Consider size effects. There is classification based on two characteristics.
h1. Electronegativity
Electronegativity increases moving diagonally up the periodic table. Why is this a good parameter to classify. Go the basics of bonding. Consider two states that approach each other in which one state is higher than another. A pertubation is proportional to overlap integral. If states are far apart they do not modify each other. Consider two cases: a strong and weak electronegativity difference. Charge transfers from one material to another in the case below. An ionic bond forms, and this is always favorable. The term <math>\Delta \chi</math> is a good parameter of charge transfer. There is a strong contribution to the enthalpy of mixing.
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!Band_diagrams_-_i_%2B_j.PNG!
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Consider when <math>\Delta \chi</math> is small. There are substantially overlapping bands. When the two are placed together, there is a full band. There is a lot of overlap, and all states are fully delocalized. <math>A</math> and <math>B</math> are neutral.
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!Band_diagrams_-_i_%2B_j_-_small_difference.PNG!
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Metals are not ionic. There is a delocalization of electrons. The electron density of <math>A</math> in a metal and <math>A</math> alone look the same.
h2. Why does chi work?
Go back to the periodic table. A goal is to find a good variable. A variable that seems to describe an effect may be related to another variable. There may be nothing to do with charge transfer or ionicity.
h2. Cohesive energy of metals
Consider a transition metal and disregard s states. States below <math>\epsilon_{atom}</math> are bonding. How does cohesive energy change with band filling. Initially electrons go into the bonding state. Adding electrons results in a decrease in energy.
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!Band_diagram_and_cohesive_energy.PNG!
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Most cohesive energy is around the <math>d</math> electrons. Consider the cohesive energy of the <math>4d</math> series. A maximum is at <math>Mo</math>. An s electron is involved. There is large cohesive energy in the <math>4d</math> and <math>5d</math> series, and <math>3d</math> elements are perturbed by magnetism.
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</p>
Why do early and late metals show large enthalpy of mixing? A small number of electrons are added to a large number of electrons. Electrons in the anti-bonding state move to the bonding state. Count electrons, if the sum of the number of <math>d</math> electrons is five, there is a strong compound former. Examples include <math>NiTi</math>, <math>PdY</math>, and <math>NiY</math>. Examples of solid solutions are <math>FeMn</math>. Electronegativity is a good predictor because it correlates with the number of valence electrons.
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!Band_diagrams_-_i_%2B_j_-_small_difference.PNG!
</center>
h2. Applications to other materials
Apply same concepts to other materials. Consider a mixture of <math>CaO</math> and <math>MgO</math>. There is an order fcc structure and cations occupy octahedral interstitials. Consider the expression below.
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<br>
<math>w \prop \left (w_{AB} - \frac{w_{AA} - w_{BB}}{2} \right )</math>
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</center>
The term <math>w</math> is small and positive. There is an elastic term due to size differences. The change in enthalpy is small and positive. Regarding the term <math>w_{AB}</math>, all electrostatic interactions are between <math>2^{+}</math> cations. Predict what the phase diagram looks like. It is below, and it shows that it is easier to put small atoms in a big host.
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!Phase_diagram_--_CaO-MgO.PNG!
</center>
h2. Semiconductors
Mix elements of groups III and V and groups II and VI.
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<table cellpadding = 10>
<tr>
<td>
<center>
<math>GaAs</math>
</center>
</td>
<td>
<center>
<math>a=0.565</math>
</center>
</td>
</tr>
<tr>
<td>
<center>
<math>GaP</math>
</center>
</td>
<td>
<center>
<math>0.545</math>
</center>
</td>
</tr>
<tr>
<td>
<center>
<math>AlP</math>
</center>
</td>
<td>
<center>
<math>0.546</math>
</center>
</td>
</tr>
<tr>
<td>
<center>
<math>AlAs</math>
</center>
</td>
<td>
<center>
<math>0.556</math>
</center>
</td>
</tr>
<tr>
<td>
<center>
<math>InP</math>
</center>
</td>
<td>
<center>
<math>0.587</math>
</center>
</td>
</tr>
<tr>
<td>
<center>
<math>InSb</math>
</center>
</td>
<td>
<center>
<math>0.648</math>
</center>
</td>
</tr>
</table>
</center>
Consider mixing <math>GaAs</math> and <math>AlAs</math>. There is a common sublattice and <math>Ga</math> and <math>Al</math> mix. The elements <math>Ga</math> and <math>Al</math> are chemically similar. Consider phase diagrams.
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!Phase_diagram_--_GaAs-AlAs.PNG!
</center>
Consider mixing with compounds of a larger parameter mismatch, such as <math>GaAs</math> and <math>GaP</math> or <math>InP</math> and <math>InSb</math>. A strain term drives separation.
