Study of diffusion.

T.Tokoro in University of Windsor

The rate of absorption was found to be linear with t^0.5 and consistent with the equation developed by Crank [1] for the case where diffusion of a liquid into a solid matrix is driven by a concentration gradient, and there are no chemical interactions between the liquid and the matrix. The theoretical curve for this model is:

M(t)/M = 4(D/l²)^0.5t^0.5[(1/p) ^0.5 + 2_{n=1 to} (-1)ⁿierfc{(nl)/(2(Dt) ^0.5)}]. (1)

In this equation, M(t) and M are the weight increase due to the absorption at time t and at very long times, D is the diffusion coefficient, and l is the thickness of the polymer solid. Absorption is occurred from both sides of the sample. The values for the diffusion coefficients were calculated from the initial linear portions of the experimental curves using following equations, which follows from equation (1) for the early stages of diffusion [1],

M(t)/M = 4{D/(pl²)}^0.5t^0.5. (2)

D = (p/64) (l²/t)_0.5 = 0.049 (l²/t)_0.5 (3)

D = (p/16) (l²/t)_1.0 = 0.196 (l²/t)_1.0 (4)

where (l²/ t)_0.5 and (l²/ t)_1.0 means the (l²/t) at the time M(t)/M = 0.5 and 1.0, respectively.

Therefore, to determine the diffusion coefficient M value has to be known. Equation (4) is used in [2] and relation of (2) to (4) are shown schematically in Fig.1. Whiles, sometime this value M is misused by M₀, the initial mass of polymer solid [3]. In some cases the thickness l is also misunderstood [4]. They used l/2 instead of l because both sides of solid absorb the liquid and therefore they think l/2 is the thickness for diffusion. If using this half of the thickness 4 in (1) should be change to 2. This is original situation of [1] to introduce (1) theoretically.

Using M provides the correct value of diffusion coefficient of liquid. However, using M₀ instead of M gives us how amount of liquid is absorbed during the immersion time of t in weight percent. Therefore, this is also useful to estimate the absorbed amount of liquid in weight percent at the time t. In other words, using M not depends on the density of polymer (g/cm³) but using M₀ depends on it.

[1] J. Crank, "Mathematics of Diffusion," 2nd Edition, Claredon, Oxford, 1975.

[2] S.L. Rice and A.F.Diaz, "Absorption of Silicone Oil by a Dimethylsiloxane Elastomer", Rubber Chemistry and Technology, Vol.61, pp.194-204, 1988.

[3]R.S.Gorur, J.W.Chang and O.G.Amburgey, "Surface Hydrophobicity of Polymers Used for Outdoor Insulation", IEEE Trans. PD., Vol.5, No.4, pp.1923-1933, 1990.

[4] H. Deng and R. Hackam, .

• Fig.1 Absorption process plotted by t^0.5. Equation (2) for any where on the linear line (i.e., same slope), (3) for at the time M(t)/M = 0.5 and (4) for at the time M(t)/M = 1.0, respectively.