After installing the software PANDAT in a personal computer (PC) and loading a thermodynamic database of interest, we can start to calculate phase diagrams. We present in this appendix a step-by-step procedure to calculate phase diagrams such as those shown in Figures 2–7 using a regular solution model for the solid and liquid phases. However, once familiar with the procedure, a user can readily proceed to calculate phase diagrams of real and more complex binary and higher-order alloy systems provided that appropriate thermodynamic databases are loaded.

The first step is to run the PANDAT program. The useful functions are mainly listed in the “File,” “Database,” “Calculation,” and “Graph” pull-down menus as shown in Figure A. The next step is to load a thermodynamic database. As shown in Figure B, this database contains only two components, A and B. Use the double arrows to move these two elements from the “Available Components” window to the “Selected Components” window. This seems trivial. However, many databases are for multicomponent alloys and we may wish to calculate the phase diagram of a specific binary A,C or that of a ternary B,D,E, etc. from, for example, a six-component database containing elements A, B, etc. to F. This option allows the user to select the components of interest such as binary A-B. Figure C shows that the components A and B have been moved to the “Selected Components” window for calculations.

Since this article is written primarily for teaching, we wish to teach the students to learn the relationship between the topological features of a phase diagram in terms of the relative stabilities of the phase involved. The obvious next step is to find out what are the solution parameters given in the binary database prior to calculating the phase diagram. Let us now refer to Figure A again. We note the folders “Components” and “Phases” in the left column. Since we know that there are only two components, A and B, there is no need to check that option. We then double-click the mouse to the folder “Phases” and subfolder “Selected Phases,” and note in Figure D that only LIQUID and FCC_A1 phases exist in this binary. By clicking to the folder “LIQUID,” we have the window shown in Figure E. In the right column of this figure, we note a complete thermodynamic description of this liquid phase. It is a disordered solution phase as indicated in line 2 from the top. Its excess Gibbs energy is represented in terms of the Redlich-Kister equation²⁸ (a modified regular solution model by adding higher-order terms in terms of composition). In the present case, the liquid is an ideal solution since the parameter is 0, valid below 6,000 K, shown in the bottom line. In the second line from the bottom we note the notation G(A,B; order = 0). The “0” indicates that only the first term L0 in the Redlich-Kister equation is used (i.e., regular solution behavior). The Gibbs energies of liquid A and B are 0; in other words, liquid A and B are used as the reference state. By clicking the mouse to the word “FCC_A1” in the left column, we see a thermodynamic description of the FCC_A1 phase shown in Figure F. In this case, the regular solution parameter is –10 kJ mol^-1 and the Gibbs energies given are relative to those of liquid A and B. In other words, they are the Gibbs energies of freezing for A and B.

The fourth step is to calculate the phase diagram (Figure G). After clicking the “OK” button, we see a calculated phase diagram with maximum congruent melting shown in Figure H. This diagram should be identical to one of the phase diagrams presented in Figure 4. In view of the difference in temperature scales used in these two figures, it is cumbersome to make a direct comparison. But the PANDAT software has an option to change the scales to the user’s taste such as those in Figure 4. In order to change the temperature scale and range to those in Figure 4, click the mouse to anywhere in the right part of graph window to activate the graph configuration option. This also activates other functions such as printing, zooming, and labeling. After clicking the mouse to the “Configure Graph Options” icon, we see a number of options (Figure I). The ones we are concerned with in this case are the “Y Max,” “Y Min,” and “# Ticks” (in the lower left corner of this figure) and the “Units” (the lower right corner). We then change the 820°C (“T Max”) and 520°C (“Y Min”) to 827°C and 427°C since the maximum and minimum temperatures in Figure 4 are 1,100 K and 700 K. We next need to change the “# Ticks” from six to eight. To change the temperature scale from Celsius to Kelvin, we click the mouse to the “Units” button of I and have the display immediately in Figure J. After changing the temperature scale to Kelvin from Celsius and click the “OK” buttons of Figures I and J, we have the diagram shown in Figure Ka. When we print this figure, we see what is shown in Figure Kb. We can use the labeling icon to label the single-phase and two-phase fields for this A-B system. Click the icon to turn the label mode on, and and then click to the two single-phase and two-phase regions. As shown in Figure Kc, we have “LIQUID,” “FCC_A1,” and “LIQUID + FCC-A1” labeled in the top and the lower single-phase regions as well as the two-phase region. We can move, delete, and change the sizes and positions of the labels, and even change the word “LIQUID” for the liquid phase to “L.” When we move the mouse to any of the labeled words such as “LIQUID + FCC_A1” and click the right button of the mouse, we have Figure Kd, which shows three options: “Move,” “Delete,” and “Edit.” Instead of going through the procedure for all three options, we will only go through the “Edit” procedure. Once the user understands the functionality of this option, the other two should be obvious. Selecting the “Edit” option shown in Figure Kd, we have the “Edit Text” window shown in Figure Ke. In this figure we see the label “LIQUID + FCC_A1” shown in the top left corner. If we want to change it L + (Al), we can simply type these on top of it. We can even change these words in terms of symbol by changing the option from “Roman” to “symbol” under the option “Font”. We also see options for “Font Size (50 300)” and “Angle (-180 – 180). In the present case as shown in Ke, the angle is 0. This means that the labeling of “LIQUID + FCC_A1” shown in Figure Kc is horizontal. If we want this labeling of this two-phase field to lie within the (LIQUID + FCC-A1) two-phase field for a more pleasing appearance, we can change the “Angle” shown in Figure Ke from 0 to 40. The results are presented in Figure Kf.

We also go through the procedure to change the solution parameters so that the students learn the relationship between the characteristics of a phase diagram and the relative stabilities. The example we just gave is for the case when the LIQUID phase is ideal and FCC_Al solid phase exhibits a regular solution parameter of –10 kJ mol^-1. Suppose that we change the liquid regular solution parameter from 0 to 20 kJ mol^-1 and that of the FCC_A1 phase from –10 kJ mol^-1 to 20 kJ mol^-1. We should get a diagram shown in Figure 6c. PANDAT builds in some special options to change the parameters. Let us go back to the PANDAT window as shown in Figure D. Click the “Database” menu at the top line to produce the window shown in Figure L. By clicking the mouse to “Edit Database” displayed in Figure L, we have another window with the thermodynamic data displayed in plain text format as shown in Figure M. We next change the regular solution parameter of the LIQUID phase from 0 to 20 kJ mol^-1 and that of the FCC_A1 from –10 to 20 kJ mol^-1, and save. The resulting data are presented in Figure N. In the PANDAT window, select the “Refresh Database” option from the “Database” menu, and then proceed to calculate the phase diagram in the same manner as for calculating the congruent melting phase diagram displayed in Figure H. It is clear that the students can carry out this exercise to learn the close relationship between thermodynamics of solutions and the phase diagrams. Moreover, once the student learns to do the calculation presented so far for binaries, she or he can readily extend the calculation to real binary and higher-order phase diagrams of real alloys such as isotherms, liquidus projection, isopleths, and solidification simulation for two extreme cases (i.e., global equilibrium and simple Scheil solidification conditions).