This lecture is about understanding relations with phase diagrams. There will likely soon be a meeting for those who haven't seen phase diagrams before. A very basic reading about phase diagrams is in Callister, Chapter 9. Chapters 4, 6, and 7 are relevant in a book by Gordon and are planned to be posted online.
Analyze two cases in which the enthalpy of mixing is either negative or positive. Below is an expression of the change in enthalpy.
<center>
<br>
<math>\Delta H = w X_A X_B</math>
<br>
</center>
The term <math>w</math> relates to atomistic bonding parameters. When the term <math>W_
</math> in the expression below is less than the others, there is mixing in all proportions. When <math>w</math> is less than zero, there is a maximization of <math>A-B</math> bonds, and this leads to compound formation. There is a basic theory and patterns of compound formation. Consider a hypothetical 2-D system. There is checkerboard ordering. In a BCC system, this pattern still works. This is called <math>B_2</math> ordering. There are intermetallics of composition 50-50. In an FCC system, there is no way to arrange with 50-50 bonds. The nearest neighbors form a tetrahedron, and there is frustration of the triangle. Anytime there is a crystal structure with triangles, the triangle is frustrated. The formation that maximizes <math>A-B</math> bonds is called <math>L_1 O</math>. The maximum percentage of <math>A-B</math> bonds that can form is two thirds. One-third of the bonds are of the same type. There is a richer phase diagram with frustrated structures.
<p>
</p>
Build a model that is better than the solution model. Show ordering in the phase diagram. Set up an energy model with <math>B_2</math> type ordering. Write down the energy in terms of the number of different types of bonds. Use a simple model. In the regular solution model, all the atoms are distributed randomly. The next level is to consider two distinct sublattices.
<center>
<br>
<math>X_A^
\right X_B^
</math>
<br>
<math>X_A^
\right X_B^
</math>
<br>
</center>
Calculate how many bonds there are from average occupancy. Below is a constraint on the composition on both sublattices.
<center>
<br>
<math>\frac{ X_B^
+ X_B^
}
= X_B</math>
<br>
</center>
There is an order parameter, <math>\eta</math>. Define concentrations.
<center>
<br>
<math>X_B^
= \frac
(1+\eta)</math>
<br>
<math>X_B^
= \frac
(1-\eta)</math>
<br>
<math>\eta = 0 \right \begin
X_B^
= \frac
X_B^
= \frac
\end
</math>
<br>
<math>\eta = 1 \right \begin
X_B^
= 1
X_B^
= 0 \end
</math>
<br>
</center>
In the case that <math>\eta</math> is equal to unity, all the <math>B</math> atoms sit on one lattice. Values of <math>\eta</math> ranging from zero to unity correspond to fully disordered systems to fully ordered systems. What is the energy? Write energy in terms of this parameter. Probabilities are expressed below.
<center>
<br>
<math>P_
= X_A^
X_B^
+ X_B^
X_A^
</math>
<br>
<math>P_
= \frac
</math>
<br>
<math>P_
= X_A^
X_A^
</math>
<br>
<math>P_
= \frac
</math>
<br>
<math>P_
= X_B^
X_B^
</math>
<br>
<math>P_
= \frac
</math>
<br>
</center>
Write the energy in terms of the number of bonds of each type.
<center>
<br>
<math>U = N_
W_
+ N_
W_
+ N_
W_
</math>
<br>
<math>U = \frac
\frac
\eta^2 \left ( w_
- \frac{w_
+ w_{BB}}
+ \right ) ... </math>
<br>
</center>
There are similar terms that scale the enthalpy compared with the solution model. The interaction goes with the square of the order parameter. Add the entropy model. In the ideal solution, there is the same energy anywhere. Redistribute atoms within one sublattice and keep all the same concentration within the sublattice. Refer to a handout that expresses entropy. Be careful how this is normalized. Write the free energy in terms of the order parameter. Minimize with respect to the order parameter. Minimize the free energy with respect to internal degrees of freedom. The solution depends on the temperature. When the temperature is zero, the order parameter, <math>\eta</math> is one.
<p>
</p>
When is the entropy maximized? Flat distributions are associated with high temperatures. Disordered systems occur at high temperature. Between extremes of temperature, the value of <math>\eta</math> varies between zero and one. Plot equilibrium order parameter. There are two types of solutions, and one shows a first order transition. There is an experimental first-order phase transition.
