Lennard-Jones Fluid Research Articles

Neural Thermodynamic Integration: Free Energies from Energy-Based Diffusion Models.

Thermodynamic integration (TI) offers a rigorous method for estimating free-energy differences by integrating over a sequence of interpolating conformational ensembles. However, TI calculations are computationally expensive and typically limited to coupling a small number of degrees of freedom due to the need to sample numerous intermediate ensembles with sufficient conformational-space overlap. In this work, we propose to perform TI along an alchemical pathway represented by a trainable neural network, which we term Neural TI. Critically, we parametrize a time-dependent Hamiltonian interpolating between the interacting and noninteracting systems and optimize its gradient using a score matching objective. The ability of the resulting energy-based diffusion model to sample all intermediate ensembles allows us to perform TI from a single reference calculation. We apply our method to Lennard-Jones fluids, where we report accurate calculations of the excess chemical potential, demonstrating that Neural TI reproduces the underlying changes in free energy without the need for simulations at interpolating Hamiltonians.

Generalized Rosenfeld–Tarazona scaling and high-density specific heat of simple liquids

The original Rosenfeld–Tarazona (RT) scaling of the excess energy in simple dense fluids predicts a ∝T3/5 thermal correction to the fluid Madelung energy. This implies that the excess isochoric heat capacity scales as Cvex∝T−2/5. Careful examination performed in this paper demonstrates that the exponent −2/5 is not always optimal. For instance, in the Lennard-Jones fluid in some vicinity of the triple point, the exponent −1/3 turns out to be more appropriate. The analysis of the specific heat data in neon, argon, krypton, xenon, and liquid mercury reveals that no single value of the exponent exists, describing all the data simultaneously. Therefore, we propose a generalized RT scaling in the form Cvex∝T−α, where α is a density- and material-dependent adjustable parameter. The question concerning which material properties and parameters affect the exponent α and whether it can be predicted from general physical arguments requires further investigation.

Coarse-Grained MD Simulations of the Capillary Interaction between a Sphere and a Binary Fluid with Truncated Lennard-Jones Potentials.

In atomic force microscopy experiments on fluid samples, a capillary bridge forms between the tip and the fluid, causing an attractive capillary force. Here, we present a computational model of the capillary interaction between a solid sphere and a coarse-grained Lennard-Jones fluid containing 10% antifreeze particles with an enlarged van der Waals radius. The capillary force acting on the sphere is obtained from the displacement of the sphere in a trap potential as the sphere is incrementally approached and then retracted from the fluid. This yields force-distance data similar to that obtained in atomic force microscopy experiments. We use this methodology to study the influence of the cutoff radius of the truncated Lennard-Jones potentials on the capillary force and its temperature dependence. The latter is found to scale with the critical temperature of the system. With the presented approach, the tip-sample interaction can be studied for a wide range of complex fluids, particle shapes, and force-probing schemes.

Determining state points through the radial distribution function of Yukawa fluids at equilibrium

Based on our previous work [Fluid Phase Equilibria, 2023, 567, 113709], we here use the radial distribution function (RDF) to determine the state points (density and temperature) of a fluid under the Yukawa potential at equilibrium. The reduced density and reduced temperature are defined as ρ*=ρσ3 and β*=1/T*=ϵ/kBT, respectively. Through the Molecular Dynamics (MD) simulations, we obtain equilibrium configurations and use these data for building models via two methods. The first method establishes two empirical correlations for each potential considered, one between the heights of the first peaks of the RDFs and state points, as well as the other between the displacements of the first peaks of the RDFs and state points. Through these empirical correlations, we can determine the state points of new Yukawa fluid systems with 100% accuracy. The second method utilizes artificial neural network models to predict state points from the heights and displacements of the first peaks of the RDFs, achieving 100% accuracy when the predicted results are rounded to one decimal place. The success of these methods again demonstrates the feasibility of determining state points solely based on equilibrium configurations, is an extension from the Lennard-Jones fluids to the Yukawa potential related fluids.

Generalization of Biasing Particle Insertion and Deletion for Grand Canonical Monte Carlo Simulation.

