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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h(r,t)} fluctuating at equilibrium at the temparature Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} . Here $t$ is time, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r } defines the d-dimensional coordinate of the interface and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d=2, r \in \cal{R}^2} .

Two assumptions are done:

• Overhangs, pinch-off are neglected, so that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle - \delta E_{pot}/\delta h(r,t) } is the elastic force trying to smooth the interface, the mobility Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle \ldots \rangle} indicates the average over the thermal noise. The diffusion constant is fixed by the Eistein relation (fluctuation-dissipation theorem):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu=1, \sigma=1/2 } we have the Edward Wilkinson equation:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h(r,t)+c } cannot be distinguished. This is a conndition os scale invariance:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h(b r, b^z t) \overset{in law}{\sim} b^{\alpha} h(r,t) }

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z=2 } in any dimension and a rough interface below $d=2$ with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_0^L d r e^{iqr}= L \delta_{q,0} } .

• Show that the EW equation writes

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat h_q(0) =0 } . Note that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle h(x,t)^2\rangle = \sum_ \langle h_q(t)h_{-q}(t) \rangle }

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