The free energy of mixing of a system, Gmix, depends on many conditions such as temperature, pressure, and composition. By fixing all variables except temperature and composition, one can generate a family of Gmix vs. composition curves:
(figure 1 is an example of a family of Gmix vs. composition curves at different temperatures. See DeHoff fig 10.1)
(Figure 2 is DeHoff figure 10.10d, which shows Gmix vs. composition for two separate phases. The two lines do not cross)
Let's assume we took the previous liquid/alpha example to a temperature high enough to melt the entire sample, then cooled it. In the melted state, the Gmix line of the liquid will be below that of the liquid (the opposite of figure 2). As we decrease the temperature, there may be compositions where Gmix curve of the alpha phase is more negative than the liquid
u phase. When this is so, it is possible to draw a line that is tangent to both curves:
(Figure 3 is a graph like the previous two, except the two curves do cross and are connected by a tangent as described above)
(Figure 4 is an MPEG of five separate images (see DeHoff 10.9a-e) which show the generation of a phase diagram. I will include instructions to click through them one image at a time rather than run it full-time, otherwise it'd be impossible to watch)
(Figure 5 is also an MPEG of the same format as figure 4 showing how the Gmix curve causes the miscibility gap to form. See DeHoff fig. 10.11)
The final paragraph will discuss how to form a binary eutectic diagram by superimposing two ideal solution graphs with a miscibility graph:
(The final figure will be Dehoff fig. 10.16d showing a binary eutectic diagram and the three sets of curves used to make it.)
This section will describe how the free energy of mixing of a system is used to create phase diagrams.
If a miscibility gap occurs in a phase that is dominant over the range of the gap, a phenomenon called Miscibility gaps can be seen in applications of all areas of materials engineering, but are especially important in polymers. Many polymers will mix into a single phase at elevated temperature, but the disassociate as the system cools.
Miscibility gaps are actually very common and occur in the majority of the systems you have studied so far. In fact, the miscibility gap is responsible for the production of one of the first phase diagrams you studied: the binary eutectic diagram. The binary eutectic is actually the combination of three separate two-tangent constructions: two two-phase constructions such as the ones in Figure 4, and a miscibility gap. Figure 6 shows how they interact to form the familiar shape of the binary eutectic:
The two-tangent construction of a phase diagram is an interesting example of how complex mathematics and delicate chemical interactions lead to the information materials engineers use every day on the job. Whether smelting several tons of steel, preparing a small batch of a specialized superalloy, or preparing a polymeric adhesive, all systems must obey the laws of free energy. And by using these laws the behavior of even the most complex systems can be predicted.
Figure 1: Plotting Gmix vs. Composition for Various Temperatures
Figure 2: Gmix Curves for Two Phases
Figure 3: Crossing Gmix Curves
For a given temperature, each phase will have its own Gmix vs. composition curve. At that particular temperature the system will seek the lowest free energy; which is to say it will become the phase with the most negative Gmix. For example suppose there existed a system which could exist as either a liquid or a solid phase called alpha:
The above figure suggests that for this particular temperature the system would rather be in the alpha phase no matter what the composition is.
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For any composition between the two tangent points, the Gmix will now lie on the tangent line between these points. It is within this range of compositions that the two phases will exist in equilibrium. As temperature continues to decline, the compositions marking the two-phase region will change. By plotting these compositions at the temperature where they occur, a phase diagram can be generated:
Under certain conditions a miscibility gap may form. A miscibility gap occurs due to a rise in a phase's Gmix vs. composition curve. The result is a region where two phases will form from a previously one-phase system:
However, in most engineering situations it is easy to eliminate many of these variables. For example, pressure is not an important consideration in the steelmaking process, since it is assumed that the plant would be operating at ambient air pressure. So, let's assume all variables can be considered constant except temperature and composition. By calculating the graphs of Gmix vs. composition for a number of temperatures,possible in a given system The system has a variety of different states it could take on: during processing a metal could be liquid or solid, ordered or disordered; Gmix determines which the system will choose. total
it is possible to calculate the change in free energy of the system as a liquid. However, Gmix of alpha is always more negative for the alpha phase, regardless of composition. Therefore the Now lNow the structure of the system does depend on composition. At certain compositions, the system will want to remain a liquid; at others, the solid phase is favoralble. When the Gmix curves do crossThe tangent seems to bridge between the two curves; it becomes the line with the lowest Gmix.
Under the right conditions, the heat of mixing can begin to dominate the free energy calculations. When this occurs, the Gmix curve of a phase can change concavity and take on a "camel-hump" appearance.
/Common Tangent Construction
favorable
. See DeHoff fig 10.9
Figure 4: Using Gmix vs. Composition Curves to Generate a Phase Diagram (this diagram is a movie that is best viewed one frame at a time rather than at full run-time)
Figure 5: The Formation of a Miscibility Gap
Figure 6: The Creation of a Binary Eutectic Diagramhe job. Whether smelting