Critical-point and coexistence-curve properties of the Lennard-Jones fluid: A finite-size scaling study

Coarse graining from variationally enhanced sampling applied to the Ginzburg-Landau model

A powerful way to deal with a complex system is to build a coarse-grained model capable of catching its main physical features, while being computationally affordable. Inevitably, such coarse-grained models introduce a set of phenomenological parameters, which are often not easily deducible from the underlying atomistic system. We present a unique approach to the calculation of these parameters, based on the recently introduced variationally enhanced sampling method. It allows us to obtain the parameters from atomistic simulations, providing thus a direct connection between the microscopic and the mesoscopic scale. The coarse-grained model we consider is that of Ginzburg-Landau, valid around a second-order critical point. In particular, we use it to describe a Lennard-Jones fluid in the region close to the liquid-vapor critical point. The procedure is general and can be adapted to other coarse-grained models.

Phase diagram of heteronuclear Janus dumbbells

Using aggregation-volume-bias Monte Carlo simulations along with successive umbrella sampling and histogram re-weighting, we study the phase diagram of a system of dumbbells formed by two touching spheres having variable sizes, as well as different interaction properties. The first sphere (h) interacts with all other spheres belonging to different dumbbells with a hard-sphere potential. The second sphere (s) interacts via a square-well interaction with other s spheres belonging to different dumbbells and with a hard-sphere potential with all remaining h spheres. We focus on the region where the s sphere is larger than the h sphere, as measured by a parameter 1 ≤ α ≤ 2 controlling the relative size of the two spheres. As α → 2 a simple fluid of square-well spheres is recovered, whereas α → 1 corresponds to the Janus dumbbell limit, where the h and s spheres have equal sizes. Many phase diagrams falling into three classes are observed, depending on the value of α. The 1.8 ≤ α ≤ 2 is dominated by a gas-liquid phase separation very similar to that of a pure square-well fluid with varied critical temperature and density. When 1.3 ≤ α ≤ 1.8 we find a progressive destabilization of the gas-liquid phase diagram by the onset of self-assembled structures, that eventually lead to a metastability of the gas-liquid transition below α = 1.2.

Critical Drying of Liquids

We report a detailed simulation and classical density functional theory study of the drying transition in a realistic model fluid at a smooth substrate. This transition (in which the contact angle θ→180°) is shown to be critical for both short-ranged and long-ranged substrate-fluid interaction potentials. In the latter case critical drying occurs at exactly zero attractive substrate strength. This observation permits the accurate elucidation of the character of the transition via a finite-size scaling analysis of the density probability function. We find that the critical exponent ν_{∥} that controls the parallel correlation length, i.e., the extent of vapor bubbles at the wall, is over twice as large as predicted by mean field and renormalization group calculations. We suggest a reason for the discrepancy. Our findings shed new light on fluctuation phenomena in fluids near hydrophobic and solvophobic interfaces.

Improved grand canonical sampling of vapour-liquid transitions

Simulation within the grand canonical ensemble is the method of choice for accurate studies of first order vapour-liquid phase transitions in model fluids. Such simulations typically employ sampling that is biased with respect to the overall number density in order to overcome the free energy barrier associated with mixed phase states. However, at low temperature and for large system size, this approach suffers a drastic slowing down in sampling efficiency. The culprits are geometrically induced transitions (stemming from the periodic boundary conditions) which involve changes in droplet shape from sphere to cylinder and cylinder to slab. Since the overall number density does not discriminate sufficiently between these shapes, it fails as an order parameter for biasing through the transitions. Here we report two approaches to ameliorating these difficulties. The first introduces a droplet shape based order parameter that generates a transition path from vapour to slab states for which spherical and cylindrical droplets are suppressed. The second simply biases with respect to the number density in a tetragonal subvolume of the system. Compared to the standard approach, both methods offer improved sampling, allowing estimates of coexistence parameters and vapor-liquid surface tension for larger system sizes and lower temperatures.

Extending the parQ transition matrix method to grand canonical ensembles

Phase coexistence properties as well as other thermodynamic features of fluids can be effectively determined from the grand canonical density of states (DOS). We present an extension of the parQ transition matrix method in combination with the efasTM method as a very fast approach for determining the grand canonical DOS from the transition matrix. The efasTM method minimizes the deviation from detailed balance in the transition matrix using a fast Krylov-based equation solver. The method allows a very effective use of state space transition data obtained by different exploration schemes. An application to a Lennard-Jones system produces phase coexistence properties of the same quality as reference data.

