Example #$2$

A nickel bicrystal is equilibrated with its vapor. Where the grain boundary intersects the free surface, a groove is formed as shown in Figure $12\mathrm{.}23.$ The surface energy of a nickel$-$vapor interface is nearly isotropic at $1400$ K and may be taken to be $2040er\frac{g}{c}{m}^2=2\mathrm{.}04\frac{J}{m^2}($Appendix F$).$ Determine the interfacial energy of this grain boundary.

The dihedral angle is ${}_{}=168$

$\frac{{}_{gb}}{{}_{sv}}=\frac{sin{}_{}}{sin(\frac{360-{}_{}}{2})}=\frac{sin(168)}{sin(96)}=\frac{0\mathrm{.}208}{0\mathrm{.}994}=0\mathrm{.}209$

$\frac{{}_{gb}}{{}_{sv}}=\frac{sin{}_{}}{sin(\frac{360-{}_{}}{2})}$

${}_{gb}=0\mathrm{.}209{}_{sv}=0\mathrm{.}2092\mathrm{.}04\frac{(J)}{m^2}=0\mathrm{.}42\frac{(J)}{m^2}$

$Y_{av}$:$2Y_{5v}cos(\frac{5}{2})$

$44$