Chiral Lennard-Jones Fluid Model

• The chiral Lennard-Jones fluid model is a two-dimensional system of Brownian disks interacting via standard Lennard-Jones potentials combined with non-conservative transverse forces.
• It quantifies phase coexistence and interface properties through methods like the Irving–Kirkwood stress tensor and Maxwell equal-area construction, revealing enhanced surface tension and persistent edge currents.
• The model demonstrates that applied chirality disrupts crystalline order, leading to melting into dynamic hexatic patches with measurable rotational viscosity and fluid-like behavior.

The chiral Lennard-Jones (LJ) fluid model is a two-dimensional statistical mechanical framework for assemblies of Brownian disks that interact via both standard LJ potentials and a non-conservative, pairwise transverse (chiral) force, designed to mimic the collective behavior of colloidal particles rotating at a prescribed angular speed. This model exhibits a rich phase diagram, including gas-liquid phase separation, interface-driven edge currents, and pronounced chirality-induced melting of solid phases, thus providing a systematic theoretical and computational foundation for understanding chiral particle ensembles (Caporusso et al., 2023).

1. Model Definition

The system is composed of $N$ two-dimensional Brownian disks, each of diameter $\sigma_d$ and mass $m$. The particles' interactions are governed by two core contributions:

• Lennard-Jones Potential:

The interaction is truncated at $r_c = 5\sigma_d$ and shifted by $\epsilon$ for continuity:

$$U(r) = \epsilon \left[ \left( \frac{\sigma_d}{r} \right)^{12} - \left( \frac{\sigma_d}{r} \right)^6 \right] + \epsilon, \quad r < r_c$$

• Transverse (“Chiral”) Force:

Each pair within $r_c$ experiences a short-range, non-conservative force:

$$\mathbf{f}_{ij} = \zeta \left( \frac{\omega}{\omega_0} \right) \hat{z} \times \frac{(\mathbf{r}_i - \mathbf{r}_j)} {|\mathbf{r}_i - \mathbf{r}_j|^3}$$

where $\omega$ is the spin rate, $\hat{z}$ is normal to the plane, $\sigma_d$0 sets the unit of time with $\sigma_d$1, and $\sigma_d$2.

• Overdamped Langevin Dynamics:

In the overdamped regime ($\sigma_d$3), the equations of motion are:

$\sigma_d$4

where $\sigma_d$5 is the friction coefficient and $\sigma_d$6 is Gaussian white noise with standard correlations.

Key Control Parameters:

Parameter	Definition	Meaning
$\sigma_d$7	Reduced temperature	Thermal energy scale
$\sigma_d$8	Surface fraction	Packing density
$\sigma_d$9	Dimensionless chirality	Chirality strength
$m$0	Rotational Péclet number ($m$1)	Chirality vs. Brownian diffusion
$m$2	Transverse-to-conservative force ratio	Relative non-conservative forcing

2. Thermodynamic Framework and Phase Coexistence

Despite explicit violation of time-reversal and parity by the transverse force, the system admits a well-defined mechanical equation of state. The pressure is computed via the Irving–Kirkwood (IK) stress tensor:

$m$3

The scalar pressure is $m$4.

Simulation yields $m$5 curves with pronounced van der Waals–type (“Mayer–Wood”) loops. Coexistence (binodal) densities and pressures can be quantitatively located via the Maxwell equal-area construction, with the rule:

$m$6

This construction, although equilibrium-based, remains applicable to the out-of-equilibrium chiral system and matches direct histogram-based density measurements.

For finite-size systems, the interface free-energy excess $m$7 shifts the area of the pressure loop:

$m$8

where $m$9 denotes the surface tension and $r_c = 5\sigma_d$0 the interface length. Increasing chirality ($r_c = 5\sigma_d$1) yields a larger $r_c = 5\sigma_d$2 and thus a higher $r_c = 5\sigma_d$3.

3. Surface Tension and Interface Structure

Surface tension is characterized using spatially resolved stress tensor components for a slab with gas–liquid interface along $r_c = 5\sigma_d$4:

$r_c = 5\sigma_d$5

Numerical evaluation shows that $r_c = 5\sigma_d$6 increases monotonically with chirality. Edge currents, generated by the transverse force, enhance tangential momentum transfer at the interface, amplifying $r_c = 5\sigma_d$7 and thus the normal–tangential stress imbalance.

These features indicate that the non-conservative chiral drive not only shifts thermodynamic coexistence but also fundamentally alters interfacial mechanics, resulting in stiffer interfaces and novel flow behavior.

