Calculation of Spinodal Curve Using User-defined Property
A Spinodal curve is where the determinant of the Hessian of Gibbs free energy with respect to composition is zero. For a phase with c-components, above condition is expressed as
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(1) |
where the molar fraction of component c is chosen as the dependent variable. The second derivative of G w.r.t. molar fractions can be calculated from the thermodynamic factors:
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(2) |
Example #1: Spinodal curve of Fcc phase in the Al-Zn binary system
In this example, the spinodal curve of the Fcc phase in the Al-Zn binary system is calculated through user-defined property. A user-defined property d2GdxZn2 for the Fcc phase is defined in AlZn_Spinodal.tdb as:
Property d2GdxZn2 298.15 ThF(Zn,Zn@Fcc)-ThF(Al,Zn@Fcc)-ThF(Zn,Al@Fcc)+ThF(Al,Al@Fcc); 6000 N !
where ThF(Zn,Zn@Fcc), ThF(Al,Zn@Fcc), ThF(Zn,Al@Fcc), and ThF(Al,Al@Fcc) are the thermodynamic factors of Fcc phase. Since the value of d2GdxZn2 is usually a large number, we define the Hessian function, HSN, as d2GdxZn2 multiplied by a factor of 10−4:
Property HSN 298.15 1e-4*d2 GdxZn2; 6000 N !
As shown in the AlZn_Spinodal.pbfx, the AlZn_Spinodal.tdb is appended to the AlMgZn.tdb. The spinodal line is calculated through contour mapping by using following conditions:
<contour name="Spinodal" property="HSN" start="0" stop="0" step="1"/>
<equilibrium_type type="individual"/>
The calculated Fcc spinodal is shown in Figure 1 with the stable Al-Zn binary phase diagram.
Example #2: Spinodal curve of Fcc phase in the X-Y-Z ternary system
In this example, the original database is XYZ.tdb, and the user-defined HSN property is defined in XYZ_Spinodal.tdb as described below:
Property d2Gdx2 298.15 ThF(Y,Y@Fcc)-ThF(X,Y@Fcc)-ThF(Y,X@Fcc)+ThF(X,X@Fcc); 6000 N !
Property d2Gdy2 298.15 ThF(Z,Z@Fcc)-ThF(X,Z@Fcc)-ThF(Z,X@Fcc)+ThF(X,X@Fcc); 6000 N !
Property d2Gdxy 298.15 ThF(Y,Z@Fcc)-ThF(X,Z@Fcc)-ThF(Y,X@Fcc)+ThF(X,X@Fcc); 6000 N !
Property HSN 298.15 1e-10*(d2Gdx2 * d2Gdy2 - d2Gdxy * d2Gdxy); 6000 N !
Note that the HSN property within the XYZ ternary system is derived and described as a function of the thermodynamic factor ThF. A factor of 10−10 is used to scale the HSN property since the numerical value of HSN is very big.
Again the spinodal lines are calculated through contour mapping. Details can be found in XYZ_Isotherm_Spinodal.pbfx and XYZ_Isopleth_Spinodal.pbfx. Figure 2 and Figure 3 show the calculated spinodal curves superimposed on the stable phase diagrams.
