1a) Draw a diagram showing the free energy versus order parameter for both : a first and second order
phase transition for T> T*, T*, and T1b) You are investigating a solid phase structural phase transition, the order of which is unknown.
Describe two examples of experiments/macroscopic behavior that you would measure to determine if a
solid phase structural transition is first order or second order.
2) Consider a magnetic material for which the free energy can be written as:
G = Go-HM+AM² CM4
where M is the magnetization, H is an applied magnetic field, Cis a positive constant, and A = a(T-Tc).
Here Teis the temperature at which a paramagnetic (T Tc) to ferromagnetic (T occurs.
a) Provide an equation for the minimum energy magnetization when this material is paramagnetic. As
M is very small in the paramagnetic state, here the CM4 term can be ignored.
b) Magnetic susceptibility is defined as dM/dH. Provide an equation for the susceptibility in the
paramagnetic state. What happens to this susceptibility as T approaches T2?
c) In the ferromagnetic state, give an expression for the rate of change in M with respect to Tin the
absence of an applied field (H=0).
d) Plot the free energy, G, as a function of M from M =0 10, for H=0, Go=0, a = 10, and C= 3.5. Let To
= 350 K and plot G from 300 to 350 in increments of 10 K. Your stepsize in M should be no larger than
e) Based on your answers for d), is this a first or second order phase transition? Why?
f) In a magnetoelastic material, there can be a coupling between strain (ES) and magnetization leading to
the following expression for free energy. Here A is coupling coefficient that indicates the strength of the
interaction between strain and magnetization.
G = Got AM² + CM4 + + (1/2)Des²
Using the equilibrium condition where dG/des =0, solve for Es in terms of M. Then use this expression
for Es to write Gin terms of 2. and M.
g) Using the same parameters as in d) and keep Tc fixed at 350 K, estimate the value of Mat 353 K in the
presence of the magnetoelastic coupling with Es =5,1=-5, and D 20.
h) What does the isign of A indicate about the role of magnetoelastic coupling on the phase transition?
3) Consider the Si-Ti phase diagram:
Atomic Percent Silicon
a) Identify all eutectic points.
b) identify all peritectoid points.
c) Identify all peritectic points.
d) List two solid binary Ti-Si phases that melt congruently.
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