Switch  to alternate form (roof
 C*   +,'  close to asymptote)'
         LEFT - 1 -  LEFT
         IF (LEFT.EQ.O) LNl - AS2
         IF (LEFT.EQ.l) LNl - AS1
         ILEFT - 0
 132      LN2  - LN1/BOA
         COT1 - l.dO/TAN(LNl)
         COT2 - l.dO/TAN(LN2)
         CSC1 - l.dO/SIN(LNl)
         CSC2 - l.dO/SIN(LN2)
         R    - COT1 + COT2/BETA
         DR   - -CSC1*CSC1  -  CSC2*CSC2/(BETA*BOA)
         DDR   - 2.dO*CSCl*CSCl*COTl + 2.dO*CSC2*CSC2*COT2/(BETA*BOA*BOA)
         LNN   - LNl  -  (R/DR)
         IF (((LEFT.EQ.O).AND.(LNN.GT.AS2)).OR.
     +              ((LEFT.EQ.l).AND.(LNN.LT.AS1))) GOTO 135
         ICODE  - 122
         IF ((ABS(LNN-LNl).LT.EPSl).AND.(ABS(R).LT.BMIN)) GOTO  140
C       Update  guess  (take one Newton step)
        LNl  -  LNN
C*       IF (IDIAG.EQ.l) WRITE  (IW,*) '    Take one Newton step	'
        GOTO  132
C       Step  in wrong  direction; switch sides
135     IF (ILEFT.EQ.O) GOTO  137
        ICODE - 123
        GOTO  120
137     ILEFT - 1
C*      IF (IDIAG.EQ.l) WRITE  (IW,*) '  Stalled—switch to  opposite side'
        LEFT -  1  - LEFT
        LNl - AS1
        IF (LEFT.EQ.O) LNl - AS2
        GOTO  132
C       Zero-th root
160     ROOT -  O.dO
        GOTO 150
C       Have located the desired root
140     ROOT -  LNN
        ROOTP - ROOT + 2.dO*EPSl
        ROOTM - ROOT  - 2.dO*EPSl
        RP - TAN(ROOTP) + BETA*TAN(ROOTP/BOA)
        RM - TAN(ROOTM) + BETA*TAN(ROOTM/BOA)
        ICODE - ICODE + 100
        IF (RP/RM.GT.O.dO) GOTO  120
                                    112

-------
150     ROOTDFF - ROOT
C*      IF (IDIAG.EQ.l)WRITE(IW,*)'   Variables N,ROOTDFF,R in routine'
C*   +  '  ROOTDFF are, respectively  ',N,ROOTDFF,R
        RETURN
        END
                                      113

-------
     PROGRAM NAME:  EQ31
                     KM DILWALI 0990'
   THIS PROGRAM MODULE SERVES AS THE I/O SHELL TO
   COMPUTE THE FRACTION OF ADDITIVE MIGRATION
   ASSUMING THE FOLLOWING CONDITIONS:   (DFI)
   - FINITE SLAB(POLYMER) IN CONTACT WITH SEMI-INFINITE SLAB(FOOD)
   - FOOD PHASE CONCENTRATION INITIALLY IS ZERO
   - MIGRANT IS UNIFORMLY DISTRIBUTED IN POLYMER
   INPUT REQUIREMENTS:
         IPOLY : 01-06 POLYMER TYPE
           01- SILICONS RUBBER   04-HDPE
           02- NATURAL  RUBBER   05=POLYSTYRENE
           03- LDPE              06-PVC(UNPLASTICIZED)
         XMW
         DP
         THRS
         XL
         NSIDE
         CINIT
         NEXT
         PC
         CSATP
         CSATA
         CSATW
         CSATE
         VPMT
         TMM
         XKOWM
         JEXT
         DEXT
C********************
      IMPLICIT REAL*8 (A-H.O-Z)
      TAU - 0.
      BETA  - 0.
C********************
      COMMON/IO/IR.IW
      CHARACTER*2 TITLE(36)
      IR - 5
      IW - 6
      OPEN(IR,FILE-'EQ31.INP')
      OPEN(IW,FILE-'EQ31.LIS
C  READ INPUT DATA
      READ(IR,'(36A2)')TITLE
      READ(IR,'(12)')IPOLY
      READ(IR,'(F10.2)')XMW
      READ(IR,'(F10.2)')DP
      READ(IR,'(F10.2)')THRS
      READ(IR,'(F10.2)')XL
      READ(IR,'(I2)')NSIDE
      READ(IR,'(F10.2)')CINIT
MOLECULAR WEIGHT OF MIGRANT              G/MOL
DIFFUSION COEFF. OF ADDITIVE IN POLYMER  CM2/S
TIME                                     HRS
POLYMER FILM THICKNESS                   CM
01-ONE-SIDED ,  02-TWO-SIDED DIFFUSION
INIT MIGRANT CONCN IN POLYMER G/CM3
EXT PHASE TYPE 01-AIR, 02-H20 ,  03-SOLID
PARTITION COEFF; 0 IF UNKNOWN
SAT CONCN OF MIGR IN POLYMER OR 0.,G/CM3
SAT CONCN OF MIGR IN AIR     OR 0.,G/CM3
SAT CONCN OF MIGR IN H20     OR 0.,G/CM3
SAT CONCN OF MIGR IN EXT.PHASE    ,G/CM3
MIGR VAPOR PRESSURE,TORR FOR CSATA CALC
MIGR MELT TEMP DEC C    FOR CSATW CALC
MIGR OCT-H20 PART COEFF FOR CSATW CALC
TYPE OF EXTERNAL PHASE-0(OTHER); 01-06(POLYMER)
DIFFUSION COEFF. IN EXTERNAL PHASE,  CM2/S
            .STATUS-'UNKNOWN')
                                    114

-------
c
15
C
               ' (I2)')NEXT
               '(F10.2)')PC
               '(F10.2)
               '(F10.2)
               '(F10.2)
               '(F10.2)
               '(F10.2)
               '(F10.2)
               '(F10.2)
               '(I2)')JEXT
               '(F10.2)')DEXT
)CSATP
)CSATA
)CSATW
)CSATE
)VPMT
)TMM
)XKOWM
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
   READ(IR,
INITIALIZE
   XLEN - XL
   IF(NSIDE.EQ.2)XLEN-XL/2.
   TSEC - THRS*3600.
 ECHO INPUT
   WRITE(IW,201)TITLE
   WRITE(IW,210)
   IF(IPOLY.EQ.1)WRITE(IW,211)
   IF(IPOLY.EQ.2)WRITE(IW,212)
   IF(IPOLY.EQ.3)WRITE(IW,213)
   IF(IPOLY.EQ.4)WRITE(IW,214)
   IF(IPOLY.EQ.5)WRITE(IW,215)
   IF(IPOLY.EQ.6)WRITE(IW,216)
   IF(XMW.NE.O.)WRITE(IW,220)XMW
   WRITE(IW,221)THRS,XL
   IF(DP.NE.O.)WRITE(IW,225)DP
   IF(DEXT.NE.O.)WRITE(IW,226)DEXT
   IF(NSIDE.EQ.1)WRITE(IW,230)
   IF(NSIDE.EQ.2)WRITE(IW,235)
   IF(NEXT.EQ.1)WRITE(IW,260)
   IF(NEXT.EQ.2)WRITE(IW,265)
   IF(NEXT.EQ.3)WRITE(IW,261)
   IF(JEXT.EQ.1)WRITE(IW,411)
   IF(JEXT.EQ.2)WRITE(IW,412)
   IF(JEXT.EQ.3)WRITE(IW,413)
   IF(JEXT.EQ.4)WRITE(IW,414)
   IF(JEXT.EQ.5)WRITE(IW,415)
   IF(JEXT.EQ.6)WRITE(IW,416)
   IF(PC.NE.O.)WRITE(IW,331)PC
     IF(PC.NE.O.)GO TO  15
   IF(CSATP.NE.O.)WRITE(IW,266)CSATP
   IF(CSATP.EQ.O.)WRITE(IW,267)CINIT
   IF(NEXT.EQ.1.AND.CSATA.NE.O.)WRITE(IW,268)CSATA
   IF(NEXT.EQ.2.AND.CSATW.NE.0.)WRITE(IW,269)CSATW
   IF(NEXT.EQ.1.AND.CSATA.EQ.O.)WRITE(IW,270)VPMT
   IF(NEXT.EQ.2.AND.CSAT¥.EQ.O.)WRITE(IW,275)TMM,XKOWM
   IF(NEXT.EQ.3)WRITE(IW,285)CSATE
 LIST OUTPUT
   WRITE(IW,310)
 CALC DIFFUSION  COEFF IN POLYMER (DP)  IF  NOT USER-SPECIFIED
   IF(IPOLY.EQ.O.AND.DP.EQ.O.)GO TO 900
   IF(DP.EQ.O.)CALL DPCALC(IPOLY,XMW,DP)
 CALC PARTITION  COEFF IF NOT USER-SPECIFIED
   VPM - VPMT/760.
                                  115

-------
      IF(PC.EQ.0.)CALL KGALC(NEXT,CSATP,CSATA,CSATW,CINIT,CSATE,
     1                       XMW.VPM.TMM.XKOWM.PC)
C   CALC DIFFUSION COEFF IN EXTERNAL PHASE(DEXT) IF NOT USER-SPECIFIED
      IF(JEXT.GT.6)GO TO 860
      IF(JEXT.EQ.O.AND.DEXT.EQ.O.)GO TO 860
      IF(JEXT.NE.O.AND.DEXT.EQ.O.)CALL DPCALC(JEXT,XMW,DEXT)
C CALC NONDIMENSIONAL PARAMETERS
      TAU - DP*TSEC/XLEN/XLEN
      BETA - PC*SQRT(DEXT/DP)
      WRITE(IW,320)TAU,BETA
      FRMIG - DOMDFI(TAU,BETA)
                36A2//)
                '** INPUT PARAMETERS **'/)
                'POLYMER CATEGORY:',T60,'   SILICONE RUBBER ')
                'POLYMER CATEGORY:
                'POLYMER CATEGORY:
                'POLYMER CATEGORY:
                'POLYMER CATEGORY:
                'POLYMER CATEGORY:
201
210
211
212
213
214
215
216
220
221

