Hard-disk Fluid Research Articles

Exploring anisotropic pressure and spatial correlations in strongly confined hard-disk fluids: Exact results.

This study examines the transverse and longitudinal properties of hard disks confined in narrow channels. Employing an exact mapping of the system onto a one-dimensional polydisperse, nonadditive mixture of hard rods with equal chemical potentials, we compute various thermodynamic properties, including the transverse and longitudinal equationsof state, along with their behaviors at both low and high densities. Structural properties are analyzed using the two-body correlation function and the radial distribution function, tailored for the highly anisotropic geometry of this system. The results are corroborated by computer simulations.

Simple and accurate expressions for radial distribution functions of hard disk and hard sphere fluids

Analytical expressions for radial distribution function (RDF) are of critical importance for various applications, such as development of the perturbation theories. Theoretically, RDF expressions for odd-dimensional fluids can be obtained by solving the Percus-Yevick integral equations. But for even-dimensional cases, such as the hard disk (2D) fluid, analytical expressions are infeasible. The only 2D RDF is a heuristic expression which provides acceptable estimations for an intermediate and low density range. In this work, we employ a simple and empirical expression for the 2D RDF and the 3D RDF based on an approach proposed for the 3D RDF. The parameters are determined in such a way that the final RDF expressions are thermodynamically consistent, namely the pressure and the isothermal compressibility constraints are both satisfied. The new RDFs for the 2D and 3D hard spheres are highly accurate for the entire density range up to the first-order phase transition points. The predictions of the first coordination numbers are consistent with simulation results for the 3D fluid. Finally, by using the 2D RDF with a primitive second-order perturbation theory, the pressure-volume-temperature relation and vapor-liquid equilibrium are calculated for the 2D Lennard-Jones fluid. Comparisons with the simulation data show promising results.

Structural properties of hard-disk fluids under single-file confinement.

The structural properties of confined single-file hard-disk fluids are studied analytically by means of a mapping of the original system onto a one-dimensional mixture of non-additive hard rods, the mapping being exact in the polydisperse limit. Standard statistical-mechanical results are used as a starting point to derive thermodynamic and structural properties of the one-dimensional mixture, where the condition that all particles have the same chemical potential must be taken into account. Analytical results are then provided for the nth neighbor probability distribution function, the radial distribution function, and the structure factor, a very good agreement being observed upon comparison with simulation data from the literature. Moreover, we have analyzed the scaling form for the disappearance of defects in the zigzag configuration for high pressure and have obtained the translational correlation length and the structural crossover in the oscillation frequency for asymptotically large distances.

Effective dynamics and fluctuations of a trapped probe moving in a fluid of active hard discs (a)

We study the dynamics of a single trapped probe surrounded by self-propelled active particles in two dimensions. In the limit of large size separation, we perform an adiabatic elimination of the small active particles to obtain an effective Markovian dynamics of the large probe, yielding explicit expressions for the mobility and diffusion coefficient. To calculate these expressions, we perform computer simulations employing active Brownian discs and consider two scenarios: non-interacting bath particles and purely repulsive interactions modeling volume exclusion. We keep the probe-to-bath size ratio fixed and vary the propulsion speed of the bath particles. The positional fluctuations of a trapped probe are accessible in experiments, for which we test the prediction from the adiabatic elimination. We find that for a passive bath the Markovian prediction that the integrated force correlations equal the drag coefficient is not fulfilled in the simulations. However, this discrepancy is small compared to the active contribution and the overall agreement between predicted and measured probe fluctuations is very good at larger speeds.

Equation of state of hard-disk fluids under single-file confinement.