A general formulation of biasing particle insertion/deletion for the grand canonical Monte Carlo (GCMC) simulation is presented. The proposed formulation unifies the concepts of the conventional cavity- and configurational-type biasing techniques and modifies a cavity biasing method to strictly satisfy the detailed balance condition. Numerical examples of this modified cavity bias method are presented for Lennard-Jones fluid and hydrogen adsorption on a platinum surface. The results obtained using this method are compared with those obtained using various GCMC methods. Furthermore, a rigorous on-the-fly on-lattice GCMC simulation for surface adsorption is formulated and performed for comparison. The numerical examples clearly demonstrate the efficiency and accuracy of the proposed formulation.

Models to predict configurational adiabats of Lennard-Jones fluids and their transport coefficients.

A comparison is made between three simple approximate formulas for the configurational adiabat (i.e., constant excess entropy, sex) lines in a Lennard-Jones (LJ) fluid, one of which is an analytic formula based on a harmonic approximation, which was derived by Heyes et al. [J. Chem. Phys. 159, 224504 (2023)] (analytic isomorph line, AIL). Another is where the density is normalized by the freezing density at that temperature (freezing isomorph line, FIL). It is found that the AIL formula and the average of the freezing density and the melting density ("FMIL") are configurational adiabats at all densities essentially down to the liquid-vapor binodal. The FIL approximation departs from a configurational adiabat in the vicinity of the liquid-vapor binodal close to the freezing line. The self-diffusion coefficient, D, shear viscosity, ηs, and thermal conductivity, λ, in macroscopic reduced units are essentially constant along the AIL and FMIL at all fluid densities and temperatures, but departures from this trend are found along the FIL at high liquid state densities near the liquid-vapor binodal. This supports growing evidence that for simple model systems with no or few internal degrees of freedom, isodynes are lines of constant excess entropy. It is shown that for the LJ fluid, ηs and D can be predicted accurately by an essentially analytic procedure from the high temperature limiting inverse power fluid values (apart from at very low densities), and this is demonstrated quite well also for the experimental argon viscosity.

Stochastic symplectic reduced-order modeling for model-form uncertainty quantification in molecular dynamics simulations in various statistical ensembles

This work focuses on the representation of model-form uncertainties in molecular dynamics simulations in various statistical ensembles. In prior contributions, the modeling of such uncertainties was formalized and applied to quantify the impact of, and the error generated by, pair-potential selection in the microcanonical ensemble (NVE). In this work, we extend this formulation and present a linear-subspace reduced-order model for the canonical (NVT) and isobaric (NPT) ensembles. The symplectic reduced-order basis is randomized on the tangent space of the Stiefel manifold to provide topological relationships and capture model-form uncertainty. Using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS), we assess the relevance of these stochastic reduced-order atomistic models on canonical problems involving a Lennard-Jones fluid and an argon crystal melt.

Free Energy Evaluation of Cavity Formation in Metastable Liquid Based on Stochastic Thermodynamics.

Nucleation is a fundamental and general process at the initial stage of first-order phase transition. Although various models based on the classical nucleation theory (CNT) have been proposed to explain the energetics and kinetics of nucleation, detailed understanding at nanoscale is still required. Here, in view of the homogeneous bubble nucleation, we focus on cavity formation, in which evaluation of the size dependence of free energy change is the key issue. We propose the application of a formula in stochastic thermodynamics, the Jarzynski equality, for data analysis of molecular dynamics (MD) simulation to evaluate the free energy of cavity formation. As a test case, we performed a series of MD simulations with a Lennard-Jones (LJ) fluid system. By applying an external spherical force field to equilibrated LJ liquid, we evaluated the free energy change during cavity growth as the Jarzynski's ensemble average of required works. A fairly smooth free energy curve was obtained as a function of bubble radius in metastable liquid of mildly negative pressure conditions.

Coordinate versus momentum cuts and effects of collective flow on critical fluctuations

We analyze particle number fluctuations in the crossover region near the critical endpoint of a first-order phase transition by utilizing molecular dynamics simulations of the classical Lennard-Jones fluid. We extend our previous study [V. A. Kuznietsov , ] by incorporating longitudinal collective flow. The scaled variance of particle number distribution inside different coordinate and momentum space acceptances is computed through ensemble averaging and found to agree with earlier results obtained using time averaging, validating the ergodic hypothesis for fluctuation observables. Presence of a sizable collective flow is found to be essential for observing large fluctuations from the critical point in momentum space acceptances. We discuss our findings in the context of heavy-ion collisions. Published by the American Physical Society 2024