FORMAT (6X
FORMAT (6X
FORMAT (6X
FORMAT (6X
FORMAT (6X
FORMAT (6X
FORMAT (6X
FORMAT (6X
FORMAT (6X
FORMAT (6X
1 6X
                                          SILICONE RUBBER
                                          NATURAL  RUBBER
                                          LDPE
                                          HOPE
                                          POLYSTYRENE
                                          PVC(UNPLASTICIZED)')
                                  ,T60,
                                  ,T60,
                                  ,T60,
                                  ,T60,
                                  ,T60,
               'MOLECULAR WEIGHT OF ADDITIVE   ',T60,1PE10.2)
               'TIME  (HRS)      ',T60,1PE10.2/
               'TOTAL POLYMER SHEET THICKNESS  (CM)',T60,1PE10.2)
225  FORMAT(6X,'USER-SPECIFIED DIFFUSION COEFFICIENT (CM2/S)',
    1       T60.1PE10.2)
226  FORMAT(6X,'USER-SPECIFIED DIFF. COEFF.IN EXTERNAL PHASE (CM2/S)'
    1       T60.1PE10.2)
     FORMAT(6X,'DIFFUSION SPECIFIED AS',T60,'  ONE-SIDED')
               'DIFFUSION SPECIFIED AS',T60,'  TWO-SIDED')
                 EXTERNAL PHASE IS        ',T60,'   AIR')
                 EXTERNAL PHASE IS        ',T60,'   WATER')
                 EXTERNAL PHASE IS        ',T60,'   SOLID')
                 SATURATION CONG.  OF MIGRANT IN POLYMER (G/CM3)',
                 T60.1PE10.2)
                 LET SATUR.  CONG.
                 T60.1PE10.2)
                                  IN AIR     (G/CM3)',T60,1PE10.2)
                                  IN WATER   (G/CM3)',T60,1PE10.2)
230
235
C
260
265
261
266
FORMAT (6X
FORMAT (6X
FORMAT (6X
FORMAT (6X
FORMAT (6X
FORMAT (6X
267  FORMAT(6X,
                                 IN POLYMER - INIT CONC.(G/CM3)',
268  FORMAT(6X,'SATURATION CONC
269  FORMAT(6X,'SATURATION CONC
270  FORMAT(6X,'MIGRANT VAPOR PRESSURE (TORR)  ',T60,1PE10.2)
275  FORMAT(6X,'MIGRANT MELT TEMP (DEC C)',T60,1PE10.2/
    1       6X,'MIGRANT OCTANOL-WATER PART.  COEF.',T60,1PE10.2)
285  FORMAT(6X,'SATURATION CONC. IN EXTERNAL PHASE  (G/CM3)',
    1           T60.1PE10.2)
310  FORMAT(//6X,'** OUTPUT VALUES (MODULE:  EQUATION 31) **'/)
     FORMAT(6X,'TAU                          ',T60,1PE10.2,/
            6X/BETA                         '  ,T60,1PE10.2)
     FORMAT(6X,'USER-SPECD PARTITION COEFFICIENT',T60,1PE10.2)
     FORMAT(6X,'EXTERNAL PHASE POLYMER CATEGORY  :',
                T60,'  SILICONE RUBBER')
               'EXTERNAL PHASE POLYMER CATEGORY  :',
                T60,'  NATURAL RUBBER')
     FORMAT (6X/EXTERNAL PHASE POLYMER CATEGORY  :',
    1           T60,'  LDPE')
     FORMAT(6X,'EXTERNAL PHASE POLYMER CATEGORY  :',
    1           T60,'  HOPE')
 320

 331
 411
412   FORMAT(6X
     1
413

414
                                     116

-------
 415  FORMAT(6X,'EXTERNAL PHASE  POLYMER CATEGORY   :',
      1         .  T60,'   POLYSTYRENE')
 416  FORMAT(6X,'EXTERNAL PHASE  POLYMER CATEGORY   :',
      1           T60,'   PVC(UNPLASTICIZED)')
      GO TO 999
860   WRITE(IW,703)
703   FORMAT(/6X,'PROGRAM STOP.  INPUT ERROR. SOLID EXTERNAL PHASE  '
      1       /6X/MUST BE SPECIFIED AS O(OTHER) OR 01-06(POLYMER
      2       /6X,'CATEGORY). FOR "OTHER" SOLIDS, THE DIFFUSION COEFF.'
      3       /6X/MUST BE INPUT  BY USER. REFER TO SEC.4.2 OF REPORT.')
      GO TO 999
 900  WRITE(IW,801)
 801  FORMAT(/6X,'PROGRAM STOP.  INPUT ERROR. FOR A USER-SPECIFIED'
      1       /6X,'POLYMER CLASS,  THE DIFFUSION COEFFICIENT OF
      2       /6X,'ADDITIVE IN POLYMER MUST BE A NON-ZERO VALUE.  ')
 999  STOP
      END
C       File DOMDFI                                     0990
C     REF: GANDER,T.P.,  PH.D. THESIS, M.I.T., CAMBRIDGE, MA 1986
C       Domain DPI:  Finite slab  (polymer) in contact with a semi-infinite
C        slab (food).  Returns fractional migration assuming the food phase
C        concentration is initially zero and the solute is uniformly
C        distributed in  the polymer.
C       Requires the subprograms  ERFCA, SHANK
C
C       Note that TAU   - Dpt/L2
C                 ALPHA - KV/AL
C                 BETA  - Ksqrt(De/Dp)
C
C  NOTE TO USER: For very low values of TAU, migration rates approach
C  zero, and this function subroutine cannot compute a value of DOMDFI.
C  In the unlikely event of certain input conditions (low Dp,low t,etc.)
C  a FRACTION MIGRATED is not reported on screen or in the output
C  file. Nor does an error message appear if the routine cannot execute
C  the mathematics.  In such instances, assume FRACTION MIGRATED is
C  negligible.
        DOUBLE PRECISION FUNCTION DOMDFI(TAU,BETA)
     IMPLICIT REAL*8 (A-H.O-Z)
      COMMON/IO/IR,IW
        NMAX - 1000
        EPS - l.Od-8
C       Solution is indeterminate if BETA - 1
        BSAVE - BETA
        IF (BETA.EQ.l.dO) BETA -  1.00001dO*BETA
        PI - 3.1415926535898dO
        RPI - SQRT(PI)
        BP - l.dO + BETA
        BM - l.dO -  BETA
        PRE - 2.dO*BETA*SQRT(TAU/PI)/BP
        SUM - l.dO
        PRET - 2.dO*BETA/BP
        PREI - BM/BP
        PREA - PRET/PREI
        N - 0
10      N - N + 1


                                     117

-------
        IF(N.GE.NMAX)GO TO 40
        PREA - ,PREA*PREI
        RN - DBLE(N)
        ARG1 - RN/SQRT(TAU)
        Bl - EXP(-ARG1*ARG1)
        B2 - ARG1*RPI*ERFCA(ARG1)
        TERM - PREA*(B1-B2)
        SUM - SUM  - TERM
        RAT - ABS(TERM/SUM)
        IF (RAT.GT.EPS) GOTO 10
        BETA - BSAVE
        DOMDFI = PRE*SUM
        WRITE(IW,100)DOMDFI
 100  FORMAT(6X,'FRACTION MIGRATED             ',T60,1PE10.2)
        RETURN
  40    WRITE(IW.lOl)
101     FORMAT(/6X,'NOTE: GIVEN THE ABOVE INPUT AND THE ESTIMATED'
     1  /6X/TAU VALUE, MIGRATION IS VERY CLOSE TO ZERO Kl.OE-04}.'
     2  //6X,'RECOMMENDATION: INCREASE EXPOSURE TIME, INCREASE
     3  /6X,'MIGRANT-POLYMER DIFFUSION COEFFICIENT, OR REDUCE POLYMER'
     4  /6X, "THICKNESS, AND RE-RUN SCENARIO.'/)
        RETURN
        END
C       ERFCA
C     REF: GANDEK.T.P., PH.D. THESIS, M.I.T., CAMBRIDGE, MA 1986
C        Function ERFCA(Z) calculates the complementary error
C         function of the real, positive argument Z.  The result
C         is accurate to about 8 significant figures at
C         worst.
C       Requires function SHANK	
        DOUBLE PRECISION FUNCTION ERFCA(Z)
     IMPLICIT REAL*8 (A-H.O-Z)
        DIMENSION S(5)
        IF (Z.LT.-Z) GOTO 30
        IF (Z.GT.12.0) GOTO 30
        Z2 - Z*Z
        RTPI - 0.17724538509055152D+01
        IF (Z.GT.4.0DO) GOTO 15
5       EPS - l.d-16
        TERM - 2.dO*EXP(-Z2)*Z/RTPI
        SUM - l.dO
        SUM - SUM - TERM
        L - 0
10      L - L + 1
        RN - FLOAT (L)
        TERM - TERM*2.dO*Z2/(2.dO*RN + l.dO)
        SUM - SUM - TERM
        RAT - ABS(TERM/SUM)
        IF (RAT.GE.EPS) GOTO 10
        ERFCA - SUM
        RETURN
15      EPS - l.d-9
        NS - 5
        SUM - EXP(-Z2)/(RTPI*Z)
        TERM - SUM
                                    118

-------
         ITERM - 0
         TERMOLD -  10. dO
         ILOC - 1
         L - 0
20       L - L + 1
         RN - FLOAT(L)
         TERM - TERM*(l.dO  - 2 .dO*RN)/(2 . dO*Z2)
         SUM - SUM  + TERM
         RAT - ABS (TERM/SUM)
         IF (ABS (TERM). GT. TERMOLD) ITERM - ITERM + 1
         TERMOLD -  ABS (TERM)
         IF ( ITERM. EQ.O) GOTO 40
         S(ILOC) -  SUM
         ILOC - ILOC + 1
         IF (ILOC.LE.NS+1) GOTO 40
         SX - SHANK(NS.S)
         ERFCA - SX
         RETURN
40       IF (RAT.GE.EPS) GOTO 20
         ERFCA - SUM
         RETURN
30       ERFCA - O.dO
         RETURN
         END
C        Function SHANK, uses Shanks transform to accelerate
C        convergence of a series.                          0990
C         S is a vector (length 25) containing N consecutive
C         terms of the series.  S is destroyed during the calculation.
C     REF:  GANDEK.T.P., PH.D.  THESIS,  M.I.T., CAMBRIDGE, MA 1986
C
        DOUBLE PRECISION FUNCTION SHANK(N,S)
     IMPLICIT REAL*8 (A-H.O-Z)
        DIMENSION S(25)
        M - N
5       DO 10 I - l.M-2
10      CONTINUE
        M - M-2
        IF (M.GE.3) GOTO 5
        IF (M.EQ.2) SHANK - S(2)
        IF (M.EQ.l) SHANK - S(l)
        RETURN
        END
                                      119

