Relaxation dynamics and long-time tails explain shear-induced diffusion of soft athermal particles near jamming

1. Problem Statement

The paper addresses the transport properties of soft athermal particles (e.g., foams, emulsions, granular materials) near the jamming transition ($\phi_J$). While the mechanical and rheological properties of these systems (such as viscosity divergence and yield stress) are well-understood through critical scaling, the mechanisms governing shear-induced diffusion ($D$) remain less clear.

Previous studies have primarily linked $D$ to spatial length scales, such as the size of collective motions ($\xi$) or system size ($L$), observing finite-size effects where $D/\dot{\gamma} \propto L^\beta$. However, the role of relaxation dynamics and the velocity auto-correlation function in determining $D$ has not been systematically explored in the context of the Green-Kubo (GK) formalism for athermal systems. Specifically, it is unknown whether the divergence of $D$ observed in finite-size simulations of sheared amorphous solids is a consequence of long-time tails in the velocity correlations, similar to phenomena observed in classical fluids.

2. Methodology

The authors employ Molecular Dynamics (MD) simulations of two-dimensional soft athermal particles under simple shear flow.

• System Setup:

  • Particles: A binary mixture of small ($d_0$) and large ($1.4d_0$) particles with a total number $N=8192$ (with finite-size checks).
  • Interactions: Repulsive elastic forces ($f_{ij}^e = k\delta_{ij}n_{ij}$) and overdamped dynamics with a damping force ($f_i^d = -\zeta[v_i - u(r_i)]$).
  • Conditions: Simulations cover packing fractions $\phi \in [0.8, 0.9]$ around the jamming point $\phi_J \approx 0.8433$ and shear rates $\dot{\gamma}t_0 \in [10^{-7}, 10^{-2}]$.
  • Boundary: Lees-Edwards boundary conditions to impose shear.

• Analytical Framework:

  • Green-Kubo (GK) Relation: The diffusion coefficient is calculated via the velocity auto-correlation function (VACF):
    $$D = \langle v_y^2 \rangle \int_0^\infty C(\tau) d\tau$$
    where $C(\tau)$ is the normalized transverse velocity auto-correlation.
  • Scaling Analysis: The authors apply critical scaling ansatzes to the mean squared velocity $\langle v_y^2 \rangle$, relaxation time $\tau^*$, and diffusion coefficient $D$ to extract critical exponents ($\nu, \lambda, \psi, \epsilon$).
  • Functional Forms: The VACF is analyzed using stretched exponential functions ($e^{-(\tau/\tau^*)^\alpha}$) and power-law decays to identify different dynamical regimes.

3. Key Contributions

1. Linking Diffusion to Relaxation Dynamics: The paper establishes a direct quantitative link between the shear-induced diffusion coefficient and the relaxation time extracted from the VACF, moving beyond purely spatial scaling arguments.
2. Identification of Long-Time Tails: The authors demonstrate that in the quasi-static regime above jamming ($\phi > \phi_J$ and very low $\dot{\gamma}$), the VACF exhibits a long-time tail (power-law decay) rather than a stretched exponential decay.
3. Explanation of Divergence: They show that this long-time tail causes the time integral in the GK formula to diverge in the thermodynamic limit ($L \to \infty$), providing a dynamical explanation for the previously observed finite-size scaling of $D$.
4. Empirical Scaling Relation: The study derives a new scaling relation connecting the critical exponents of diffusion, velocity fluctuations, and relaxation time:
   $$\nu + \psi - \epsilon = \lambda$$

4. Key Results

A. Critical Scaling of Mean Squared Velocity ($\langle v_y^2 \rangle$)

• The mean squared transverse velocity follows a critical scaling law: $\langle v_y^2 \rangle \sim |\Delta\phi|^\psi \mathcal{F}(\dot{\gamma}/|\Delta\phi|^\lambda)$.
• Below Jamming ($\phi < \phi_J$): The system is Newtonian; $\langle v_y^2 \rangle \propto \dot{\gamma}^2$ (consistent with constant viscosity).
• Above Jamming ($\phi > \phi_J$): In the quasi-static limit, $\langle v_y^2 \rangle \propto \dot{\gamma}$ (consistent with yield stress behavior).
• Exponents: $\psi \approx 5.39$ and $\lambda \approx 3.87$.

