Lennard-Jones Model Research Articles

Consistency of likelihood estimation for Gibbs point processes

Strong consistency of the maximum likelihood estimator (MLE) for parametric Gibbs point process models is established. The setting is very general. It includes pairwise pair potentials, finite and infinite multibody interactions and geometrical interactions, where the range can be finite or infinite. The Gibbs interaction may depend linearly or non-linearly on the parameters, a particular case being hardcore parameters and interaction range parameters. As important examples, we deduce the consistency of the MLE for all parameters of the Strauss model, the hardcore Strauss model, the Lennard-Jones model and the area-interaction model.

Efficient measurement of point-to-set correlations and overlap fluctuations in glass-forming liquids.

Cavity point-to-set correlations are real-space tools to detect the roughening of the free-energy landscape that accompanies the dynamical slowdown of glass-forming liquids. Measuring these correlations in model glass formers remains, however, a major computational challenge. Here, we develop a general parallel-tempering method that provides orders-of-magnitude improvement for sampling and equilibrating configurations within cavities. We apply this improved scheme to the canonical Kob-Andersen binary Lennard-Jones model for temperatures down to the mode-coupling theory crossover. Most significant improvements are noted for small cavities, which have thus far been the most difficult to study. This methodological advance also enables us to study a broader range of physical observables associated with thermodynamic fluctuations. We measure the probability distribution of overlap fluctuations in cavities, which displays a non-trivial temperature evolution. The corresponding overlap susceptibility is found to provide a robust quantitative estimate of the point-to-set length scale requiring no fitting. By resolving spatial fluctuations of the overlap in the cavity, we also obtain quantitative information about the geometry of overlap fluctuations. We can thus examine in detail how the penetration length as well as its fluctuations evolve with temperature and cavity size.

Charge-dependent many-body exchange and dispersion interactions in combined QM/MM simulations.

Accurate modeling of the molecular environment is critical in condensed phase simulations of chemical reactions. Conventional quantum mechanical/molecular mechanical (QM/MM) simulations traditionally model non-electrostatic non-bonded interactions through an empirical Lennard-Jones (LJ) potential which, in violation of intuitive chemical principles, is bereft of any explicit coupling to an atom's local electronic structure. This oversight results in a model whereby short-ranged exchange-repulsion and long-ranged dispersion interactions are invariant to changes in the local atomic charge, leading to accuracy limitations for chemical reactions where significant atomic charge transfer can occur along the reaction coordinate. The present work presents a variational, charge-dependent exchange-repulsion and dispersion model, referred to as the charge-dependent exchange and dispersion (QXD) model, for hybrid QM/MM simulations. Analytic expressions for the energy and gradients are provided, as well as a description of the integration of the model into existing QM/MM frameworks, allowing QXD to replace traditional LJ interactions in simulations of reactive condensed phase systems. After initial validation against QM data, the method is demonstrated by capturing the solvation free energies of a series of small, chlorine-containing compounds that have varying charge on the chlorine atom. The model is further tested on the SN2 attack of a chloride anion on methylchloride. Results suggest that the QXD model, unlike the traditional LJ model, is able to simultaneously obtain accurate solvation free energies for a range of compounds while at the same time closely reproducing the experimental reaction free energy barrier. The QXD interaction model allows explicit coupling of atomic charge with many-body exchange and dispersion interactions that are related to atomic size and provides a more accurate and robust representation of non-electrostatic non-bonded QM/MM interactions.

A model metallic glass exhibits size-independent tensile ductility

Metallic glasses (MGs) usually suffer brittle fracture under uniaxial tension due to size-dependent shear band cavitation. Here we developed a concurrent multi-scale simulation method to describe the uniaxial tension of a binary Lennard-Jones (BLJ) model glass up to 88 microns in length. No cavitation or brittle fracture was found even for the longest BLJ sample. As the length increases, the shear band temperature increases, then saturates, while the elastic unloading from shear-off diminishes. We conjecture that the shear band of a BLJ sample, even with a macroscopic length, cannot reach the herein estimated critical cavitation conditions. Thus, BLJ samples appear to be free of size-induced tensile brittleness. Based on the shear band evolution and the critical cavitation conditions, we propose three classes of MGs in terms of tensile ductility: brittle-MGs (brittle for all lengths), normal-MGs (ductile for short samples, brittle for long samples), cohesive-MGs (ductile for all lengths). Our simulation results illustrate limitations of existing molecular models, and suggest that certain experimental metallic glasses may be free of size-induced tensile brittleness.

