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.

In the narrow wells approximation, the randomness of the disordered potential reduces to two random quantities: the distance between wells $\Delta$ and the threshold $\sigma _{i}^{\rm {th}}$ that must be overcome for the interface to escape the trapping well.

The universal properties of the depinning transition remain unchanged if one of these two quantities is taken as constant. Here, we choose: Uniform thresholds: All local thresholds are taken equal to one. The only remaining random variable is $\Delta$.

Extensions of the 1D model

The system’s dimensionality is encoded in the elastic force acting on each block. In spatial dimension $d$, the local force on block $i$ is written as a sum over its nearest neighbours:

$\sigma _{i}={\frac {1}{z}}\sum _{j\in \mathrm {nn} (i)}{\bigl (}h_{j}-h_{i}{\bigr )}+f,$

where $z$ is the coordination number, i.e. the number of nearest neighbours of each block. The value of $z$ increases with the spatial dimension (e.g. $z=4$ for a square lattice in $d=2$, $z=6$ in $d=3$, and so on).

This form of the elastic force ensures that when a block becomes unstable and advances by an amount $\Delta$, its $z$ neighbours each receive an extra stress $\Delta /z$.

To describe the model in the limit of high dimension, it is convenient to replace the discrete Laplacian by a fully connected elasticity, corresponding to $z=L$. In this case, the force becomes:

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

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

In the last part of the lecture we will solve the fully connected model explicitly. However, other elastic kernels are widely studied.

1. Long-range depinning kernels:

$\sigma _{i}=\sum _{j\neq i}{\frac {h_{j}-h_{i}}{|j-i|^{d+\alpha }}}+f,$

Here the sum extends over all sites, but the contribution decays with distance. The parameter $\alpha$ controls the interaction range and typically lies between $2/d$ and $2$. For these values, the critical exponents depend continuously on $\alpha$. For $\alpha \geq 2$, one recovers the short-range results, while for $\alpha \leq 2/d$ one recovers the fully connected (mean-field) behavior. Many physical systems exhibit a long-range depinning transition; for instance, a 1D crack front corresponds to $\alpha =1$. Importantly, the transition remains a depinning transition, and in particular the no-passing rule continues to hold.

2. Kernels that violate the no-passing rule:

In some systems, such as the yielding transition of amorphous solids, the elastic interactions are described by Eshelby kernels. These interactions are long-ranged, anisotropic, and have a quadrupolar symmetry with zero spatial sum (the stress released in one region is redistributed so that the net force on the system remains unchanged). Such kernels break the no-passing rule and lead to qualitatively different critical behavior, which we will discuss in the conclusions of the next lecture.

Velocity-Force Caracteristics

We define the interface velocity at time $t$:

$v^{t}=h_{\mathrm {CM} }(t)-h_{\mathrm {CM} }(t-1).$

In the fully connected model, the blocks have no spatial structure, and therefore there are no privileged interactions (such as nearest- or next-nearest-neighbor couplings). For this reason, the state of the system at time $t$ is entirely characterized by the distribution of the distance to instability of a single block, $P_{t}(x)$.

To derive its evolution equation, we write the dynamics for a single block:

$x(t+1)=1-F-h_{\mathrm {CM} }(t+1)+h(t+1).$

We must distinguish two cases:

If $x(t)>0$:

$x(t+1)=1-F-h_{\mathrm {CM} }(t)-v^{t+1}+h(t)=x(t)-v^{t+1}.$

If $x(t)<0$:

$x(t+1)=1-F-h_{\mathrm {CM} }(t)-v^{t+1}+h(t)+\Delta =x(t)-v^{t+1}+\Delta .$

Using the Heaviside function $H(x)$, the evolution equation for $P_{t+1}(x)$ can be written as the sum of these two contributions:

$P_{t+1}(x)=P_{t}(x+v^{t+1})\,H(x+v^{t+1})\;+\;\int _{0}^{\infty }d\Delta \;P_{t}(x+v^{t+1}-\Delta )\,g(\Delta )\,H(-x-v^{t+1}+\Delta ).$

This equation fully describes the dynamics of the system, given an initial condition $P_{0}(x)$ and a distribution of threshold distances $g(\Delta )$.

We are now interested in stationary solutions, which become independent of the initial condition and are characterized by a constant stationary velocity $v$. In the stationary state, the equation reads:

$P(x)=P(x+v)\,H(x+v)\;+\;\int _{0}^{\infty }d\Delta \;P(x+v-\Delta )\,g(\Delta )\,H(-x-v+\Delta ).$

From this self-consistent equation, we want to derive a relation that expresses the stationary velocity $v$ as a function of the external force $F$. To do this, we consider the first and second moments of the left- and right-hand sides.

It is useful to verify the following identity for a generic test function $\phi (x)$:

$\int _{-\infty }^{\infty }dx\,\phi (x)P(x)=\int _{0}^{\infty }dx\,\phi (x-v)P(x)+\int _{-\infty }^{0}dx\,P(x)\int _{0}^{\infty }d\Delta \;\phi (x-v+\Delta )\,g(\Delta ).$

First moment

Using $\phi (x)=x$ we obtain the equation for the first moment:

${\overline {x}}={\overline {x}}-v+{\overline {\Delta }}\int _{-\infty }^{0}P(x)\,dx,$

from which we derive the relation connecting the stationary velocity to the fraction of unstable sites:

$v={\overline {\Delta }}\int _{-\infty }^{0}P(x)\,dx.$

This result shows that the mean velocity is proportional to the probability of finding an unstable site, with the proportionality factor given by the average jump size ${\overline {\Delta }}$.

Second moment

Using $\phi (x)=x^{2}$ and ${\overline {x}}=1-F$ we obtain the equation for the second moment:

$v^{2}+2v(1-F-{\frac {\overline {\Delta ^{2}}}{2{\overline {\Delta }}}})-2{\overline {\Delta }}\int _{-\infty }^{0}dx\,xP(x)=0$

One has to show that

Failed to parse (syntax error): {\displaystyle \int_0^{\infty} dx \, x P(x) = \frac{\overline{\Delta^2}}{2 \overline{\Delta}}}{2} }

and observe ${\overline {\Delta }}\int _{-\infty }^{0}dx\,xP(x)={\overline {\Delta }}{\overline {x}}-{\overline {\Delta }}\int _{0}^{\infty }dx\,xP(x)$ to get the final equation:

$v^{2}+2v(1-F-{\frac {\overline {\Delta ^{2}}}{2{\overline {\Delta }}}})-2{\overline {\Delta }}(1-F--{\frac {\overline {\Delta ^{2}}}{2{\overline {\Delta }}}})\int _{-\infty }^{0}dx\,xP(x)=0$