Small Systems Which Approach Complete Homogenization Solution for such a case is assumed to be of the form C(x,t) = X(x) T(t) (1) [Note: Solutions for infinite systems were of the form C(x,t) = f(x/  t)] Substituting equation in Fick’s second law equation, we get Only way this equation can be true is if both L.H.S and R.H.S. of the equation are equal to a constant. In addition, the constant must be negative, for the inhomogeneities to disappear with time.

 T =T 0 exp ( - 2 Dt)  X= A’ Sin x + B’ Cos x A n, B n, and n are obtained from the boundary conditions. A n example of a problem involving small systems is diffusion out of a slab.

Diffusion Out of a Slab Boundary conditions: For 0<x<h, C = C o t = 0 For x=0 and x= h, C = 0 at t>0 and C = C o at t=0 C=0 at x=0,t >0  B n =0 C=0 at x=h, t>0  Sin n x = Sin n h = 0  n h = n   n = (n  /h)

C = Co at t = 0 for 0 < x < h  Multiply both sides by Sin (p  x/h) and integrate from 0 to h. All the terms in the integral on the R.H.S. with n  p are zero. With n=p,

A n = 0 for all even n for all odd n So A n can be written in terms of an integer variable “j” instead of ‘n’ as follows

Each successive term is smaller than the preceding one. For longer times, h 2 <16 Dt, one can consider only the first term. Average concentration in the slab is given by, For longer times,

Homogenization At x = 0, l, 2l, 3l, … Flow occurs from regions of –ve curvature to regions with +ve curvature.

At t=0, where the relaxation time,  = (l 2 /  2 D B ) Amplitude  decreases exponentially with time. Longer the , longer the homogenization process.

Thin Film Solution In the previous two cases dealing with infinite systems, we had infinite sources of solute. We assumed that the interface concentration remains constant. When we had a fixed amount of solute at the interface to diffuse out to the adjacent regions, the form of the solution is different. For a case where a fixed amount of solute  kg/m 2 (or other appropriate units)) is placed at the interface between two long rods with initial concentration of 0, the solution is given by The total quantity of solute is conserved 