MASSACHUSETTS INSTITUTE OF TECHNOLOGY

Kinetic Processes in Materials

3.21 Spring 2002

Samuel M. Allen and W. Craig Carter

Department of Materials Science and Engineering

Massachusetts Institute of Technology

77 Massachusetts Ave.

Cambridge, MA 02139

Problem Set 7: Due Wed. April 17, 2002, Before 5PM in 4-049

Exercise 7.1

Use the symmetric molar regular free energy of mixing for a binary alloy of $A$ and $B$ at fixed pressure.

$$\begin{split}\ensuremath{\overline{\ensuremath{{\Delta G}^{\m... ...ath{X_A}+ \ensuremath{X_B}\log \ensuremath{X_B}}) ] \end{split}$$ (1)

and do the following.

1. Determine the critical temperature $T_c$ in terms of $\ensuremath{{\Omega}^{\mathcal{RS}}}$ and $R$.
2. Plot the equilibrium compositions from $T=0$ to $T=1.1 T_c$.
3. Plot $\mu_A(X_B)$ for $0 \leq X_B \leq 1$.
4. Plot $\mu_A(X_B)$ versus $\mu_B(X_B)$ for $0 \leq X_B \leq 1$.

Exercise 7.2

Consider the stagnation problem associated with the disappearance of a nearly cylindrical grain in a thin sheet with thickness $h$.

Figure 7-2-i: Illustration of disappearing grain in a thin sheet. The circular boundary groove, radius $R_g$, which forms on each surface creates a pinning force resisting boundary motion.

If a groove develops as shown, the grain boundary can become ``pinned.''

1. Show that, for a pinned boundary, $r(z) = R_w \cosh (z/R_w)$ is the equilibrium shape of the grain boundary if all interfaces are isotropic.

2. Calculate the net force on the groove due to the grain when the radius of the groove $R_g = h$. Note that $\alpha \beta = \cosh (\beta/2)$ has two solutions when $\alpha = 1$, $\beta = 1.1787$ and $\beta = 4.2536$.

3. $\alpha \beta = \cosh (\beta/2)$ ceases to have any solutions when $\alpha < 0.75$. What happens to the grain when $R_g$ deceases to about 3/4 $h$?

Exercise 7.3

Calculating the fastest growing and smallest unstable wavelengths for a cylinder which is evolving due to surface diffusion.

Start with a uniform cylinder and perturb with an infinitessimal pertubation $R(z,t) = R_o + \epsilon(t) \sin 2 \pi z/\lambda$. Use the small slope approximation for the surface diffusion equation:

$$\frac{\partial R}{\partial t} = D_S^\star \frac{\partial^2 \kappa}{\partial z^2}$$

Find an expression for $\epsilon(t)$ and maximize with respect to $\lambda$.

Exercise 7.4

Determine the fastest growing and smallest unstable wavelengths (if they exist) for:

1. a nonconserved order parameter, $\eta(x)$ with homogeneous free energy density:
   $$f(\eta) = f_s ((1+\eta)(1-\eta))^4$$

2. a conserved order parameter, $c(x)$ with homogeneous free energy density:
   $$f(c) = \frac{2^8 f_s}{c_\beta - c_\alpha)^8} ((c-c_\alpha)(c-c_\beta))^4$$