EPA/6Q0/A-97/015
DESCRIPTION OF A METHOD FOR MEASURING THE DIFFUSION COEFFICIENT
OF THIN FILMS TO mRn USING A TOTAL ALPHA DETECTOR
Ronald B. Mosley
U.S. Environmental Protection Agency
National Risk Management Research Laboratory
Research Triangle Park, NC 27711
ABSTRACT
The present paper describes a method for using a total alpha detector to measure the diffusion
coefficient of a thin film by monitoring the accumulation of radon that penetrates the film. It will be
demonstrated that a virtual steady state condition exists in the thin film during the early stages of
accumulation that allows reliable measurements of the diffusion coefficient without having to wait
for the final condition of equilibrium or having to analyze the complex transient solutions. In some
cases, the final condition of equilibrium would require the measurement to last 3 or more weeks
rather than 3 days.
INTRODUCTION
While it has been accepted for some time that exposure to indoor radon constitutes a
potentially serious health threat, it has become increasingly apparent that the construction industry
prefers a passive mitigation method of preventing entry of radon into the indoor environments. One
such method, applicable to new construction, consists of installing passive barriers such as a thin
membrane to prevent ingress of radon gas into the indoor environment. Such a barrier would need
to control both advective and diffusive transport of radon. Use of a membrane as a barrier has the
advantage over other approaches of serving multiple purposes. Membranes are currently specified
in many localities for moisture control. In order to investigate the applicability of new materials for
use as membranes, a simple and convenient method of measuring the diffusivity of thin films is
needed. The present paper discusses a laboratory method for measuring the mRn diffusion
coefficient using a total alpha detector. The apparatus is described by Perry and Snoddy (1996) and
will not be discussed in detail here. A number of studies ~ Nielson et al. (1981), Nielson, Rich, and
Rogers (1982), Jha, Raghavayya, and Padmanabhan (1982), Rogers and Nielson (1984), Hafez and
Somogyi (1986), and Nielson, Holt, and Rogers (1996) -- have addressed the measurement of 222Rn
diffusion through barriers including films, soils, and concrete. These methods used either the steady
state solution for diffusion or a very complex transient solution. The present paper proposes a
simpler mathematical solution which describes a virtual steady state that exists when the
concentration at one interface of the film increases very slowly with time.
This paper has been reviewed in accordance with the U.S. Environmental Protection Agency's
peer and administrative review policies and approved for presentation and publication.
1

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MATHEMATICAL MODEL
In order to test a film's resistance to radon transport, the film will be placed between a
chamber containing a source of radon and a chamber that accumulates the radon transported through
the film. The tests will be performed under ambient conditions. It is assumed that no advective
transport through the film occurs. A schematic of this arrangement is illustrated in Figure 1. Region
1 represents the radon source in which the radon concentration is assumed to remain constant during
the measurements. Region 2 corresponds to the film to be tested. The transport equation that applies
in Region 2 is given by:
ac na2c , ~
where C is the radon concentration (atoms m*3) in the film, t is the time (s), D is the diffusion
coefficient (m2 s"1) in the film, x is the position (m) within the film, and Ar,, is the decay constant (s'1)
for 222Rn. In general, the radon concentration [ C(x,t) J within the film is a function of both position
and time. The non-steady solution of Equation 1 can be expressed as an infinite sum of position-
dependent trigonometric functions multiplied by an exponentially decreasing time function (Crank,
1994). Colle' et al. (1981) and Crank (1994) have shown that the relaxation time, associated
with the approach to steady state is given approximately by t, = + n2 D d'2)"', where d is the
thickness (m) of the film. When the film is 1.27 x 10"4 m (5 mils) thick, the relaxation time is about
0.3 minute for a diffusion coefficient of 10'10 m2s*1 and about 4 hours for a diffusion coefficient of 10'
13 m2 s'1. This three order-of-magnitude range in diffusion coefficient is believed to include most of
the commonly used construction films. After a time corresponding to several multiples of x„ the film
can be assumed to be in a steady state provided the concentrations at the boundaries remain constant.
In fact we define the condition in the film in which the concentrations at the boundaries do not change
significantly during times that are long compared to the relaxation time as a condition of virtual steady
state. During a virtual steady state, the flux is nearly constant during times comparable with xr
Approximate solutions to Equation 1 corresponding to the condition of a virtual steady state will be
used to avoid the very complex analysis associated with non-steady state solutions.
Region 3 is a closed volume in which M2Rn accumulates. Consequently, the concentration
at the surface of the film, Cd, will slowly increase with time to match the increasing concentration in
Region 3, C,(t). The condition of virtual steady state in the film will continue to apply so long as the
fractional change in C,(t) is small during time intervals comparable to the relaxation time. The
appropriate boundary conditions for the virtual steady state are C(0) = C„ the concentration in
Region 1, and C(d) = CJ(t).
The virtual steady state condition is determined by letting the time derivative of C go to zero.
Equation 1 then becomes;
2