A hybrid atomistic-continuum framework for multiscale simulations of boiling

Boiling is a multiscale physics process where the nucleation of vapour bubbles occurs due to molecular-scale interactions between the fluid and a heated wall, but it also depends on the larger-scale hydrodynamics and thermal boundary layers determined by the outer system boundary conditions. Modelling boiling from the nanometre up to the millimetre scales at which bubble departure occurs is not possible via state-of-the-art simulation methods: Molecular Dynamics (MD) simulations can capture nucleation from first principles but are limited to nanometre scales due to their computational cost, whereas computational fluid dynamics (CFD) simulations based on the continuum Navier-Stokes equations cannot capture nucleation. Here, we present a novel multiscale simulation method which merges MD and CFD descriptions into a single modelling framework, where MD resolves the near-wall region where molecular interactions are important, and a CFD solver resolves the bulk flow. We model the progressive heating of a Lennard-Jones fluid via contact with a solid wall until a vapour bubble nucleates in the MD region of the domain and grows by entering in the CFD domain. Our results show that an incompressible CFD flow model based on the Volume Of Fluid method with interphase mass transfer calculated via the Hertz-Knudsen-Schrage equation is sufficient to obtain seamless coupling of phase fraction, velocity and temperature fields, with the hybrid MD-CFD framework yielding bubble dynamics closely matching those of MD alone.

Unmasking quantum effects in the surface thermodynamics of fluid nanodrops.

The focus of our study is an in-depth investigation of the quantum effects associated with the surface tension and other thermodynamic properties of nanoscopic liquid drops. The behavior of drops of quantum Lennard-Jones fluids is investigated with path-integral Monte Carlo simulations, and the test-area method is used to determine the surface tension of the spherical vapor-liquid interface. As the thermal de Broglie wavelength, λB, becomes more significant, the average density of the liquid drop decreases, with the drop becoming mechanically unstable at large wavelengths. As a consequence, the surface tension is found to decrease monotonically with λB, vanishing altogether for dominant quantum interactions. Quantum effects can be significant, leading to values that are notably lower than the classical thermodynamic limit, particularly for smaller drops. For planar interfaces (with infinite periodicity in the direction parallel to the interface), quantum effects are much less significant with the same values of λB but are, nevertheless, consequential for values representative of hydrogen or helium-4 at low temperatures corresponding to vapor-liquid coexistence. Large quantum effects are found for small drops of molecules with quantum interactions corresponding to water, ethane, methanol, and carbon dioxide, even at ambient conditions. The notable decrease in the density and tension has important consequences in reducing the Gibbs free-energy barrier of a nucleating cluster, enhancing the nucleation kinetics of liquid drops and of bubble formation. This implies that drops would form at a much greater rate than is predicted by classical nucleation theory.

Transport properties of 2D Lennard–Jones fluid: effect of particle size polydispersity

ABSTRACT We performed molecular dynamics simulation to investigate the transport coefficients of 2D size polydisperse Lennard–Jones fluids considering two size distributions, namely, Gaussian (G) and Uniform (U). In particular, we investigate how transport coefficients like viscosity (η), self-diffusion (D) and thermal conductivity (κ) are affected by particle size polydispersity (δ). For both systems, it is found that η is a non-monotonic function of δ, i.e. η gradually decreases and then increases as one vary δ across terminal polydispersity δ t . The re-entrant freezing at large δ due to increase fractionation may describe this behaviour. Dynamics of different size particles indicate that small particles tend to be more diffusive and contribute significantly to the collective diffusion according to SE relation. The self-diffusion coefficient obtained from the collective dynamics of the particles and the fluid's viscosity agree well with one another. It is noted that the thermal conductivity, κ, exhibits a decreasing trend with both δ and T as a result of the decreasing correlation in particle arrangement and the increase in defect such as vacancies or interstitials in the structure or composition due to polydispersity. Finally, it is shown that the particle size polydispersity of the developed material may be tuned to obtain the desired κ value.