-------
      SUBROUTINE DPCALC(IPOLY,DMW,DPOLY)
C SUB-PROGRAM NAME: DPCALC             KM DILWALI 0990
C THIS ROUTINE CALCULATES THE POLYMER DIFFUSION COEFF.
C AS A FUNCTION OF THE POLYMER TYPE AND MIGRANT MOLECULAR WT.
C MAIN PROGRAM : AMEM CALLS EQUATION MODULES, WHICH CALL DPCALC.
      IMPLICIT REAL*8 (A-H.O-Z)
      COMMON/IO/IR.IW
      GO TO  (100,200,300,400,500,600),IPOLY
C   SILICONE RUBBER
100    IF(DLOG10(DMW) .LE. 1.5) DLPOLY- -.384*DLOG10(DMW)-4.12
       IF((DLOG10(DMW) .GT. 1.5)  .AND. (DLOGIO(DMW) .LE. 2.125))
     1     DLPOLY- -1.31*DLOG10(DMW)-2.73
       IF(DLOG10(DMW) .GT. 2.125) DLPOLY—2.73*DLOG10(DMW)
          GOTO 690
C   NATURAL RUBBER
200    IF(DLOG10(DMW) .LE. 1.375) DLPOLY - -.94*DLOG10(DMW)-4.53
       IF((DLOG10(DMW) .GT. 1.375) .AND.  (DLOGIO(DMW) .LE.  2.31))
     1     DLPOLY- -1.58*DLOG10(DMW)-3.64
       IF(DLOG10(DMW) .GT. 2.31) DLPOLY- -3.49*DLOG10(DMW)+.76
           GOTO 690
C   LDPE
300    IF(DLOG10(DMW) .LE. 1.83) DLPOLY- -1.40*DLOG10(DMW)-5.12
       IF((DLOG10(DMW) .GT. 1.83) .AND. (DLOGIO(DMW) .LE. 2.55))
     1   DLPOLY- -2.01*DLOG10(DMW)-4.01
       IF(DLOG10(DMW) .GT. 2.55) DLPOLY- -4.22*DLOG10(DMW)+1.61
            GOTO 690
C   HOPE
400    IF(DLOG10(DMW) .LE.
       IF(DLOG10(DMW) .GT.
            GOTO 690
C   POLYSTYRENE
500    IF(DLOG10(DMW)
       IF(DLOG10(DMW)
                       2.05) DLPOLY- -2.0*DLOG10(DMW)-4.50
                       2.05) DLPOLY- -4.0*DLOG10(DMW)-.40
C
600

690

900
PVC
IF(DLOG10(DMW)
     GOTO 690

IF(DLOG10(DMW)
IF(DLOG10(DMW)
                  .LE.
                  .GT.
                  .GT.
                    1.625) DLPOLY- -4.0*DLOG10(DMW)-2.5
                    1.625) DLPOLY- -10.4*DLOG10(DMW)+7.95
                  .LE.
                  .GT.
                           1.825) DLPOLY- -11.104*DLOG10(DMW)+9.88
1.43) DLPOLY- -7.8*DLOG10(DMW)-I-1.70
1.43) DLPOLY- -9.41*DLOG10(DMW)+4.16
   DPOLY- 10.0**DLPOLY
  WRITE(IW,900)DPOLY
  FORMAT(6X/ESTD. DIFFUSION COEFFICIENT IN POLYMER (CM2/S)',
 1          T60.1PE10.2)
   RETURN
   END
                                      120

-------
      SUBROUTINE KCALC(NEXT,CSATP,CSATA,CSATW,CINIT,CSATE,
      1                      XMW,VPM,TMM,XKOWM,CPART)
C  K2M DILWALI                               0990
C  REVISIONS BY RCREID  INCORPORATED  IN ORIGINAL EQUATIONS.
C  THIS ROUTINE CALCS THE PARTITION  COEFFICIENT DEFINED AS THE RATIO
C  OF THE SATURATION CONCN OF MIGRANT IN THE EXTERNAL PHASE TO THE
C  CONCN OF MIGRANT IN  THE POLYMER.
C  INPUT REQMTS ARE AS  FOLLOWS:
C   CSATP =SAT CONCN OF MIGRANT IN POLYMER      G/CM3
C   CSATA OR CSATW -SAT CONCN OF MIGRANT IN EXT PHASE  G/CM3
C     EXTERNAL PHASE- AIR> VPM -MIGRANT VAP PRESSURE @ 25C(ATM)
C     EXTERNAL PHASE- H20> TMM -MIGRANT MELT TEMP  (DEC C )
C                        XKOWM -MIGRANT OCTANOL-H20 PART COEFF.
C     EXTERNAL PHASE- SOLID> CSATE
      IMPLICIT REAL*8 (A-H.O-Z)
      COMMON/IO/IR.IW
      CPART - 0.
      IF(CSATP.EQ.O.)CSATP=CINIT
      GOTO (100,200,300).NEXT
C EXT PHASE IS AIR
100   IF(CSATA.NE.O.)GO TO 110
      CSATA - VPM*XMW*40.626 *1.E-06
      WRITE(IW,510)CSATA
510   FORMAT(6X,'ESTD.  SATUR.  CONC. OF MIGRANT IN AIR (G/CM3)',
      1           T60, 1PE10.2)
110   CEXT  - CSATA
      GO TO 400
C EXT PHASE IS H20 — USE REVISED EQN FOR TLOG FROM RC REID
200   IF(CSATW.NE.O.)GO TO 210
      TLOG-ALOGIO(XMW)-1.123*ALOG10(XKOWM)-0.0099*TMM-2.067
      CSATW - 10.**TLOG
      WRITE(IW,520)CSATW
520   FORMAT(6X,'ESTD.  SATUR.  CONC. OF MIGRANT IN WATER (G/CM3)',
     1           T60, 1PE10.2)
210   CEXT  - CSATW
      GO TO 400
C EXT PHASE IS SOLID
300   CEXT - CSATE
      IF(CSATE.EQ.O.)WRITE(IW,540)
540   FORMAT(6X,'SATURATION CONCENTRATION IN EXTERNAL PHASE SPECIFIED'
     1      /6X,'AS 0.0 G/CM3. RECHECK INPUT.')
400   CPART - CEXT/CSATP
      WRITE(IW,530)CPART
530   FORMAT(6X,'ESTIMATED PARTITION COEFFICIENT'.T60.1PE10.2)
      RETURN
      END
                                       121

-------
      SUBROUTINE RKCALC(NEXT,KHORZ,NPLC,AIRVEL,RSURFA,VPM,
     1 HEIGHT,WATVEL,IRWAT,RSURFW,DPIPE,XMW,RK)
C K2M DILWALI                               0990
C REVISIONS BY RCREID INCORPORATED IN ORIGINAL EQUATIONS.
C THIS ROUTINE CALCULATES THE MASS TRANSFER COEFFICIENT RK
C THRU AIR OR WATER. FOR AIR, THE FLOW VELOCITY, POLYMER
C POSITION HORIZ OR VERT, POLYMER LOCATION INDOORS OR OUTDOORS
C AND POLYMER DIMENSIONS ARE REQD; FOR WATER, THE FLOW VELOCITY,
C AND POLYMER TYPE PLATE OR PIPE, AND DIMENSION ARE REQD INPUT.
      COMMON/IO/IR,IW
      IMPLICIT REAL*8 (A-H.O-Z)
      IF(NEXT.EQ.2)GO TO 200
C AIR FLOW TRANSFER CASES
C CALC LAMINAR BULK FLOW CONVECTION TERM
       RKA-2.0*(AIRVEL**Q.5)/((((2.5 + (XMW**0.333))**!.33)
     + *(RSURFA**0.5)))
C CALC THERMAL CONVECTION TERM FOR INDOORS ONLY
       IF(NPLC.EQ.l)DAIR-3.3/((2.5 + (XMW)**.333)**2.0)
       IF(NPLC.EQ.1)RKB-I.3*DAIR
C CALC DENSITY DRIVEN CONVECTION TERM FOR VERTICAL SURFACES ONLY
       IF(KHORZ.EQ.2)RKC-(0.41*VPM/HEIGHT)**.25
C USE APPROPRIATE MASS TRANS COEFF EQN BASED ON POSN & LOCN
       IF((KHORZ.EQ.l).AND.(NPLC.EQ.2))GO TO 110
       IF((KHORZ.EQ.l).AND.(NPLC.EQ.1))GO TO 120
       IF((KHORZ.EQ.2).AND.(NPLC.EQ.2))GO TO 130
       IF((KHORZ.EQ.2).AND.(NPLC.EQ.1))GO TO 140
110    RK-RKA
       RETURN
120    RK-(1.0/RKA + 1.0/RKB)**(-1.0)
       RETURN
130    RK-U.O/RKA + I.O/RKC)**(-I.O)
       RETURN
140    RK-(1.0/RKA + 1.0/RKB + 1.0/RKC)**(-1.0)
       RETURN
C  WATER FLOW TRANSFER CASES
200   IF(IRWAT.EQ.l)GO TO 300
C   WATER FLOW OVER A PLATE
       RE-100.0*RSURFW*WATVEL
       IF(RE .GT. 1000000.0) GOTO 250
C  LAMINAR PLATE FLOW
       RK - 5.2E-04*SQRT(WATVEL/RSURFW)
       RETURN
C  TURBULENT PLATE FLOW
250    RK- 1.2E-04*(WATVEL**0.8)/(RSURFW**0.2)
       RETURN
C   WATER FLOW THROUGH A PIPE
300    RE-100.0*WATVEL*DPIPE
       DMIG- 7.45E-05/(XMW**0.408)
       IF(RE .GT. 2100.0) GOTO 350
C   LAMINAR FLOW SITUATION
       RK-3.65*DMIG/DPIPE
       RETURN
C   TURBULENT FLOW SITUATION
350    RK- 7.2E-05*(WATVEL**0.8)/(DPIPE**0.2)
       RETURN
       END
                                        122

-------
           Worst Case Scenario
                                                 Select External Phase
                                            Air
                                            Water
                                            Solid
                                                    Press Enter To Continue J
                                           	 Exposure To External Phase
                                            One Side
                                            Two Sides
 Modify Input Values
 Consider Partitioning / Mass Transfer
 Quit
	 Press Enter To Continue J
	 Press Enter To Continue J
	 Identify Polymer Type 	
 SILICONS RUBBER
 NATURAL RUBBER
 LDPE
 HOPE
 POLYSTYRENE
 PVC (UNPLASTICIZED)
	 Press Enter To Select
             AMEM: Arthur D. Little, Inc. Polymer Migration  Estimation Model
                      Figure A-2.  Example of AMEM menu screen.

-------
                                 File = EQ23.LIS
  [  BEGINNING OF FILE  ]

        TITLE>


        ** INPUT PARAMETERS  **

        POLYMER CATEGORY:                                        HOPE
        MOLECULAR WEIGHT  OF  ADDITIVE                            1.50E+02
        TIME  (HRS)                                              l.OOE+02
        TOTAL POLYMER  SHEET  THICKNESS   (CM)                     5.00E-02
        DIFFUSION SPECIFIED  AS                                  TWO-SIDED


        ** OUTPUT VALUES  (MODULE: EQUATION  23)  **

        ESTD. DIFFUSION COEFFICIENT IN  POLYMER (CM2/S)          7.86E-10
        TAU                                                      4.53E-01
        FRACTION MIGRATED                                       7.35E-01
L[0000/0020]-
               AMEM: Arthur D.  Little, Inc.  Polymer Migration  Estimation Model
      Figure A-3.  Example of AMEM output screen for worst case migration prediction using Eq. (3-23).