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D
£C
dx2
" C = 0
(2)
with boundary conditions: C(0) = C, and C(d) = Cd = C,, where C, is the concentration in Region
3. We assume that Region 3 remains well mixed. The solution to Equation 2 is:
C. Sinh
C(x) =

"¦Rn
D
(d-x) + Cd Sinh
A
~F
Swift
A
"Rn
D
(3)
Also note that:
dCt
dx
*x=d

Rn
D
Cs - Cd Cosh ^
D
Sinh.
\
Kn
D
d
<«)
Region 3 is an accumulation chamber. The radon concentration in this chamber will be
measured as a function of time. Measuring the rate of increase of radon in Region 3 gives a direct
measurement of the flux from the surface of the film. This flux is easily computed when transport
through the film remains in a virtual steady state condition. Mass balance in Region 3 requires that:
£-
dt
=m ~
p	Jin 
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ADC
Po =

Sinh
\
Rn
D
D
(7)
Equation 5 then becomes:
dC.
dt
(V » V c„ - 4
(8)
where
P0 Cosh,
K =

nRn
D
(9)
Vfs
The solution of Equation 8 is:
Ca(t) = P° (1 - exp(-[^ + Ap]r)}	(10)
VtAARn + V
After substituting Equation 10 into 6 and rearranging, we may write:
PC) = P. + (Po ~ PJ exP(~[^n + *p!0	(11)
where
0 nCosh.
p.. p, [,. _L_J
X
'Rn d
D ]	(12)
w*+ v
Equation 10 expresses the 222Rn concentration in Region 3, while Equation 11 gives the rate
of transport of atoms through the film. If the ^Rn concentration were being measured directly in
this experiment, these equations would be sufficient to yield a value of diffusion coefficient.
However, in the current set of measurements, the total alpha activity was measured. These measured
values contain contributions from mRn, 218Po, and 214Po. Since the third alpha particle is emitted
during the fourth decay step following radon, it is necessary to solve all the decay rate equations
(Bateman equations) in sequence. These equations are given by: 	
4
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^Rn:
-f = - + m
(13)
2l*Po:
dNA
(14)
dt
214Pb:
dNB

