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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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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!

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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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<math>w \prop \left (w_{AB} - \frac{w_{AA} - w_{BB}}{2} \right )</math>

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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!

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h2. Semiconductors

Mix elements of groups III and V and groups II and VI.

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<table cellpadding = 10>

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<math>GaAs</math>
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</td>


<td>
<center>
<math>a=0.565</math>
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</td>

</tr>

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<td>
<center>
<math>GaP</math>
</center>
</td>


<td>
<center>
<math>0.545</math>
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</td>

</tr>


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<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>
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</td>

</tr>


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<td>
<center>
<math>InSb</math>
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</td>


<td>
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<math>0.648</math>
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</td>

</tr>

</table>

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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!

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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.

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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.

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<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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There is no optimization possible of <math>\underline H_{mixed}</math>.  It doesn't drive phase separation, and the unmixed state is unhappy.

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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.  

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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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<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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Volume of one mer is equal to the volume of one solvent molecule.

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<math>V_p = n V_s</math>

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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.  

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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!

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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.

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<math>\omega_1 = N-ni</math>

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<math>\omega_2 = z \frac{N-ni}{N}</math>

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<math>\omega_3 = z-1 \left ( \frac{N-ni}{N} \right )</math>

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<math>\omega_{i+1} = z (z-1)^{n-z} \left ( \frac{N-ni}{N} \right )^n</math>

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<math>z = \mbox{coordination number}</math>

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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. 

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<math>\Omega = \frac{\omega_1 \omega_2 ...\omega_{N_p}}{N_p!}</math>

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<math>k_B \ln (\Omega) \right \mbox{entropy}</math>

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<math>\frac{\Delta S_{mixing} }{ k_B } = -N_p \ln \phi_p - N_s \ln \phi_s</math>

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<math>\phi_p = \frac{nN_p}{N}</math>

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<math>\phi_s = \frac{N_s}{N}</math>

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<math>\phi_p + \phi_s = 1</math>

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Normalize a number of different ways, such as by the number of different molecules.  Divide by <math>N_p + N_s</math>.

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<math>\Delta S_{mixing} = -k_B [x_p \ln \phi_p + x_s \ln \phi_s]</math>

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Divide by <math>N = n N_p +N_s</math>.

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<math>\Delta S = -k \left [\frac{\phi_p}{n} \ln \phi_p + \phi_s \ln \phi_s \right]</math>

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</center>

There is a standard term of <math>x \ln x</math>. Volume fractions correspond to mole fraction.

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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.