🔬 condensed matter

A Smoluchowski equation for a sheared suspension of frictionally interacting rods

This paper develops a Smoluchowski equation and stress tensor for dense, sheared suspensions of frictionally interacting rods by applying Doi's Onsager variational method to incorporate both solid and lubricated friction, while also proposing a generalized model for the average number of inter-rod contacts.

Original authors: Genevieve Quiñones, Peter D. Olmsted

Published 2026-07-20

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Genevieve Quiñones, Peter D. Olmsted

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a world where the stuff we drink, paint, or print isn't just a simple liquid, but a chaotic dance of tiny, stiff sticks floating in a fluid. This is the realm of suspensions, a branch of physics that studies how particles move when mixed into a liquid. For over a century, scientists have understood how these sticks behave when they are far apart, drifting lazily like leaves in a gentle stream. They also know how they act when they are crowded but still sliding past each other on a thin layer of slippery fluid, like ice skaters on a frozen pond. But there is a third, messier scenario: what happens when the crowd gets so dense that the "ice" breaks? When the sticks crash into each other, lose their slippery coating, and start grinding against one another with friction? This is the question that drives modern rheology (the study of flow). Understanding this is crucial because it explains why some materials suddenly turn from a runny liquid into a solid block when you stir them too fast, a phenomenon that can clog industrial pipes or jam 3D printers.

In this paper, Genevieve Quiñones and Peter D. Olmsted tackle the messy math of these frictional crashes. They build a new set of rules, called a Smoluchowski equation, to predict how a dense crowd of frictional rods behaves when sheared (stirred). Think of it as upgrading a traffic simulation from a model where cars only drive on smooth, frictionless roads to one where they can actually skid, grind, and lock up. The authors suggest that when these rods get crowded, they don't just slide; they sometimes stick and scrape, creating a "frictional torque" that changes how they rotate. Their work suggests that this friction acts like a brake, suppressing the wild, tumbling motions the rods usually make and forcing them to line up more neatly with the flow. They also propose a new way to count how many times a rod bumps into its neighbors, finding that the more orderly the crowd is, the fewer bumps occur. While they haven't built a physical machine to prove this yet, their computer simulations show that adding friction changes the dance steps of the rods, keeping them from spinning out of control and potentially explaining why some thick fluids suddenly get even thicker when you push them harder.

The Story of the Grinding Sticks

Imagine you are at a crowded dance floor. If the room is spacious, everyone can spin and twirl freely. This is like a dilute suspension of rods, where the particles are far apart and only bump into each other occasionally. Physicists have known for a long time how to predict this dance; they use equations (like Jeffery's equation) that describe how a single stick rotates in a flowing liquid.

Now, imagine the dance floor gets packed. The dancers are no longer just spinning; they are bumping into each other. In a semi-dilute crowd, they might still slide past one another because of a thin layer of fluid (like sweat or oil) acting as a lubricant. But in a concentrated crowd, that lubricant layer can break. Suddenly, the dancers aren't sliding; they are grinding their shoes against each other. This is solid friction.

For a long time, scientists struggled to write the math for this grinding. Previous attempts assumed that even when rods were grinding, they were still following the same "dance steps" (Jeffery's equation) as if they were in a sparse crowd, just with an extra "push" added on. But when the authors of this paper tried to calculate the average effect of these pushes, they found the math canceled out to zero. It was as if the friction was invisible. This didn't make sense to them, because we know from real life that friction slows things down and changes how they move.

The New Approach: The "Energy Bill"

Quiñones and Olmsted decided to try a different strategy. Instead of guessing how the rods move and then adding friction, they started with the energy. They used a method called the Rayleighian, which is essentially a way of calculating the "energy bill" for the whole system.

Think of the Rayleighian as a total cost calculator. It adds up two things:

The Free Energy: How much the rods want to arrange themselves in a certain way (like magnets aligning).
The Dissipation: How much energy is lost to heat due to friction and drag.

The authors realized that to find the true motion of the rods, you have to minimize this total "energy bill." When they included the energy lost to solid friction (the grinding) and boundary lubricated friction (the sliding with a thin film) in this bill, something magical happened. The math naturally produced a new "torque" (a twisting force) that wasn't zero.

