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Kinetic derivation of thermal viscous models for nematic liquid crystal dynamics

This paper derives a macroscopic thermodynamic theory for nematic liquid crystals by applying Chapman-Enskog expansions and constrained maximization to a kinetic model with BGK-type collisions, thereby generalizing existing inviscid theories to include viscous, thermal, and spin-diffusive effects for both compressible and incompressible fluids.

Original authors: P. E. Farrell, J. Málek, O. Souček, U. Zerbinati

Published 2026-07-01

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CC BY 4.0

Original authors: P. E. Farrell, J. Málek, O. Souček, U. Zerbinati

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 box full of tiny, rigid sticks (like matchsticks) floating in a fluid. Sometimes, these sticks float randomly in every direction (like matches shaken in a box). Other times, they suddenly line up and point in the same direction, creating a "liquid crystal" state. This paper is a mathematical recipe for predicting exactly how these sticks move, rotate, and interact with heat and pressure, especially when they are starting to line up.

Here is the story of the paper, broken down into simple concepts:

1. The Starting Point: The "Microscopic" View

The authors start by looking at the fluid from the perspective of individual molecules. They use a mathematical tool called Kinetic Theory, which is like tracking the position and speed of every single matchstick in the box.

The Problem: Tracking billions of sticks is impossible.
The Solution: They use a "collision operator" (a rule for how sticks bump into each other). Instead of calculating every tiny bump, they use a simplified rule called the BGK operator. Think of this as a "relaxation rule": if the sticks are out of alignment, this rule gently nudges them back toward a comfortable, average state, much like a crowd of people naturally finding a rhythm when walking together.

2. The Big Leap: From Micro to Macro

The paper's main achievement is translating the chaotic dance of individual sticks into smooth, large-scale laws (like the laws of fluid dynamics you might know from weather forecasts).

The Method: They use a technique called Chapman–Enskog expansion. Imagine zooming out from a high-resolution photo of a crowd to a blurry view where you only see the flow of the crowd. This method allows them to derive equations for mass, momentum, energy, and entropy (disorder) from the microscopic rules.
The Twist: Previous models assumed the sticks were already lined up. This paper is special because it explains how they get lined up in the first place. It accounts for the "emergent" behavior where the sticks spontaneously decide to align due to their interactions, rather than being forced to do so.

3. The "Traffic Cop": Entropy Maximization

Once they have the equations for how the fluid moves, they need to figure out the specific "constitutive relations"—the rules that tell us how the fluid resists flow (viscosity) or conducts heat.

The Strategy: They use a principle called Maximization of Entropy Production.
The Analogy: Imagine a river flowing down a hill. The water wants to get to the bottom as fast as possible, but it has to deal with rocks and friction. Nature seems to choose the path that creates the most "friction" (entropy) in the most efficient way possible. The authors use this idea to mathematically "lock in" the correct formulas for how the liquid crystal flows and conducts heat.

4. The New Ingredients: Viscosity, Heat, and Spin

The paper generalizes an older, simpler model (which only worked for "inviscid" or frictionless fluids) to include real-world effects:

Viscosity (Friction): The fluid resists flow. The paper derives a specific formula for this resistance based on the stick alignment.
Heat Conduction: The fluid carries heat. The paper shows how the alignment of the sticks affects how heat moves through the fluid.
Spin Diffusion: This is a unique feature of liquid crystals. Because the sticks have a direction, they can "spin" or rotate relative to the flow. The paper derives a new rule for how this spinning motion spreads out and slows down, similar to how a spinning top eventually wobbles and stops.

5. The Result: A Complete "Game Plan"

The authors end up with a complete set of equations (a "closed system") that can describe:

Compressible fluids: Where the fluid can be squeezed (like a gas).
Incompressible fluids: Where the fluid cannot be squeezed (like water).

These equations are a more advanced version of the famous Leslie–Ericksen equations used to model liquid crystals. The new version is "thermodynamically consistent," meaning it strictly obeys the laws of physics regarding energy and heat.

Summary in a Nutshell

Think of this paper as upgrading the instruction manual for a complex fluid.

Old Manual: "Assume the sticks are already lined up. Ignore friction and heat."
New Manual (This Paper): "Here is exactly how the sticks decide to line up on their own. Here is how they rub against each other (friction), how they carry heat, and how their spinning motion slows down. We derived these rules from the ground up, starting with the physics of individual collisions."

The paper does not claim to solve specific real-world engineering problems (like making better screens) or medical applications. Instead, it provides the rigorous mathematical foundation and the "source code" for how these complex fluids behave, ensuring that any future simulations or models built on this work will be physically accurate.

Technical Summary: Kinetic Derivation of Thermal Viscous Models for Nematic Liquid Crystal Dynamics

Problem Statement
Recent work established a compressible, inviscid variant of the Leslie–Ericksen equations for nematic liquid crystals derived from kinetic theory. However, that approach relied on a moment closure around an already-aligned state, effectively imposing collective orientational behavior rather than allowing it to emerge from the dynamics. Furthermore, the resulting model was strictly inviscid, failing to capture the dissipative, thermal, and spin-diffusive effects inherent in real liquid crystal flows. This paper addresses these limitations by deriving a compressible viscous variant of the Leslie–Ericksen equations that accounts for the emergence of alignment and includes thermal and viscous effects.