Effect of distribution shape on the melting transition, local ordering, and dynamics in a model size-polydisperse two-dimensional fluid

Abstract We study the effect of particle size polydispersity (δ) on the melting transition (T *), local ordering, solid–liquid coexistence phase and dynamics of two-dimensional Lennard–Jones fluids up to moderate polydispersity by means of computer simulations. The particle sizes are drawn at random from the Gaussian (G) and uniform (U) distribution functions. For these systems, we further consider two different kinds of particles, viz., particles having the same mass irrespective of size, and in the other case the mass of the particle scales with its size. It is observed that with increasing polydispersity, the value of T * initially increases due to improved packing efficiency (ϕ) followed by a decrease and terminates at δ ≈ 8% (U-system) and 14% (G-system) with no significant difference for both mass types. The interesting observation is that the particular value at which ϕ drops suddenly coincides with the peak of the heat capacity (CP ) curve, indicating a transition. The quantification of local particle ordering through the hexatic order parameter (Q 6), Voronoi construction and pair correlation function reveals that the ordering decreases with increasing δ and T. Furthermore, the solid–liquid coexistence region for the G-system is shown to be comparatively wider in the T–δ plane phase diagram than that for the U system. Finally, the study of dynamics reveals that polydisperse systems relax faster compared to monodisperse systems; however, no significant qualitative differences, depending on the distribution type and mass polydispersity, are observed.

Freezing density scaling of transport coefficients in the Weeks-Chandler-Andersen fluid.

It is shown that the transport coefficients (self-diffusion, shear viscosity, and thermal conductivity) of the Weeks-Chandler-Andersen (WCA) fluid along isotherms exhibit a freezing density scaling (FDS). The functional form of this FDS is essentially the same or closely related to those in the Lennard-Jones fluid, hard-sphere fluid, and some liquefied noble gases. This proves that this FDS represents a quasi-universal corresponding state principle for simple classical fluids with steep interactions. Some related aspects, such as a Stokes-Einstein relation without a hydrodynamic diameter and gas-to-liquid dynamical crossover, are briefly discussed. Simple fitting formulas for the transport coefficients of the dense WCA fluid are suggested.

Entropy approximations for simple fluids.

Liquid state entropy formulas based on configurational probability distributions are examined for Lennard-Jones fluids across a range temperatures and densities. These formulas are based on expansions of the entropy in a series of n-body distribution functions. We focus on two special cases. One, which we term the "perfect gas" series, starts with the entropy of an ideal gas; the other, which we term the "dense liquid" series, removes a many-body contribution from the ideal gas entropy and reallocates it among the subsequent n-body terms. We show that the perfect gas series is most accurate at low density, while the dense liquid series is most accurate at high density. We propose empirical interpolation methods that are capable of connecting the two series and giving consistent predictions in most situations.

Stochastic thermodynamics of Brownian motion in temperature gradient

We study stochastic thermodynamics of a Brownian particle which is subjected to a temperature gradient and is confined by an external potential. We first formulate an over-damped Ito-Langevin theory in terms of local temperature, friction coefficient, and steady state distribution, all of which are experimentally measurable. We then study the associated stochastic thermodynamics theory. We analyze the excess entropy production both at trajectory level and at ensemble level, and derive the Clausius inequality as well as the transient fluctuation theorem (FT). We also use molecular dynamics to simulate a Brownian particle inside a Lennard-Jones fluid and verify the FT. Remarkably we find that the FT remains valid even in the under-damped regime. We explain the possible mechanism underlying this surprising result.

Structure of liquid-vapor interfaces: Perspectives from liquid state theory, large-scale simulations, and potential grazing-incidence x-ray diffraction.

Grazing-incidence x-ray diffraction (GIXRD) is a scattering technique that allows one to characterize the structure of fluid interfaces down to the molecular scale, including the measurement of surface tension and interface roughness. However, the corresponding standard data analysis at nonzero wave numbers has been criticized as to be inconclusive because the scattering intensity is polluted by the unavoidable scattering from the bulk. Here, we overcome this ambiguity by proposing a physically consistent model of the bulk contribution based on a minimal set of assumptions of experimental relevance. To this end, we derive an explicit integral expression for the background scattering, which can be determined numerically from the static structure factors of the coexisting bulk phases as independent input. Concerning the interpretation of GIXRD data inferred from computer simulations, we extend the model to account also for the finite sizes of the bulk phases, which are unavoidable in simulations. The corresponding leading-order correction beyond the dominant contribution to the scattered intensity is revealed by asymptotic analysis, which is characterized by the competition between the linear system size and the x-ray penetration depth in the case of simulations. Specifically, we have calculated the expected GIXRD intensity for scattering at the planar liquid-vapor interface of Lennard-Jones fluids with truncated pair interactions via extensive, high-precision computer simulations. The reported data cover interfacial and bulk properties of fluid states along the whole liquid-vapor coexistence line. A sensitivity analysis shows that our findings are robust with respect to the detailed definition of the mean interface position. We conclude that previous claims of an enhanced surface tension at mesoscopic scales are amenable to unambiguous tests via scattering experiments.

