Changeset 349

Timestamp:

03/27/08 11:28:08 (8 months ago)

Author:

powell

Message:

Parameterized free energy with temperature support.

Location:

trunk/matml/src/ternary

Files:

• ChangeLog (modified) (1 diff)
• freenergy.c (modified) (1 diff)
• ternary.c (modified) (3 diffs)
• ternary.h (modified) (1 diff)

trunk/matml/src/ternary/ChangeLog

@@ -3 +3 @@
   * Documentation update: qhull reference, other small changes.
   * Check for qhull library in configure.in.
+  * Parameterized free energy function with temperature support.
   * TODO: better single-phase check in ternary.c,
     multiple free energy functions; later refine qhull concave facets.

trunk/matml/src/ternary/freenergy.c

@@ -19 +19 @@
   double C3 Concentration of species 3.
 
-  double T Temperature (ignored for now).
+  double T Temperature.
+
+  energy_params *eparams Free energy function parameters.
   ++++++++++++++++++++++++++++++++++++++*/
 
-double free_energy (double C2, double C3, double T)
+double free_energy (double C2, double C3, double T, energy_params *eparams)
 {
-  double C1 = (1.-C2-C3 < 0.) ? 0. : 1.-C2-C3;
-  double m1=1., m2=2., m3=5.;                  /* Flory-Huggins molar volume */
-  double chi12 = -.5, chi13 = 2., chi23 = .2;      /* Interaction parameters */
+  double C1 = (1.-C2-C3 < 0.) ? 0. : 1.-C2-C3,
+    G1 = eparams->G1_T0 + eparams->G1_C*(T-eparams->T0),
+    G2 = eparams->G2_T0 + eparams->G2_C*(T-eparams->T0),
+    G3 = eparams->G3_T0 + eparams->G3_C*(T-eparams->T0);
 
   /*+ This uses the Flory-Huggins free energy formula for a polymer solution,
     which divides each species' entropy term by its molar volume and adds
     binary interaction parameters.
-    +latex+The resulting expression for free energy is:
-    +latex+\begin{equation}
+    +latex+The resulting regular solution expression for free energy is:
+    +latex+\begin{eqnarray}
     +latex+  \label{eq:flory-huggins}
-    +latex+  G = \frac{1}{m_1}C_1\log(C_1) + \frac{1}{m_2}C_2\log(C_2)
-    +latex+  + \frac{1}{m_3}C_3\log(C_3)
-    +latex+  + \chi_{12}C_1C_2 + \chi_{13}C_1C_3 + \chi_{23}C_2C_3.
-    +latex+\end{equation}
+    +latex+  G &=& G_1 + (G_2-G_1) C_2 + (G_3-G_1) C_3 \\\ \nonumber
+    +latex+  &+& RT \left(\frac{1}{m_1}C_1\log C_1
+    +latex+    + \frac{1}{m_2}C_2\log C_2
+    +latex+    + \frac{1}{m_3}C_3\log C_3\right) \\\ \nonumber
+    +latex+  &+& \Omega_{12}C_1C_2 + \Omega_{13}C_1C_3
+    +latex+  + \Omega_{23}C_2C_3 +\Omega_{123}C_1C_2C_3.
+    +latex+\end{eqnarray}
     +*/
-  return  ((C1==0.)?0.:C1/m1*log(C1)) + ((C2==0.)?0.:C2/m2*log(C2)) +
-    ((C3==0.)?0.:C3/m3*log(C3)) +
-    chi12*C2*C1 + chi13*C3*C1 + chi23*C2*C3;
+  return G1 + (G2-G1)*C2 + (G3-G2)*C3
+    + eparams->R*T*(((C1==0.)?0.:C1/eparams->M1*log(C1))
+                    + ((C2==0.)?0.:C2/eparams->M2*log(C2))
+                    + ((C3==0.)?0.:C3/eparams->M3*log(C3)))
+    + eparams->Omega12*C2*C1 + eparams->Omega13*C3*C1 + eparams->Omega23*C2*C3
+    + eparams->Omega123*C1*C2*C3;
 }

trunk/matml/src/ternary/ternary.c

@@ -33 +33 @@
   double Gmin, Gmax;
   ternary_point *points;
+  energy_params eparams={1.,1., 0.2,0.1,0., 1.,1.,1., 1.,2.,5., -.5,2.,.2, 0.};
 
   if (argc>1)
@@ -52 +53 @@
 
   /* Calculate free energies, including upper and lower bounds for coloring */
-  Gmin = Gmax = free_energy (0., 0., 0.);
+  Gmin = Gmax = free_energy (0., 0., 1., &eparams);
   for (i=0; i<=loop_max; i++)
     for (j=0; j<=loop_max-i; j++)
@@ -60 +61 @@
         points[index].C2 = (double)i/loop_max;
         points[index].C3 = (double)j/loop_max;
-        points[index].G = free_energy (points[index].C2, points[index].C3, 0.);
+        points[index].G = free_energy (points[index].C2, points[index].C3, 1., &eparams);
         Gmin = (points[index].G<Gmin) ? points[index].G : Gmin;
         Gmax = (points[index].G>Gmax) ? points[index].G : Gmax;

trunk/matml/src/ternary/ternary.h

@@ -70 +70 @@
 } ternary_point; /*+ Ternary "point" structure. +*/
 
-double free_energy (double C2, double C3, double T);
+typedef struct {
+  double R;        /*+ Ideal gas law constant +*/
+  double T0;       /*+ Reference temperature +*/
+  double G1_T0;    /*+ Free energy of species 1 at reference temperature +*/
+  double G2_T0;    /*+ Free energy of species 2 at reference temperature +*/
+  double G3_T0;    /*+ Free energy of species 3 at reference temperature +*/
+  double G1_C;     /*+ Species 1 heat capacity +*/
+  double G2_C;     /*+ Species 2 heat capacity +*/
+  double G3_C;     /*+ Species 3 heat capacity +*/
+  double M1;       /*+ Species 1 Flory-Huggins relative molar volume +*/
+  double M2;       /*+ Species 2 Flory-Huggins relative molar volume +*/
+  double M3;       /*+ Species 3 Flory-Huggins relative molar volume +*/
+  double Omega12;  /*+ Species 1-2 regular solution interaction parameter +*/
+  double Omega13;  /*+ Species 1-3 regular solution interaction parameter +*/
+  double Omega23;  /*+ Species 2-3 regular solution interaction parameter +*/
+  double Omega123; /*+ Species 1-2-3 regular solution interaction parameter +*/
+} energy_params;
+
+double free_energy (double C2, double C3, double T, energy_params *eparams);
 
 int GeomviewBegin (FILE **geompipe, char *version);