Ternary Diffusion: Restrictions on the ( 2 D) Matrix
- PDF / 624,769 Bytes
- 7 Pages / 594 x 774 pts Page_size
- 82 Downloads / 233 Views
(20)
Matrix
LOUIS S. CASTLEMAN An analysis has been made of ternary diffusing systems to determine the possibility that individual diagonal interdiffusion coefficient elements of the ,(2D) (k = 1,2,3) interdiffusion coefficient matrices can be negative. It is shown that in ternary diffusing systems, in which the concentration-penetration curves are monotonic, as many as three of the six diagonal coefficients can be negative. It is further shown that several ternary diffusing systems having a negative diagonal interdiffusion coefficient matrix element have already in fact been determined experimentally.
A
phenomenological description of one-dimensional, isothermal, isobaric diffusion in a ternary diffusing system is based on the principles of Nonequilibrium Thermodynamics. For one primarily concerned with theory, it usually starts with the phenomenological equations of Onsager Ji "~ -
3 E 3tik(OUk/OX) k=l
satisifed in the case of ternary diffusion:
Ji = -
(i = 1,2,3)
[1]
i, Z k =
3(2Dll)3(2D22) -- 3(2D,E)3(ED2,) > 0
[5]
~___413(2Dll)3(2D22) - 3(2DI2) 3(2D21)]
[6]
These relationships arise when the diffusion flux equations are rewritten in matrix form, and the problem of obtaining the solution is regarded as an eigenvalue problem: The expression for the two eigenvalues is as follows: 1
u = 5 [3(2D") + 3(2D22)] (i = 1,2,3)
[2] +l
where j is the flux of the ith component in the x-direction, 3Di, is an element of the chemical diffusion or interdiffusion coefficient matrix (also called a partial diffusion coefficient2), and c k is the concentration of the kth component. In this paper, these elements will be referred to simply as diffusion coefficients. Usually, a volume-fixed system 3 is chosen, and in such a system c~ is expressed in moles per unit volume, at. pct per unit volume, or in some other convenient unit. There is a well-known procedure (to be reviewed later) whereby the (3D) matrix of Eqs. [2] is converted to the k(3D) matrix, in which an element is defined as follows: ~(3Dij)=3Dij-3Dik
[41
[3(2Dll) + 3(2D22)] 2
where J i is the molar flux of the ith component in the x-direction, Lik is an element of the Onsager (L) matrix (commonly called a phenomenological coefficient') and uk is the molar chemical potential of the kth component. An alternative description favored by experimentalists is provided by the generalized Fick's equations 3 ~ 3Dik(OCk/OX) k=l
3(2D11) + 3(2D22) ~ 0
1,2,3;i4:kq:j
[3]
The ,(3D) matrix, however, is replaced in experimental work by the ~(2D) matrix, which is simply the k(3D) matrix with one row removed (the kth row). No loss in diffusional information results from this since one of the flux equations in Eqs. [2] is r e d u n d a n t : Kirkaldy, et al 4~ have examined the effects of thermodynamic restrictions on the 3(2D) matrix, and have shown that the following inequalities must be
LOUIS S. CASTLEMAN is Professor of Physical Metallurgy, Department of Physical and Engineering Metallurgy, Polytechnic Institute of New York, Brooklyn, NY 11201. Manuscript submitted
Data Loading...
