🔬 materials science

Frenkel line of Yukawa fluids within the self-consistent relaxation theory

This paper utilizes self-consistent relaxation theory to define the Frenkel line in Yukawa fluids as the thermodynamic boundary where the roton minimum in the longitudinal excitation dispersion relation vanishes, demonstrating that this dynamic crossover can be directly determined from the static structure factor and aligns with molecular dynamics simulation results.

Original authors: Ilnaz I. Fairushin, Anatolii V. Mokshin

Published 2026-09-04

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CC BY 4.0

Original authors: Ilnaz I. Fairushin, Anatolii V. Mokshin

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

In the world of matter, there is a state that exists beyond the familiar boundaries of liquid and gas. When a substance is heated and pressurized past a certain critical point, the sharp line that usually separates a boiling liquid from a rising vapor disappears. The result is a supercritical fluid, a unique state where the substance behaves like a hybrid, possessing the density of a liquid but the ability to diffuse like a gas. Scientists have long been interested in mapping this hidden territory, looking for invisible lines that separate different kinds of behavior within this fluid. One such line, known as the Frenkel line, marks a fundamental shift in how the particles inside the fluid move. On one side of this line, the particles are locked in a rhythmic, vibrating dance, moving together in a way that resembles a solid. On the other side, they break free, moving more like a gas where they drift and collide without a coordinated rhythm. Understanding where this line lies is crucial because it tells us when a fluid stops acting like a solid and starts acting like a gas, a transition that governs everything from the behavior of deep-earth minerals to the efficiency of industrial cooling systems.

Researchers Ilnaz Fairushin and Anatolii Mokshin have taken a fresh look at this problem using a specific type of fluid called a Yukawa fluid. This is a theoretical model used to describe systems where particles push each other away with a force that gets weaker as they get farther apart, a behavior seen in things like dusty plasmas. Instead of trying to watch every single particle move in a computer simulation, which is a common but computationally heavy method, the authors used a mathematical framework called self-consistent relaxation theory. This approach allows them to predict how the fluid behaves by looking at its structure—specifically, how the particles are arranged relative to one another. They focused on the sound waves that travel through the fluid, which are actually vibrations of the particles moving together. In a solid or a dense liquid, these waves have a specific pattern: their speed changes as their wavelength changes, creating a curve that dips down to a low point before rising again. This dip is called a roton minimum.

The team discovered that this roton minimum acts as a reliable signpost for the Frenkel line. When the fluid is in the solid-like regime, where particles are tightly packed and vibrating together, this dip in the wave pattern is clearly visible. However, as the fluid becomes hotter or less dense, the particles begin to move more freely and the collective rhythm breaks down. At a specific point, the dip in the wave pattern disappears entirely. The researchers found that the exact moment this roton minimum vanishes corresponds consistently with the Frenkel line. By calculating the conditions under which this dip disappears, they were able to map out the line on the fluid's phase diagram. Their results matched closely with previous studies that used massive computer simulations to track individual particles, confirming that this structural feature is a consistent indicator of the transition, with the numerical procedure showing an error not exceeding 5%.

What makes this finding particularly powerful is that it offers a direct way to identify the Frenkel line using only static data. Usually, determining where this line lies requires complex measurements of how the fluid moves over time. The authors showed that if you know the static structure of the fluid—essentially a snapshot of how the particles are arranged—you can calculate the behavior of the sound waves and predict exactly where the transition happens. They also proposed a physical explanation for what the roton minimum represents. They suggest that the frequency of this minimum is directly tied to how long it takes for a particle to escape its temporary cage of neighbors. When the particles are stuck in a solid-like rhythm, this time is long, and the minimum is present. When they break free, the time becomes short, and the minimum vanishes.

The study also addressed a competing idea that had been proposed recently, which suggested the Frenkel line could be found by looking at the height of the peak in the particle arrangement graph. The authors found that their method, based on the disappearance of the wave dip, produced a different result, noting a significant deviation from the alternative criterion. However, since their work mainly focused on a specific range of fluid states, they concluded that it is difficult to draw a definite conclusion as to how the alternative approach relates to theirs. Their work confirms that the disappearance of the roton minimum is a consistent and measurable marker for the shift from solid-like to gas-like dynamics. By linking the invisible line of the Frenkel transition directly to a visible feature in the fluid's wave patterns, the researchers have provided a clearer, more direct tool for scientists to understand the complex behavior of supercritical fluids. This approach not only validates previous findings but also simplifies the process of mapping these dynamic boundaries, offering a new perspective on how matter changes its nature under extreme conditions.

