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 $d=1,r\in {\cal {R}}$, in a solid instead $d=2,r\in {\cal {{R}^{2}}}$.

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

$\langle \eta (r,t)\rangle =0,\;\langle \eta (r',t')\eta (r,t)\rangle =2D\delta ^{d}(r-r')\delta (t-t')$

The symbol $\langle \ldots \rangle$ 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:

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

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

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

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:

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

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

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

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:

${\hat {h}}_{q}(t)={\frac {1}{L}}\int _{0}^{L}e^{iqr}h(r,t),\quad h(r,t)=\sum _{q}e^{-iqr}{\hat {h}}_{q}(t)$

Here $q=2\pi n/L,n=\ldots ,-1,0,1,\ldots$ and recall $\int _{0}^{L}dre^{iqr}=L\delta _{q,0}$.

• Show that the EW equation writes

$\partial _{t}{\hat {h}}_{q}(t)=-\mu \sigma q^{2}{\hat {h}}_{q}(t)+\eta _{q}(t),\quad {\text{with}}\;\langle \eta _{q_{1}}(t')\eta _{q_{2}}(t)\rangle ={\frac {D}{L}}\delta _{q_{1},-q2}\delta (t-t')$

The solution of this first order linear equation writes

${\hat {h}}_{q}(t)={\hat {h}}_{q}(0)+\int _{0}^{t}dse^{-\mu \sigma q^{2}s}\eta _{q}(s)$

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

• Compute $\langle {\cal {{E}_{q}(t)\rangle }}$ which describes how the noise injects the energy on the different modes. Comment about equipartition and the dynamical exponent
• Compute the width $\langle h(x,t)^{2}\rangle =\sum _{q}\langle h_{q}(t)h_{-q}(t)\rangle$. Comment about the roughness of the interface and the growth at short times.