DESCRIPTION OF A METHOD FOR MEASURING THE DIFFUSION COEFFICIENT
OF THIN FILMS TO 22% USING A TOTAL ALPHA DETECTOR
Ronald B. Mosley
U.S. EPA, National Risk Management Research Laboratory, Air Pollution Prevention and Control Division
Research Triangle Park, NC
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 three or more weeks rather than three 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 banier 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 measurin the diffisivity of thin films is needed. The
present paper discusses a laboratory method for measuring the ^Rn 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, K.K. at el. (1981), Nielson, K.K., Rich, D.C., and Rogers, V.C. (1982), Jha, G., Raghavayya, M,
and Padrnanabhan, H. (1982), Rogers, V.C., and Nielson, K.K. (1984), Hafez, A. And Somogyi, G. (1986), and
Nielson, K.K., Holt, R.B., and Rogers, V.C. (1996) have addressed the measurement of 2 2 2 ~ ndiffusion 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 to use which describes
a virtual steady state that exist 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.
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 I 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
1996 International Radon Symposium 11  2.1
where C is the radon concentration (Atoms m'3) in the film, t is the time (s), D is the diffusion coefficient (m2 s")
in the film , x is the position (m) within the film, and ^.R,, is the decay constant (s"') for ^ ~ n . In general, the
concentration [ C(x,t) ] within the film is a function of both position and time. The nonsteady 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). C o l l e et al(1981) and Crank (1994) have shown that the relaxation time,
t,, associated with the approach to steady state is given approximately by x, = (ARn + !t2 D d'2 )" , where d is the
thickness (m) of the film. When the film is 1.27 x lo4 m (5 mils) thick, the relaxation time is about 0.3 minutes for a
diffusion coefficient of 1 0 " ~m2s' and about 4 hours for a diffusion coefficient of l0'I3 m2 s". This three orderofmagnitude range in diffusion coefficient is believed to include most of the commonly used construction films. After
a time corresponding to several multiples of T,, 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 T,. 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 nonsteady state solutions.
Region 3 is a closed volume in which ^ ~ naccumulates. Consequently, the concentration at the surface of
the film, Cd, will slowly increase with time to match the increasing concentration in region 3, CÃˆ(t) The condition of
virtual steady state in the film will continue to apply so long as the fractional change in CJt) is small during time
intervals comparable to the relaxation time. The appropriate boundary conditions for the virtual steady state are
C(0) = Cs, the concentration in region 1, and C(d) = C/t).
The virtual steady state condition is determined by letting the time derivative of C go to zero. Equation 1
then becomes
with boundary conditions: C(0) = C, and C(d) = Cd = C, , where Ca is the concentration in region 3. We assume
that region 3 remains well mixed. The solution to equation 2 is
(dx)
C(x) =
+ C d Sinh
r
Also note that
I996 International Radon Symposium I1

2.2
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
where C,(t) is the ^ ~ nconcer ation (atoms m'3) in region 3, p(t) is the rate of transport (atoms s" ) of ^ ~ n
atoms through the film, and V, is the volume (m3) of region 3. In a steady state P(t) is given by
where A is the cross sectional area (m2) of the film and
=
Sinh d
Equation 5 then becomes
where

1996 International Radon Symposium I1 2.3
The solution of equation 8 is
After substituting equation 10 into 6 and rearranging, we may write
W)=
+
tAÃ  PJ
expt[Aa, +
/id/:
where
Equation 10 expresses the ^RU concentration in region 3, while equation 1 1 gives the rate of transport of
atoms through the film. If ^ ~ nconcentration 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 ^ ~ n ,"'PO, and ^PO. 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

1996 International Radon Symposium 11 2.4
where Npn is the number of ^ ~ natoms present, Ann (2.1 x lo4 s" ) is their decay constant, NA is the number of
"'PO atoms, ?LA (3.80 x lu3s") is their decay constant, NB is the number of *14pbatoms, AB (4.32 x lo4 s") is their
i
and \c (5.87 x lo4 s") is their decay constant. Because ^PO
decay constant, Nc is the number of 2 1 4 ~atoms,
which produces the third alpha particle has a halflife of only 1.6 x lo4 s, it is assumed to occur simultaneously with
^ ~ i . 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 ^RII concentration is increasing with time.
Decays that give rise to alpha activity are represented by
1996 International Radon Symposium I1  2.5
As explained above, the third alpha particle is actually emitted by ^PO. However, it occurs so shortly after the 2 ' 4 ~ i
decay that we consider them to be simultaneous.
The total measured activity, MA, (decays s") is given by
where Epn is the efficiency of the detector for the first alpha particle, Ei is the efficiency for the second alpha
particle, and Ecis the efficiency for the third alpha particle. Keeping only the dominant terms in equations 17 19,
equation 20 becomes

When \'\ << t, the last three terms in equation 21 can be neglected, so that 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

