Interactions and Elastic Moduli

A crystal's linear elastic response to small distortions is described by its Lamé coefficients $\lambda$ and $\mu$ [32]. If we consider only nearest-neighbor interactions in an FCC crystal characterized by isotropic pair interactions, U(r), we find

$$\lambda = \frac{\sqrt{2}}{a} \, \left( \left. \frac{\partia... ...eft. \frac{\partial U}{\partial r}\right\vert _{r = a} \right)$$ (27)

and

$$\mu = \frac{1}{\sqrt{2}{a}} \left( \left. \frac{\partial^2 ... ...ft.\frac{ \partial U}{\partial r}\right\vert _{r = a} \right).$$ (28)

The bulk modulus is related to $\lambda$ and the shear modulus $\mu$through

$$B = \lambda + {2 \over 3} \mu.$$ (29)

For the screened-Coulomb FCC crystal,

  

$$\frac{2\sqrt{2}}{3} \frac{U(a)}{a^3} [2 (\kappa a)^2 + \kappa a + 1],$$ B = (30)

$$\mu \frac{1}{\sqrt{2}} \frac{U(a)}{a^3} [(\kappa a)^2 - \kappa a - 1].$$ = (31)

The shear modulus appears to vanish when $\kappa a = 1.618 \cdots$, the golden mean. However, interactions would be sufficiently long ranged under these conditions that second- and higher-nearest-neighbor interactions ought to be taken into account.

Substituting values for the dense crystal into Eqs. (30) and (31) we obtain $B = 0.52 \pm 0.05~\mathrm{dyne/cm}^2$ and $\mu = 0.17 \pm 0.04~\mathrm{dyne/cm}^2$. Even these fabulously small elastic moduli are enormous compared to the values for the dilute crystal, $B = 0.05 \pm 0.02~\mathrm{dyne/cm}^2$ and $\mu = 0.02 \pm 0.01~\mathrm{dyne/cm}^2$. Since we have ignored next-nearest-neighbor interactions and entropic contributions to the isotropic compressibility, these values should be considered lower bounds on the elastic moduli for defect-free colloidal crystals.