They found that friction acts like a rotational brake. When the rods grind against each other, it becomes harder for them to spin. This changes the Smoluchowski equation, which is the master equation that predicts how the orientation of all the rods changes over time.

What They Found: The Dance Changes

The authors ran computer simulations to see what happens when you stir these frictional rods. Here is what they discovered:

The "Kayaking" Stops: In a frictionless crowd, the rods often perform a move called "kayaking," where they tilt and spin in a complex, out-of-plane orbit. The authors found that when friction is turned on, this wild spinning is suppressed. The rods stop kayaking and instead tend to stay aligned with the flow, like a school of fish swimming in a straight line.
The "Jamming" Connection: They suggest that this frictional braking is a key ingredient in Discontinuous Shear Thickening (DST). This is the weird phenomenon where a liquid (like cornstarch and water) suddenly turns into a solid when you hit it hard. The authors' model suggests that as you stir faster, the friction between rods increases, the "brakes" lock up, and the material jams.
Counting the Bumps: They also came up with a new formula to estimate how many times a rod touches its neighbors. They found that the number of contacts depends on how ordered the crowd is. If the rods are perfectly aligned (high order), they touch fewer neighbors. If they are messy and disordered, they touch more. This is a generalization of a previous idea that only worked for messy, disordered crowds.

The Limits of the Model

It is important to note what this paper does not claim. The authors are careful to say that their model assumes the rods are always sliding past each other. They cannot yet model the moment when rods get so stuck that they form a giant, unmoving cluster (static friction). In the real world, these stuck clusters are likely the main reason materials jam completely. The authors admit their model misses this "stuck" part because static friction doesn't generate heat (dissipation), so it doesn't show up in their energy bill.

Furthermore, their results are based on simulations and mathematical derivations, not on new physical experiments. They suggest that their findings explain why friction matters, but they haven't measured it in a lab yet. They also assume the suspension is perfectly mixed and uniform, ignoring the possibility that friction might cause the rods to clump together in specific spots.

The Takeaway

In simple terms, Quiñones and Olmsted have built a new mathematical toolkit that finally accounts for the "grind" between particles in a thick fluid. They showed that if you ignore friction, you miss the brakes that stop the rods from spinning wildly. By including this friction, they found a self-consistent way to predict how these crowded rods move and how much stress they put on the container holding them. While they haven't solved the mystery of why fluids jam completely (that requires understanding the "stuck" state), they have provided a crucial piece of the puzzle: the frictional torque that slows the dance down before the jam happens.

Technical Summary: A Smoluchowski Equation for a Sheared Suspension of Frictionally Interacting Rods

Problem Statement
The rheology of dense suspensions of rigid rod-like particles under shear is not fully captured by existing theories, particularly in regimes exhibiting discontinuous shear thickening (DST), shear jamming, and vorticity tilting. While classical models (e.g., Jeffery, Doi, Advani-Tucker) successfully describe orientational dynamics in dilute and semi-dilute regimes via hydrodynamic interactions and excluded volume effects, they fail to account for solid inter-particle contacts. Recent experimental and computational evidence suggests that frictional contacts between rods are the primary driver of these unexplained rheological behaviors. However, previous attempts to incorporate friction into rod dynamics (e.g., Sandstrom & Tucker, Bounoua et al.) yielded null results for the average frictional torque, largely because they assumed the rods' motion followed Jeffery's equation and simply added a torque term, leading to cancellations upon averaging.

Methodology
The authors develop a constitutive framework for dense, sheared suspensions of frictionally interacting rods by applying Onsager's variational method as formulated by Doi. The core of the methodology involves constructing a Rayleighian ( $R$ ), defined as the sum of the rate of change of free energy ( $\dot{A}$ ) and a dissipation function ( $\Phi$ ) dependent on the rod angular velocity ( $\omega$ ).