Spinodals and critical point using short-time dynamics for a simple model of liquid

We have applied the short-time dynamics method to the gas-liquid transition to detect the supercooled gas instability (gas spinodal) and the superheated liquid instability (liquid spinodal). Using Monte Carlo simulation, we have obtained the two spinodals for a wide range of pressure in sub-critical and critical conditions and estimated the critical temperature and pressure. Our method is faster than previous approaches and allows studying spinodals without needing equilibration of the system in the metastable region. It is thus free of the extrapolation problems present in other methods, and in principle could be applied to systems such as glass-forming liquids, where equilibration is very difficult even far from the spinodal. We have also done molecular dynamics simulations, where we find the method again able to detect the both spinodals. Our results are compared with different previous results in the literature and show a good agreement.

Nested sampling of isobaric phase space for the direct evaluation of the isothermal-isobaric partition function of atomic systems

Nested Sampling (NS) is a powerful athermal statistical mechanical sampling technique that directly calculates the partition function, and hence gives access to all thermodynamic quantities in absolute terms, including absolute free energies and absolute entropies. NS has been used predominately to compute the canonical (NVT) partition function. Although NS has recently been used to obtain the isothermal-isobaric (NPT) partition function of the hard sphere model, a general approach to the computation of the NPT partition function has yet to be developed. Here, we describe an isobaric NS (IBNS) method which allows for the computation of the NPT partition function of any atomic system. We demonstrate IBNS on two finite Lennard-Jones systems and confirm the results through comparison to parallel tempering Monte Carlo. Temperature-entropy plots are constructed as well as a simple pressure-temperature phase diagram for each system. We further demonstrate IBNS by computing part of the pressure-temperature phase diagram of a Lennard-Jones system under periodic boundary conditions.

Phase separation and self-assembly in a fluid of Mickey Mouse particles

Recent developments in the synthesis of colloidal particles allow for control over shape and inter-particle interaction. One example, among others, is the so-called "Mickey Mouse" (MM) particle for which the self-assembly properties have been previously studied yielding a stable cluster phase together with elongated, tube-like structures. Here, we investigate under which conditions a fluid of Mickey Mouse particles can yield phase separation and how the self-assembly behaviour affects the gas-liquid coexistence. We vary the distance between the repulsive and the attractive lobes (bond length), and the interaction range, and follow the evolution of the gas-liquid (GL) coexistence curve. We find that upon increasing the bond length distance the binodal line shifts to lower temperatures, and that the interaction range controls the transition between phase separation and self-assembly of clusters. Upon further reduction of the interaction range and temperature, the clusters assume an increasingly ordered tube-like shape, ultimately matching the one previously reported in literature. These results are of interest when designing particle shape and particle-particle interaction for self-assembly processes.

Understanding the sensitivity of nucleation free energies: The role of supersaturation and temperature

Simulations are used to investigate the vapor-to-liquid nucleation of water for several different force fields at various sets of physical conditions. The nucleation free energy barrier is found to be extremely sensitive to the force field at the same absolute conditions. However, when the results are compared at the same supersaturation and reduced temperature or the same metastability parameter and reduced temperature, then the differences in the nucleation free energies of the different models are dramatically reduced. This finding suggests that comparisons of experimental data and computational predictions are most meaningful at the same relative conditions and emphasizes the importance of knowing the phase diagram of a given computational model, but such information is usually not available for models where the interaction energy is determined directly from electronic structure calculations.

Tuning the Liquid-Liquid Transition by Modulating the Hydrogen-Bond Angular Flexibility in a Model for Water

We propose a simple extension of the well known ST2 model for water [F. H. Stillinger and A. Rahman, J. Chem. Phys. 60, 1545 (1974)] that allows for a continuous modification of the hydrogen-bond angular flexibility. We show that the bond flexibility affects the relative thermodynamic stability of the liquid and of the hexagonal (or cubic) ice. On increasing the flexibility, the liquid-liquid critical point, which in the original ST2 model is located in the no-man's land (i.e., the region where ice is the thermodynamically stable phase) progressively moves to a temperature where the liquid is more stable than ice. Our study definitively proves that the liquid-liquid transition in the ST2 model is a genuine phenomenon, of high relevance in all tetrahedral network-forming liquids, including water.

