Missing term in ternary quadratic expression

[tkphd]

Hi @jeanlucf22,

The ternary free energy approximation for a ternary phase,

Thermo4PFM/src/QuadraticFreeEnergyFunctionsTernaryThreePhase.cc

Lines 66 to 67 in 43c1611

	double fe = A[0] * (conc[0] - ceq[0]) * (conc[0] - ceq[0])
	+ A[1] * (conc[1] - ceq[1]) * (conc[1] - ceq[1]);

implements the sum of independent parabolas:

$$ \mathscr{F}(c_{\mathrm{a}}, c_{\mathrm{b}}) \approx A (c_{\mathrm{a}} - c_{\mathrm{a}}^{\mathrm{eq}})^2 + B (c_{\mathrm{b}} - c_{\mathrm{b}}^{\mathrm{eq}})^2 $$

The quadratic approximation (second-order Taylor expansion in two variables) for a ternary free energy ought to include the cross-term:

$$ \begin{align*} \mathscr{F}(c_{\mathrm{a}}, c_{\mathrm{b}}) \approx &A (c_{\mathrm{a}} - c_{\mathrm{a}}^{\mathrm{eq}})^2 + 2 X (c_{\mathrm{a}} - c_{\mathrm{a}}^{\mathrm{eq}})(c_{\mathrm{b}} - c_{\mathrm{b}}^{\mathrm{eq}}) + B (c_{\mathrm{b}} - c_{\mathrm{b}}^{\mathrm{eq}})^2\\ & A = \frac{\partial^2 F}{\partial c_{\mathrm{a}}^2}\\ & B = \frac{\partial^2 F}{\partial c_{\mathrm{b}}^2}\\ & X = \frac{\partial^2 F}{\partial c_{\mathrm{a}} \partial c_{\mathrm{b}}} \end{align*} $$

Is this a simplifying assumption for KKS or grand-potential inversion, or is it an oversight? Some free energy landscapes are significantly rotated relative to the coordinate axes!

[jeanlucf22]

@tkphd This was implemented for a (sintering) model were this simplified form was sufficient. So far, I have not needed the more general case, but it may come in the near future. Are you interested in using the general form?