-------
                                   APPENDIX B

                               AMEM EVALUATION


                                                                          Page No.
1.   INTRODUCTION 	  126
2.   CONCLUSIONS AND RECOMMENDATIONS	  127
3.   EXAMPLE CASES	  130
     Example #1 - Rigid Poly(vinyl Chloride) Sheet/Stearyl Alcohol	  130
     Example #2 - Rigid Poly(vinyl Chloride) Sheet/Dioctyl Tin Stabilizer 	  131
     Example #3 - Rigid Poly(vinyl Chloride) Pipe/Calcium Stearate  	  132
     Example #4 - Rigid Poly(vinyl Chloride) Pipe/Tin Stabilizer	  134
     Example #5 - Low Density Polyethylene/Dihydroxybenzophenone	  135
     Example #6 - High Density Polyethylene/Dihydroxybenzophenone	  137
     Example #7 - Isotactic Polypropylene/Dihydroxybenzophenone	  138
     Example #8 - High Density Polyethylene/3,5-Di-t-Butyl-4-Hydroxytoluene ....  139
     Example #9 - High Density Polyethylene/Irganox 1076  	  140
     Example #10 - Polystyrene/Styrene  	  141
     Example #11 - Impact Polystyrene/3,5-Di-t-Butyl-4-Hydroxytoluene  	  143
     Example #12 - Plasticized Poly(vinyl Chloride)/Di-2-Ethylhexyladipate  	  144
     Example #13 - Plasticized Poly (vinyl Chloride)/Dibutyl Phthalate  	  146
References	  148
                                        125

-------
1.     INTRODUCTION

       The methods for estimating migration from polymeric materials provided in
Volume 11 and the computer program AMEM (Arthur D. Little, Inc., Migration Estimation
Model) were derived from diffusion and mass transfer theory. They require inputs of
physical property data and values describing physical characteristics of the system.  In many
cases, the properties or characteristics are not known and must be estimated.  Furthermore,
there were several  important assumptions and constraints used to develop the AMEM. These
factors lead to questions of the validity and limitations of the methods as practical means for
predicting migration. In Section 5 of Volume 11, example calculations were presented to
illustrate  use of the predictive model.   Several examples were based on conditions of actual
migration testing.  Other examples, however, were unrealistic although they did demonstrate
application of the methods developed.

       The objective of this appendix is to evaluate the range of applicability and the
limitations of the methods  by comparing predicted migration with migration data published in
the technical literature.  The focus is data for situations in which either air or water  was  the
external phase.  Furthermore, there was the constraint that the experimental conditions
associated with the published data be sufficiently described so that the model inputs could be
specifically stated,  readily deduced, or  closely approximated. Based on these criteria, 13
example cases were selected.  Descriptions of the 13 cases, and the application of the
methodology to predict migration for each case  follow.
                                         126

-------
2.     CONCLUSIONS AND RECOMMENDATIONS

       The methods for estimating migration were validated by comparing predictions
generated with the AMEM computer program with those measured experimentally.  The
thirteen examples documented herein as summarized in Table B-l. In most of the examples
the agreement between the model and the experimental results was within an order of
magnitude. Those cases in which good agreement is not achieved appear to be in two
categories:

•      The first category is those migrants for which the diffusion coefficient, D  , in the
       polymer is not known and the molecular weights (MW) are greater than those  used to
       develop the D  versus MW correlation (i.e., Figures 5 and 6 in Volume 11). For these
       migrants, it appears that the D  predicted by the AMEM is too low, especially in the
       case of rigid polyvinyl chloride (PVC). Another possibility is that the very low
       migration levels reported in some examples may be accounted for by release from the
       surface rather  than a diffusional process.

•      The second category involves migrants, such as an antioxidant, that migrate rapidly
       but have a low solubility and degrade in the external phase. In its present form, the
       AMEM model does not take into account chemical reaction of the migrating species.

       As areas for continued improvement of the AMEM model, we recommend better
quantification of mass transfer coefficients for conditions of particular concern to  the EPA.
These conditions might include conditions in rooms and buildings under various forms of
ventilation, in water storage containers, and in automobile interiors.  There is also the  need to
develop and validate procedures  for applying the AMEM methodology at temperatures outside
the 20-30°C range of the present model. We also recommend continued monitoring of the
technical literature for migration data that can be used to further validate and establish limits
for application of the  AMEM model.  Finally, we recommend tailoring the AMEM model
interface to address exposure  scenarios of particular interest to the EPA.
                                         127

-------
                            TABLE B-l.  MIGRATION DATA EXAMPLES USED TO VALIDATE AMEM MODEL
K)
oo
Example
No. Polymer
1 Rigid PVC
2 Rigid PVC
3 Rigid PVC
4 Rigid PVC
5 LDPE
6 HDPE
7 PP
8 HDPE
Migrant, External Temp.
Initial Loading (wt %) Phase (°C)
Stearyl alcohol, Water 20
0.6%
Irgastab 17 MOK, Water 20
1%
Calcium stearate, Water 25
1.4%
Dioctyl tin stablizer, Water 25
1.1%
Dihydroxybenzophenone, Water 44
0.03%
Dihydroxybenzophenone, Water 44
0.003%
Dihydroxybenzophenone, Water 44
0.07%
3,5-Di-t-butyl-4-hydroxytoluene, Water 40
0.2%
Time Fraction Migrated
(days)
Measured AMEM
60 0.00005 <0.0001
60 0.00008 <0.0001
42 0.0005 <0.0001
42 7.0xlO-7 <0.0001
10 1.0 1.0
10 0.8 0.97
10 0.7 0.64
.10 0.005 0.014-
0.12
Comments
Good agreement
Good agreement
Fair agreement,
inadequate D value
Good agreement,
inadequate D value
Good agreement
Good agreement,
inadequate Dp value
Good agreement
Prediction brackets
measured value
                                                                                                         depending on

                                                                                                         external volume input
                                                                                                                  (continued)

-------
                                                         TABLE B-l .(Continued)
to
Example
No.
9

10

11
12
13
Migrant, External
Polymer Initial Loading (wt %) Phase
HOPE Irganox 1076, Water
0.1%

Polystyrene Styrene monomer, Water
0.1%

Impact Polystyrene 3,5-Di-t-butyl-4-hydroxytoluene, Water
1%
Plasticized PVC Diethylhexyl adipate, Air
39%
Plasticized PVC Dibutyl phthalate, Water
33%
Temp. Time
(°C) (days)
40 10

40 0.04
7
42
49 5
35
58 .17
25 5
15
Fraction
Measured
0.00065

0.002
0.007
0.008
0.002
0.005
0.01
0.004
0.006
Migrated
AMEM
0.007-
0.051

0.003
0.009
0.01
0.002
0.004
0.005
0.003
0.003
Comments
Prediction brackets
measured value
depending on
external volume input
Good agreement

Good agreement
Fair agreement,
partitioning/mass
transfer important
Good agreement
partition equilibrium
predicted

-------
3.     EXAMPLE CASES

Example #1 - Unplasticized Polv(vinyl Chloride) Sheet/Stearvl Alcohol

       To better understand migration from packaging materials into foods, cosmetics and
pharmaceutical products, Figge, Koch, and Freytag (1978) studied the migration of stearyl
alcohol and a dioctyl tin stabilizer from unplasticized or rigid polyvinyl chloride  (PVC).
They also measured the migration of antioxidants from high density polyethylene.  Their work
on the  polyethylene is used later in Examples #8 and #9.

       In one series of tests, room temperature water was the external  phase. The additives
were radiolabelled (14C)  and incorporated into a PVC compound having the following
composition:

                                                              wt %

            PVC resin                                         98.0
            Irgastab 17 MOK                                    1.0
            Stearyl alcohol                                       0.6
            Loxiol E 10                                         0.4

       Irgastab  17 MOK® is a trade name for 2-ethylhexyl di-n-octyltindithiogiycollate, a
dioctyl tin stabilizer. Two samples of the PVC blend were produced:   one with the dioctyl tin
stabilizer radiolabelled and the other with the stearyl alcohol  carrying the radiolabel.  Test
samples were pressed to a thickness of 0.035 cm. The sample size was such that the two-
sided exposure surface area was 9 cm .  The samples were immersed in 1 liter of water at
20°C for 60 days after which the concentration of the migrant in the water was measured.
The  results for the migration of stearyl alcohol follow while the results for the dioctyl tin
stabilizer are reported in Example #2.

       The results for the 60-day migration experiment were reported as a percentage of the
initial additive amount so that only a direct conversion is required to compare the measured
result with the fraction migrated value predicted by the AMEM model.  Figge et al. found
that 0.005 weight percent of  the stearyl  alcohol originally present in the PVC migrated into
the water.  This is equivalent to a fraction migrated of 0.00005 or 5 x  10"5.

       Prediction of migration under these conditions using AMEM requires the  following
input data and conditions:

       Migration:                        To water
       Exposure:                        Two  sides
       Thickness:                        0.035 cm
       Time:                            1440 hours (60 days)
       Polymer:                         PVC (Unplasticized)
       Migrant:                          Stearyl alcohol, MW = 270.48
       Migrant  Initial Concentration:       0.6 wt% or 8.4 x 10"3  g/cm3
                                        [(0.006 g/g PVC)(1.4 g/cm3) at a polymer density
                                        of 1.4 g/cm3]

                                          130

-------
       External Phase:                   Stagnant water at 20°C
                                        Volume:  0.001 m3 (1  liter)
       Surface Area:                     9 cm2
       Surface Length:                   2 cm

       Because no value was reported, the diffusion coefficient for stearyl alcohol in the
polymer was estimated using AMEM.  This was done by entering 1 at the diffusion
coefficient entry screen (to estimate), selecting PVC (Unplasticized) from the polymer type
screen, and entering the migrant molecular weight. The D  estimated by AMEM is
1.88 x 10"19 cm2/s.  With these minimum inputs of Dp = 1.9 x 10"19 cm2/s, time =
1440 hours, and thickness = 0.035 cm, the worst case fraction migrated was predicted to be
< 0.0001 using Eq. (3-23)*.  (Note:  As reported in Section 6 of Volume 11, AMEM does
not report fraction migrated values lower than 0.0001 for Eq. (3-23) prediction involving
small values of i.) This prediction is in good agreement with the measured value of 5 x 10
fraction migrated.

       Because the migrant is essentially insoluble in water and the degree of mixing was not
reported, partitioning and mass transfer resistances may act to reduce migration .even further.
In this example,  however, the worst case migration is already predicted to be quite low so
that further reductions in the migration prediction will not be apparent.  Figge et al. gives no
information as to the degree of mixing in the water, so a good estimate for a mass transfer
coefficient is difficult to make. However, using the polymer surface length of 2 cm and an
essentially stagnant water velocity of 10 cm/s, the mass transfer coefficient for flow over  a
plate was  estimated by AMEM as 1.2 x 10  cm/s. The saturation concentration for stearyl
alcohol in water is reported as less than 1 ppm or 1 x 10"6 g/cm3.  On the basis of the initial
stearyl alcohol concentration in the polymer (because the saturation concentration of stearyl
alcohol in the polymer is not known) and an aqueous solubility of 1 x 10   g/cm , the
partition coefficient was estimated by AMEM as 1.2 x 10"4. Considering these partitioning
and mass  transfer conditions, AMEM again predicts < 0.0001  of the stearyl alcohol initially
present to migrate in 1,440 hours using Eq. (3-17).

Example  #2 - Rigid Pory(vinyl Chloride) Sheet/Dioctvl Tin Stabilizer

       The migration of Irgastab  17  MOK, a dioctyl tin stabilizer, also was measured from
the same PVC sheet samples to water after 60 days at 20°C. The result was that 0.008 wt%
or 0.00008 of the stabilizer initially present migrated to the water.  Migration under these
conditions was estimated using AMEM and the following input data:

       Migration:                        To water
       Exposure:                        Two sides
       Thickness:                        0.035 cm
       Time:                            1440 hours (60 days)
       Polymer:                         PVC (Unplasticized)
       Migrant:                          Irgastab 17 MOK, MW = 750.69 g/mol
       In this addendum, the equations referred to are those in Volume 11.

