Ternary dissolution kinetics in the Fe-Ni-P system
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which is in e q u i l i b r i u m with a n o t h e r s e m i - i n f i n i t e phase before solution t r e a t m e n t . For the F e - P b i n a r y , where the phosphide F e s P is d i s s o l v i n g in (~, the c o m position of Fe or P in FeaP is d e s c r i b e d by the g e n e r a l solution: s
cPh=APh+BPherf[~ 1 2
[i]
where C ph is the P or Fe concentration at any time, t, and any d i s t a n c e , x, f r o m the t w o - p h a s e i n t e r f a c e within the FeaP phase. A Ph and B TM a r e c o n s t a n t s d e t e r m i n e d by the i n i t i a l and b o u n d a r y conditions, and DPh is the c h e m i c a l diffusion coefficient in FeaP, a s s u m e d to be independent of c o m p o s i t i o n . A s i m i l a r equation d e s c r i b e s the P or Fe c o n c e n t r a t i o n in the a phase: C c~ = A s + B c~erf
2 ~
where A ~ and B e a r e also c o n s t a n t s d e t e r m i n e d by the a p p r o p r i a t e i n i t i a l and b o u n d a r y conditions, and J5 c~ is the c h e m i c a l diffusion coefficient in c~. The v e l o c i t y of the c~/Fe3P i n t e r f a c e can be d e t e r m i n e d f r o m the following equation:
jFeaP
-
joe
= ~d~ -
(C" -
c')
[s~
where the r i g h t - h a n d side of the equation is the m a s s m o v e m e n t r e s u l t i n g f r o m the i n e q u a l i t y of the fluxes, ] FeaP and J a , of e i t h e r component in the two p h a s e s . The t e r m ~ is the position of the a / F e a P i n t e r f a c e , with r e s p e c t to the o r i g i n a l two-phase i n t e r f a c e at x = ~ = 0 and at t i m e t = 0. The v e l o c i t y of the i n t e r f a c e is d~/dt, and C" and C' a r e a s s u m e d to be the e q u i l i b r i u m phase d i a g r a m c o m p o s i t i o n s . A s c h e m a t i c d i a g r a m of the d i s s o l u t i o n p r o c e s s is shown in Fig. 1. If tbe d i s s o l u t i o n p r o c e s s is diffusion c o n t r o l l e d and the p h a s e s a r e e f f e c t i v e l y infinite in extent, the m o v e m e n t of the i n t e r f a c e follows a p a r a b o l i c law: VOLUME 6A, APRIL 1975-891
where 7 is a constant determined by the initial and boundary conditions as well as the diffusion coefficients. 5 The initial and boundary conditions shown in Fig. 1 are: C ph = C~h for x < 0,
t =0
Ca = C~ f o r x > 0 ,
t =O
C"
cPh=
for x = ~-, t > 0
Ca = C '
f o r x = ~*,t > 0
With these conditions and Eq. [4], Eq. [1] can be written:
cPh :
C'+C~, h e r f ( - ~ - )
+
erf
[51
1 + erf ( ~ - ) where d~ = ~Ph/~a. Similarly, Eq. [2] can be written: Ca = C ' - Ca' erf (7) Ca' - C ' 1 - err (7) + 1 - erf (7)
err ( ~ \ \2~/JO" .,:~---/at/
[6] Differentiating Eqs. [4], [5], and [6] and substituting into Eq. [3], results in the following equation for either component:
c' (C'-C'/4-~
C" - C' ) exp ( - 7 5)
edge of the diffusion coefficients, initial concentrations, and interface compositions in each phase is required. As stated earlier, Eqs. [4], [5], and [6] can be solved analytically as long as the boundary conditions remain infinite in extent.
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