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h1.  Lecture 13 
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*Clapeyron equation*

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The [Clausius-Clapeyron relation|http://en.wikipedia.org/wiki/Clausius-Clapeyron_equation] was introduced for one-component systems.  It can be used to calculate the shift in [transition|http://en.wikipedia.org/wiki/Phase_transition] temperature.

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<math> \frac{dT}{dY_j}  =  \frac{- \Delta X_j}{\Delta S} </math>
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Below this is written in terms of the [enthalpy|http://en.wikipedia.org/wiki/Enthalpy] difference

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<math> \frac{dT}{dY_j}  =   \frac{-T \Delta X_j}{\Delta H} </math>
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The above equation could be generalized in terms of any two conjugate pairs, with a phase curve relating their [intensive variables|http://en.wikipedia.org/wiki/Intensive_quantity].

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<math> \frac{dY_i}{dY_j}  =   \frac{- \Delta X_j}{\Delta X_i} </math>
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*Super-elasticity*

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Imagine two [phases|http://en.wikipedia.org/wiki/Phase_%28matter%29], a high temperature phase, <math>P</math>, and a low temperature, matensite phase, <math>M</math>.  The martensite phase is larger in one direction, and this fact can be used to control the [free energy|http://en.wikipedia.org/wiki/Gibb%27s_free_energy] difference.  Look at the equations of energy and find a differential relating length, [force|http://en.wikipedia.org/wiki/Force], and free energy.

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<math> dU = ... + F dl </math>

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<math> G = U -TS + PV -Fl </math>

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<math>l = \left (\frac{-\partial G}{\partial F} \right )_{T,P,...}</math>
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The derivative of free energy with respect to force is equal to the length.  A larger length corresponds to a larger decrease in free energy with the application of force.  This fact can be used to influence phase transition.  When the states of a sample are close in free energy, elongation could be induced by a phase transition.  The martensite phase would decrease more in free energy, and there could be a transition from the <math>P</math> phase to the <math>M</math> phase.

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Imagine a sample at temperature <math>T</math> between <math>T_o</math> and <math>T(F)</math>.  With the application of a force, there can be a phase transition.  We would want a fairly reversible transition in a useful temperature range.

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!Phase_transition_shift.PNG!

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This example is based on a one-dimensional change.  In three dimensions, there would be a change in [crystal symmetry|http://en.wikipedia.org/wiki/Crystal_symmetry], and all lattice vectors would change.  Calculations would involve more than one crystal and would be more complicated.

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*Example calculation*

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Calculate how much [stress|http://en.wikipedia.org/wiki/Stress_%28physics%29] is needed to be applied to induce a phase transition.

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<math> T_o = 300 K</math>

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<math> \Delta H = 300 \frac{J}{mol \cdot K}</math>

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<math> \Delta \epsilon = -0.075 </math>

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Below is a Clausius-Clapeyron equation, and the [extensive variables|http://en.wikipedia.org/wiki/Extensive_variable] are changed to intensive quantities.


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<math> \frac{dF}{dT}  =  \frac{- \Delta S}{\Delta l} </math>

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<math> \Delta l = \Delta \epsilon l </math>

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<math> \frac{1}{A} \left ( \frac{dF}{dT} \right )  =  \frac{1}{A} \left ( \frac{- \Delta S}{\Delta \epsilon l} \right )</math>

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<math> \frac{d \sigma}{dT}  =  \frac{- \Delta S}{\Delta \epsilon V} </math>

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<math> \frac{d \sigma}{dT}  = 1.66 \frac{MPa}{K} </math>

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It is important to stay in the elastic regime.  [Plastic deformation|http://en.wikipedia.org/wiki/Deformation] is an [irreversible process|http://en.wikipedia.org/wiki/Irreversibility].  Plastic deformation in metals usually occurs at a few hundred MPa, so in the example above, there could be a phase transition before plastic deformation.  It's important to push the temperature from <math>T_o</math> far enough to avoid random transitions due to temperature fluctuations.  With a <math>\Delta T</math> value of <math>20^\circ </math>, <math>33 Mpa</math> would need to be applied.

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The amount of stress needed to be added to the induce a phase transition is very temperature dependent.  An issue is that there is a disontinuity in the length during the phase transition.  We want to spread out the transformation.  A way to do this is to compositionally grade the material.  The composition affects <math>T_o</math>, and this shift can be calculated.  Write the Clausius-Clapeyron equation and the differential form of Gibbs free energy for this system.  The temperature shift can be calculated from a known chemical potential.

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<math> \frac{dT}{dN_B}  =  \pm \frac{\mu_B}{\Delta S} </math>

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The transformation first occurs where there is a higher <math>T_o</math>.