                                          131

-------
       Migrant Initial Concentration:      1.0 wt% or 0.014 g/cm3
                                        (at a polymer density of 1.4 g/cm3)
       External Phase:                   Stagnant water at 20°C
                                        Volume:  0.001 m3
       Surface Area:                     9 cm2
       Surface Length:                   2 cm

       Again, a value was not reported for the diffusion coefficient of the dioctyl tin
stabilizer in the PVC so it was estimated by AMEM using the migrant molecular weight and
the correlation for PVC (Unplasticized).  D  was estimated as 1.3 x 10   cm/s.  Note that
the migrant molecular weight is at the upper bound of the diffusion coefficient versus
log(molecular weight) correlation shown  in Figure 5 of Volume 11. Thus, the estimate D
value is uncertain and may underestimate the actual value.  The worst case migration
predicted using Eq. (3-23) and the values for D , polymer thickness, and time was < 0.0001
fraction migrated. This prediction is in good agreement with the measured value of 8 x 10"5.

       Again, partitioning and mass transfer resistances can be considered but will not result
in a further reduction of the fraction migrated prediction. For the same Example #1
conditions of a 10 cm/s water velocity and a 2 cm polymer surface length, the mass transfer
coefficient, k, was estimated as  1.2 x 10   cm/s.  The water solubility of Irgastab 17 MOK is
reported as less than 25 ppm or 2.5 x 10"5 g/cm3.  The partition coefficient K was estimated
by AMEM as 1.8 x  10"3 using the aqueous solubility value and the initial concentration in the
polymer of 1.4 x  10   g/cm .  Although partitioning and mass transfer resistances in the
external phase may act to reduce the migration rate, the Eq. (3-17) prediction by AMEM was
again a fraction migrated of < 0.0001.

Example #3 - Rigid Poly(vinyl Chloride) Pipe/Calcium Stearate

       Dietz, Banzer,  and Miller (1979) studied the migration of additives from  pipe
fabricated from a PVC compound having the following composition:
            PVC resin
            Dioctyl tin stabilizer
            Titanium dioxide
            Wax
            Calcium stearate

In compounding, phr stands for parts per hundred resin on a weight basis.  For example, in
the above formulation there are 1.2 grams of stabilizer for every 100 grams of PVC resin.

       Sections of a one-inch diameter, schedule 40 pipe were used as the test specimens.
They were 14-cm in length and eight such lengths were placed in a glass jar containing
two liters of water.  This surface to volume ratio (pipe surface to water volume), according to
the authors, corresponds to 4 mL of water per square inch of pipe surface as recommended by
the National Sanitation  Foundation.  The temperature was held at 25°C.  Migration of the

                                          132

-------
calcium stearate and dioctyl tin stabilizer was measured.  The results are presented here for
calcium stearate and in Example #4 for the dioctyl tin stabilizer.

       Results for the calcium stearate migration were reported as a part per million
concentration found in the two-liter volume of water.  To convert this to a fraction migrated,
the polymer sample volume and initial migrant concentration in the polymer must be
calculated. To calculate the sample volume, the following dimensions of schedule 40 pipe
were required:

                    Inner Diameter             2.664 cm
                    Outer Diameter            3.340 cm
                    Wall thickness             0.338 cm

On the basis of these dimensions, the exposed surface area (inner and outer) of each pipe
specimen was 264.1  cm2, and its volume 44.63 cm3.  Thus, for the eight samples used in the
experiment, the total surface area was 2112.8 cm2 and the total pipe volume was 357 cm3 in
the two liters of water.

       The initial concentration of the calcium stearate in the PVC compound was 1.5 phr.
To convert this to a concentration, the weight fraction of the calcium stearate in the PVC is
multiplied by the density of the compound, which is assumed to be that of PVC or 1.4 g/cm3:

                         (1.5 g/104.8  g)(1.4 g/cm3) = 0.02 g/cm3

Therefore, there was (0.02 g/cm3)(357 cm3) or 7.15 g of calcium  stearate initially present in
the 357 cm3 of the pipe samples.

       After six  weeks (1,008 hours) immersion of the pipe samples, the concentration of
calcium stearate in the water was measured to be 1.91 ppm. To convert this to a mass of
calcium stearate, one must multiply by the mass of water present:

             (1.91 g calcium stearate/1 x 106 g H2O)(2000 g H2O) = 0.0038 g

Therefore, (0.0038 g/7.15 g) or a 0.00053 fraction of the calcium stearate initially present in
the PVC pipe migrated into the water at the end of 1,008  hours.

       To predict migration under these conditions, the inputs to  AMEM were:

       Migration:                       To water
       Exposure:                        Two sides
       Thickness:                       0.338 cm
       Time:                           1,008 hours
       Polymer:                         PVC (Unplasticized)
       Migrant:                         Calcium stearate, MW = 607 g/mol
       Migrant Initial Concentration:      0.02 g/cm3
       External Phase:                   Stagnant water at 20°C
                                       Volume:  0.002 m3

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       Surface Area:                      2112.8cm2
       Surface Length:                    14 cm

       Since no diffusion coefficient was reported by the researchers, one was estimated by
AMEM using the migrant molecular weight.  For PVC (unplasticized), the diffusion
coefficient was estimated as 9.34 x  10"23 cm2/s.  The worst case migration predicted by
Eq. (3-23) was < 0.0001 fraction migrated, which is lower than the measured value of
5 x 10~4.

       Mass transfer resistance and partitioning effects were also considered. With a water
flow velocity set equal to 10 cm/s and a pipe surface length of 14 cm (treated as flow over
surface and not flow through pipe because the samples are small  sections and involve 2-sided
exposure), the mass transfer coefficient estimated by AMEM is 4.4 x 10"4 cm/s.    The
saturation concentration of calcium stearate in water is reported as 25 ppm or
2.5 x 10"5 g/cm3. The fraction migrated predicted by Eq. (3-17) again was < 0.0001.

       From both Eq. (3-23) and Eq. (3-17), the predicted value is lower than the measured
fraction migrated. The reason for the lack of agreement is not known. For Examples #1  and
#2, Figge et al. commented that  the migration may  be due to a surface release rather than a
diffusional process. If this were the case in this example, the  model would not be applicable.
Another possibility is that the correlation by which  the diffusion coefficient is estimated is
inadequate. With reference to Figure 5 in Volume  11, the molecular weight of the stabilizer
is higher than that for which data are available.   Perhaps the correlation cannot be
extrapolated.

Example #4 - Rigid Polyfvinvi Chloride) Pipe/Tin Stabilizer

       In a separate experiment under the  same conditions as  Example #3, the migration of
the dioctyl tin stabilizer, di(n-octyl)tin s,s'-bis(iso-octylmercaptoacetate), was measured from
eight, schedule 40 PVC pipe specimens into a two-liter volume of water.  In this case the
initial concentration of the migrant was 1.2 phr  or:

                         (1.2 g/104.8 g)(1.4 g/cm3) = 0.016  g/cm3
       	               <3         ft
       Thus, (0.016 g/cm )(357 cm ) or 5.7 g of the dioctyl tin stabilizer were initially
present in the 357 cm3 PVC pipe specimens.  After the six weeks, the concentration of the
stabilizer measured in the water was less than the detection limit  of 0.002 ppm.  In 2 liters or
2,000 g of water, this concentration corresponds to  a stabilizer mass of less than 4.0 x 10"6 g
or 4 jig.  Converting this mass to a fraction of the initial migrant concentration indicates that
less than a 7 x 10"7 fraction  migrated in  the 1,008 hours.

       For this example, the inputs to AMEM were:

       Migration:                         To water
       Exposure:                         Two sides
       Thickness:                         0.338 cm
       Time:                             1,008 hours

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       Polymer:                         PVC (Unplasticized)
       Migrant:                         Dioctyl tin stabilizer, MW = 751.8 g/mole
       Migrant Initial Concentration:      0.016 g/cm3
       External Phase:                   Stagnant water at 25°C
                                        Volume:  0.002 m3
       Surface Area:                     2112.8cm2

       Again, because no diffusion coefficient was reported for the stabilizer in the PVC, a
value was estimated using the migrant molecular weight. The estimate for Dp was
1.25 x 10"23 cm2/s.  The worst case migration predicted  by AMEM was again < 0.0001
fraction migrated.

       Similar to Examples #1 through #3, partitioning and mass transfer resistances can be
considered but will not reduce  the fraction migration prediction because of the very low
prediction under worst case conditions. For a 10  cm/s water flow velocity and a 14 cm
polymer surface length, the mass transfer coefficient, k,  was estimated as 4.4 x 10"4 cm/s.
The solubility of the dioctyl tin stabilizer in water at 25°C is reported as less than 25 ppm or
2.5 x 10  g/cm3. For this solubility in water and the initial  concentration in the polymer, the
partition coefficient, K, was estimated as 1.6 x 10 . The fraction migrated predicted by
Eq. (3-17), which considers partitioning and mass transfer resistances, is again the same as
that predicted with Eq. (3-23).

Example #5 - Low Density Polyethylene/Dirivdroxybenzophenone

       The migration of 2,4-dihydroxybenzophenone from three polyolefins was studied by
Westlake and Johnson (1975).  They performed both diffusion and migration studies to
characterize the migration process.  In the diffusion experiments, they measured the diffusion
coefficient and the saturation concentration of the stabilizer in each polymer. Then, in
migration  tests, they measured  the fraction migrated into water after 10 days. Both
experiments, however, were conducted at 44°C, a temperature higher than the 20-30°C range
for which  the AMEM estimation procedures were developed. The results for low density
polyethylene (LDPE) follow, while the results for high density polyethylene (HDPE) are
reported in Example #6 and for polypropylene (PP) in Example #7.

       The LDPE diffusion study was performed  at 44°C using a coupon 0.15-cm thick and
3.5-cm in  diameter. One side of the coupon was  saturated with radiolabelled 2,4-
dihydroxybenzophenone.  The increase in the radioactive counting rate on the downstream
side of the coupon, initially free of stabilizer, was monitored. The counting rate was
monitored until an equilibrium rate was attained.  At this point, the diffusion process had
reached equilibrium and the polymer was saturated with the  migrant. From the counting rate
data, the diffusion coefficient was calculated to be 2.7 x 10"9 cm2/s. The saturation
concentration of 2,4-dihydroxybenzophenone in the LDPE was measured as 0.03 wt%.

       In  the migration experiment, a saturated LDPE coupon, 0.022-cm in thickness, was
immersed  in 10 mL of water at 44°C with both surfaces in contact with the water.  The
amount of stabilizer lost from the coupon after  10 days was  determined from the
concentration of the stabilizer in  the water, which was changed periodically over the test

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duration.  An important factor not reported was the frequency at which the 10 mL volume of
water was changed. The authors only state that "the volume of water was well-mixed and
changed at appropriate intervals" and that "the solubility of the stabilizer in water must be
high relative to the concentrations encountered during the experiment."