The exact transfer-matrix solution for the longitudinal equilibrium properties of the single-file hard-disk fluid is used to study the limiting low- and high-pressure behaviors analytically as functions of the pore width. In the low-pressure regime, the exact third and fourth virial coefficients are obtained, which involve single and double integrals, respectively. Moreover, we show that the standard irreducible diagrams do not provide a complete account of the virial coefficients in confined geometries. The asymptotic equation of state in the high-pressure limit is seen to present a simple pole at the close-packing linear density, as in the hard-rod fluid, but, in contrast to the latter case, the residue is 2. Since, for an arbitrary pressure, the exact transfer-matrix treatment requires the numerical solution of an eigenvalue integral equation, we propose here two simple approximations to the equation of state, with different complexity levels, and carry out an extensive assessment of their validity and practical convenience vs the exact solution and available computer simulations.

Augmented scaled particle theory for a hard disk fluid

The success of scaled particle theory (SPT) can be certainly ascribed to the appealing physical idea embedded in it, i.e., accounting for the work needed to insert a particle into a liquid by a bulk contribution and a surface contribution. So, it can describe bulk and surface thermodynamic properties in the same framework. The surface tension term in SPT involves one flat-surface contribution and two additional contributions due to surface curvatures. High-order curvature terms (also called non-Hadwiger terms) were evidenced. Recently, we have developed an augmented scaled particle theory (ASPT) by adding one high-order curvature term for a hard sphere fluid. Here, we extend ASPT to a two-dimensional hard disk fluid. As for the 3D case, our 2D ASPT allows for improving both bulk and surface thermodynamic properties of a hard disk fluid with respect to the original SPT.

Inside and out: Surface thermodynamics from positive to negative curvature.

To explore the curvature dependence of solid-fluid interfacial thermodynamics, we calculate, using Grand Canonical Monte Carlo simulation, the surface free energy for a 2d hard-disk fluid confined in a circular hard container of radius R as a function of the bulk packing fraction η and wall curvature C̄=-1/R. (The curvature is negative because the surface is concave.) Combining this with our previous data [Martin et al., J. Phys. Chem. B 124, 7938-7947 (2020)] for the positive curvature case (a hard-disk fluid at a circular wall, C̄=+1/R), we obtain a complete picture of surface thermodynamics in this system over the full range of positive and negative wall curvatures. Our results show that γ is linear in C̄ with a slope that is the same for both positive and negative wall curvatures, with deviations seen only at high negative curvatures (strong confinement) and high density. This observation indicates that the surface thermodynamics of this system is consistent with the predictions of so-called morphometric thermodynamics at both positive and negative curvatures. In addition, we show that classical density functional theory and a generalized scaled particle theory can be constructed that give excellent agreement with the simulation data over most of the range of curvatures and densities. For extremely high curvatures, where only one or two disks can occupy the container at maximum packing, it is possible to calculate γ exactly. In this limit, the simulations and density functional theory calculations are in remarkable agreement with the exact results.

Molecular hints of two-step transition to convective flow via streamline percolation.

Convection is a key transport phenomenon important in many different areas, from hydrodynamics and ocean circulation to planetary atmospheres or stellar physics. However, its microscopic understanding still remains challenging. Here we numerically investigate the onset of convective flow in a compressible (non-Oberbeck-Boussinesq) hard disk fluid under a temperature gradient in a gravitational field. We uncover a surprising two-step transition scenario with two different critical temperatures. When the bottom plate temperature reaches a first threshold, convection kicks in (as shown by a structured velocity field) but gravity results in hindered heat transport as compared to the gravity-free case. It is at a second (higher) temperature that a percolation transition of advection zones connecting the hot and cold plates triggers efficient convective heat transport. Interestingly, this picture for the convection instability opens the door to unknown piecewise-continuous solutions to the Navier-Stokes equations.

Phase diagram of SALR fluids on spherical surfaces.

We investigate the phase diagram of a fluid of hard-core disks confined to the surface of a sphere and whose interaction potential contains a short-range attraction followed by a long-range repulsive tail (SALR). Based on previous work in the bulk we derive a stability criterion for the homogeneous phase of the fluid, and locate a region of instability linked to the presence of a negative minimum in the spherical harmonics expansion of the interaction potential. The inhomogeneous phases contained within this region are characterized using a mean-field density functional theory. We find several inhomogeneous patterns that can be separated into three broad classes: cluster crystals, stripes, and bubble crystals, each containing topological defects. Interestingly, while the periodicity of inhomogeneous phases at large densities is mainly determined by the position of the negative minimum of the potential expansion, the finite size of the system induces a richer behavior at smaller densities.