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!Composition_gradient.PNG!
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There is real [hysterisis|http://en.wikipedia.org/wiki/Hysterisis] when this is done [adiabatically|http://en.wikipedia.org/wiki/Adiabatically].  Boundary movement results in pure [dissipation|http://en.wikipedia.org/wiki/Dissipation].

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*Generalize*

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What is the heat effect?

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<math>H=U+PV</math>

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<math>dH=dU+pdV+VdP</math>

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<math>dU = \delta Q - pdV + Fdl</math>

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<math>(dH)_p = (\delta Q)_p + Fdl</math>

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There is an extra work term in the free energy differential.  The change in enthalpy at constant pressure is not just due to heat flow.  Perform a [Legendre transform|http://en.wikipedia.org/wiki/Legendre_transformation].

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<math>H=U+PV-Fl</math>

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<math>dH=\delta Q + VdP -ldF</math>

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<math>(dH)_{p,F} = (\delta Q)_{p,F}</math>

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Below is the generalized enthalpy.  It is the free energy minus all the work terms.

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<math>H=U-\sum_{i} Y_i X_i</math>

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<math>(dH)_{Y_i} = (\delta Q)_{Y_i}</math>

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An error in calculations can result from using the enthalpy for heat reaction when subjected to non-PdV work.  The full heat release is not the standard enthalpy.  The error can be large when doing electrochemistry.

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*Magnetically induced transition*

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Consider an example of transition between a [ferromagnetic|http://en.wikipedia.org/wiki/Ferromagnet] state and a [paramagnetic|http://en.wikipedia.org/wiki/Paramagnet] state.  Can there be a shift in transition with the application of a magnetic field?  Yes, anytime there are phases that differ in extensive quantities, essentially implying a first order phase transition, there can be a shift by applying intensive variable.  Look at the differentials and Clausius-Clapeyron equation.

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<math>dU=TdS+HdM</math>

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<math>dG=-SdT+...-MdH</math>

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<math>\frac{dT}{dH} = -\frac{\Delta M}{\Delta S} </math>

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How large or small is the change in transition per atom due to magnetization?  How large is <math>\Delta M</math>?  Consider a few aligned electron spins.  A large estimate of <math>\Delta M</math> is a few [Bohr magnetons|http://en.wikipedia.org/wiki/Bohr_magneton] per atom.  Yet <math>\Delta M</math> is still small in energetic terms.  Use the [Boltzmann constant|http://en.wikipedia.org/wiki/Boltzmann_constant], <math>k_B</math>, for the <math>\Delta S</math> term.   This is the atomic version of <math>R</math>.  The result of the calculation is that there is a change in the transition temperature of 0.66 K per applied [Tesla|http://en.wikipedia.org/wiki/Tesla_%28unit%29].  A giant field is required to shift temperature.  In this case the effect of magnetism on the transition temperature is small, and the only hope for getting in a reasonable region is if there is a small heat effect in the transition, which corresponds to a small value of <math>\Delta S</math>.

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*Example: <math>(La, Ca)MnO_3</math>*

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Consider an example of an <math>ABO_3</math> material.  In the case of <math>(La, Ca)MnO_3</math>, the amount of <math>Mn^{3+}</math> and <math>Mn^{4+}</math> ions can be controlled by varying the amount of <math>La</math> and <math>Ca</math>, which are of charge <math>3+</math> and <math>2+</math>, respectively.  Look at the electrical and magnetic phase diagram.

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!Magnetic_phase_diagram.PNG!
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Ferromagnetic materials are better conductors than [anti-ferromagnetic|http://en.wikipedia.org/wiki/Antiferromagnetic_interaction].  Electrons are able to hop between atoms and preserve spin.  Paramagnetic materials are not as bad conductors as anti-ferromagnetic.  This is due to [Hund's Rule|http://en.wikipedia.org/wiki/Hund%27s_rule_of_Maximum_Multiplicity].  

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!Ferromagnetic%2C_anti-ferromagnetic%2C_paramagnetic.PNG!
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Imagine one is at a point above the transition temperature in the phase diagram.  The material is in the paramagnetic field, and it is possible to be in the magnetized phase by applying a field and shifting the transition temperature.  Conductivity increases by four orders of magnitude, and this effect is termed the [Giant Magnetoresistive Effect (GMR)|http://en.wikipedia.org/wiki/Giant_magnetoresistive_effect].  There is not much changing intrinsically and is complicated in reality; there is a percolation issue.  IBM discovered a bigger change and termed it the [Colossal Magnetoresistance Effect (CMR)|http://en.wikipedia.org/wiki/Colossal_magnetoresistance].  

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Remember that there are different extensive properties that result from applying a field