Question

An orthorhombic crystal has three orthogonal low energy surface orientations with surface energies of γ(100)=500; γ(010)=300; and γ(001)=100 (all in units of mJ/m2 ). Assume a critical nucleus forms that is fully facetted with these orientations. Lengths X, Y and Z are the distances from the center of the nucleus to the planes (100), (010) and (001), respectively. a) (5%) Draw a schematic of the nucleus showing its shape and position in coordinate system x-y-z. b) (5%) What is ΔGsurface in terms of these surface energies and variables x, y and z? c) (5%) What is ΔGsystem in terms of these surface energies and variables x, y and z? d) (10%) Demonstrate that the shape of this nuclei will obey the relation γ (100)/X = γ(010)/Y = γ(001)/Z e) (5%) Explain why this critical nucleus has a very different geometry than that of a sphere, which has a minimum surface area per equivalent volume.

Answer 1

An expression for ∆G for forming the brick-shaped particle with edge lengths x, y, and z:

∆G = xyz∆g_B + γ¹⁰⁰yz + +γ⁰¹⁰zx + γ⁰⁰¹xy

Determine the equilibrium shape by minimizing with respect to x, y, and z:

∂∆G /∂x= yz∆g_B + γ⁰¹⁰z + γ⁰⁰¹y

∂∆G /∂y= xz∆g_B + γ¹⁰⁰z + γ⁰⁰¹x

∂∆G /∂z= xy∆g_B + γ¹⁰⁰y + γ⁰¹⁰x

Setting these derivatives equal to zero and denoting the critical sizes as xc = X, yc = Y , and zc = Z gives the desired relation

γ¹⁰⁰/X =γ⁰¹⁰ /Y=γ⁰⁰¹/Z = −∆g_B

(d)

A spherical shape will have smaller surface area than the fully-facetted brickshaped particle of equivalent volume. Even though the brick has a larger surface area, the total surface free energy is made low by forming the critical nucleus with its largest faces in the orientation with the lowest surface free energy and with its smallest faces in the orientation with the highest surface free energy.