LBan-IV

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:

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

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

The No-Passing Rule

Interfaces obey the so-called no-passing rule. Consider two interfaces $h_{\alpha }(x,t)$ and $h_{\beta }(x,t)$ such that $h_{\alpha }(x,t=0)<h_{\beta }(x,t=0)$ for every $x$. In the overdamped case, $\alpha$ will never overtake $\beta$.

To see why, assume for contradiction that at some time $t^{*}$, $\alpha$ reaches $\beta$ at a point $x^{*}$, i.e., $h_{\alpha }(x^{*},t^{*})=h_{\beta }(x^{*},t^{*})$. At this point, it can be shown that the local velocity of $\alpha$, denoted by $v_{\alpha }(x^{*},t^{*})$, is strictly less than the local velocity of $\beta$, $v_{\beta }(x^{*},t^{*})$.

This contradiction implies that the no-passing rule holds: $\alpha$ cannot overtake $\beta$. An important consequence of the no-passing rule is that the value of the critical force $f_{c}$ is independent of the initial condition. Indeed, if at a given force $f$ the configuration $\beta$ is a dynamically stable state, it will act as an impenetrable boundary for all configurations preceding it.

When $f=f_{c}$, the system possesses a single dynamically stable configuration. For $f>f_{c}$, no metastable states exist, and the system transitions into a fully moving phase.

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 its definition.

The 1D model

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 up to a local threshold force $\sigma _{i}^{th}$ is overcomed. The distance between two consecutive pinning centers is a positive random variable $\Delta$, drawn from a distribution $g(\Delta )$.

The total force acting on block $i$ is:

$\sigma _{i}={\frac {1}{2}}{\bigl (}h_{i+1}+h_{i-1}-2h_{i}{\bigr )}+f.$

As $f$ is slowly increased, each block experiences a gradually increasing pulling force. An instability occurs when:

$\sigma _{i}\geq \sigma _{i}^{th},$

When this condition is met, block $i$ jumps to the next available well, and the forces are updated as:

${\begin{cases}\sigma _{i}\;\to \;\sigma _{i}-\Delta ,\\[6pt]\sigma _{i\pm 1}\;\to \;\sigma _{i\pm 1}+{\dfrac {\Delta }{2}},\end{cases}}$

After such an instability, one of the neighboring blocks may also become unstable, initiating a chain reaction.

Extensions of the 1D model

Step 3: Mean-Field (Fully Connected) Limit

Let us now study the mean-field version of the cellular automata introduced above. We make two approximations:

• Fully connected interaction: Replace the short-range Laplacian with a mean-field coupling to the center-of-mass height:

$\sigma _{i}=h_{CM}-h_{i}+f,$

where $h_{CM}={\frac {1}{L}}\sum _{j=1}^{L}h_{j}$ is the center-of-mass height.

• Uniform thresholds: All local thresholds are taken equal to one

In the limit $L\to \infty$, the statistical properties of the system can be described by the distribution of local stresses $\sigma _{i}$. It is convenient to introduce the distance to threshold:

$x_{i}=1-\sigma _{i}.$

An instability occurs when a block reaches $x_{i}=0$. This is followed by its stabilization and by a uniform redistribution over all blocks:

${\begin{cases}x_{i}=0\;\to \;x_{i}=\Delta &{\text{(stabilization)}}\\[6pt]x_{j}\;\to \;x_{j}-{\dfrac {\Delta }{L}}&{\text{(redistribution)}},\end{cases}}$