<p>
</p>
Through epitaxial growth, a lattice parameters of deposited elements are the same as a substrate. Phase diagrams are different. Consider the free energy of mixing in the bulk at constant pressure and epitaxially. The parameter <math>a'</math> is controlled and is set by substrate.
<center>
<br>
<math>\mbox{Bulk}</math>
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<math>\Delta H_{mix} = \underline H_{mixed} (a_{mixture}) - x_A \underline H_A (a_A) - x_B \underline H_B (a_B)</math>
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<math>\mbox{Epitaxial}</math>
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<math>\Delta H_{mix} = \underline H_{mixed} (a') - x_A \underline H_A (a') - x_B \underline H_B (a')</math>
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</center>
There is no optimization possible of <math>\underline H_{mixed}</math>. It doesn't drive phase separation, and the unmixed state is unhappy.
<p>
</p>
Consider weakly separating systems. Epitaxially deposit and remove the driving force of mixing. Compounds form. In the 1980's there was work on III-V semiconductors.
<p>
</p>
Almost everything is determined by thermodynamics. A kinetic state is a locally stable thermodynamic state.
h1. Polymer mixing
Consider polymers or biological species in solution. The volume of the molecule could be a hundred times a solvent. Below are relevant terms.
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<br>
<math>N_s = \mbox{number of solvent molecules}</math>
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<math>N_p = \mbox{number of polymer molecules}</math>
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<math>n = \mbox{degree of polymerization}</math>
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<math>V_s = \mbox{volume of solvent molecule}</math>
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<math>V_p = \mbox{volume of polymer molecule}</math>
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</center>
Volume of one mer is equal to the volume of one solvent molecule.
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<br>
<math>V_p = n V_s</math>
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</center>
A lot of thermodynamics is derived by considering discrete state models. Calculations are started in class, and there is handout that provides details. Lay polymer along a cubic lattice. A mer is at the center of points.
<p>
</p>
Consider whether the box contains a mer or solution and the number of ways to put down a polymer on a lattice after a polymer is already down. It is a function of the product of combinations. There are <math>i</math> chains of length <math>n</math> and <math>N-ni</math> sites open.
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!Polymer_solvent_-_discrete_state_model.PNG!
</center>
In laying down the first site, there are <math>N-ni</math> sites from which to choose. The number of ways to lay down a mer on a second site is related to a coordination number. Account for the probability whether a site is open, which is expressed by <math>(N - ni)/N</math>. The number of ways to lay down a mer on third site and sites thereafter are expressed the same. Consider critical assumptions. No correlation is assumed regarding whether a site is free. Polymers are uncorrelated, and this is typical of mean field theory.
<center>
<br>
<math>\omega_1 = N-ni</math>
<br>
<math>\omega_2 = z \frac{N-ni}{N}</math>
<br>
<math>\omega_3 = z-1 \left ( \frac{N-ni}{N} \right )</math>
<br>
<math>\omega_{i+1} = z (z-1)^{n-z} \left ( \frac{N-ni}{N} \right )^n</math>
<br>
<math>z = \mbox{coordination number}</math>
<br>
</center>
The number of ways to lay down a chain is give by the product of all probabilities. Find entropy by taking natural log and applying Stirling's approximation.
<center>
<br>
<math>\Omega = \frac{\omega_1 \omega_2 ...\omega_{N_p}}{N_p!}</math>
<br>
<math>k_B \ln (\Omega) \right \mbox{entropy}</math>
<br>
<math>\frac{\Delta S_{mixing} }{ k_B } = -N_p \ln \phi_p - N_s \ln \phi_s</math>
<br>
<math>\phi_p = \frac{nN_p}{N}</math>
<br>
<math>\phi_s = \frac{N_s}{N}</math>
<br>
<math>\phi_p + \phi_s = 1</math>
<br>
</center>
Normalize a number of different ways, such as by the number of different molecules. Divide by <math>N_p + N_s</math>.
<center>
<br>
<math>\Delta S_{mixing} = -k_B [x_p \ln \phi_p + x_s \ln \phi_s]</math>
<br>
</center>
Divide by <math>N = n N_p +N_s</math>.
<center>
<br>
<math>\Delta S = -k \left [\frac{\phi_p}{n} \ln \phi_p + \phi_s \ln \phi_s \right]</math>
<br>
</center>
There is a standard term of <math>x \ln x</math>. Volume fractions correspond to mole fraction.
<p>
</p>
The number of ways to interchange a polymer molecule provides entropy. There are a lot fewer degrees of freedom when <math>n</math> is large. The number can be very small. Polymers do strange things thermodynamically. Most driven by configurational entropy. There is more phase space. This is the same of polymers, but a scale is different. Other terms are not necessarily positive when mixing. This is important when the configurational entropy is small. Go from disordered state to ordered state.
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