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

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

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

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

There are usually two kind of disorder: (i) Random Bond (RB) if $V(x,h)$ is short range correlated. Hence, the area below $\mathrm{\Delta }(h)$ is zero, (ii) Random field if $V(x,h)$ is a Brownian motion along $h$.Hence, $\mathrm{\Delta }(h)$ 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 of the center of mass of the interface. It is vanishing as

$v_{CM}\sim |f-f_c{|}^{\beta }.$

• Two point correlation function:

$\overline{v(y+x,t)v(y,t)}\sim \frac{e^{-x/\xi (f)}}{|x{|}^{\eta }},\xi (f)\sim |f-f_c{|}^{-\nu }$

• The interface is rough at $f_c$

$u(bx)\sim b^{\zeta }u(x),{\hat{u}}_{bq}\sim b^{\zeta -d}{u}_q$

$\overline{{\hat{u}}_q{\hat{u}}_{q^{\prime }}}={\delta }^d(q+q^{\prime })S(q),S(q)\sim \frac{1}{|q{|}^{d+2\zeta }}$

• The motion is intermittent with avalanches even below $f_c$. Their size and duration is scale free up to a cut-off:

$P(S)=S^{-\tau }f(S/S_{\max }),S_{\max }\sim \xi (f{)}^{d+\zeta }\sim |f-f_c{|}^{-(d+\zeta )\nu }$ $P(T)=T^{-\alpha }f(T/T_{\max }),T_{\max }\sim \xi (f{)}^z\sim |f-f_c{|}^{-(z)\nu }$

To work below threshold it is useful to disretize the equation of motion and introduce the following protocol

$v_i(t)={\partial }_t{h}_i(t)=h_{i+1}(t)+h_{i-1}(t)-2h_i(t)+m^2(w-h_i(t))+F_i(h_i(t)),$

scaling arguemnts

Depinning exponents

Exponent	Observable	Mean field	d=1
$z$	$\ell (t)\sim t^z$	2	$1.43\pm 0.01$
$\zeta$	$h(bx)\sim b^{\zeta }h(x)$	0	$1.25\pm 0.01$
$\nu$	$\xi (f)\sim |f-f_c{|}^{-\nu }$	1/2	$\nu =\frac{1}{2-\zeta }$
$\beta$	$v_{CM}\sim |f-f_c{|}^{\beta }$	1	$\beta =\nu (z-\zeta )$
$\tau$	$P(S)\sim S^{-\tau }$	3/2	$\tau =2-\frac{2}{d+\zeta }$

Cellular Automata

We consider a discrete version of the equation of motion:

• We call ${\sigma }_i=(h_{i+1}+h_{i-1}-2h_i)+m^2(w-h_i)$.
• The disorder can be imagined as a sequence of narrow wells and the point of the interface is trapped in the well
• Increasing $w$, ${\sigma }_i$ reaches a local random threshold ${\sigma }_i^{th}$.
• The point of the interface moves to the next well