       After 240 hours (10 days), all of the stabilizer initially present in the LDPE had
migrated into the water.  For comparison purposes, the diffusion coefficient calculated from
the migration data was reported as 4.2 x  10  cm/s, which is slightly higher than the value
measured in the diffusion experiment.

       Thus, the data input to predict migration under these conditions were:

       Migration:                        To water
       Exposure:                         Two sides
       Thickness:                        0.022 cm
       Time:                            240 hours
       Polymer:                         LDPE
       Migrant:                         2,4-dihydroxybenzophenone, MW =, 214.21 g/mol
       Migrant Initial  Concentration:       0.03 wt% or 2.76 x 10"4 g/cm3)
                                        (at a polymer density of 0.92 g/cm3)
       External Phase:                   Water at 44°C
                                        Volume: 1 x 10"5 m3 (10 cm3) that is frequently
                                        changed and well-mixed
       Surface  Area:                     19.24 cm2

In this example, we used the Dp value measured in the diffusion experiment, 2.7 x 10"9 cm2/s,
as input to AMEM. Under worst case conditions using Eq.  (3-23), AMEM predicts that all of
the dihydroxybenzophenone migrates to the water in 240 hours, as was found in the migration
experiment.

       If the diffusion coefficient value was not reported, D  could be estimated using
AMEM, however, the  value will underestimate the actual D  in this case because the
estimation technique applies only to the 20-30°C temperature range.  The value predicted by
AMEM from the molecular weight diffusion coefficient correlation for LDPE was
        Q   O
2.0 x 10  cm /s, which is only slightly lower than the value measured in the diffusion
experiment. Using this estimated Dp value, AMEM also predicts a fraction migration of 1.0.

       Although the external phase was reported to be well-mixed and frequently changed,
partitioning and mass transfer effects can be considered if certain assumptions are made
regarding the external  phase. Although reported as well-mixed, we estimated a mass transfer
coefficient  value using a 10 cm/s water flow rate and a 3.5 cm surface length.  The estimated
value, 8.8  x 10"4 cm/s, thus represents a value approximating that for stagnant water.  For
partitioning effects,  a value for K was estimated using the reported saturation concentration in
the polymer, 2.76 x 10  g/cm , and a value for the saturation concentration in  water,
1 x 10"4 g/cm3 (Merck Index, 1990). Thus, K was estimated as 0.36. The volume of water
                             c  q        ^
was assumed to be only 1  x 10   m  (10 cm ), which may underestimate the actual total
volume because the 10 cm3 volume was changed repeatedly.  With these input values,

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AMEM predicts a fraction migrated of 0.95 using Eq. (3-17); thus partitioning and mass
transfer resistances contribute only a very small effect. Because of the good agreement
between the worst case model prediction and the data, mass transfer and partitioning factors
apparently are not relevant. Even when using conservative assumptions, these conditions only
reduce the fraction migrated prediction from 1.0 to 0.95.

Example #6 - High Density Polyethylene/Dihydroxybenzophenone

       Similar diffusion and migration experiments were performed with HDPE coupons.
From the diffusion experiment at 44°C, the diffusion coefficient was measured to be
5.8 x 10"10 cm2/s and the saturation concentration of the 2,4-dihydroxybenzophenone in
HDPE was reported as 0.003 wt%. In the migration experiment,  a 0.8 fraction of the
stabilizer initially present in the  HDPE migrated into the water after 240 hours. Again,
details regarding the  mixing, total volume, and frequency of changing the water were not
reported.

       The following input data were used with AMEM:

       Migration:                        To water
       Exposure:                        Two sides
       Thickness:                        0.022 cm
       Time:                            240 hours
       Polymer:                         HDPE
       Migrant:                          2,4-Dihydroxybenzophenone, MW = 214.21 g/mol
       Migrant Initial Concentration:       0.003 wt% or 2.88 x 10"5 g/cm3
                                        (at a polymer density  of 0.96 g/cm3)
       External Phase:                   Water at 44°C
                                        Volume:  1 x 10"5 m3 (10 cm3) that is frequently
                                        changed and well-mixed
       Surface Area:                     19.24 cm2

       As in the previous example, the diffusion coefficient value measured in the diffusion
experiment,  5.8 x 10"10 cm2/s, was first used to estimate the worst case migration.  With this
value, the worst case prediction  using Eq. (3-23) would be that all of the
dihydroxybenzophenone migrates to the water in the 240 hours.  This prediction is close to
but exceeds  the reported value of 0.8 fraction migrated. Possibly, mass transfer or
partitioning effects are factors in this case.

       Again, although all values  were not reported, some  approximations can be  made to
judge whether partitioning or mass transfer affect migration.  Although reported as well-
mixed, we estimated  a mass transfer coefficient value  using a 10  cm/s surface flow rate and a
3.5 cm surface length.  The estimated value was 8.3 x 10  cm/s.  For partitioning effects, we
estimated a value for K using the reported saturation concentration in the polymer and the
value for the saturation concentration in water.  Thus K was  estimated as 3.47, which  is quite
high. The volume of water was assumed to be only 10 cm3, which may underestimate the
actual volume.  With these input values,  AMEM predicts a fraction migrated  of 0.99 using
Eq. (3-17); thus partitioning and mass transfer resistances are predicted to contribute only a

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small reduction in the fraction migrated.  Consequently, the slight lack of agreement is
attributed to the D value used, which overestimates the migration behavior.  In fact,
Westlake and Johnson report that a lower value of D  was calculated from  the migration test
results as compared with the value from the diffusion experiment

       If a value  for D  had not been reported, one could be estimated using AMEM. The
estimation technique, however, was developed for the temperature range from 20-30°C.  At
the molecular weight of 214.21 g/mol, AMEM estimates a value for D  equal to
1.9 x 10   cm /s.  For this value for estimated value of D , the fraction migrated prediction
was 0.971 using Eq.  (3-23) for worst case conditions, and a fraction migrated of 0.966 using
Eq. (3-17) to consider mass transfer and partitioning limitations.

Example #7 - Isotactic  Polypropvlene/Dihydroxybenzophenone

       Data were also reported by Westlake and Johnson for diffusion and  migration
experiments with  an isotactic PP sample. The polypropylene polymer was reported to be 65%
crystalline with a density of 0.9 g/cm .  From the diffusion study at 44°C, the saturation
concentration of the 2,4-dihydroxybenzophenone in the polypropylene coupon was reported as
0.07% by weight  and the diffusion coefficient was reported as 5.5 x 10"11 cm2/s.  In the
migration experiment, 0.7 of the stabilizer initially present migrated into the water after
240 hours.  The input data to AMEM are:

       Migration:                        To water
       Exposure:                         Two sides
       Thickness:                        0.022 cm
       Time:                            240 hours
       Polymer:                         Isotactic PP, 65% crystalline
       Migrant:                          2,4-Dihydroxybenzophenone, MW = 214.21 g/mol
       Migrant Initial Concentration:       0.07 wt% or 6.3 x 10"4  g/cm3
                                        (at a polymer density of 0.90 g/cm3)
       External Phase:                    Water at 44°C
                                        Volume:  1 x 10"5 m3 that is frequently changed
                                        and well-mixed
       Surface Area:                     19.24 cm2

       The diffusion coefficient measured in the diffusion experiment, 5.5 x 10"11 cm2/s, was
used to predict migration.  In this case, AMEM predicts the worst case migration using
Eq. (3-23) as 0.69 fraction migrated, which is in very good agreement with the measured
value of 0.7.

       As in Examples #5 and #6, partitioning and mass transfer effects can be considered if
some assumptions are made regarding the experimental conditions.  If the stagnant water
scenario again assumed, the mass transfer coefficient is estimated as 8.8 x 10"4 cm/s.  For
partitioning effects, K was estimated as 0.16.  Again, the total water volume was  assumed to
be 1 x 10  m3. For these input values, AMEM predicts a fraction  migrated of 0.64 using Eq.
(3-17). The fraction  migrated prediction is reduced somewhat when partitioning and mass
transfer effects are considered.  The difference, however, is quite small considering that

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several of the assumptions made regarding the required input values would exaggerate the
influence of these effects.

       Migration values could also be predicted using AMEM to estimate a value for D .  In
this case, however, one must consult Section 4.2, Table 7, and Figure 6, in Volume 11
because polypropylene is not one of the polymers for which D  can be estimated directly by
AMEM. To estimate D , one must select the polymer group representative of polypropylene
in Table 3 and use Figure 6 to locate the value of D  corresponding to the migrant's
molecular weight.  Isotactic polypropylene with a high degree of crystallinity  (65%) is found
at the bottom of the Polyolefms-H group.  For MW = 214.21, and log (MW)  = 2.33, the
estimated value of D is 1  x 10"11 cm2/s, which is close to the measured value of Dp.  The
measured value is higher because it was measured at 44°C whereas the data plotted in Figure
6 are for 20-30°C. With this value of D , AMEM  predicts a worst case fraction migrated of
0.3 using Eq. (3-23), only approximately one half of the measured value.  Again, this lower
prediction is the result of using an estimated value  for Dp that applies at lower temperatures.

Example #8 - High Density Polvethvlene/3,5-Di-t-Butvl-4-HvdroxytoIuene

       In addition to their study of additive migration from rigid PVC, Figge, Koch and
Freytag (1978) also investigated the migration  of antioxidants from HDPE. The migration of
two antioxidants, 3,5-di-t-butyl-4-hydroxy-toluene (BHT) and Irganox 1076®, was measured
into water after 10 days  at 40°C.  The study involved a radiotracer technique  in which the
additives were synthesized with a 14C label and incorporated into a HDPE compound that was
then formed into sheet.   The composition of the HDPE sheet specimens was:

                                                    wt %

      HDPE resin                                   99.7
      BHT                                          0.2
      Irganox  1076                                   0.1

      Irganox  1076  is a trade name for octadecyl 3-(3',5'-di-tert-butyl-4'-hydroxyphenyl)
propionate.  Two samples of this blend were produced; one with the  BHT radiolabelled and
the other with the Irganox  1076 carrying the radiolabel.  Test samples were pressed to a
thickness of 0.035 cm and sample size was such that the exposed surface area was 9 cm.
The exposure of the HDPE coupon was  two-sided.  The volume of the water used in the
migration experiments, however, was not reported.  The results for BHT follow and the results
for Irganox 1076 are reported in Example #9.

      The results for the 10-day migration experiment are given as a percentage of the initial
additive amount. Figge  et al.  found that 0.48 weight percent or a fraction of  0.0048 of the
BHT originally present had migrated into the water. Prediction of the migration under these
conditions using AMEM requires the following data:

      Migration:                       To water
      Exposure:                        Two sides
      Thickness:                       0.035 cm

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       Time:                            240 hours (10 days)
       Polymer:                         HDPE
       Migrant:                         BHT, MW = 220.4 g/mol
       Migrant Initial Concentration:       0.2 WL% or 1.92 x 10"3 g/cm3
                                        (at a polymer density of 0.96 gm/cm3)
       External Phase:                   Stagnant water at 40°C
       Surface Area:                     9 cm2
       Surface Length:                   1.2 cm

       A value for the diffusion coefficient of BHT in the polymer was not reported but was
estimated by AMEM based on the migrant molecular weight.  The estimated Dp value was
1.7 x 10"10 cm2/s.  The fraction migrated estimated by the model is 0.75 under worst case
conditions using Eq. (3-23).  This estimate is more than two orders of magnitude higher than
the 0.0048 fraction migrated reported by the authors and indicates that partitioning or mass
transfer resistances may be controlling the migration rate. Mass transfer effects were
considered by  assuming that the stagnant volume of water has a low flow rate of about 10
cm/s over the  1.2-cm polymer surface length.  The  estimated mass transfer coefficient was
1.5 x 10'3 cm/s.