Carnahan-Starling type equations of state for stable hard disk and hard sphere fluids

The well-known Carnahan-Starling (CS) equation of state (EoS) [N.F. Carnahan and K.E. Starling. J. Chem. Phys. 51 (2), 635–636 (1969). doi:10.1063/1.1672048] for hard sphere (HS) fluid was derived from a quadratic relationship between the integer portions of the virial coefficients, , and their orders, . In this paper, the method is extended to cover the full virial coefficients, , for the general D-dimensional case. We propose a (D-1)th order polynomial for the virial coefficients starting from the 4th order and EoS’s are derived from it. For the hard rod (D = 1) case, the exact solution is obtained. For the stable hard disk fluid (D = 2), the most recent virial coefficients, up to the 10th order, [N. Clisby and B.M. McCoy. J. Stat. Phys. 122 (1), 15–57 (2006). doi:10.1007/s10955-005-8080-0] and accurate compressibility data [J. Kolafa and M. Rottner. Mol. Phys. 104 (22–24), 3435–3441 (2006). doi:10.1080/00268970600967963; J.J. Erpenbeck and M. Luban. Phys. Rev. A. 32 (5), 2920–2922 (1985). doi:10.1103/PhysRevA.32.2920] are employed to construct and to test the EoS. For the stable hard sphere (D = 3) fluid, a new CS-type EoS is obtained by using the most recent virial coefficients [N. Clisby and B.M. McCoy. J. Stat. Phys. 122 (1), 15–57 (2006). doi:10.1007/s10955-005-8080-0; A.J. Schultz and D.A. Kofke. Physical Review E. 90 (2) (2014). doi:10.1103/PhysRevE.90.023301], up to the 11th order, along with highly-accurate compressibility data [S. Pieprzyk et al. Phys. Chem. Chem. Phys. 21 (13), 6886–6899 (2019). doi:10.1039/C9CP00903E; M.N. Bannerman et al. J. Chem. Phys. 132 (8), 084507 (2010). doi:10.1063/1.3328823; J. Kolafa, et al. Phys. Chem. Chem. Phys. 6 (9), 2335–2340 (2004). doi:10.1039/B402792B]. The simple new EoS’s prove to be as accurate as the Padé approximations based on all available virial coefficients, which significantly improve the accuracy of the CS-type EoS in the hard sphere case. This research also reveals that as long as the virial coefficients obey a polynomial function, any EoS derived from it will diverge at the non-physical packing fraction, .

Predicting the slope of the radial distribution function of a hard disk fluid

The slope of the radial distribution function at contact is investigated for a pure component two-dimensional hard disk fluid. Using the framework of scaled particle theory (SPT), an exact equation for g'(σ +) is derived in terms of SPT quantities. This new expression requires the value of the average of a three-particle correlation function for which an ideal gas approximation is developed. The resultant predictions for g'(σ +) are compared to those generated from literature equations and to simulation data generated for this work. In addition to simulation data for g'(σ +), simulation values for the three-particle correlation function are presented for comparison to the approximation. The simple fourth-order integer coefficient polynomial is an accurate predictor of g'(σ +), on the order of or better than what is calculated by existing equations.

Surface Free Energy of a Hard-Disk Fluid at Curved Hard Walls: Theory and Simulation.

