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.
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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.
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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.
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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.
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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.
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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.
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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)
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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)
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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.
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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
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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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
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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
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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
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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
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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
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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
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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
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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
-------
• 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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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.
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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.
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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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
-------
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
133
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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
135
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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
137
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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
139
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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:
140
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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.
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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.
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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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