Dissipation Functions: The authors derive dissipation functions for two distinct frictional mechanisms:
- Boundary Lubricated (BL) Friction: Scales linearly with the relative tangential contact velocity ( $v_c$ ), modeled as $F \propto v_c$ .
- Solid Amontons-Coulomb Friction: Scales with the magnitude of the normal force ( $|F_N|$ ) and depends only on the direction of $v_c$ , modeled as $F \propto |F_N|$ .
Mean Field Approach: To handle the many-body nature of contacts, the authors employ a mean field approximation. They derive an expression for the average number of contacts per rod, $\langle c \rangle$ , as a function of volume fraction ( $\phi$ ), aspect ratio ( $L/D$ ), and the nematic order parameter ( $S$ ). This generalizes Philipse's random contact equation for disordered packings to include orientational order.
Variational Derivation: Unlike previous approaches that assumed a pre-determined equation of motion, the authors minimize the total Rayleighian with respect to $\omega$ . This yields a self-consistent equation of motion where the frictional torque is naturally a function of the angular velocity itself.
Smoluchowski Equation and Stress: The minimization of the Rayleighian produces a new Smoluchowski equation for the orientational distribution function $\psi(\hat{u})$ and a corresponding stress tensor $\sigma$ . The model incorporates a switching function $\Theta(p)$ to transition between lubricated and solid friction regimes based on the reduced pressure.

Key Contributions

Non-Zero Frictional Torque: The paper demonstrates that by deriving the dynamics from a dissipation function rather than adding a torque to Jeffery's equation, a non-zero average frictional torque is obtained. This resolves the "null result" found in prior literature (e.g., Bounoua et al.) which arose from averaging odd powers of contact position variables.
Generalized Contact Number: The authors propose a form for the average number of contacts $\langle c \rangle$ that depends on the nematic order parameter $S$ , specifically $\langle c \rangle \approx \phi (L/D) (1 - S^2)$ . This links the microstructure of contacts directly to the macroscopic orientational order.
Self-Consistent Dynamics: The resulting Smoluchowski equation is self-consistent; the effective rotational drag coefficient depends on the contact velocity, which in turn depends on the angular velocity. This non-linearity is inherent to the Coulomb friction model.
Stress Tensor Formulation: A new stress tensor is derived that includes contributions from both hydrodynamic drag and frictional dissipation, with the frictional component scaling with the average frictional torque.

Results and Numerical Study
The authors perform a numerical study of the derived Smoluchowski equation (approximated via a truncated series expansion and Langevin dynamics) for a simple shear flow.

Lubricated Limit: In the absence of solid friction, the model reduces to the Doi-Edwards theory with a renormalized rotational drag constant, effectively slowing rod rotation.
Solid Friction Limit: When solid friction is active ( $\mu_k = 1$ $μ_{k} = 1$ ), the dynamics change significantly. The model predicts a suppression of director motion in the flow-gradient plane.
- Flow Alignment: Unlike the frictionless case where the director may exhibit "kayaking" (out-of-plane tumbling) or periodic orbits at intermediate Peclet numbers, the inclusion of friction stabilizes the director in a flow-aligned state.
- Initial Conditions: The frictional model shows that even starting from an isotropic distribution, the system tends to align with the flow and remain there, with only sporadic excursions into kayaking orbits.
- Vorticity Tilt: At low Peclet numbers, the vorticity component of the director drifts near zero, contrasting with the erratic wandering seen in the frictionless case.

Significance
The paper claims to provide a theoretical description of dense rod suspensions that explicitly accounts for frictional contacts, a factor previously neglected or treated incorrectly in constitutive models. By utilizing Onsager's variational principle, the authors establish a self-consistent framework where frictional dissipation naturally generates the necessary torques to alter orientational dynamics. The work suggests that frictional contacts are a critical mechanism for the suppression of complex orientational motions (like kayaking) and may be responsible for the onset of shear thickening and jamming in dense rod suspensions. The authors note that while their model assumes spatial homogeneity and sliding friction (excluding static sticking clusters), it offers a rigorous foundation for understanding the rheological anomalies observed in concentrated rod suspensions.

The Story of the Grinding Sticks

The New Approach: The "Energy Bill"

What They Found: The Dance Changes

The Limits of the Model

The Takeaway

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