Liquid-vapor interface of the Stockmayer fluid in a uniform external field

The effect of a uniform (nonspatially varying) external field on the liquid-vapor interface of the Stockmayer fluid (Lennard-Jones particles embedded with a point dipole) has been investigated by molecular-dynamics simulations. The long-ranged parts of both the dipole and Lennard-Jones interactions are treated using an Ewald summation, which removes the effects of the cutoff. The direction of the field shifts the critical point and interfacial properties in different directions. For an external field parallel to the interface, the critical temperature increases, while for a field applied perpendicular to the interface, it decreases. The effects of the field on surface tension and interfacial width are also investigated. For zero field, dipoles near the liquid-vapor interface show a weak orientation parallel to the interface. For fields parallel to the interface, ordering in the liquid phase is greater than the vapor, while for fields perpendicular to the interface, the opposite is true.

Quantum Gibbs ensemble Monte Carlo

We present a path integral Monte Carlo method which is the full quantum analogue of the Gibbs ensemble Monte Carlo method of Panagiotopoulos to study the gas-liquid coexistence line of a classical fluid. Unlike previous extensions of Gibbs ensemble Monte Carlo to include quantum effects, our scheme is viable even for systems with strong quantum delocalization in the degenerate regime of temperature. This is demonstrated by an illustrative application to the gas-superfluid transition of (4)He in two dimensions.

Gas-liquid coexistence for the boson square-well fluid and the (4)He binodal anomaly

The binodal of a boson square-well fluid is determined as a function of the particle mass through a quantum Gibbs ensemble Monte Carlo algorithm devised by R. Fantoni and S. Moroni [J. Chem. Phys. (to be published)]. In the infinite mass limit we recover the classical result. As the particle mass decreases, the gas-liquid critical point moves at lower temperatures. We explicitly study the case of a quantum delocalization de Boer parameter close to the one of (4)He. For comparison, we also determine the gas-liquid coexistence curve of (4)He for which we are able to observe the binodal anomaly below the λ-transition temperature.

A nucleation-based method to study hydrophobic interactions under confinement: enhanced hydrophobic association driven by energetic contributions

A novel simulation approach was developed and applied to the study of hydrophobic interactions for a small hydrophobic solute pair under confinement. In this method, the aggregation-volume-bias Monte Carlo algorithm, developed originally for nucleation studies, is used to evaluate the association free energy with water molecules for a methane pair through the gradual addition of water molecules into a nanometer-sized sphere. Through a thermodynamic cycle, this method allows for a convenient examination of the free energy difference between two different solvated configurations without sampling any of the configurations in between. The potential of mean force (PMF) for a methane pair under confinement obtained from this method reveals that the stability of the contact pair configuration can be enhanced compared to that in bulk water, which is in agreement with previous studies. Also, constraining the center of this methane pair at the center of this confined volume yields a PMF with a metastable solvent separated configuration, resembling more closely the PMF from the bulk-phase system compared to previous studies in which this solvent-separated minimum was found to be completely absent. A combination with histogram reweighting enables the study of this association behavior at different thermodynamic conditions without additional simulations. From a comprehensive thermodynamic analysis, it is evident that such hydrophobic association, known to be entropically driven in the bulk-phase system at ambient conditions, is entropically favorable only when a suitable range of solvent molecules is added to the confined system. More importantly, the energetic contributions are a favorable factor that explains the enhanced hydrophobic association toward the high number of solvent molecules.

Enhanced sampling algorithms

In biomolecular systems (especially all-atom models) with many degrees of freedom such as proteins and nucleic acids, there exist an astronomically large number of local-minimum-energy states. Conventional simulations in the canonical ensemble are of little use, because they tend to get trapped in states of these energy local minima. Enhanced conformational sampling techniques are thus in great demand. A simulation in generalized ensemble performs a random walk in potential energy space and can overcome this difficulty. From only one simulation run, one can obtain canonical-ensemble averages of physical quantities as functions of temperature by the single-histogram and/or multiple-histogram reweighting techniques. In this article we review uses of the generalized-ensemble algorithms in biomolecular systems. Three well-known methods, namely, multicanonical algorithm, simulated tempering, and replica-exchange method, are described first. Both Monte Carlo and molecular dynamics versions of the algorithms are given. We then present various extensions of these three generalized-ensemble algorithms. The effectiveness of the methods is tested with short peptide and protein systems.