Technical Summary: Frenkel line of Yukawa fluids within the self-consistent relaxation theory

Problem Statement
The paper addresses the definition and determination of the Frenkel line, a dynamic crossover line in the phase diagram of supercritical fluids. This line separates two distinct regimes of particle dynamics: a solid-like regime characterized by collective vibrational motion (where the Frenkel relaxation time $\tau_R$ exceeds the Debye period $\tau_D$ ) and a gas-like regime dominated by diffusive motion ( $\tau_R < \tau_D$ ). While previous studies have identified this line using molecular dynamics (MD) simulations and various thermodynamic criteria, the authors seek to establish a theoretical framework that defines the Frenkel line directly from structural characteristics, specifically the static structure factor, without relying solely on extensive dynamic simulations. The study focuses on the Yukawa fluid, a model system with purely repulsive interactions that lacks a liquid-vapor phase transition, making it an ideal candidate for studying supercritical dynamics.

Methodology
The authors employ the self-consistent relaxation theory to model the collective dynamics of the Yukawa fluid. The core of the methodology involves the following steps:

Theoretical Framework: The dynamic structure factor $S(k, \omega)$ is derived using a continued fraction expansion of the time correlation function of density fluctuations. This expansion relies on frequency moments $\langle \omega^{(l)}(k) \rangle$ and frequency relaxation parameters $\Delta_n(k)$ .
Approximation Scheme: The theory invokes a reduction to a limited set of dynamical variables by assuming the equalization of time scales for higher-order variables ( $1/\sqrt{\Delta_4} = 1/\sqrt{\Delta_5} = \dots$ ). This allows the dynamic structure factor to be expressed analytically in terms of the first four relaxation parameters ( $\Delta_1$ through $\Delta_4$ ).
Dispersion Relation: Using the derived $S(k, \omega)$ , the authors obtain an analytical expression for the dispersion relation of longitudinal collective excitations, $\omega_L(k)$ . This relation is determined by the static structure factor $S(k)$ and the radial distribution function $g(r)$ .
Numerical Implementation: The static structure factor $S(k)$ is calculated via the Fourier transform of $g(r)$ , where $g(r)$ is obtained from equilibrium molecular dynamics simulations (NVT ensemble, 64,000 particles, LAMMPS package).
Frenkel Line Criterion: The condition for the Frenkel line is defined by the behavior of the $\omega_L(k)$ dispersion curve. Specifically, the line is identified as the set of thermodynamic states $(\Gamma, \kappa)$ where the roton minimum in the dispersion relation disappears. Mathematically, this occurs when the frequency of the local maximum $\omega_L^{(m)}$ equals the frequency of the roton minimum $\omega_L^{(r)}$ (i.e., $\omega_L^{(m)} = \omega_L^{(r)}$ ).

Key Contributions and Results

Structural Determination of the Frenkel Line: The paper demonstrates that the Frenkel line for Yukawa fluids can be determined directly from the static structure factor $S(k)$ . By analyzing the dispersion relation $\omega_L(k)$ derived from $S(k)$ , the authors identify the thermodynamic states where the roton minimum vanishes.
Quantitative Agreement with MD Simulations: The calculated Frenkel line, defined by the condition $\omega_L^{(m)} = \omega_L^{(r)}$ , yields a ratio of coupling parameters $\Gamma_F(\kappa)/\Gamma_m(\kappa) \approx 0.057$ . This result is shown to be consistent with previous MD simulation results, which reported a ratio of approximately 0.05.
Physical Interpretation of Frequencies: The authors propose a physical interpretation for the characteristic frequencies near the Frenkel line:
- The local maximum frequency $\omega_L^{(m)}$ is identified with the Debye frequency ( $1/\tau_D$ ), representing the limit of longitudinal phonon propagation.
- The roton minimum frequency $\omega_L^{(r)}$ is proposed to be directly related to the inverse Frenkel relaxation time ( $1/\tau_R$ ). This is supported by the observation that $\omega_L^{(r)}$ exhibits strong sensitivity to the coupling parameter (temperature), whereas $\omega_L^{(m)}$ is more sensitive to the screening parameter (interaction softness).
Validation of Relaxation Theory: The study confirms that the correlation relations for higher-order relaxation parameters, previously derived for specific ranges of $\Gamma$ and $\kappa$ , remain valid for determining the Frenkel line in broader thermodynamic states.

Significance
The paper claims that its primary significance lies in providing a theoretical method to locate the Frenkel line using only structural data (the static structure factor), bypassing the need for complex dynamic simulations to track relaxation times directly. By linking the disappearance of the roton minimum to the dynamic crossover, the work offers a unified view where structural characteristics dictate dynamic regimes. Furthermore, the proposed relation $\tau_R \approx 1/\omega_L^{(r)}$ offers a new physical interpretation of the roton minimum in classical liquids, suggesting it characterizes the averaged frequency dispersion of local particle cages rather than strict collective excitations. The consistency of these findings with MD results validates the self-consistent relaxation theory as a robust tool for analyzing the dynamic properties of supercritical fluids.

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