Methodology
The authors develop a macroscopic thermodynamic theory starting from a kinetic theory of ordered fluids. The methodology proceeds through the following stages:

Kinetic Framework: The starting point is the kinetic theory of ordered fluids proposed in [7], which incorporates mean-field alignment interactions via a Vlasov potential. The system assumes a separation of time scales where orientational relaxation occurs faster than translational momentum relaxation. The distribution function $f(x, p, \nu, \varsigma, t)$ depends on position, linear momentum, microstructural configuration (orientation $\nu$ ), and conjugate momentum $\varsigma$ .
Collision Operator: To make the analysis tractable, the full Boltzmann collision operator is replaced by a single-relaxation Bhatnagar–Gross–Krook (BGK) operator. This operator preserves the conservation laws and the entropy production structure of the full kinetic theory.
Balance Laws: The authors derive the macroscopic balance equations for mass, linear momentum, angular momentum, energy, and entropy by taking moments of the kinetic equation. A key feature is the identification of a mean-field torque term in the angular momentum balance, which drives the emergence of alignment.
Thermodynamic Structure: The Helmholtz free energy is identified as the thermodynamic potential. The authors prove that the equilibrium distribution (Maxwellian) minimizes this free energy and derive the associated Gibbs distribution for the orientational profile. They demonstrate that under specific interaction potentials (Kuramoto and Maier–Saupe types), the system undergoes a phase transition from an isotropic to a nematic state as temperature decreases, with the Vlasov torque vanishing at the aligned equilibrium.
Chapman–Enskog Expansion: A first-order Chapman–Enskog expansion is performed around the local Maxwellian equilibrium. Unlike standard derivations, the time derivatives in the expansion are substituted using the zeroth-order balance laws (which include Ericksen stress and couple stress terms), rather than the standard compressible Euler equations. This ensures that the orientational elasticity propagates correctly into the first-order corrections.
Constitutive Closure: The authors employ the constrained maximisation principle of Rajagopal and Srinivasa. They combine the entropy production rate derived from the kinetic expansion with the thermodynamic structure inferred from the Helmholtz free energy. By maximizing the entropy production subject to the balance laws, they derive constitutive relations for the Cauchy stress, couple-stress, and heat flux.

Key Contributions and Results

Emergent Alignment: Unlike previous inviscid models that assumed alignment, this derivation allows the isotropic-to-nematic transition to emerge naturally from the kinetic dynamics via the Vlasov torque. The analysis confirms that the mean-field torque vanishes at the aligned Gibbs equilibrium, ensuring consistency with the macroscopic balance laws.
Viscous and Thermal Effects: The paper derives a complete set of constitutive equations for a compressible, viscous, heat-conducting, and spin-diffusive nematic fluid. The resulting stress tensor includes:
- A Newtonian shear viscosity.
- An isotropic bulk viscosity arising from translational-rotational energy imbalance.
- An anisotropic viscous stress coupled to the Ericksen tensor and the velocity gradient.
- A couple-stress tensor that includes spin-diffusion effects (an Eringen-type micropolar term) and coupling to the director gradient.
Transport Coefficients: Explicit expressions for the transport coefficients (shear viscosity, thermal conductivity, bulk viscosity, anisotropic viscosity, and spin viscosity) are derived in terms of the BGK relaxation time $\tau$ and the Frank modulus $K$ . The thermal conductivity satisfies the Eucken relation for diatomic gases.
Incompressible Limit: The framework naturally yields the incompressible counterpart by enforcing a divergence-free constraint via a Lagrange multiplier during the maximisation procedure, without requiring separate derivation.
Structural Novelty: The derived viscous stress tensor is governed by a single viscosity parameter (due to the isotropic BGK approximation) and does not recover the full five-coefficient Leslie viscous stress tensor. The authors note that recovering the full Leslie tensor would require an anisotropic BGK operator with multiple relaxation timescales, which is a subject of ongoing work.

Significance and Claims
The paper claims to generalise the recent inviscid kinetic theory of Farrell, Russo, and Zerbinati [1] to account for viscous, thermal, and spin-diffusive effects. The primary significance lies in providing a thermodynamically consistent derivation of these dissipative effects directly from kinetic theory, bridging the gap between microscopic particle interactions and macroscopic continuum mechanics for ordered fluids.

The authors emphasize that their approach overcomes the limitation of previous models by not imposing alignment a priori; instead, the alignment emerges from the minimisation of the free energy and the vanishing of the Vlasov torque at equilibrium. The resulting system (Equation 8.21) represents a compressible, viscous, heat-conducting, and spin-diffusive variant of the Leslie–Ericksen equations. In the inviscid limit ( $\tau \to 0$ ), the model reduces to the inviscid system derived in [1].

The work is presented as a step towards a more complete kinetic theory of liquid crystals, with the authors modestly noting that the recovery of the full Leslie viscous stress tensor with its five independent viscosities remains an open problem requiring a more complex, anisotropic collision operator.