       BHT is considered "insoluble" in water with a saturation concentration less than
1 ppm or 1 x 10"6 g/cm3.  A  value for the saturation concentration at 40°C was estimated
based upon the migrant molecular weight, melt temperature, and octanol water partition
coefficient.  For BHT, with a melt temperature equal to 70°C and log K^ equal to 5.98, the
water solubility was estimated by AMEM as 8.5 x 10"8 g/cm3 at 25°C and 4.2 x 10"7 g/cm3 at
40°C using the temperature extrapolation method (Eq. 8.15.5) in Reid et al.  (1977). For this
external phase  saturation concentration and the initial concentration of BHT in the HDPE,  the
partition coefficient was estimated as 2.2 x  10"4.  This  low value suggests that partitioning
may affect the  migration rate unless a very large volume of water is used as the external
phase.  Unfortunately, the authors do not report the volume of water used.

       If an external phase volume of 1  x 10"4 m3 (100 cm3) is assumed, mass transfer and
partitioning effects  are predicted to reduce the fraction migrated to 0.12 when Eq. (3-17) was
used. If an external phase  volume of 1 x 10"5 m3 (10 cm ) was assumed, Eq. (3-17) predicts
a fraction migrated 0.014.  Thus, a change of one order of magnitude in external phase
volume results  in a reduction in the predicted migration by a factor of ten.  This example
demonstrates how the migration behavior can be strongly influenced by the  external
conditions and illustrates the importance of considering the specific conditions under which
polymer products may be used when performing exposure assessments.

Example #9 - High Density Polvethylene/Irganox 1076

       The migration of Irganox 1076 from the same HDPE sheet samples to water also was
measured over a period of  10 days at 40°C. The result was that 0.065 wt% or 0.00065 of the
antioxidant initially present migrated to the  water. The following input data were used with
AMEM to estimate migration under these conditions:
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       Migration:                        To water
       Exposure:                        Two sides
       Thickness:                        0.035 cm
       Time:                            240 hours (10 days)
       Polymer:                         HDPE
       Migrant:                          Irganox  1076, MW = 530.9 g/mol
       Migrant Initial Concentration:       0.1 wt% or 9.6 x 10"4 g/cm3
                                        (at a polymer density of 0.96 g/cm3)
       External Phase:                   Stagnant water at 40°C
       Surface Area:                     9 cm2

       As in Example #8, the volume of the external phase was not reported for this
experiment.  Neither was a value reported for the diffusion coefficient of Irganox 1076 in
HDPE.  A value at 20-30°C was estimated by AMEM using the migrant molecular weight
and the correlation for HDPE.  D was estimated as 5.0 x 10"12 cm2/s. The worst case
migration predicted using Eq. (3-23) was 0.13, which greatly exceeds the 0.00065 fraction
migrated measured experimentally. However, if partitioning and mass transfer resistances are
considered, the predicted migration is much lower.

       For a 10 cm/s stagnant water flow velocity and the  1.2-cm surface length, the mass
coefficient, k, was estimated as 1.5 x  10~3 cm/s. The solubility of Irganox 1076 is listed in
the manufacturer's  literature as less than 0.2 ppm or 2.2 x 10   g/cm at 25°C and
1.1 x 10"6 g/cm3 at 40°C. As for the estimation of partitioning effects, the partition
coefficient, K, was estimated by AMEM  as 1.15 x 10"3 using the saturation concentration in
water at 40°C and the initial concentration of Irganox 1076 in the HDPE. A volume of
       A   "\       "\
1 x 10  cm  (10 cnr) was assumed because a value was not reported.  For this volume, the
fraction migrated predicted using Eq. (3-17) is  0.0508. If the external phase volume is further
reduced to 1 cm3, the Eq. (3-17)  fraction migrated prediction is reduced to 0.007 which
approaches the measured value.

Example #10 - Polvstyrene/Styrene

       General purpose polystyrene is widely used in food packaging applications.
Commercial, food-grade polystyrene used in contact with food normally has a residual
monomer concentration of less than 0.1% by weight. Arthur D. Little, Inc. (1981) measured
the migration of styrene from crystalline polystyrene specimens into water at 40°C.
Commercial  polystyrene beads were blended with radiolabelled styrene and pressed  into sheet
at a thickness of 0.0254-cm.  The monomer was incorporated into the polymer sheet at a level
of 800 ppm.  In the experiment, 13 sample coupons were immersed in 31 cm  of water, for a
two-sided exposure, at 40°C.  The total surface  area of the coupons was reported as 100 cm .
The test cell was agitated using a temperature controlled shaker bath. The concentration of
styrene in the water was measured as a function of time and reported as a percentage of the
styrene originally available for migration.
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       Data were reported at the following time intervals:

                    Time               Percent             Fraction
                    (hrs)                Migrated            Migrated

                      1                 0.16                0.0016
                     168                0.66                0.0066
                    1013                0.75                0.0075

       Use of AMEM to predict the migration of styrene requires the following inputs:

       Migration:                        To water
       Exposure:                        Two sides
       Thickness:                        0.0254 cm
       Time:                            1  hour
                                        168 hours
                                        1013 hours
       Polymer:                         Polystyrene
       Migrant:                          Styrene, MW =104 g/mol
       Migrant Initial Concentration:      800 ppm or 8.32 x  10"4 g/cm3
                                        (at a polymer density  of 1.04 g/cm3)
       External Phase:                    Agitated water at 40°C
                                        Volume: 3.1  x  10'5 m3 (31  cm3)
                                              fj
       Surface Area:                     100 cm
       Surface Length:                   2  cm

       The diffusion coefficient of the monomer in the polystyrene was reported as
3.3 x 10   cm/s, which was noted as being in  good agreement with values reported in the
literature.  The measured value was used as input to AMEM to eliminate possible errors
resulting from a value estimated at 25°C.

       In this experiment the external phase was agitated but a flow velocity was not
reported, so a value of 100 cm/s was used to  approximate the water flow velocity over the
2.0-cm polystyrene surface length. Under these  conditions, AMEM  estimates the mass transfer
coefficient as 3.7 x  10'3 cm/s.  The solubility of styrene  monomer in water at 40°C was
determined experimentally by Lane (1946)  and  was reported as 400 ppm or 4.0 x 10"4 g/cm3.
Since styrene monomer is infinitely soluble in polystyrene, the  saturation concentration in the
polymer was set equal to the monomer density  or 0.9 g/cm3. This results in an estimated
partition coefficient equal to 4.4 x 10"4.

       The predicted fractions migrated as  a function of time under both worst case
conditions and when considering mass transfer and partitioning effects  were:
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                                        Fraction Migrated
                    Time        Worst Case   Mass Trans./Part.
                    (hrs)         Eq. (3-23)       Eo. (3-17)

                      1           0.0033             0.0025
                     168          0.040               0.009
                    1013          0.098               0.010

       Although AMEM did not predict that complete partitioning had occurred using
Eq. (3-17), it does indicate that the migration rate is so slow that little migration occurs
during the period from 168 hours to  1,013 hours.  The Eq. (3-17) predicted values again are
in good agreement with the measured migration values.

Example #11 - Impact Polystyrene/3,5-Di-t-Butvl-4-Hydroxytoluene

       Impact polystyrene (EPS) is used to package food products and has been investigated
as pipe for water transport.  IPS derives its impact resistance from rubber particles (usually
polybutadiene) that are uniformly dispersed in the polystyrene matrix. Additives, such as
antioxidants, lubricants, and processing aids, are also added to retard thermal degradation and
to facilitate processing.

       The migration of radiolabelled BHT from IPS into water at 49°C was measured by
Arthur D. Little, Inc. (1981).  The labelled antioxidant was incorporated into the
commercially obtained polymer at a level of 1090 ppm. Sample sheets were pressed to a
thickness of 0.045-cm.  Six sample coupons, approximately 2-cm in diameter, were immersed
in 16.3 cm3 of water so that the exposure was two-sided.  The test cell was agitated using a
temperature controlled shaker bath. The concentration of the BHT in the water was measured
after 5 and 35 days. The amount migrated was reported as p.g of BHT per dm2 of polymer
surface area.
                                       «      ____
       After 5 days  (119 hours), 4.0 p.g/dm  of BHT were measured in the water. To convert
this number to a fraction of the initial BHT present, one must calculate the amount of BHT
initially in the test sample. The total, two-sided surface area of the six sample coupons was
reported as 46 cm2 which is equivalent to a single-sided surface area per sample coupon of
3.83 cm2. Multiplying this area by the 0.045-cm sample thickness results in an individual
coupon volume of 0.1725 cm , and a total polymer volume of 1.035 cm . In this volume,
there initially was 1,090 ppm of the BHT or:

       [(1090 g BHT)/(1 x 106 g E?S)](1.04 g IPS/cm3)(1.035 cm3) = 0.001173 g BHT

at an IPS density of 1.04 g/cm3.  Given the total surface area of 46 cm2 (or 0.46 dm2), the
4.0 fig/dm2 measured in the water represents 1.84 p.g of BHT migrated and a fraction
migrated of 0.0016.  Similarly, the results after 35 days (840 hours) were reported as
12.5 jig/dm2. This corresponds  to 5.75 ^.g of BHT and a fraction migrated of 0.005.
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       With AMEM, the required input data are:

       Migration:                       To water
       Exposure:                        Two sides
       Thickness:                       0.045 cm
       Time:                           119 hours
                                       840 hours
       Polymer:                         Impact Polystyrene
       Migrant:                         BHT, MW = 220 g/mol
       Migrant Initial Concentration:      1090 ppm or 1.13 x 10~3 g/cm3
                                       (at a polymer density of  1.04 g/cm3)
       External Phase:                   Agitated water at 49°C
                                       Volume: 1.63 x 10'5 m3 (16.3 cm3)
       Surface Area:                    46 cm2
       Surface Length:                   2 cm

       The diffusion coefficient of the BHT in the impact polystyrene was reported as
3.0 x 10   cm2/s. This reported value was used as an input to AMEM.  If a value had not
been reported, one would estimate the diffusion coefficient in polystyrene using the molecular
weight of the BHT. For the best accuracy, one would also need to account for the temperature
difference between 49°C and the 20-30°C temperature range for which the D  correlation was
developed; a capability not now provided by AMEM.

