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 examine how such systems can also be pinned and resist external deformation.

Disorder creates a complex energy landscape with many minima, maxima, and saddle points. When an external force is applied, the landscape is tilted. Local minima remain stable until a finite threshold is reached.

This gives rise to dynamical phase transitions.

Depinning vs Yielding

Two important transitions associated with pinning are:

• Depinning transition.

Interfaces pinned by impurities appear in many contexts: magnetic domain walls, crack fronts, dislocations, vortices. Above a critical force $f_c$, steady motion sets in. Close to threshold, motion is intermittent and proceeds via avalanches (e.g. Barkhausen noise).

• Yielding transition.

Amorphous materials (foams, emulsions, pastes) deform elastically at small stress and flow at large stress. The analogue of the depinning threshold is the yield stress ${\sigma }_y$, separating solid-like from flowing behavior.

Both the critical force per unit length $f_c$ and the yield stress ${\sigma }_y$ are self-averaging quantities, analogous to the free-energy density in equilibrium disordered systems. Sample-to-sample fluctuations are universal but subleading in system size. The two transitions share many phenomenological features (threshold, intermittency, avalanches) but differ in an important respect:

• Depinning obeys monotonic dynamics (no-passing rule).
• Yielding generally does not, due to stress redistributions of mixed sign.

Equation of Motion for Depinning

At zero temperature, in the overdamped regime, the interface evolves as $${\partial }_{t}h(x,t)={\mathrm{\nabla }}^{2}h(x,t)+f+F(x,h(x,t)),F(x,h)=-\frac{\delta V(x,h)}{\delta h}.$$

Here:

• $f$ is the external driving force,
• $F(x,h)$ is the quenched disorder force.

The No-Passing Rule

Consider two interfaces evolving under the same disorder: $${\partial }_{t}h={\mathrm{\nabla }}^{2}h+f+F(x,h).$$

Let $$h_{\alpha }(x,0)<h_{\beta }(x,0)\mathrm{\forall }x.$$

Define their difference: $$\delta h(x,t)=h_{\beta }(x,t)-h_{\alpha }(x,t).$$

Assume that at some first contact point $(x^{*},t^{*})$, $$\delta h(x^{*},t^{*})=0.$$

Subtracting the equations of motion gives $${\partial }_t\delta h={\mathrm{\nabla }}^2\delta h+F(x,h_{\beta })-F(x,h_{\alpha }).$$

At the first contact:

• $\delta h=0$,
• ${\mathrm{\nabla }}^2\delta h\ge 0$ (minimum),
• the disorder force is identical because it is quenched.

One finds that the velocity of the lower interface is strictly smaller than that of the upper one: $$v_{\alpha }(x^{*},t^{*})<v_{\beta }(x^{*},t^{*}).$$

Thus crossing is impossible and ordering is preserved.

Consequences

• Metastable states are totally ordered.
• The critical force $f_c$ is independent of initial conditions.
• For $f>f_c$, no metastable states survive.

This monotonic structure is specific to depinning and does not hold for yielding systems.

Outlook

The depinning transition is simpler and better understood. In the following we focus on depinning and introduce minimal cellular automata capturing its physics. We will later return to yielding and discuss how similar automata can describe plastic flow.

Cellular Automaton for Depinning

We now introduce a discrete model in the depinning universality class. Time is discrete and the interface evolves through jumps between narrow pinning wells. The model captures threshold dynamics and avalanche propagation while remaining analytically tractable.

Degrees of freedom

The interface is represented by blocks of height $h_1,\dots ,h_N$.

Elastic interactions in finite dimension

In spatial dimension $d$, each block interacts with its nearest neighbours: $$F_i^{\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{t}}=\frac{1}{z}\sum _{j\in nn(i)}(h_j-h_i),$$ where $z$ is the coordination number.

When a block jumps forward by $\mathrm{\Delta }$, each neighbour receives an additional stress $\mathrm{\Delta }/z$.

Narrow-well disorder

Each block is trapped in a sequence of narrow pinning wells along the $h$-axis. Different blocks have independent trap sequences (translationally invariant disorder).

