Description of a Method for Measuring the Diffusion Coefficient of Thin Films to 22% Using a Total Alpha Detector

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 ban-ier 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 (1 996) 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 ~ n 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 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 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). Co 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 10"~ m2s-' and about 4 hours for a diffusion coefficient of l0'I3 m2 s". This three order-ofmagnitude 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 non-steady state solutions. Region 3 is a closed volume in which ^ ~ n accumulates. Consequently, the concentration at the surface of the film, Cd, will slowly increase with time to match the increasing concentration in region 3, CÈ(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 (d-x) + C d Sinh C(x) = r