L-2

Stochastic Interfaces and growth processes

The physical properties of many materials are controlled by the interfaces embedded in it. This is the case of the dislocations in a crystal, the domain walls in a ferromagnet or the vortices in a supercoductors. In the next lecture we will discuss how impurities affect the behviour of these interfaces. Today we focus on thermal fluctuations and introduce two important equations for the interface dynamics: the Edward Wilkinson euqation and the Kardar Parisi Zhang equation.

An interface at Equilibrium: the Edward Wilkinson equation

Consider domain wall $h(r,t)$ fluctuating at equilibrium at the temparature $T$. 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.

The Langevin equation of motion is

Failed to parse (syntax error): {\displaystyle \partial_t h(r,t)= - \mu \frac{\delta E_{pot}}{\delta h(r,t)} + \eta(r,t) \\ \langle \eta(r,t) \rangle =0 \; \langle \eta(r',t')\eta(r,t) \rangle = 2 D \delta^d(r-r') \delta(t-t') }

The first term $-\delta E_{pot}/\delta h(r,t)$ is the elastic force trying to smooth the interface, the mobility $\mu$ is inversily proportional to the viscosity. The symbol $⟨\dots ⟩$ indicates the average over the thermal noise. The diffusion constant is fixed by the Eistein relation (fluctuation-dissipation theorem):

$D=\mu K_{B}T$

The potential energy of surface tension is

Failed to parse (unknown function "\grad"): {\displaystyle E_{pot} = \sigma \int d^d r\sqrt{1 +(\grad h)^2} \sim \text{const.} + \frac{\sigma}{2} \int d^d r (\grad h)^2 }