B. Relaxation Time and VACF Behavior

The behavior of the VACF $C(\tau)$ and the relaxation time $\tau^*$ depends on the regime:

• Stretched Exponential Regime: For $\phi < \phi_J$ or $\phi > \phi_J$ with sufficiently high $\dot{\gamma}$, $C(\tau) \approx \exp[-(\tau/\tau^*)^\alpha]$. Here, $\alpha \to 1$ in the Newtonian limit.
• Critical Scaling of $\tau^*$: The relaxation time scales as $\tau^* \sim |\Delta\phi|^\epsilon \mathcal{G}(\dot{\gamma}/|\Delta\phi|^\lambda)$ with $\epsilon \approx 2.57$.
• Long-Time Tail Regime: In the quasi-static regime above jamming ($\phi > \phi_J, \dot{\gamma}t_0 \le 10^{-6}$), the VACF cannot be fitted by a stretched exponential. Instead, it exhibits a power-law decay:
  $$C(\tau) \sim (\dot{\gamma}\tau)^{-1}$$
  This "long-time tail" implies that the integral $\int_0^\infty C(\tau) d\tau$ diverges.

C. The Reduced GK Formula and Scaling Relation

• When $C(\tau)$ is a stretched exponential, the GK formula reduces to:
  $$D = \frac{\tau^* \langle v_y^2 \rangle}{\alpha} \Gamma\left(\frac{1}{\alpha}\right)$$
  The authors verify this relation numerically, showing excellent agreement between $D$ calculated from Mean Squared Displacement (MSD) and the reduced GK formula.
• By substituting the critical scalings of $D$, $\langle v_y^2 \rangle$, and $\tau^*$ into this reduced formula, they derive the scaling relation:
  $$\nu + \psi - \epsilon = \lambda$$
  Using their estimated values ($\nu \approx 0.939, \psi \approx 5.39, \epsilon \approx 2.57, \lambda \approx 3.87$), the relation holds ($3.76 \approx 3.87$), confirming the internal consistency of the critical exponents.

D. Divergence of Diffusion Coefficient

• In the quasi-static regime above jamming, the power-law decay $C(\tau) \sim \tau^{-1}$ leads to a logarithmic divergence of the GK integral.
• This explains why numerical simulations show $D/\dot{\gamma} \propto L^\beta$ (with $\beta \ge 1$): the diffusion coefficient diverges in the thermodynamic limit due to the long-time tail, a phenomenon analogous to the breakdown of the GK relation in 2D classical fluids.

5. Significance

• Theoretical Unification: The work unifies the understanding of shear-induced diffusion by connecting spatial scaling (length scales) with temporal scaling (relaxation dynamics). It demonstrates that the "finite-size effects" observed in previous studies are actually manifestations of long-time tails in the velocity correlations.
• Dimensionality Dependence: The findings suggest that the divergence of $D$ in 2D jammed systems is a direct consequence of the long-time tail, similar to classical fluids, whereas 3D systems may not exhibit this divergence (as noted in recent literature).
• Predictive Power: The derived scaling relation ($\nu + \psi - \epsilon = \lambda$) provides a rigorous constraint for future theoretical models and simulations of jammed matter.
• Methodological Advance: By validating the reduced GK formula for athermal systems, the paper offers a robust method to extract diffusion coefficients from velocity correlations, provided the system is not in the singular quasi-static jammed regime.

In conclusion, Saitoh and Kawasaki provide a comprehensive dynamical explanation for shear-induced diffusion near jamming, identifying the transition from stretched exponential relaxation to power-law long-time tails as the critical mechanism driving the divergence of the diffusion coefficient in two-dimensional athermal systems.