L-5

Goal : This is the first lecture about the dynamics of a disordered system. We will see that different systems display pinning until a critical threshold. We will revisit Larkin arguments and discuss the spectrum of excitation of the instabilities.

Pinning and depininng of a disordered material

In the first lectures we saw that a disorder system can be trapped in deep energy states and form a glass. Today we will see that it can be also pinned and resist external deformation. Indeed disorder is at the origin of a complex energy landscape characterized by many minima (more or less deep), maxima and saddle points. An external force tilts this multidimensional landscape in a direction, but local minima survives until a finite threshold that unveils spectacular critical dynamics.

Experiments

We will discuss two examples of transition induced by pinning:

• The depinning transition: interfaces pinned by impurities are ubiquitous and range from magnetic domain walls to crack fronts, from dislocations in crystals or vortices in superconductors. Above a critical force, interfaces depin, their motion is intermittent and a Barkhausen noise is detected.

• The yielding transition: Everyday amorphous materials such as mayonnaise, toothpaste or foams display a behaviour intermediate between solid and liquid. They deform at small stress (as a solid) and flow  at large stress (as a liquid). In between we observe intermittent plastic events.

Equation of motion

We focus on zero temperature and on the overdamped regime. In presence of an external force $f$ the equation of motion of the interface is

$\partial _{t}h(x,t)=\nabla ^{2}h+f+F(x,h(x,t)),\quad {\text{with}}\;F(x,h(x,t))=-{\frac {\delta V(x,h(x,t))}{\delta h(x,t)}}$

The disorder force $F(x,h(x,t))$ is a stochastic function:

${\overline {F(x,h)F(x',h')}}=\delta ^{d}(x-x')\Delta (h-h')$

There are usually two kind of disorder: (i) Random Bond (RB) if $V(x,h)$ is short range correlated. Hence, the area below $\Delta (h)$ is zero, (ii) Random field if $V(x,h)$ is a Brownian motion along $h$.Hence, $\Delta (h)$ is short range corraleted.

• The velocity - force characteristic

• Existence of a unique critical force $f_{c}$: no-passing rule and Larkin model

• Large force behaviour

Scaling behaviour of the depinning transition

The order parameter of the transition is the velocity vanishing as

$v\sim |f-f_{c}|^{\beta }.$

The interface is rough at $f_{c}$

$u(bx)\sim b^{\zeta }u(x),\quad {\hat {u}}_{bq}\sim b^{\zeta -d}u_{q}$ ${\overline {{\hat {u}}_{q}{\hat {u}}_{q'}}}=\delta ^{d}(q+q')S(q),\quad S(q)\sim {\frac {1}{|q|^{d+2\zeta }}}$

above $f_{c}$ :