Modelling the Effect of Composition on the Stability of Equilibrium Intergranular Films with Diffuse Interfaces
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deviations from bulk compositions when film thicknesses approached a characteristic length [4]. We also use a phase field method to model changes in order parameters within a finite volume. Previous work considered a glassy film at a grain boundary [5]. Here O(x) is the order parameter and is a measure of the orientation of the structure at a point with respect to an arbitrary axis. Orientation is assumed to be continuous and has fixed values at the boundary of the film indicating that crystalline order is induced into the glassy film by the neighboring grains. The equation for the excess interfacial free energy is formulated as below. u+
f(h) h/
jj[2.2(x)
+ lV~lj dx
A12__1
12- h
(1)
Where the first term in the integrand represents the difference in the homogeneous free energy between the lowest energy state (O(x) = 0 for all x or a totally amorphous film) and the state of the system. The second term in the integrand is a gradient energy penalty. The last term on the right hand side represents the integral of the van der Waals forces between the two grains over the area of the grain boundary, where A 121 is the Hamaker constant for the system. This formulation is incorrect because the homogeneous free energy term must not depend on the orientation. Furthermore, there is ambiguity in the interpretation of ¢ = 0 75 Mat. Res. Soc. Symp. Proc. Vol. 586 @2000 Materials Research Society
because it represents both an amorphous state and one in a particular orientation with respect to a reference state. In addition, it has been shown that the gradient energy penalty contribution from an orientation parameter must vary as IV0I for its contribution to the interfacial free energy to be finite at the sharp interface limit [6]. Adsorption of a minority component was observed by Huang et al. in numerical simulations of symmetric ternary systems using the following expression for the total free energy of the system [7].
F
=
00
[Ao+
A
i
i=A,B,C
ii(Vi)C dx
(2)
I
Here OA and EB are the mole fractions of the majority components and qc the minority component subject to OA + OB ± OC = 1. Aifo is the difference between the lowest energy and the symmetric regular solution model free energy of the current state per lattice site and rii is the constant gradient energy coefficient for each component. The first term in the integrand tends to thin the interface while the gradient terms tend to thicken the interface. For an initial condition defined by Eqs. 3 and 4, the minority component was observed to adsorb to the diffuse interface between two phases of bulk equilibrium composition.
±tanh(-4X)
(3)
2
2
and
iX(l +-
Where 4 is a non-dimensional decay length, here C = 0.5136, and X is a non-dimensional measure of length. We have duplicated the result in Fig. 1 using gradient flow techniques to search for an approximation to steady state.
Composition Profile at Long Times 1.0
0
t0.6
A B
0 0.4 0.2 0.0
-60
,
,
,
-40
0
-20
20
40
60
X Figure 1. A plot of OA, •B, and qc, the mole fraction of each component,
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