Pinning and Depinning of a Disordered Material

In earlier lectures, we discussed how disordered systems can become trapped in deep energy states, forming a glass. Today, we will examine how such systems can also be pinned and resist external deformation. This behavior arises because disorder creates a complex energy landscape with numerous features, including minima (of varying depth), maxima, and saddle points.

When an external force is applied, it tilts this multidimensional energy landscape in a specific direction. However, local minima remain stable until a finite critical threshold is reached. Two important dynamical phase transitions are 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 to vortices in superconductors. Above a critical force, interfaces depin, their motion becomes intermittent, and a Barkhausen noise is detected.

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

Depinning tranition: the equation of motion

In the following we focus on the depinning trasition. At zero temperature and in the overdamped regime, where $\rho \partial _{t}^{2}+{\frac {1}{\mu }}\partial _{t}\approx {\frac {1}{\mu }}\partial _{t}$ , the equation of motion for 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)}}

Here we set $\mu =1$ , $f$ the external force and the disorder force is $F(x,h(x,t))$ .

Cellular Automata

We now introduce a discrete version of the interface equation of motion. These cellular automata belong to the same universality class as the original model, and they are straightforward to implement numerically. For clarity, we first discuss the case of one spatial dimension, $d=1$ . We then extend the definition to the fully connected case, which can be solved analytically.

Step 1: Discretization along the x-direction

The interface is represented as a collection of blocks $i=1,\ldots ,L$ connected by springs with spring constant set to unity. The velocity of the $i$ -th block is given by:

v_{i}(t)=\partial _{t}h_{i}(t)={\frac {1}{2}}{\bigl [}h_{i+1}(t)+h_{i-1}(t)-2h_{i}(t){\bigr ]}+f+F_{i}\!{\bigl (}h_{i}(t){\bigr )}.

Here, $h_{i}(t)$ is the position of block $i$ at time $t$ , $f$ is the external driving force, and $F_{i}$ is the quenched random pinning force.

Step 2: Discretization along the h-direction

The key simplification is the narrow-well approximation for the disorder potential. In this approximation, impurities act as pinning centers, each trapping a block of the interface at discrete positions $h_{i}$ . The distance between two consecutive pinning centers is a positive random variable $\Delta$ , drawn from a distribution $g(\Delta )$ .