In this work, we examine the surface thermodynamics of a hard-disk fluid at curved hard walls using Monte Carlo (MC) simulation and a generalized scaled particle theory (gSPT). The curved walls are modeled as hard disks of varying radii, R. The surface free energy, γ, and excess surface volume, vex, for this system are calculated as functions of both the fluid packing fraction and the wall radius. The simulation results are used to test, for this system, the assumptions of morphometric thermodynamics (MT), which predicts that both γ and vex are linear functions of the surface curvature, 1/R, for a two-dimensional system. In addition, we compare the simulation results to the gSPT developed in this work, as well as with virial expansions derived from the known virial coefficients of the binary hard-sphere fluid. At low to intermediate packing fractions, the non-MT terms (terms of higher order than 1/R in a expansion of γ and vex) of γ are zero within the simulation error; however, at the highest densities, deviations from MT become significant, similar to what was seen in our earlier simulation work on the three-dimensional hard-sphere/hard-wall system. In addition, the new gSPT gives improved results for both γ and vex over standard scaled particle theory (SPT) but underestimates the deviations from MT at high density.

Analytical evaluation of third and fourth virial coefficients for hard disk fluids in narrow channels and equation of state

The third and fourth virial coefficients of hard disks in narrow channels have been evaluated as a series in powers of the channel width deviations from the one dimensional limit. We have obtained analytically the coefficients of the power series to very high order. These virial coefficients are used in truncated virial series to provide approximate analytical equation of state (EOS). The EOS is shown to accurately describe Monte Carlo simulations over a wide range of pressures and densities near the one dimensional limit. Explicit expressions for the free energy, chemical potential, compressibility and EOS as a function of the channel widths are given. These results represent progress toward understanding the long standing problem of the thermodynamic crossover from a one dimensional to quasi one dimensional fluid. They are also useful in a number of applications. We mention the examples of thermodynamic perturbation theory and stochastic dynamic mobility in anomalous single-file diffusion.

Heat conduction in a hard disc system with non-conserved momentum

We describe results of computer simulations of steady state heat transport in a fluid of hard discs undergoing both elastic interparticle collisions and ‘pseudo collisions’ which do not conserve momentum. The latter are done by picking particles at random and randomizing the directions of their velocities. The system consists of N discs of radius r in a unit square, periodic in the y-direction and having thermal walls with different temperatures at x = 0 and at x = 1. We extrapolate results from different N, to N → ∞, r → 0, such that πr2N = 1/2. We find that in the (extrapolated) hydrodynamic limit N → ∞, the systems’ local density and temperature profiles are those of local thermodynamic equilibrium (LTE), the corresponding pressure is constant independent of position and the heat flux obeys Fourier’s law. The variance of global quantities, such as the total energy, deviates from its local equilibrium value in a form consistent with macroscopic fluctuation theory.

Equations of state for the hard disk fluids

By using the published accurate new virial coefficients B11∼B14 for the hard disk fluids [C. Zhang and B.M. Pettitt, Mol. Phys., 2014, 112, 1427], we here propose a new updated version of Tian-Gui-Mulero equation of state [J.X. Tian, Y.X. Gui and A. Mulero, Phys. Chem. Chem. Phys., 2010, 12, 13597]. Compared with other proposals, the new version stands strongly to be the only which can reproduce the known virial coefficients B2∼B13 at the same time that can describe the relation of the compressibility factor versus the packing fraction for the hard disk fluids with high accuracy.

Transport dynamics of complex fluids

Thermal motion in complex fluids is a complicated stochastic process but ubiquitously exhibits initial ballistic, intermediate subdiffusive, and long-time diffusive motion, unless interrupted. Despite its relevance to numerous dynamical processes of interest in modern science, a unified, quantitative understanding of thermal motion in complex fluids remains a challenging problem. Here, we present a transport equation and its solutions, which yield a unified quantitative explanation of the mean-square displacement (MSD), the non-Gaussian parameter (NGP), and the displacement distribution of complex fluids. In our approach, the environment-coupled diffusion kernel and its time correlation function (TCF) are the essential quantities that determine transport dynamics and characterize mobility fluctuation of complex fluids; their time profiles are directly extractable from a model-free analysis of the MSD and NGP or, with greater computational expense, from the two-point and four-point velocity autocorrelation functions. We construct a general, explicit model of the diffusion kernel, comprising one unbound-mode and multiple bound-mode components, which provides an excellent approximate description of transport dynamics of various complex fluidic systems such as supercooled water, colloidal beads diffusing on lipid tubes, and dense hard disk fluid. We also introduce the concepts of intrinsic disorder and extrinsic disorder that have distinct effects on transport dynamics and different dependencies on temperature and density. This work presents an unexplored direction for quantitative understanding of transport and transport-coupled processes in complex disordered media.

