predicting the occurrence of zero flux planes
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I.
INTRODUCTION
A zero flux plane (ZFP) is the position in a diffusion couple where the flux of one component is zero. Dayananda and Kim first observed ZFP's in single phase diffusion couples prepared from Cu-Ni-Zn alloys. ] Continued investigation by Dayananda's group has shown that ZFP's occur in many ternary alloy systems 2'3 as well as in quaternary systems. 4'5 In addition, they occur in two-phase couples, including CuZn-Sn vapor-solid couples. 2 However, ZPF's are not found in all diffusion couples. Dayananda et al. 1-5 have suggested, based on empirical evidence, that they occur only when the diffusion path crosses one of the constant activity lines which passes through a terminal alloy composition. Recently, Thompson and Morral 6 have shown, based on a theoretical analysis which assumes constant kinetic properties, that ZFP's occur only when the initial concentration differences fall in a specific range that is determined by the square root diffusivity 7 of the system. The objective of the following communication is twofold: first, to test the theoretical predictions by comparing them to previous experimental observations; and second, to give a general discussion of the conditions when zero flux planes are expected. The latter discussion will be given first and will focus on geometric characteristics of concentration profiles and on how they dictate the conditions for ZFP occurrence. II.
THE BOLTZMANN-MATANO INTEGRATION AND ZFP's
The Boltzmann-Matano analysis s gives the flux of a component at any position in a diffusion couple as being proportional to the integral of its concentration profile as shown for position x* in Figure 1. It follows 1'6 that a ZFP occurs in ternary diffusion couples only when the concentration profile contains an extremum like the one in the figure. Thompson and Mortal 6 designate this a Type I extremum in order to differentiate it from a second type (Type II, illustrated in Figure 3) that is never associated with the formation of ZFP's. A ZFP occurs at a position x where the sum of areas A and B on Figure 1 is zero. It can be seen M.S. THOMPSON, Graduate Student, and J.E. MORRAL, Professor, are with the Department of Metallurgy and Institute of Materials Science, University of Connecticut, U-136, Room 111, 97 North Eagleville Road, Storrs, CT 06268. Manuscript submitted January 28, 1987. METALLURGICAL TRANSACTIONS A
t
a(e6
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0
c
C (.,o)
/
B~
c(-~)
_
_
X=l
X :~: X ZFP
Fig. 1--Concentration profile for component i in fusion couple. The zero flux plane, XzFp,, is at A + areaB = 0. The origin, x = 0, is the Matano C, ( - ~ ) are initial concentrations of the diffusion
a multicomponent difa position where area interface. C, (+oo) and couple alloys.
on the figure that at positions to the right of x* the areas change so that areaA + B > 0, while to the left of x*, A + B < 0. Therefore x* is the only location where A + B = 0 and no more than one ZFP is expected on this side of the Matano interface.
I I I . C O M P O S I T I O N RANGES FOR T H E O C C U R R E N C E OF
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