(15)
dt
dNc
(16)
dt
where Nr,, is the number of 222Rn atoms present, A*, (2.1 x 10"6 s"1) is their decay constant, NA is the
number of21®Po atoms, XA (3.80 x 10"3 s"1) is their decay constant, NB is the number of 214Pb atoms,
Ab (4.32 x 10"4 s"1) is their decay constant, Nc is the number of 214Bi atoms, and Ac (5.87 x 10"4 s*1)
is their decay constant. Because 214Po which produces the third alpha particle has a half-life of only
1.6 x 10"4 s, it is assumed to occur simultaneously with 214Bi. Consequently, only four rate equations
will be solved. Note that these equations must be solved sequentially and the resulting solutions
substituted into the next equation. Equations 13 through 16 differ from the traditional Bateman
equations in that the ^Rn concentration is increasing with time.
Decays that give rise to alpha activity are represented by:
^Rn:
K X*. ' <1 -	01
ARn* H
"Rn Rn
(17)
5
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218po;
X" N" ' T7T 11 * (1 1 1 ^P(-V) - (. ¦ >e*PHV * ApW
ARn AP	AA~AJbt AP	AA Aito~Ap
214
'Bi:
xc Nc. -Mlu ~ WW	exp(_X/)
O^A ^Rn ^p)(^4 ^j)(^4 ^c)
+ j- 	 	^A^B^C	^	^B^C$Rn+^ p)
(\< ~^Rn	^pX^C~^s) ^A ~^Rn~^&A ~^XAC" A£)
¦,	/ i	ri .	^A^B^C
]exp(~XJ) ~ [1+-
^C~^B &A ^Rn~^p)(^S~^Rn~ ApXAc " A^) ....
	^Agfo^+Ap)	 ^ 	AgA^A^+Ap)	
&A ~^Rn ~hXXA -A5)(Ac-Ab) (A,-A^ -Ap)(Ay4-A£)(A/1 -Ac)
K	K^c
]exp(-Ac/)
0"C~^b)	(^A ^Rn ^P)(^B ^Rn~^f)(^C~^Rn~^p)
As explained above, the third alpha particle is actually emitted by 2I4Po. However, it occurs so shortly
after the 214Bi decay that we consider them to be simultaneous.
The total measured activity, MA (decays s"1), is given by:
MA - Nr„ Er, * XA Na Ea * Xc Nc Ec,	(20)
where is the efficiency of the detector for the first alpha particle, EA is the efficiency for the
second alpha particle, and Ec. is the efficiency for the third alpha particle. Keeping only the dominant
terms in Equations 17-19, Equation 20 becomes:
MA -	pHV-a,,];))
(21)
+5iexDf-A t) + X^cEc> fexP(-V) exp(-AcT)]
K	h&c-h) ih-K) (V*c)
When A"'a « t, the second line in Equation 21 can be neglected, so that:
6
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MA = a[l-exp(-y/)]
(22)
Equation 22 is just a convenient mathematical form in which the constants a and y are parameters
to be chosen to yield the best fit to the data. By comparing Equations 21 and 22, along with using
7 and 9, it follows that:
D = (y -XJ dl
tanh,
>
hid
D
>
hsLd
D
(23)
Equation 23, which contains one of the fitting parameters, y, is transcendental. It must be solved
numerically or iterated to obtain the diffusion coefficient, D. However, when (k^ i D)H d « 1, the
last equation becomes:
D = (y~XRn)dL	(24)
Equation 23 or 24 provides a measured value of the diffusion coefficient whose accuracy depends
upon the degree to which the measured activity fits the expression in Equation 22. For a highly
accurate fit, one needs to extend the measurements until the curve begins to approach its maximum
value. For films with low values of diffusion coefficient, these measurements can require many days
or even a few weeks. Since shorter measurement times would be convenient, we choose to analyze
the early stages of the measurements. In the range that X"'A «t « (A^ + Ap)*!, Equation 21
reduces to:
MA ~ XRn^Q(ERn+ Ea+ Ec)t	(25)
which is linear with time. The diffusion coefficient is related to the slope, SR, of the linear portion
of the curve by:
D =
SRdL
sinh.
%Rn
\ D
Ea+ £c)
(26)
d
D
The slope, SR, can be determined by a regression fit to the linear portion of the curve. Equation 26
is transcendental and cannot be solved explicitly for D. Simple numerical methods will provide a
solution of this equation. However, when (Ar,, / D)% d « 1, Equation 26 reduces to:
7
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D =
SRdL
Ea+ Ec)
(27)
This approximation is typically valid for 1.27 x 10"4 m (5 mil) thick films whose diffusion coefficients
are greater than 1.0 x 10'12 m2 s'1. Note that both Equations 26 and 27 contain the total efficiency
of the alpha detector. In general this quantity will be determined by an independent calibration and
depends on the geometry of both the detector and the chamber. For the present set of measurements,
the total efficiency can be determined from a series of longer measurements using Equation 22. In
terms of the parameters used to fit Equation 22, the efficiency becomes:
Er»+ Ea+ Ec,
ay
cosh^
tn d
D
VWy
^¦Ri)
D
(28)
When / D)'4 « 1, the efficiency becomes:
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materials, Equation 27 yields a reasonable estimate of the diffusion coefficient. An improved value
is obtained when the initial estimate from Equation 27 is used to evaluate the right-hand side of
