Discontinuous change of viscosity in a sheared granular gas with velocity-dependent restitution

This study demonstrates that a sheared granular gas with velocity-dependent restitution exhibits a discontinuous, S-shaped transition in shear viscosity between low- and high-shear Bagnold-type regimes, driven purely by kinetic effects rather than frictional contacts or jamming.

The Gist

Imagine a giant, invisible ball pit filled with billions of tiny, hard marbles. In a normal ball pit, if you shake it gently, the marbles bounce around and eventually settle down. But in this paper, the authors are studying a special kind of "granular gas" where the marbles don't just bounce; they lose energy when they hit each other, like a basketball that gets a little flatter with every bounce.

Here is the simple story of what they discovered, using some everyday analogies.

1. The Special Rule: The "Bounciness" Switch

Usually, when two objects collide, they lose a bit of energy. Scientists call this the "restitution coefficient" (how bouncy they are).

• Normal marbles: They always lose the same amount of energy, no matter how fast they hit.
• The authors' marbles: These are tricky. They have a speed-dependent bounciness.
  • If they hit slowly, they are less bouncy (they lose a lot of energy).
  • If they hit fast, they are more bouncy (they keep more energy).

Think of it like a door with a spring: If you push the door gently, the spring is stiff and stops you dead (low bounciness). But if you run into the door at full speed, the spring gives way, and you bounce back further (high bounciness).

2. The Experiment: Shaking the Box

The researchers put these special marbles in a box and started shaking it sideways (shearing it) at different speeds. They wanted to see how "thick" or "sticky" the gas felt (its viscosity) as they changed the shaking speed.

In normal fluids (like water or honey), if you stir them faster, they usually get thinner (shear thinning) or stay the same. But these marbles did something weird.

3. The Discovery: The "S-Shape" Surprise

When the marbles were set up so that slow collisions were very sticky and fast collisions were very bouncy, something magical happened as they increased the shaking speed:

1. Slow Shaking: The marbles are moving slowly, so they hit each other gently. Because slow hits are "sticky," the whole gas feels thick and resistant.
2. Medium Shaking: As they shake it faster, the marbles start hitting harder. Suddenly, the "sticky" rule switches to the "bouncy" rule. But here's the catch: the system gets confused. It's trying to decide whether to act like the slow, sticky gas or the fast, bouncy gas.
3. The Jump: At a certain point, the resistance (viscosity) doesn't just change smoothly. It jumps. The gas suddenly snaps from being "thick" to being "thin" (or vice versa, depending on how you look at it).

If you were to draw a graph of this, it wouldn't be a straight line. It would look like an S. In the middle of that "S," there is a range where three different thicknesses could exist for the exact same shaking speed.

4. The Analogy: The Traffic Jam vs. The Highway

To understand why this is special, imagine traffic:

• Normal Traffic (Dense Suspensions): Usually, traffic jams happen because cars get too close and bump into each other, getting stuck (jamming). This is like the "Wyart-Cates" scenario mentioned in the paper, where friction and crowding cause the jump.
• This Granular Gas (The Paper's Discovery): There are no traffic jams here! The marbles are far apart (a "dilute gas"). They aren't touching or rubbing against each other. The "jump" happens purely because of how they bounce. It's like if a highway suddenly switched from "slow traffic rules" to "fast lane rules" based on how fast your car is going, causing a sudden, chaotic shift in how the whole road flows, even though no cars are actually crashing into each other.

5. Why Does This Matter?

The authors found that this "S-shaped" jump only happens if the difference between the "slow bounciness" and "fast bounciness" is big enough.

• If the marbles are bouncy at both speeds, nothing happens.
• If they are sticky at both speeds, nothing happens.
• The magic happens only when the two states are in direct competition.

The Big Takeaway

This paper shows that you don't need friction, crowding, or jamming to create sudden, dramatic changes in how a material flows. You just need particles that change their "personality" (how much energy they lose) depending on how hard they hit each other.

It's a reminder that even in a simple gas of bouncing balls, if you give them a rule that changes based on their speed, the whole system can behave in a surprisingly complex, "jumping" way. This could help us understand everything from industrial powders to charged particles in space, where different types of energy loss compete with each other.

Technical Summary

1. Problem Statement

Granular flows composed of dissipative particles exhibit complex non-equilibrium behaviors. While the rheology of granular gases with constant restitution coefficients is well-understood (typically following Bagnold scaling where viscosity is proportional to the shear rate), the behavior of systems with velocity-dependent restitution coefficients under steady shear remains largely unexplored.

The specific problem addressed is whether a granular gas, where the energy dissipation during collisions changes based on the relative impact velocity, can exhibit discontinuous rheological transitions (specifically, a jump in viscosity) similar to the "discontinuous shear thickening" (DST) observed in dense suspensions. The authors investigate a model where the restitution coefficient switches between two values ($e_1$ and $e_2$) depending on whether the collision velocity is below or above a threshold ($v_c$).

