Analytical Derivation of the Sauer-Freise Flux Equation for Multicomponent Multiphase Diffusion Couples with Variable Partial Molar Volumes

Abstract

The well-known Sauer-Freise flux equation is derived analytically for the general case of a multiphase, multicomponent diffusion couple with variable molar volume. Discontinuities in concentration versus distance data are specifically treated. Fluxes with respect to the local center of volume are employed if the partial molar volumes are constants. Fluxes with respect to the local center of moles are employed for the more general case of variable partial molar volumes.

Appendix

Further insight regarding the effect of a variable molar volume can be gained by integrating Eq 51 at constant t to obtain

$$ x(n,t) - x_0(t) = \int _{n_0}^n \frac{dn'}{c(n',t)} = t^{1/2}\int _0^{\uplambda ^*(n,t)}\frac{d\uplambda ^{*'}}{c(\uplambda ^{*'})}=:t^{1/2}h(\uplambda ^*). $$

(71)

Here we have defined $x_0(t) = x(n_0,t)$, which gives the trajectory of the material marking the initial discontinuity of the diffusion couple at $n = n_0$. Since $h(\uplambda ^*)$ is a monotonically increasing function of $\uplambda ^*$, we can define an equivalent similarity variable $\uplambda _x := h(\uplambda ^*)$ in which case $(x-x_0(t))/t^{1/2} = \uplambda _x$. At constant t, $d\uplambda ^* = c d\uplambda _x = (c/t^{1/2})dx$. When $n=n_0$, $\uplambda ^* =0$ and $h(\uplambda ^*) = \uplambda _x =0$.

By taking the total derivative of Eq 71 with respect to time, we obtain

$$ \frac{d x}{d t} - \frac{d x_0(t)}{d t} = \frac{1}{2t^{1/2}}\int _0^{\uplambda ^*(n,t)}\frac{d\uplambda ^{*'}}{c(\uplambda ^{*'})}+ \frac{t^{1/2}}{c(\uplambda ^*)}\frac{d \uplambda ^*(n,t)}{d t}. $$

(72)

But

$$ \frac{d \uplambda ^*}{d t} = \frac{d }{d t}\frac{(n-n_0)}{t^{1/2}} = -\frac{\uplambda ^*}{2t} +\frac{c}{t^{1/2}} \left[ \frac{d x}{d t} -v^*\right] . $$

(73)

Substitution into Eq 72 results in

$$ v^*(x,t) - \frac{d x_0}{d t} = \frac{1}{2t^{1/2}}\int _0^{\uplambda ^*}\frac{d\uplambda ^{*'}}{c(\uplambda ^{*'})} - \frac{\uplambda ^*}{2c\,t^{1/2}}. $$

(74)

These last two terms can be combined by integration by parts, resulting in

$$ v^*(x,t) - \frac{d x_0}{d t} = - \frac{1}{2t^{1/2}}\int _0^{\uplambda ^*}\uplambda ^{*'}\frac{d V_m(\uplambda ^{*'})}{d \uplambda ^{*'}}d\uplambda ^{*'}. $$

(75)

Equation 75 shows that the formerly eliminated velocity $v^*(x,t)$ can be related directly to the fact that the molar volume is variable. For markers near the right end of the diffusion couple, we have $v^* = d x_R/d t$ and we can take $\uplambda ^* = \infty$ to obtain

$$ \frac{d x_R}{d t} - \frac{d x_0}{d t} = -\frac{1}{2t^{1/2}}\int _0^{\infty }\uplambda ^{*'}\frac{d V_m(\uplambda ^{*'})}{d \uplambda ^{*'}}d\uplambda ^{*'} =\frac{1}{2t^{1/2}}\int _0^\infty [V_m(\uplambda ^{*'})-V_m(\infty )]d\uplambda ^{*'}:= \frac{1}{2t^{1/2}} g^+, $$

(76)

where we have integrated by parts with $d V_m/d \uplambda ^{*'} = d [V_m- V_m(\infty )]/d \uplambda ^{*'}$. Similarly, at the left end,

$$ \frac{d x_L}{d t} - \frac{d x_0}{d t} = \frac{1}{2t^{1/2}}\int _{-\infty }^{0}\uplambda ^{*'}\frac{d V_m(\uplambda ^{*'})}{d \uplambda ^{*'}}d\uplambda ^{*'} =-\frac{1}{2t^{1/2}}\int _{-\infty }^0 [V_m(\uplambda ^{*'})-V_m(-\infty )]d\uplambda ^{*'}:= -\frac{1}{2t^{1/2}} g^-. $$

(77)

By eliminating $d x_0/d t$ and integrating on time, we find

$$ x_R(t) -x_L(t) = x_R(0) -x_L(0) + t^{1/2}(g^- + g^+), $$

(78)

which shows how the distance between distant markers changes with time. Similarly, by integrating Eq 77 we obtain

$$ x_0(t) -x_L(t) = x_0(0) -x_L(0) + t^{1/2}g^- . $$

(79)

Note that Eq 78-79 only give relative values of $x_L(t)$, $x_0(t)$ and $x_R(t)$. Values with respect to an observer depend on forces that act on the diffusion couple in the x direction. If such forces were zero, the center of mass of the couple would not change its state of motion and could remain fixed with respect to the reference frame of an observer. Alternatively, one end, say the left end of the couple, could be clamped to the reference frame of the observer. Then $x_L(0)= x_L(t) =0$ and Eq 79 becomes simply $x_0(t) = x_0(0) + t^{1/2}g^-$.

When the molar volume is a constant, the right hand side of Eq 75 vanishes and $v^*(x,t) = d x_0(t)/d t$ becomes independent of position x. In that case, the diffusion couple is rigid and one can work in a reference frame in which $d x_0(t)/d t =0$.

We also note that in the case of constant partial molar volumes it can be shown that the quantity g ⁺ + g ⁻ in Eq 78 vanishes by virtue of Eq 64, which is equivalent to the usual equal area construction for the Matano interface position in this case. Thus the length of the couple, as measured by the distance between distant markers that track the center of moles, does not change if the partial molar volumes are constant. Finally, we note that for constant partial molar volumes the center of mole velocity is the same as the center of volume velocity in the ends where the fluxes vanish.[13,14]