Equation 26. The new value of D obtained in Equation 26 can be used iteratively to compute an
improved solution of Equation 26. In the present case, convergence is adequate after only two
iterations. The linear segment corresponding to the initial data in Figure 3 is shown in Figure 5.
Once again, linear regression analysis yields the slope and the coefficient of determination. Equation
26 yields the diffusion coefficient. The diffusion coefficients for all four figures are given in Table
1. It can also be seen from Table 1 that the values of diffusion coefficient computed from the initial
data differ by only 9 and 12% from the values computed from the full curves.
Figures 6 and 7 show the accumulation curves for total alpha activity when 222Rn diffuses
through two similar films of natural latex rubber 1.225 x 10"4 m thick. The curve through the data
represents a least-square fit. The fit parameters and the coefficient of determination are shown.
Figure E illustrates the early portion of the data in Figure 6. Note that the curve is quite linear. The
regression slope and coefficient of determination are shown on the figure. Figure 9 illustrates the
early data in Figure 7. The diffusion coefficients computed from these fits are given in Table 1. Note
that the agreement between the diffusion coefficients computed by the two methods is not as good
for the latex films as for the polyethylene. This may be due, in part, to the fact that much less data
is used in the calculation for latex. The linear portion of the curve exists for a much shorter time.
However, because of the rapid approach to equilibrium for this case, it is quite practical to extend
the curve sufficiently to obtain an excellent fit to Equation 22.
CONCLUSIONS
While Equation 23 works quite well for determining the diffusion coefficient of thin films, it
may require very long times to sufficiently complete the shape of the curve to yield good accuracy.
It has been demonstrated that shorter measurements along with the use of Equation 26 yield an
adequate determination of the diffusion coefficient in some cases. This method was demonstrated to
work for diffusion coefficients in the range 10*10 to 10"12 m2s"1. It is estimated, based on instrument
sensitivity, that the method should be applicable for values that are at least two orders-of-magnitude
lower.
These measured diffusion coefficients appear to be largely consistent with values reported for
similar materials. For instance, the average value for polyethylene, 8.81 x 10'12 m2 s'1, differs by only
12% from the value, 7.8 xl0'12m2 s"1, reported by Hafez and Somogyi (1986). The average value
for latex, 1.43 10"10 m2 s*1, differs by 127% from the value, 6.36 x 10"10 m2 s"1, reported by Jha,
Raghavayya, and Padmanabhan (1982). While these results are relatively consistent, little is known
about just how similar the materials really are.
REFERENCES
Colle", R., Rubin, R.J., Knab, L.I., and Hutchinson, J.M.R. Radon transport through and exhalation
from building materials: A review and assessment. National Bureau of Standards Technical Note
9
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1139, September 1981.
Crank, J, The mathematics of diffusion. Clarendon Press, Oxford, 1994.
Hafez, A. and Somogyi, G. Determination of radon and thoron permeability through some plastics
by track technique. Nuclear Tracks, Vol 12, Nos 1-6, pp 697-700, 1986.
Jha, G., Raghavayya, M., and Padmanabhan, H. Radon permeability of some membranes. Health
Physics, Vol 42, No 5, pp 723-725, 1982.
Nielson, K.K., Rogers, V.C., Rich, D.C., Nederhand, P.A., Sandquist, G.M., and Jensen, C.M.
Laboratory measurements of radon diffusion through multilayered cover systems for uranium tailings.
Department of Energy Report UMT/0206, December 1981.
Nielson, K.K., Rich, D.C., and Rogers, V.C. Comparison of radon diffusion coefficients measured
by transient diffusion and steady-state laboratory methods. Report to U.S. Nuclear Regulatory
Commission, Washington, D.C., NUREG/CR-2875. 1982.
Nielson, K.K., Holt, R.B., and Rogers, V.C. Residential radon resistant construction features
selection system. EPA-600/R-96-005 (NTIS PB96-153473), February 1996.
Perry, R. and Snoddy, R. A method for testing the diffusion coefficient of polymer films.
Proceedings: The 1996 International Radon Conference, Haines City, FL, September 29 - October
2, 1996.
Rogers, V.C. and Nielson, K.K. Radon attenuation handbook for uranium mill tailings cover design.
U.S. Nuclear Regulatory Commission, Washington, D.C., NUREG/CR-3533, 1984.
Table 1. Comparison of Diffusion Results
Film	A^C,	d	E	D*,^	D" ^ %diff

