Consider size effects. There is classification based on two characteristics.

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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</center>

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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</center>

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.

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.

Cohesive energy of metals

Consider a transition metal and disregard s states. States below <math>\epsilon_

Unknown macro: {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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</center>

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.

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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</center>

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.

<math>w \prop \left (w_

Unknown macro: {AB}

\frac{w_
Unknown macro: {AA}
- w_{BB}}
Unknown macro: {2}
\right )</math>

</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_

</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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</center>

Semiconductors

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

<tr>

</tr>

<tr>

</tr>

<tr>

</tr>

<tr>

</tr>

<tr>

</tr>

<tr>

</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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</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.

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.

<math>\mbox

Unknown macro: {Bulk}

</math>

<math>\Delta H_

Unknown macro: {mix}

= \underline H_

Unknown macro: {mixed}

(a_

Unknown macro: {mixture}

) - x_A \underline H_A (a_A) - x_B \underline H_B (a_B)</math>

<math>\mbox

Unknown macro: {Epitaxial}

</math>

<math>\Delta H_

= \underline H_

Unknown macro: {mixed}

(a') - x_A \underline H_A (a') - x_B \underline H_B (a')</math>

</center>

There is no optimization possible of <math>\underline H_

</math>. It doesn't drive phase separation, and the unmixed state is unhappy.

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.

Almost everything is determined by thermodynamics. A kinetic state is a locally stable thermodynamic state.

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.

<math>N_s = \mbox

Unknown macro: {number of solvent molecules}

</math>

<math>N_p = \mbox

Unknown macro: {number of polymer molecules}

</math>

<math>n = \mbox

Unknown macro: {degree of polymerization}

</math>

<math>V_s = \mbox

Unknown macro: {volume of solvent molecule}

</math>

<math>V_p = \mbox

Unknown macro: {volume of polymer molecule}

</math>

</center>

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

</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.

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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</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.

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

<math>\omega_2 = z \frac

Unknown macro: {N-ni}

Unknown macro: {N}

</math>

<math>\omega_3 = z-1 \left ( \frac

Unknown macro: {N}

\right )</math>

<math>\omega_

Unknown macro: {i+1}

= z (z-1)^

Unknown macro: {n-z}

\left ( \frac

Unknown macro: {N-ni}

\right )^n</math>

<math>z = \mbox

Unknown macro: {coordination number}

</math>

</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.

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

Unknown macro: {N_p!}

</math>

<math>k_B \ln (\Omega) \right \mbox

Unknown macro: {entropy}

</math>

<math>\frac{\Delta S_

Unknown macro: {mixing}

}

Unknown macro: { k_B }

= -N_p \ln \phi_p - N_s \ln \phi_s</math>

<math>\phi_p = \frac

Unknown macro: {nN_p}

Unknown macro: {N}

</math>

<math>\phi_s = \frac

Unknown macro: {N_s}

</math>

</center>

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

<math>\Delta S_

= -k_B [x_p \ln \phi_p + x_s \ln \phi_s]</math>

</center>

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

<math>\Delta S = -k \left [\frac

Unknown macro: {phi_p}

Unknown macro: {n}

\ln \phi_p + \phi_s \ln \phi_s \right]</math>

</center>

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

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.

Child pages

Lecture 21

Electronegativity

Why does chi work?

Cohesive energy of metals

Applications to other materials

Semiconductors

Polymer mixing