Influence of repulsion on entropy scaling and density scaling of monatomic fluids.

Entropy scaling is applied to the shear viscosity, self-diffusion coefficient, and thermal conductivity of simple monatomic fluids. An extensive molecular dynamics simulation series is performed to obtain these transport properties and the residual entropy of three potential model classes with variable repulsive exponents: n, 6 Mie (n = 9, 12, 15, and 18), Buckingham's exponential-six (α = 12, 14, 18, and 30), and Tang-Toennies (αT = 4.051, 4.275, and 4.600). A wide range of liquid and supercritical gas- and liquid-like states is covered with a total of 1120 state points. Comparisons to equations of state, literature data, and transport property correlations are made. Although the absolute transport property values within a given potential model class may strongly depend on the repulsive exponent, it is found that the repulsive steepness plays a negligible role when entropy scaling is applied. Hence, the plus-scaled transport properties of n, 6 Mie, exponential-six, and Tang-Toennies fluids lie basically on one master curve, which closely corresponds with entropy scaling correlations for the Lennard-Jones fluid. This trend is confirmed by literature data of n, 6 Mie, and exponential-six fluids. Furthermore, entropy scaling holds for state points where the Pearson correlation coefficient R is well below 0.9. The condition R > 0.9 for strongly correlating liquids is thus not necessary for the successful application of entropy scaling, pointing out that isomorph theory may be a part of a more general framework that is behind the success of entropy scaling. Density scaling reveals a strong influence of the repulsive exponent on this particular approach.

Polymeric surfactants at liquid–liquid interfaces: Dependence of structural and thermodynamic properties on copolymer architecture

Polymeric surfactants are amphiphilic molecules with two or more different types of monomers. If one type of monomer interacts favorably with a liquid, and another type of monomer interacts favorably with another, immiscible liquid, then polymeric surfactants adsorb at the interface between the two liquids and reduce the interfacial tension. The effects of polymer architecture on the structural and thermodynamic properties of the liquid–liquid interface are studied using molecular simulations. The interface is modeled with a non-additive binary Lennard-Jones fluid in the two-phase region of the phase diagram. Block and gradient copolymer surfactants are represented with coarse-grained, bead-spring models, where each component of the polymer favors one or the other liquid. Gradient copolymers have a greater concentration at the interface than do block copolymers because the gradient copolymers adopt conformations partially aligned with the interface. The interfacial tension is determined as a function of the surface excess of polymeric surfactant. Gradient copolymers are more potent surfactants than block copolymers because the gradient copolymers cross the dividing surface multiple times, effectively acting as multiple individual surfactants. For a given surface excess, the interfacial tension decreases monotonically when changing from a block to a gradient architecture. The coarse-grained simulations are complemented by all-atom simulations of acrylic-acid/styrene copolymers at the chloroform-water interface, which have been studied in experiments. The agreement between the simulations (both coarse-grained and atomistic) and experiments is shown to be excellent, and the molecular-scale structures identified in the simulations help explain the variation of surfactancy with copolymer architecture.

Reassessing the transport properties of fluids: A symbolic regression approach.

The viscosity and thermal conductivity coefficients of the Lennard-Jones fluid are extracted through symbolic regression (SR) techniques from data derived from simulations at the atomic scale. This data-oriented approach provides closed form relations that achieve fine accuracy when compared to well-established theoretical, empirical, or approximate equations, fully transparent, with small complexity and high interpretability. The novelty is further outlined by suggesting analytical expressions for estimating fluid transport properties across the whole phase space, from a dilute gas to a dense liquid, by considering only two macroscopic properties (density and temperature). In such expressions, the underlying physical mechanisms are reflected, while, at the same time, it can be a computationally efficient alternative to costly in time and size first principle and/or molecular dynamics simulations.