1996 International Radon Symposium I1 2.6
comparing equations 2 1 and 22, it follows that
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, it follows that
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 ( A h 1 D)^d << 1, the last equation becomes
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 \'\ Ã t Ã (Xh
+ A,,)" ,equation 2 1 reduces to
which is linear with time. The diffusion coefficient is related to the slope, SR,of the linear portion of the curve by
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,
1996 International Radon Symposium I1  2.7
when (ARn 1 D)^ d << 1, equation 26 reduces to
This approximation is typically valid for 1.27 x lo4 m (5 mil) thick films whose diffusion coefficients are greater
than 1.0 x 10"~m2 s". 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
When (Ab I D)" << 1, the efficiency becomes
Once an average value of efficiency has been established using equation 28 or 29, then shorter runs can be used to
compute the diffusion coefficient using either equation 26 or 27.
DATA ANALYSIS
Measurements have been performed on a large number of films. Several of these measurements are
considered preliminary and are not reported here. In an effort to evaluate the feasibility of performing short term
tests to measure diffusion coefficients of thin films, we will analyze duplicate measurements on two materials,
polyethylene and natural latex rubber. Figures 2 and 3 show measurements on two polyethylene films 1.524 x 1 0 m
(6 mil) thick. Background counts and the initial data prior to the virtual steady state have been subtracted. More
than 8,000 oneminute counts are represented by tiny squares in the figures. The line through the data points
represents a leastsquare fit to equation 22, utilizing the methods of Levenberg and Marquardt to determine the
fitting parameters. The parameters yielding the best fit to equation 22 are shown in the figures. For convenience the
time is plotted in units of hours, however, the equations illustrated in the figures use seconds. The coefficients of
1996 International Radon Symposium I1
 2.8
determination for these fits are also shown. As is illustrated by the values of R ,the degree of agreement with the
mathematical equation is excellent. Less than 0.1% of the variation is unexplained.
Figure 4 shows the early data in Figure 2. The tiny squares represent measurements while the line
represents a linear regression analysis. The regression slope and the coefficient of determination for the fit are given
in the figure. Once again the degree of fit is very good. The regression slope can be used in equation 26 or 27 to
calculate the diffusion coefficient. For present 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 diffision
coefficient computed from the initial data differ by only 9% and 12% from the values computed from the fall curves.
Figures 6 and 7 show the accumulation curves for total alpha activity when ^ ~ ndiffuses through two
similar films of natural latex rubber 1.225 x 1o4 m thick. The curve through the data represents a leastsquare fit.
The fit parameters and the coefficient of determination are shown. Figure 8 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 diffision 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. 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 exist for a much shorter time. In this case, however, it is quite practical to extend the
curve sufficiently to obtain a reliable 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 equation 26 yield an adequate determination of the diffision coefficient in
some cases. This method appears to work for diffision coefficients in the range 1 0 " ' ~m2 s" to 1 0 " ~m2s"' . It is
estimated that the method should be applicable for values that are twoordersofmagnitude 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"~
m2 s"' ,differs by only 12% from the value,
7.8 xl0I2m2 s", reported by Hafex and Somogyi (1986). The average value for latex, 1.43 10"~
m2 s"', differs by
127% form the value, 6.36 x 1 0 " ~ m2 s", reported by Jha, Raghavayya, and Padmanabhan (1982). While these
results are relatively consistent, little is known about just how similar the materials really were.
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. NBS Technical Note 1139, Sept. I98 1.
Crank, J. The mathematics of diffision. 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 16, pp 697700, 1986.
Jha, G., Raghavayya, M., and Padmanabhan, H. Radon permeability of some membranes. Health Physics, Vol42.
No 5, pp 723725, 1982.
1996 International Radon Symposium I1

2.9
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 UMTl0206, December, 1981.
Nielson, K.K., Rich, D.C., and Rogers, V.C. Comparison of radon diffusion coefficients measured by transient
diffusion and steadystate laboratory methods. Report to U.S. Nuclear Regulatory Commission, Washington, D.C.,
NUREGICR2875. 1982.
Nielson, K.K., Holt, R.B., and Rogers, V.C. Residential radon resistant construction features selection system.
EPA600/R96005, 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,Sept. 29 Oct. 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., NUREGICR3533, 1984.
Table 1. Comparison of Diffusion Results
D' "an
(m2 s')
D
% diff
(m2 s')
12
7.79xl0~~
6.87~10l2
9.83~
10l2
8.91~10'~~
9
latex I
2.90xlo6
1.225~10"
0.61 1
.55x10'~
l.l0xl0~~
29
latex 2
2 . 9 3 ~1o6
1.225x lo''
0.680
1.31~10'~
9.37~10"
28
*
**
Dnon= Diffusion coefficient computed from nonlinear curve
Dim = Diffusion coefficient computed from linear curve
1996 International Radon Symposium I1

2.10
Film
Region 1
3
Source
Accumulation
Chamber
v
Figure 1. Schematic o f the measurement system
200
0
0
80
120
. Time (hours)
40
0
160
Rgure 2. Accumulated activity for Poly 1
Slope = 9.29 X
0
lo4
1
0
MA 180 [ l  ~ x p (  5 . 6 9 ~ 1t)]
0~
40
80 120
Time (hours)
160
Figure 3. Accumulated activity for poly 2 Him
Bq so'
0
0 2 4 6 8 10121416
Time (hours)
floure 4. Unear portion of poly 1 activity curve
0 2 4 6 8 10121416
Time (hours)
Rgure 6. Linear portion of p l y 2 activity curve
1996 International Radon Symposium I1  2.1 1
0
40
80
120
Time (hours)
160
Rgure 6. Accumulated activity for latex 1 film
0
1
2
Time (hours)
3
flgure 8. Linear portion of latex 1 activity curve
4
0 10 20 3040 50 60 70 80
Time (hours)
flgure 7. Accumulated acthdty for latex 2 film
1
2
Time (hours)
0
3
Rgure 9. Linear portion of latex 2 activity curve

1996 International Radon Symposium I1 2.12