L-2

Goal: 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.

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 inversily proportional to the viscosity. The second term is the Langevin Gaussian noise defined by the correlations

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

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 can be expanded at the lowest order in the gradient:

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

Setting $\mu =1,\sigma =1/2$ we have the Edward Wilkinson equation:

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

Scaling Invariance

The equation enjoys of a continuous symmetry because $h(r,t)$ and $h(r,t)+c$ cannot be distinguished. This is a conndition os scale invariance:

$h(br,b^{z}t)\stackrel{inlaw}{\sim }{b}^{\alpha }h(r,t)$

Here $z,\alpha$ are the dynamic and the roughness exponent rispectively. From dimensional analysis

$b^{\alpha -z}{\partial }_{t}h(r,t)=b^{\alpha -2}{\mathrm{\nabla }}^{2}h(r,t)+b^{-d/2-z/2}\eta (r,t)$

From which you get $z=2$ in any dimension and a rough interface below $d=2$ with $\alpha =(2-d)/2$.

Exercise L2-A: Solve Edward-Wilkinson

For simplicity, consider a 1-dimensional line of size L with periodic boundary conditions. It is useful to introduce the Fourier modes:

${\hat{h}}_q(t)=\frac{1}{L}\underset{0}{\overset{L}{\int }}{e}^{iqr}h(r,t),h(r,t)=\sum _q{e}^{-iqr}{\hat{h}}_q(t)$

Here $q=2\pi n/L,n=\dots ,-1,0,1,\dots$ and recall $\underset{0}{\overset{L}{\int }}dr{e}^{iqr}=L{\delta }_{q,0}$.

• Show that the EW equation writes

${\partial }_t{\hat{h}}_q(t)=-\mu \sigma q^2{\hat{h}}_q(t)+{\eta }_q(t),\text{with}⟨{\eta }_{q_1}(t^{\prime }){\eta }_{q_2}(t)⟩=\frac{D}{L}{\delta }_{q_1,-q2}\delta (t-t^{\prime })$

The solution of this first order linear equation writes

${\hat{h}}_q(t)={\hat{h}}_q(0)+\underset{0}{\overset{t}{\int }}ds{e}^{-\mu \sigma q^{2}s}{\eta }_q(s)$

Assume that the interface is initialy flat, namely ${\hat{h}}_q(0)=0$. Note that $E_{pot}(t)=\sum _q{{ℰ}}_q(t)=\frac{L\sigma }{2}\sum _q{q}^2{h}_q(t)h_{-q}(t)$

• Compute $⟨{\mathrm{E}}_{{𝓆}}{(}{𝓉}{)}{⟩}$

which describes how the noise injects the energy on the different modes. Comment about equipartition and the dynamical exponent

• Compute the width

$⟨h(x,t{)}^2⟩=\sum _{⟨}{h}_q(t)h_{-q}(t)⟩$

Comment about the roughness of the interface and the growth at short times.