POLYMER L E T T E R S

A NOTE ON CRANK'S CORRESPONDENCE PRINCIPLE IN DIFFUSION WITH A VARIABLE DIFFUSION COEFFICIENT

Solut ions of the diffusion equation with a variable diffusion coeffi- c ient can, as a rule, be obtained only by lengthy procedures. There- fore , it s e e m s worthwhile to point to a general correspondence principle, briefly mentioned by Crank (l), according to which e a c h such solution descr ibes in fact not only one, but two different diffusion problems with different dependence of the diffusion coefficient on concentration. Al- though Crank res t r ic t s h i s brief remark to uni-dimensional sorption and desorption under simple ini t ia l and boundary conditions, h i s correspond- ence prihciple is actual ly much more general.

A simple example will b e considered first. T h e diffusion equation for one species, i, in one dimension and in the absence of convection, sources and s inks , and thermal diffusion may be written a s

where the diffusion coefficient wil l first be assumed to depend on con- centration only:

where f t i a l and boundary conditions ate

is any known function of concentration. Suppose that the ini-

t = 0, x > o , ci = f*(X) ( 3 )

t 2 0, x = o , ci = fg( t ) (4)

where f , and f , are known functions of the s p a c e coordinate and t i m e , respect ively. The solut ion, in general terms, may be written as

C i = F(x,t) ( 5 )

giving the concentration a s a function of s p a c e and t i m e . It is e a s y to verify that the solution of the problem remains the

s a m e if C i is replaced throughout by Co - C,, where C , is variable and C o is a constant equal to or greater than the maximum concentration CI occurring in the problem. Accordingly, the solution F(x,t), once calcu- la ted, c a n a l s o be used to obtain, in a simple manner, the solution of t h e following different problem:

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dC ,/at = div (Dj grad C ,)

t = 0, x > 0, cj = g2(x) = co - fZ(X)

t 2 0, x = 0, c, = g3(t) = co - f3(t)

the solution G(x,t) being

C j = G(x,t) = C o - F(x,t) (5a)

Here, g,, g 2 , and g 3 are functions related to f , , f 2 , and f 3 as indicated.

pendences f and g l of diffusion coefficients corresponding to one an- other in sorption and desorption. In Table I , f ,(O) and gl(0) are the dif- fusion coefficients a t zero concentration of the spec ies , and the con- s tan t parameters a and b are the s a m e for corresponding cases. The headings, sorption and desorption, can, of course, be interchanged.

It is evident from the form of eq. (1) that the correspondence princi- ple a l so applies to diffusion in two and three dimensions. Furthermore, the diffusion coefficient may a l so depend on t ime, location, and abso- lute value of the concentration gradient, provided that these depend- ences are the same in corresponding functions f , and g,. The diffusion coefficient may a l so depend on temperature and on s t ress exerted by the medium, provided that the profiles of both temperature and s t r e s s are the same in both corresponding cases . Las t , but not leas t , the correspond- ence principle a l so holds for anisotropic media, provided that each of the diffusion coefficients along the principal axes obey a correspond- ence relation of the type of eq. (2a).

A physical interpretation of the correspondence principle is readily given and may serve to explain the bas i s and generality without re- course to mathematics. Any redistribution of one spec ies i by diffusion can be considered formally as a simultaneous redistribution of two spe- c ies i and j , where the fictitious second species j is complementary to i so that, a t any location and t ime, the sum of the concentrations of both spec ies is constant:

A s an illustration, Table I lists various typical concentration de-

For example, spec ies j can be thought of as the “void” that can be oc- cupied by i, and Co would then correspond to a maximum “capacity”. [The second spec ies may be real rather than fictitious. Examples are binary interdiffusion of counterions in ion exchangers where eq. (6) (with concentrations given in equivalents) is the electroneutrality condi-

POLYMER LETTERS 91

tion, and binary interdiffusion with a constant-volume restriction (with concentrations given a s volume fractions).] Vith the condition (6) the flux of j is automatically equal in magnitude (and opposite in sign) to that of i , and the same is true for the concentration gradients. If both species are to obey the diffusion equation (l), their diffusion coeffi- cients Di and D, must be identical a t any location and time (C, and C j , of course, being normally different). From Di = Dj and condition (6) one obtains at once the interrelation of the concentration dependences f ,(C,) and g,(Cj) a s given in eq. (2a). Essentially the same argument applies to more complex systems, a s discussed before.

Symbols

C = concentration. D = diffusion coefficient. x = space coordinate. t = t ime. Indices i and j refer to the respective diffusing species.

Reference

(1) Crank, J., The Mathematics of Diffusion, Claredon Press , Oxford, 1956, p. 270.

F. Helfferich

She11 Development Company Emeryville, California

Received November 13, 1962