(Bq)
(m)

(m2 s"1)
(m2 s'1)

poly 1
2.80xl06
1.524x10"4
0.742
7.79xl0"'2
6.87xl0"12
12
poly 2
2.95xl06
1.524X10"4
0.651
9.83xl0*12
8.91xl0*12
9
atex 1
2.90x10s
1.225x10"*
0.611
1.55xl0"10
l.lOxlO"10
29
latex 2
2.93x10*
1.225x10"*
0.680
1.31xl0"10
9.37x10'"
28
Doon = Diffusion coefficient computed from nonlinear curve
Dfo = Diffusion coefficient computed from linear curve
10
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Film
Region 1 2/
Accumulation
Chamber
Source
—120
MA" 191[1-exp(-5.51x10 t)]
R2 - 0,889
-h	x = o x = d	x
Figure 1. Schematic of the measurement system
200
40 80 120 160
Time (hours)
Figure 2, Accumulated activity for poly 1 film
MA" 180 M-exp{-5.69x10 t)]
0.989
-120
40 80 120 160
Time (hours)
Figure 3. Accumulated activity for poly 2 film
Slope =9.29x1 (T4 Bq s*1
0.996
2 4 6 8 10 12 14 16
Time (hours)
Figure 4. - Linear portion of poly 1 activity curve
Slope = 9.25 x 10** Bq «"1.
R* = 0.997
0 2 4 6 8 10 12 14 16
Time (hours)
figure I. Linear portion of poly 2 activity curve
11
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A- 2SS [ 1 -«xp(-0.00007251)]
IT - 0.892
*150
p 100
<
300
120 161
Time (hours)
Figure 6. Accumulated activity for latex 1 film
A* 286 [1 -exp(-0.0000614tj]
*150
0.994
o 100
<
0 10 20 30 40 50 60 70 80
Time (hours)
figure 7. Accumulated activity for latex 2 film
0.0133 Bq s
0 1 2
Time (hours)
figure 8. Linear portion of latex 1 activity curve
140
120
*5100
CD
r so
*
> 60
S 40
Slope = 0.0125 Bq s"
R2 • 0.999
Time (hours)
Hgure 9. Linear portion of latex 2 activity curve
12
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OTMDI -arm -n m:a TECHNICAL REPORT DATA ,
IN ri JVJ K Lj~ K 11 ~ r I04 (Please read Instructions on the reverse before compl
1. REPORT NO. 2.
EPA/600/A-97/015

4. TITLE AND SUBTITLE
Description of a Method for Measuring the Diffusion
Coefficient of Thin Films to Using a Total
Alpha Detector
S. REPORT DATE
6. PERFORMING ORGANIZATION CODE
7. AUTHOR(S)
R. B. Mosley
8. PERFORMING ORGANIZATION REPORT NO.
9. PERFORMING ORGANIZATION NAME AND ADDRESS
See Block 12
10. PROGRAM ELEMENT NO.
11. CONTRACT/GRANT NO.
68-D4-0006, Task 2-051
(A eurex)
12. SPONSORING AGENCY NAME AND ADDRESS
EPA, Office of Research and Development
Air Pollution Prevention and Control Division
Research Triangle Park, NC 27711
13. TYPE OF REPORT AND PERIOD COVERED
Published paper; 6-9/96
14. SPONSORING AGENCY CODE
EPA/600/13
15.supplementary notes aPPTD project officer is Ronald B. Mosley, Mail Drop 54, 919/
541-7865. Presented at International Radon Symposium, Haines City, FL, September
29-October 2, 1996.
i6. abstract jhe paper describes a method for using a total alpha detector to measure
the diffusion coefficient of a thin film by monitoring the accumulation of radon that
penetrates the film. It demonstrates that a virtual steady state condition exists in the
thin film during the early stages of accumulation that allows reliable measurements
of the diffusion coefficient without having to wait for the final condition of equilibrium
or having to analyze the complex transient solutions. In some cases, the final con-
dition of equilibrium would require the measurement to last 3 or more weeks rather
than 3 days.
17. KEY WORDS AND DOCUMENT ANALYSIS
a. DESCRIPTORS
b.IDENTIFIERS/OPEN ENDEO TERMS
c. COSATI Field/Group
Pollution
Radon ,
Measurement
Diffusion Coefficient
Thin Films
Alpha Particle Detectors
Pollution Control
Stationary Sources
Total Alpha Detectors
13	B
07B
14	B
20M
20L.09A
18 D
18. DISTRIBUTION STATEMENT
Release to Public
19. SECURITY CLASS (ThisReport)
Unclassified
21. NO. OF PAGES
20. SECURITY CLASS (Thispage)
Unclassified
22. PRICE
EPA Form 2220-1 (9-73)
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