Analytic results for the three- and four-particle correlation functions of the fluid of hard disks.

Analytic expressions for the three- and four-particle correlation functions of the hard disk fluid in terms of geometric functionals have been derived using the combined geometric/thermodynamic idea of Boublik [Fluid Phase Equilib. 316, 1 (2012)]. To assess the approximate result, extensive Monte Carlo simulations were performed to obtain the triplet correlation function, g3, over a wide range of densities. We then focussed on the rolling contact configurations that are required in the second-order thermodynamic perturbation theory of patchy colloid models. The analytic approximation is shown to perform very well, and the simulation data have therefore been parametrized using Boublik's functional form for their subsequent use.

Scaling with a fractional power: Overcoming the shortage of scaled-particle variables for the hard-disk fluid.

Within scaled-particle theory, we construct an equation of state (EOS) for hard-disk mixtures by making use of an additional scaled-particle variable which weighs the densities of the different components by its radii to the power χ. This allows us to simultaneously respect exact results pertaining to the cases of a large particle or a point particle being added to the mixture. In the limit χ → 2, the mixture EOS of Santos et al. [Mol. Phys. 96, 1 (1999)] is recovered, while the limit χ → 0 yields the accurate expression for the interfacial free energy of Martin et al. [J. Chem. Phys. 149, 084701 (2018)]. From the low-density limit of the EOS, the value χ ≈ 0.8 is extracted, which is shown to yield a mixture EOS that is significantly more accurate than the expressions due to Santos et al. and Martin et al. In particular, the systematic deviation inherent to these prior results is remedied.

Efficient Computation of Entropy and Other Thermodynamic Properties for Two-Dimensional Systems Using Two-Phase Thermodynamic Model.

We present a method based on the two-phase thermodynamic model (2PT) to calculate the entropy and free energy of various molecular systems in two dimensions (2D) using molecular dynamics (MD) simulations. The 2PT method has been used widely to calculate absolute entropy in a variety of molecular systems in three dimensions. When applying the idea to 2D systems, we found that the fluidicity that determines the decomposition of the vibrational density of states (DoS) into a solidlike and a gaslike component needs to be revised. The solid part is treated using quantum statistics, and the gas part is treated as a hard-disk fluid. We validate this method by computing thermodynamic properties of a two-dimensional Lennard-Jones fluid over a range of densities and temperatures and find excellent agreement with these quantities computed from the equation of state. More importantly, this method allows for the calculation of the entropy and free energy of 2D systems efficiently from a single MD trajectory of less than 50 ps; therefore, it can be a new, powerful way of assessing the thermodynamic properties in 2D problems.

Thermodynamics of the hard-disk fluid at a planar hard wall: Generalized scaled-particle theory and Monte Carlo simulation.

A generalized scaled-particle theory for the uniform hard-disk mixture is derived in the spirit of the White Bear II free energy of the hard-sphere fluid [H. Hansen-Goos and R. Roth, J. Phys. C: Condens. Matter 18, 8413 (2006)]. The theory provides a very simple result for the interfacial free energy γ of the hard-disk fluid at a planar hard wall (which in d = 2 is a line) in terms of the equation of state. To complement and assess the theory, we perform Monte Carlo simulations from which we obtain γ using Gibbs-Cahn integration. While we find excellent overall agreement between theory and simulation, it also becomes apparent that the set of scaled-particle variables available in d = 2 is too limited, prohibiting a quasi-exact result for γ. Furthermore, this is reflected in the mixture equation of state resulting from our theory, which, similar to a previous attempt by Santos et al. [Mol. Phys. 96, 1 (1999)], displays a small but systematic deviation from simulations.