Consider the following image from the grain growth simulation: $(27$ points$)$

a$)$ The boundary marked with an $\text{'}$X$\text{'}$ separates grains $1$ and $2.$ Do you expect this boundary to

move toward grain $1$ or grain $2$ during the process of grain growth? Provide explanations for

your answer.

b$)$ Suppose that the interface marked above is the cross section through a grain boundary in

copper $(Cu),$ and that this section of the grain boundary has a spherical shape with a radius of

curvature of $2m.$ Assuming a grain boundary energy of $0\mathrm{.}62\frac{J}{m^2},$ calculate the chemical

potential difference, $$ between Cu atoms on the $\text{'}1\text{'}$ and $\text{'}2\text{'}$ sides of the grain boundary.

c$)$ On the schematic below, indicate which grain is grain $1$ and which one is grain $2.$

d$)$ Suppose $J_{12}$ is the flux of $Cu$ atoms jumping from grain $1$ to grain $2,$ and $J_{22}$ is the flux

at which atoms jump from grain $2$ to grain $1.$ Calculate the ratio, $\frac{J_{12}}{J_{12}}$ at $T=700K.$