       Under worst case conditions, Eq. (3-23) predicts  a fraction migrated of 0.0025 after
119 hours and a fraction migrated of 0.0048 after 840 hours.  These estimates are both in
good agreement with the measured values. Mass transfer and partitioning effects may also be
considered. As in Example #10, a 100 cm/s  flow velocity was assumed for the agitated water
flow over the 2-cm polymer surface length. Under these conditions, k was estimated as
3.7 x 10"3 cm/s by AMEM. The manufacturer's literature reports BHT as "insoluble" but a
value in g/cm3 may be estimated at 25°C using the  AMEM estimation procedure. BHT has a
melt temperature of 70°C and a log (octanol/water partition coefficient) equal to 5.98.  Using
these parameters, the solubility  in water was  estimated by  AMEM as 8.5 x 10"8 g/cm3 at
25°C.  This value was extrapolated to 49°C using Eq. 8.15.5 in Reid et aL (1977) with the
estimated value equal to 1 x 10"6 g/cm3. The partition coefficient, K, was estimated by
AMEM as  8.9 x  10"4.

       When mass transfer and partitioning effects  were considered, the fractions migrated
predicted by AMEM were 0.0016 after 119 hours and 0.0037 after 840 hours. In this
example, mass transfer and partitioning considerations reduce the migration  predictions only
by a small  amount and remain in good  agreement with the measured values.

Example #12 - Plasticized Poly(vinyl Chloride)/Di-2-ethylhexvladipate

       Migration of a plasticizer was measured at 58°C from plasticized PVC into an air
stream moving at about 1 ft^/min by Quackenbos (1954).  Quackenbos measured the
migration of di-2-ethylhexyladipate (DEHA)  from one side of a 0.1-cm thick PVC film of
surface area 1000 cm2 over a period of 400 hours.  The plasticizer was initially present at a

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concentration of 65 phr. To calculate the concentration of plasticizer initially present in the
polymer sample, parts per hundred resin was converted to g/cm3 using the density of the
plasticized PVC:

                        [(65)/(100+65)](1.22 g/cm3) = 0.48 g/cm3

Thus, initially there were 48 g of DEHA present in the 100 cm3 of PVC sample. After
400 hours, Quackenbos measured the migration loss as 500 |J.g/cm2 or:

                      (500 ng/cm2)(1000 cm2) = 5 x 105 jig = 0.5 g

This mass represents  a fraction migrated of 0.01.

       To estimate migration under these conditions, AMEM was used with the following
inputs:

       Migration:                       To air
       Exposure:                       One-side
       Polymer:                        Plasticized poly(vinyl chloride)
       Thickness:                       0.1 cm
       Time:                           400 hours
       Migrant:                         Di-2-ethylhexyladipate, MW = 371 g/mol
       Migrant  Initial Concentration:      65 phr or 0.48 g/cm
       External Phase:                  Air at 58°C, flowing at  1 ft3
                                       Volume: 680 m3 (24,000 ft3 after 400 hours)
       Surface Area:                      1000 cm2

The diffusion coefficient for DEHA, at the reported initial concentration and 58°C, was
                                      Q   v
calculated by Quackenbos to be 4.4 x 10  cm /s.

       In a separate test, Quackenbos measured the mass transfer coefficient. A piece of
cloth saturated with DEHA  was placed in the warm air flow in place of the PVC film.  After
400 hours, the cloth was weighed and a plasticizer loss of 0.0008 g/cm2 was reported.  Thus,
the average loss rate was:

                (0.0008 g/cm2)/[(400 hr)(3,600 s/hr)] = 5.6 x lO'10 g/cm2-s

From this  loss rate, one may calculate the mass transfer coefficient, k, using the following
relationship:

                                    Rate = k x Csat

where Csat is the saturation  concentration of the migrant in the air external phase.
Quackenbos reports a vapor pressure of 6.6 x 10"3 Pa  for DEHA at 58°C.  The saturation
concentration was estimated using the vapor pressure,  molecular weight, and temperature, as
follows:
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       Cst = (PyXMWVCRT) = (6.6 x 10-3)(371)/(8.314)(331)(106) = 8.9 x 1040 g/cm3
       Using this saturation concentration and the average evaporation rate, the mass transfer
coefficient was calculated:

            k = Rate/Csal = (5.6 x 10'10 g/cm2-s)/(8.9 x 10'10 g/cm3) = 0.63 cm/s

AMEM was first used to estimate migration under worst case conditions.  The measured
value of D equal to 4.4 x 10"9 cm2/s was used.  The fraction migrated predicted using
Eq. (3-23) was 0.83, which is much higher than the measured value of 0.01.

       Partitioning and mass transfer effect were  then considered.  The measured value for k
was used with Quackenbos' value of D .  AMEM estimated the partition coefficient as
9.6 x 10"10 using the value for Csat and the saturation concentration of DEHA in the polymer
set equal to the DEHA density of 0.925 g/cm3 (because of its infinite solubility in PVC).  K
is low due to the low solubility of the plasticizer in air.  However,  the total volume of air to
which the polymer surface is exposed is quite large, 680 m3, since fresh air was continuously
flowing over the surface. For these conditions, the fraction migrated predicted by AMEM
using Eq.  (3-17) is 0.005, much closer to the measured value, of 0.01. In this example, mass
transfer and partitioning limitations strongly influence the migration rate.

Example #13 - Plasricized Poly(vinyl ChlorideVDibutyl Phthalate

       Kampouris (1975) measured the migration of the plasticizer dibutyl phthalate (DBP)
from a PVC film to water.  The PVC film was initially plasticized at a level of 50 phr.
Migration tests using a radiolabel technique were  performed using 2-cm wide by 5-cm long
by 1-mm thick samples, exposed on one side to 250 cm3 of water.  The amount of DBP that
migrated into the water was measured after periods of 5 and 15 days.

       After 5 days (120 hours), Kampouris measured 6.0 x 10"6 g/cm3 in the 250 cm3 of
water.  After  15 days (360 hours), the concentration was measured  as 1.0 x 10~5 g/cm3 in the
250 cm3 of water.  These concentrations correspond to a plasticizer mass of 0.0015 g  after
120 hours and 0.0025 g after 360 hours. To calculate the fraction of DBP that  migrated, one
must calculate the amount of plasticizer initially present in the polymer sample  at 50 phr:
                        [(50 g)/(150 g)](1.22 g/cm3) = 0.41 g/cmj

and

                             (2 cm)(5 cm)(0.1 cm) = 1 cm3

Thus, there were initially 0.41  g of DBP in the PVC samples.  The resulting fractions
migrated were 0.0037 at 120 hours and 0.0062 at 360 hours.
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       Migration under these conditions was estimated by AMEM using the following input
 data:

       Migration:                        To water
       Exposure:                        One-side
       Thickness:                        0.1 cm
       Time:                            120 hrs
                                        360 hrs
       Polymer:                         Plasticized poly(vinyl chloride)
       Migrant:                         Dibutyl phthalate, MW = 278 g/mol
       Migrant Initial Concentration:       50 phr or 0.41 g/cm3
                                        (at a polymer density of 1.22 g/cm3)
       External Phase:                   Stagnant water at 25°C
                                        Volume: 2.5 x 10'4 m3 (250 cm3)
       Surface Area:                     10 cm2

       No value for the diffusion coefficient was reported by the researchers. As indicated in
 Table 7 of Volume 11, the few experimentally measured values of D  for plasticizer
 migration from PVC  at 25°C reported in the literature fall within the Polyolefins-n category.
 Thus, an approximate value for Dp was estimated using Figure 6 and the DBP molecular
 weight. The estimated value for D was 2 x 10'11 cm2/s.  Migration under worst case
 conditions was first predicted by AMEM as 0.033 after 120 hours and 0.057 after 360 hours
 using Eq. (3-23), which exceed the measured values by about one order of magnitude.

       Mass transfer  resistances and partitioning limitations were then considered.  A stagnant
 water flow velocity of 10 cm/s  was again assumed and the 5-cm polymer plate surface length,
 was used to estimate  the mass transfer coefficient.  A value of 7.4 x 10  cm/s was estimated
 by AMEM.  Because plasticizers are infinitely soluble in PVC, the density of DBP at 25°C,
 1.046 g/cm3, was used as the saturation concentration of the migrant in the polymer. If the
 density of the migrant or the true saturation concentration in the polymer was not known,
 then the 0.41 g/cm3 initial concentration of  DBP in the PVC compound could be used.  Use
 of the initial concentration instead of the saturation concentration results in an estimate of K,
 the partition coefficient, greater than or equal to the true K, because additives are usually
 present at concentrations below their solubility limits.

       The saturation concentration of the DBP in the water was reported by Haward (1943)
 and Monsanto  (1983) as 0.0011 wt% or 1.1 x 10"5 g/cm3 at 25°C. The partition coefficient
 was estimated  by AMEM as  1.05 x 10"5 using the saturation concentration in water and the
 density of DBP. Based on these input values, AMEM predicts a fraction migrated of 0.0025
 after 120 hours and 0.0026 after 360 hours  using Eq. (3-17). Both these estimates are in fairly
 good agreement with  the measured values, but underestimate the fraction migrated after
 360 hours.  Over the  period from 120 to 360 hours, AMEM predicts that a partition
equilibrium is approached and the migration process essentially stops.
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REFERENCES

Arthur D. Little, Inc., "Migration from Styrene-Based Polymers: Styrene from Crystal
Polystyrene and BHT from Impact Polystyrene," Summary report on FDA Contract No. 223-
77-2360, 1981.

Bellobono, I.R., B. Marcandalli, E. Selli and A. Polissi, "A Model Study for Release of
Plasticizers from Polymer Films Through Vapor Phase," J. Appl. Polym, Sci.,  Vol. 29,
pp. 3185-3195, 1984.

Dietz, G.R., J.D. Banzer and E.M. Miller, "Water Extraction of Additives from PVC Pipe," J.
Vinyl Tech., Vol.  1 (3), pp. 161-163, 1979.

Figge, K., J.  Koch, and W. Freytag, "The Suitability of Simulants for Foodstuffs, Cosmetics
and Pharmaceutical Products in Migration Studies," Rd. Cosmet. Toxicol, Vol. 16, pp. 135-
142, 1978.

Haward, R.N., "Determination of the Solubility  of Plasticizers in Water," The  Analyst,
Vol. 68, pp. 303-305, 1943.

Kampouris, E.M.,  "Study of Plasticizer Migration Using Radioactive Labelling,"  Rev. Gen.
Caoutch. Plast, Vol. 54 (4), pp. 289-292, 1975.

Lane, W.H.,  "Determination of the Solubility of Styrene in Water and of Water in Styrene,"
Ind. and Eng. Chem., Vol. 18 (5), 295-296, 1946.

The Merck Index,  llth Ed., S. Budavari, Ed., Merck Co., Inc., Rahway, New  Jersey, 1989.

Monsanto Co., "Plasticizer Performance in Polyvinyl Chloride Resin," Plasticizers and Resin
Modifiers Book, 1983.

Quackenbos, H.M., Jr., "Plasticizers in Vinyl Chloride Resins, Migration of Plasticizer," Ind.
and Eng. Chem., Vol. 46 (6), pp. 1335-1344, 1954.

Reid, R.C., J.M. Prausnitz and T.K.  Sherwood, The Properties of Gases and Liquids, 3rd Ed,
McGraw-Hill Book Co., New York, 1977.
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