4. Edge Currents and Rotational Viscosity

At liquid-gas coexistence, the interface sustains persistent edge currents: droplets or strips of chiral liquid display a unidirectional flow along their perimeters. The radial profile of the azimuthal velocity $r_c = 5\sigma_d$8 near the edge follows

$r_c = 5\sigma_d$9

where $\epsilon$0 is the droplet radius and $\epsilon$1 is the edge current penetration depth; here, $\epsilon$2 is shear viscosity, $\epsilon$3 is rotational viscosity, and $\epsilon$4 is substrate friction. The edge velocity magnitude is

$\epsilon$5

Rotational viscosity $\epsilon$6 is central to emergent chiral hydrodynamics. It is determined microscopically either by inverting the steady-state velocity profile,

$\epsilon$7

or directly from the stress tensor in a homogeneous chiral liquid,

$\epsilon$8

Both estimates yield consistent values within approximately $\epsilon$9.

5. Chirality-Induced Melting and Hexatic Patch Formation

At reduced temperatures $$U(r) = \epsilon \left[ \left( \frac{\sigma_d}{r} \right)^{12} - \left( \frac{\sigma_d}{r} \right)^6 \right] + \epsilon, \quad r < r_c$$0, the equilibrium 2D LJ system forms a solid with quasi–long-range orientational (hexatic) order. Introduction of the chiral force induces internal stresses that disrupt global crystalline order above a critical chirality $$U(r) = \epsilon \left[ \left( \frac{\sigma_d}{r} \right)^{12} - \left( \frac{\sigma_d}{r} \right)^6 \right] + \epsilon, \quad r < r_c$$1. The local hexatic order parameter is

$$U(r) = \epsilon \left[ \left( \frac{\sigma_d}{r} \right)^{12} - \left( \frac{\sigma_d}{r} \right)^6 \right] + \epsilon, \quad r < r_c$$2

where the sum is over nearest neighbors. The global and spatial correlations are:

$$U(r) = \epsilon \left[ \left( \frac{\sigma_d}{r} \right)^{12} - \left( \frac{\sigma_d}{r} \right)^6 \right] + \epsilon, \quad r < r_c$$3

In the chiral solid, $$U(r) = \epsilon \left[ \left( \frac{\sigma_d}{r} \right)^{12} - \left( \frac{\sigma_d}{r} \right)^6 \right] + \epsilon, \quad r < r_c$$4 decays exponentially, $$U(r) = \epsilon \left[ \left( \frac{\sigma_d}{r} \right)^{12} - \left( \frac{\sigma_d}{r} \right)^6 \right] + \epsilon, \quad r < r_c$$5, with hexatic correlation length $$U(r) = \epsilon \left[ \left( \frac{\sigma_d}{r} \right)^{12} - \left( \frac{\sigma_d}{r} \right)^6 \right] + \epsilon, \quad r < r_c$$6. This $$U(r) = \epsilon \left[ \left( \frac{\sigma_d}{r} \right)^{12} - \left( \frac{\sigma_d}{r} \right)^6 \right] + \epsilon, \quad r < r_c$$7-dependence indicates that stronger spinning fragments the solid into finer “hexatic patches,” each retaining local sixfold order but exhibiting swirling dynamics.

Defects localize at the boundaries of these rotating patches, and particle displacements predominantly occur along domain walls, generating a mosaic of dynamically rotating hexatic domains that fluidize the dense phase without loss of local orientational order.

6. Theoretical Significance and Outlook

The chiral Lennard-Jones fluid model provides a statistically robust framework for non-equilibrium chiral fluids, revealing that:

• Phase coexistence can be captured by equilibrium-inspired thermodynamic constructs augmented by interface stress contributions.
• Surface tension increases monotonically with applied chirality due to persistent edge currents driven by non-conservative transverse forces.
• The emergence of rotational viscosity $$U(r) = \epsilon \left[ \left( \frac{\sigma_d}{r} \right)^{12} - \left( \frac{\sigma_d}{r} \right)^6 \right] + \epsilon, \quad r < r_c$$8 is intrinsic and quantifiable both at the hydrodynamic and microscopic levels.
• Chirality melts otherwise stable 2D solids into dynamic hexatic patchworks, with the characteristic domain size set by the inverse of chirality.

These insights lay the foundation for advanced theoretical treatment and simulation of chiral particle assemblies, bridging equilibrium and non-equilibrium statistical mechanics in active matter and soft condensed phases (Caporusso et al., 2023).