2. Methodology

The authors employ a combination of kinetic theory and event-driven molecular dynamics (EDMD) simulations.

• System Model: A dilute ($\phi = 0.01$), frictionless granular gas of monodisperse hard spheres.
• Restitution Law: The normal restitution coefficient $e(v_n)$ is defined as a step function:
  $$e(v_n) = e_1 - (e_1 - e_2)\Theta(v_n - v_c)$$
  where $v_n$ is the normal relative velocity and $v_c$ is the threshold.
• Theoretical Framework:
  • The system is analyzed using the Boltzmann equation for a uniformly sheared state.
  • The evolution of the stress tensor (specifically the kinetic contribution) is derived from the second velocity moment of the distribution function.
  • To close the system of equations, Grad's approximation (13-moment approximation) is used, assuming the velocity distribution function is a Maxwellian distribution corrected by stress anisotropy terms.
  • Collision integrals ($\Omega$) are evaluated analytically for the specific step-function restitution law.
• Simulation: EDMD simulations are performed to validate the theoretical predictions, specifically checking the steady-state temperature, anisotropy, and viscosity against the derived constitutive relations.

3. Key Contributions

• Derivation of S-Shaped Constitutive Curves: The study theoretically derives the shear viscosity for a granular gas with velocity-dependent restitution and demonstrates that under specific conditions ($e_1 < e_2$), the viscosity vs. shear rate curve becomes S-shaped.
• Identification of Discontinuous Transition: The authors show that this S-shape implies a region where three viscosity values exist for a single shear rate, leading to a discontinuous jump in viscosity as the shear rate is increased.
• Mechanism Differentiation: The paper establishes that this transition is driven purely by kinetic effects (competition between two dissipation regimes) rather than by frictional contacts, jamming, or hydrodynamic interactions, which are the standard mechanisms for DST in dense suspensions.
• Phase Diagram: A phase diagram in the $(e_1, e_2)$ plane is constructed, identifying the precise conditions (specifically $e_1 < e_2$ with a sufficiently large difference) required for the discontinuous transition to occur.

4. Results

• Bagnold Scaling Limits: In the limits of very low and very high shear rates, the system behaves as a gas with a constant restitution coefficient ($e_1$ for low shear, $e_2$ for high shear). In both regimes, the system follows Bagnold scaling (Temperature $T \propto \dot{\gamma}^2$, Viscosity $\eta \propto \dot{\gamma}$).
• The S-Shaped Curve:
  • When $e_1 > e_2$ (dissipation decreases with velocity), the viscosity curve is monotonic. No discontinuity occurs.
  • When $e_1 < e_2$ (dissipation increases with velocity, e.g., low-velocity collisions are more inelastic), the viscosity exhibits an S-shaped dependence on the shear rate.
  • In the intermediate shear regime, the system transitions between the low-shear branch (dominated by $e_1$) and the high-shear branch (dominated by $e_2$).
• Discontinuous Jump: As the shear rate increases along the lower branch, the slope $\partial \eta / \partial \dot{\gamma}$ diverges at a critical point. The system cannot sustain a stable state beyond this point and jumps discontinuously to the upper branch (higher viscosity).
• Validation: The theoretical curves derived from Grad's approximation show excellent agreement with EDMD simulation results, confirming the accuracy of the kinetic theory approach for this problem.
• Phase Boundary: The discontinuous transition only occurs when the low-velocity restitution coefficient is significantly smaller than the high-velocity one ($e_1 < e_2$). If the difference is too small, the S-shape disappears, and the viscosity remains a single-valued function of the shear rate.

5. Significance

• New Mechanism for Discontinuous Rheology: This work reveals a novel mechanism for discontinuous shear thickening that does not rely on particle friction, jamming, or dense packing. It proves that competing dissipation channels based on collision velocity alone can induce nontrivial rheological transitions.
• Relevance to Complex Systems: The condition $e_1 < e_2$ (higher dissipation at low velocities) is counter-intuitive for dry granular materials but is physically realizable in systems with:
  • Charged particles (electrostatic dissipation).
  • Adhesive particles (van der Waals forces).
  • Suspensions where hydrodynamic lubrication effects modify collision dynamics.
• Minimal Model: The study provides a "minimal framework" to understand how microscopic dissipation rules can lead to macroscopic instability. It suggests that discontinuous rheological responses may be more general than previously thought, arising in dilute systems if the dissipation mechanism is velocity-dependent in the correct manner.
• Theoretical Advancement: It extends the application of kinetic theory to non-trivial constitutive behaviors in granular gases, bridging the gap between simple cooling states and complex sheared rheology.

In conclusion, the paper demonstrates that a granular gas with a specific velocity-dependent restitution law can undergo a discontinuous transition in viscosity, driven by the competition between two distinct Bagnold-type regimes. This finding challenges the notion that discontinuous shear thickening requires dense, frictional, or jammed states.