LBan-II

Interfaces: thermal shaking

Consider domain wall $h(r,t)$ fluctuating at equilibrium at the temperature $T$. Here $t$ is time, $r$ defines the d-dimensional coordinate of the interface and $h$ is the scalar height field. Hence, the domain wall separating two phases in a film has $d=1,r\in {ℛ}$, in a solid instead $d=2,r\in {{ℛ}}^{{𝟐}}$.

Two assumptions are done:

• Overhangs, pinch-off are neglected, so that $h(r,t)$ is a scalar univalued function.
• The dynamics is overdamped, so that we can neglect the inertial term.

Derivation

The Langevin equation of motion is

${\partial }_{t}h(r,t)=-\mu \frac{\delta E_{pot}}{\delta h(r,t)}+\eta (r,t)$

The first term $-\delta E_{pot}/\delta h(r,t)$ is the elastic force trying to smooth the interface, the mobility $\mu$ is the inverse of the viscosity. The second term is the Langevin noise. It is Guassian and defined by

$⟨\eta (r,t)⟩=0,⟨\eta (r^{\prime },t^{\prime })\eta (r,t)⟩=2dD{\delta }^d(r-r^{\prime })\delta (t-t^{\prime })$

The symbol $⟨\dots ⟩$ indicates the average over the thermal noise and the diffusion constant is fixed by the Einstein relation $D=\mu K_{B}T$. We set $\mu =K_B=1$

The potential energy of surface tension ($\nu$ is the stiffness) can be expanded at the lowest order in the gradient:

$E_{pot}=\nu \int d^{d}r\sqrt{1+(\mathrm{\nabla }h{)}^2}\sim \text{const.}+\frac{\nu }{2}\int d^{d}r(\mathrm{\nabla }h{)}^2$

Hence, we have the Edwards Wilkinson equation:

${\partial }_{t}h(r,t)=\nu {\mathrm{\nabla }}^{2}h(r,t)+\eta (r,t)$