[N]pT ensemble and finite-size-scaling study of the critical isostructural transition in the generalized exponential model of index 4

First-order transitions of system where both lattice site occupancy and lattice spacing fluctuate, such as cluster crystals, cannot be efficiently studied by traditional simulation methods, which necessarily fix one of these two degrees of freedom. The difficulty, however, can be surmounted by the generalized [N]pT ensemble [J. Chem. Phys. 136, 214106 (2012)]. Here we show that histogram reweighting and the [N]pT ensemble can be used to study an isostructural transition between cluster crystals of different occupancy in the generalized exponential model of index 4 (GEM-4). Extending this scheme to finite-size scaling studies also allows us to accurately determine the critical point parameters and to verify that it belongs to the Ising universality class.

Phase diagram and universality of the Lennard-Jones gas-liquid system

The gas-liquid phase transition of the three-dimensional Lennard-Jones particles system is studied by molecular dynamics simulations. The gas and liquid densities in the coexisting state are determined with high accuracy. The critical point is determined by the block density analysis of the Binder parameter with the aid of the law of rectilinear diameter. From the critical behavior of the gas-liquid coexisting density, the critical exponent of the order parameter is estimated to be β = 0.3285(7). Surface tension is estimated from interface broadening behavior due to capillary waves. From the critical behavior of the surface tension, the critical exponent of the correlation length is estimated to be ν = 0.63(4). The obtained values of β and ν are consistent with those of the Ising universality class.

Determination of the solid-fluid coexistence of the n - 6 Lennard-Jones system from free energy calculations

The solid-fluid coexistence properties of the n - 6 Lennard-Jones system, n from 7 to 12, are reported. The procedure relies on determining Helmholtz free energy curves as a function of volume for each phase independently, from several NVT simulations, and then connecting it to points of known absolute free energy. For n = 12 this requires connecting the simulated points to states of very low densities on the liquid phase, and to a harmonic crystal for the solid phase, which involves many extra simulations for each temperature. For the reference points of the remaining systems, however, the free energy at a given density and temperature can be calculated relative to the n = 12 system. The method presented here involves a generalization of the multiple histogram method to combine simulations performed with different potentials, provided they visit overlapping regions of the phase space, and allows for a precise calculation of relative free energies. The densities, free energies, average potential energies, pressure, and chemical potential at coexistence are presented for up to T∗ = 5.0 and new estimations of the triple points are given for the n - 6 Lennard-Jones system.

A numerical coarse-grained description of a binary alloy

We employ Monte Carlo simulation in the semi-grand canonical ensemble to obtain the coarse-grained free energy corresponding to an embedded-atom method description of a binary alloy. In particular, the Ginzburg-Landau free energy for a Cu-Ni alloy was determined from a tabulated histogram of the joint probability density of composition, energy, and volume. Using histogram reweighting techniques, the free energy is extrapolated to a range of points in parameter space from a small number of simulations. The results are interpreted by comparing the free energy with that corresponding to a regular solution model of an alloy. In addition, we obtain expressions for thermodynamic quantities in terms of the joint cumulants of the probability density at a given temperature and chemical potential difference. These expressions may then be likewise extrapolated to obtain the dependence of the composition on the temperature and the chemical potential difference over a wide range of parameter space.

Widom line and noise-power spectral analysis of a supercritical fluid

We have performed extensive molecular dynamics simulations to study noise-power spectra of density and potential energy fluctuations of a Lennard-Jones model of a fluid in the supercritical region. Emanating from the liquid-vapor critical point, there is a locus of isobaric specific heat maxima, called the Widom line, which is often regarded as an extension of the liquid-vapor coexistence line. Our simulation results show that the noise-power spectrum of the density fluctuations on the Widom line of the liquid-vapor transition exhibits three distinct 1/f^{γ} behaviors with exponents γ=0, 1.2, and 2, depending on the frequency f. We find that the intermediate frequency region with an exponent γ∼ 1 appears as the temperature approaches the Widom temperature from above or below. On the other hand, we do not find three distinct regions of 1/f^{γ} in the power spectrum of the potential energy fluctuations on the Widom line. Furthermore, we find that the power spectra of both the density and potential energy fluctuations at low frequency have a maximum on the Widom line, suggesting that the noise power can provide an alternative signature of the Widom line.