Each well has a local depinning threshold. Disorder may affect both the threshold values and the distances between wells. Here, for simplicity, we set all thresholds equal: $$f_Y=1.$$ The distances $\mathrm{\Delta }>0$ between consecutive wells are random variables drawn from a distribution $g(\mathrm{\Delta })$. A common choice is exponential wells: $$g(\mathrm{\Delta })=e^{-\mathrm{\Delta }}.$$

Open circles: trap positions. Filled circles: instantaneous interface configuration in $d=1$.

Driving protocols

Two drivings will be used in the course.

• Constant force

$$F_i^{\mathrm{d}\mathrm{r}\mathrm{i}\mathrm{v}\mathrm{e}}=F.$$

• Displacement control

$$F_i^{\mathrm{d}\mathrm{r}\mathrm{i}\mathrm{v}\mathrm{e}}=k_0(w-h_i).$$

In this page we focus on constant force. Displacement control will be introduced later to study avalanches.

Distance to instability

Define $$x_i=f_Y-F_i^{\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{t}}-F.$$

Interpretation:

• $x_i>0$: block stable.
• $x_i\le 0$: block unstable.

The dynamics can be written entirely in terms of the variables $x_i$.

Update rule

If a block $i$ becomes unstable, it jumps to the next well: $$h_i\to h_i+\mathrm{\Delta }.$$

In finite dimension, this induces an elastic redistribution of stress to its neighbours. Each neighbour receives an additional stress $\mathrm{\Delta }/z$.

Fully connected limit

In high spatial dimension, elasticity becomes mean-field: $$F_i^{\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{s}\mathrm{t}}=h_{\mathrm{C}\mathrm{M}}-h_i,h_{\mathrm{C}\mathrm{M}}=\frac{1}{N}\sum _i{h}_i.$$

When block $i$ jumps by $\mathrm{\Delta }$: $$x_i\to x_i+\mathrm{\Delta }(1-\frac{1}{N}),x_{j\ne i}\to x_j-\frac{\mathrm{\Delta }}{N}.$$

The jumping site tends to stabilize, while all other sites are shifted uniformly toward instability. This homogeneous redistribution of stress is the origin of avalanche propagation.

Thermodynamic limit

In the fully connected model, there is no spatial structure. All blocks are statistically equivalent.

In the thermodynamic limit $N\to \mathrm{\infty }$, the state of the system at time $t$ is completely characterized by the distribution $$P_t(x),$$ the probability density of distances to instability.

Define the interface velocity $$v^t=h_{\mathrm{C}\mathrm{M}}(t)-h_{\mathrm{C}\mathrm{M}}(t-1).$$

The evolution of $x$ for a single block is:

If $x(t)>0$: $$x(t+1)=x(t)-v^{t+1}.$$

If $x(t)<0$: $$x(t+1)=x(t)-v^{t+1}+\mathrm{\Delta }.$$

Using the update rule for stable and unstable sites separately, one obtains: $$P_{t+1}(x)=P_t(x+v^{t+1})H(x+v^{t+1})+\underset{0}{\overset{\mathrm{\infty }}{\int }}d\mathrm{\Delta }{P}_t(x+v^{t+1}-\mathrm{\Delta })g(\mathrm{\Delta })H(-x-v^{t+1}+\mathrm{\Delta }).$$

This equation fully describes the dynamics of the force-controlled model.

Stationary solutions

In the stationary state the velocity becomes constant $v$, and $P_t(x)\to P(x)$.

Solving this equation in the thermodynamic limit (see exercise) yields:

Deterministic critical force

$$F_c=1-\frac{\overline{{\mathrm{\Delta }}^2}}{2\bar{\mathrm{\Delta }}}.$$

The critical force is self-averaging.

Velocity–force relation

The stationary velocity satisfies the implicit quadratic relation $$v^2+2v(2F_c-F-1)-2\bar{\mathrm{\Delta }}(F_c-F)=0.$$

This equation determines the full $v-F$ characteristic curve.