<p>
</p>
Do all compounds disorder? Consider the order parameter above the melt. Consider fully-ordered compounds going into a melt.
<p>
</p>
Consider the free energy of compounds. There is a compettition between ordered and disordered states. The free energy curves are much steeper. There isn't a tolerance of stoichiometry. How the curves move depends on stoichiometry. The one with the largest entropy moves down relative to the other.
<center>
<br>
<math>\frac
= -S</math>
<br>
</center>
Consider a solid solution that is associated with larger entropy. The curve associated with the solid solution moves up relative to the others, and the common tangents move inward. Consider the phase diagrams, which are provided in a handout. Consider the solid state of copper and gold. They are both FCC compounds. There are two different compounds, one that maximizes composition at <math>\frac
</math> and the other that maximizes at <math>\frac
</math>, which is associated with layering of <math>A</math> and <math>B</math>.
<p>
</p>
Consider a simple energetic model. All phases come to a point in composition space. The role of <math>A</math> and <math>B</math> could be inverted. There is a different composition going into an ordered state. There is an important flaw and problematic assumption in mean-field theory; correlations are neglected. A problem is that probabilities are considered to be independent. The first two expressions below consider the probabilities to be independent. The third term is an expression that takes into account conditional probability.
<center>
<br>
<math>P_
= P_A P_B</math>
<br>
<math>P_
= X_A^
X_A^
</math>
<br>
<math>P_
= P_A^
P_B^
|_
</math>
<br>
</center>
Sites may not act independently. The likelihood of <math>B</math> next to <math>A</math> may be higher due to a correlation. The occupancies are related. This is what causes the mean-field theory to be inaccurate. This is more severe in a frustrated system.
<p>
</p>
After introducing correlations, the phase diagram looks topologically correct. There is purely entropic problem.
<p>
</p>
A compound at zero Kelvin is ordered, and the value of the order parameter minimizes. Approach the order-disorder from below disorder and there is long-range order. Above order-disorder there is solid solution.
<p>
</p>
There are short-range order (SRO) parameters.
<center>
<br>
<math>\alpha_
= \frac{P_
- X_B^2}
.</math>
<br>
</center>
Consider disorder at <math>87^
</math>. The neares neighbor is given by <math>\frac
(011)</math>. Below are values
<center>
<br>
<math>\frac
(011) = -0.16</math>
<br>
<math>\frac
(002) = 0.17</math>
<br>
</center>
Consider a system of <math>Pd_3 V</math>. Calculate the <math>PdV</math> bond.
<center>
<br>
<math>P_
= (x_v)^2 + x_v (1-x_v) \alpha</math>
<br>
<math>P_
= (0.25)^2 + 0.25 \cdot 0.75 \cdot -0.16</math>
<br>
<math>P_
= 0.0625 + 0.03</math>
<br>
<math>P_
= 0.0325</math>
<br>
</center>
The short-range correlation is one-half of what it would be if the system were completely random. The affinity of <math>VV</math> bonds is reflected in the solid solution state. The energetics can be learned from short-range order. Tell the probabilities of all the states to determine energetics. Measure the probabilities of all the states. Consider if the entropy of a state is larger of smaller than the solution state. If <math>VD</math> bonds are less than random, the entropy of the state is smaller.
<p>
</p>
There are only two types of systems: immiscible and compound forming. These are associated wtih positive and negative enthalpy of mixing, respectively. If in a solid solution, how can it be determine whether the system is a compound former? Measure the correlation in the disordered state with neutron diffraction or anything that probes short-range contacts or environment. There are subtle effects in the enthalpy of mixing. Consider the lattice parameter of intermediate states.
<p>
</p>
Vegar's law is shown graphically below. Which way do deviations from the linear relationship go? Consider the relation of the energy of a bond with the length of a bond. With compound formers, there is a negative deviation 90% of the time, but this is not always true. If <math>AB</math> bonds are not favored, there is a positive deviation.
<p>
</p>
Consider two phase equilibrium of liquid and solids. Consider liquid and solid ideal solutions. There is a lens type phase diagram. The free energy of mixing is the same. The curves are the same but shifted. At a temperature between the two melting points, the free energy curves must cross over. Above the melting point, reverse the graphs shown in <math>T_1</math>. Assume liquid-solid ideal solutions. Lower the temperature and there is a miscibility gap. Solid solution reflects the energetics at low temperatures. Consider real systems. Phase diagrams are included in a handout.