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In order to integrate the partial differential equations (2.5.1) or (2.5.3) it
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is necessary to choose suitable initial and boundary conditions, i.e. to define a model of a certain experiment. The initial condition describes the concentration conditions of the system at the beginning of the diffusion process. The values of the concentrations or material fluxes (which are proportional to concentration gradients) at the boundary surfaces of the system or at concentration discontinuities inside the system are described by the boundary conditions. For illustration, consider the simplest type of diffusion, described by the partial differential equation (2.5.3), also called linear diffusion. The system will be represented by an infinite tube closed at one end (for x = 0) and initially filled with a solution with concentration c . Diffusion is produced by very fast removal (e.g. by precipitation or an electrode reaction) of the dissolved substance at the x = 0 plane (the reference plane). The initial concentration c is retained at large distances from this reference plane (x >oo). The initial condition is thus x > 0, and the boundary conditions are
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(2.5.4)
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Solution of the differential equation (2.5.3) together with the initial and boundary conditions (2.5.4) and (2.5.5) yields the relationship (2.5.6) where the error function erf y is defined by the equation erf y = Table 2.3 Error function erf (x) = (2/VJT) Jj e"y2 dy
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For x < 0.1 in sufficient approximation erf (x) = 1. 128JC - 0.376*
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Fig. 2.11 Concentration distribution in the case of linear diffusion with c = 0 for x = 0 (see Eq. 2.5.6), where D = 10"' s , time in seconds is indicated at each curve and the effective diffusion layer thickness is shown for t = 100 s
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(see Table 2.3). Figure 2.11 depicts the concentration distribution. It can be seen from Eq. (2.5.6) that the ratio c/c is independent of the concentration but only depends on the distance from the reference plane, on the time and on the diffusion coefficient. Assuming that the solution of the diffusing substance is dilute, the diffusion coefficient does not depend on its concentration. The material flux through the reference plane is given by the concentration gradient for x = 0. In the present case the concentration gradient is dc dx'~
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4Dt)
(2.5.7)
and the concentration gradient at the reference plane is then (2.5.8) Obviously, the material flux decreases with increasing time and, finally, for t ><*> approaches zero. If a tangent to the curves in Fig. 2.11 is drawn at the origin, it intersects the straight line c = c at a distance of
6 D = (nDt)1/2
(2.5.9)
108 termed the diffusion layer thickness. It is a measure of the region that is depleted by the diffusion process. Another example is linear diffusion, with a prescribed concentration gradient at the reference plane, i.e. a prescribed material flux through the reference plane. This type of diffusion transport is important mainly for electrode processes (see Section 5.4). The point of interest in this case is the concentration at the reference plane. In the simplest case, the material flux is constant, so that the boundary condition for x=0 (Eq. 2.5.5) can be replaced by = 0, t>0, D^ = dx (2.5.10)
The resultant concentration distribution (Fig. 2.12) is given by the equation
The error function complement erfcy is defined by the relationship erfcy = 1 erf y. The concentration at the reference plane c* is c* = c - 2Ktm(jzD)-m (2.5.12)