Modeling sympathetic cooling of molecules by ultracold atoms

We model sympathetic cooling of ground-state CaF molecules by ultracold Li and Rb atoms. The molecules are moving in a microwave trap, while the atoms are trapped magnetically. We calculate the differential elastic cross sections for CaF-Li and CaF-Rb collisions, using model Lennard-Jones potentials adjusted to give typical values for the s-wave scattering length. Together with trajectory calculations, these differential cross sections are used to simulate the cooling of the molecules, the heating of the atoms, and the loss of atoms from the trap. We show that a hard-sphere collision model based on an energy-dependent momentum transport cross section accurately predicts the molecule cooling rate but underestimates the rates of atom heating and loss. Our simulations suggest that Rb is a more effective coolant than Li for ground-state molecules, and that the cooling dynamics are less sensitive to the exact value of the s-wave scattering length when Rb is used. Using realistic experimental parameters, we find that molecules can be sympathetically cooled to 100$\mu$K in about 10s. By applying evaporative cooling to the atoms, the cooling rate can be increased and the final temperature of the molecules can be reduced to 1$\mu$K within 30s.

Vapor-phase metastability and condensation via the virial equation of state with extrapolated coefficients

Recently reported virial coefficients for the Lennard-Jones model are extrapolated to very high order, and the results are used to study the behavior of virial equation of state (VEOS). Convergence of the VEOS is examined in the context of gas-phase metastability and condensation. Comparison to molecular simulation data shows that the VEOS can accurately describe the equation of state over much of the metastable region, and the stability limits of very low-order isotherms correspond well with simulation-based estimates of the spinodal densities. However, as higher-order terms are added to the density series, the VEOS becomes less capable of characterizing metastable states, and instead appears to be moving toward a description of condensation. The fully-summed VEOS based on virial coefficients extrapolated to infinite order abruptly ends in the metastable region with a branch-point singularity. This form represents the culmination of a sequence of curves in which the pressure reaches a maximum before turning downward, both more sharply and at lower density with increasing series order; the corresponding sequence of maxima converge to the point where the fully-summed VEOS diverges. Thus, the extrapolation-based fully-summed VEOS exhibits the qualitative features of condensation, but it fails to provide quantitative agreement with condensation densities established by molecular simulation. The shortcomings point to a need to better understand the behavior of the virial coefficients with increasing order, perhaps with consideration of the volume dependence of the cluster integrals on which the VEOS is based.

Eighth to sixteenth virial coefficients of the Lennard-Jones model.

We calculated virial coefficients BN, 8 ≤ N ≤ 16, of the Lennard-Jones (LJ) model using both the Mayer-sampling Monte Carlo method and direct generation of configurations, with Wheatley's algorithm for summation of clusters. For N = 8, 24 values are reported, and for N = 9, 12 values are reported, both for temperatures T in the range 0.6 ≤ T ≤ 40.0 (in LJ units). For each N in 10 ≤ N ≤ 16, one to four values are reported for 0.6 ≤ T ≤ 0.9. An approximate functional form for the temperature dependence of BN was developed, and fits of LJ BN(T) based on this form are presented for each coefficient, 4 ≤ N ≤ 9, using new and previously reported data.

How osmolytes influence hydrophobic polymer conformations: A unified view from experiment and theory

It is currently the consensus belief that protective osmolytes such as trimethylamine N-oxide (TMAO) favor protein folding by being excluded from the vicinity of a protein, whereas denaturing osmolytes such as urea lead to protein unfolding by strongly binding to the surface. Despite there being consensus on how TMAO and urea affect proteins as a whole, very little is known as to their effects on the individual mechanisms responsible for protein structure formation, especially hydrophobic association. In the present study, we use single-molecule atomic force microscopy and molecular dynamics simulations to investigate the effects of TMAO and urea on the unfolding of the hydrophobic homopolymer polystyrene. Incorporated with interfacial energy measurements, our results show that TMAO and urea act on polystyrene as a protectant and a denaturant, respectively, while complying with Tanford-Wyman preferential binding theory. We provide a molecular explanation suggesting that TMAO molecules have a greater thermodynamic binding affinity with the collapsed conformation of polystyrene than with the extended conformation, while the reverse is true for urea molecules. Results presented here from both experiment and simulation are in line with earlier predictions on a model Lennard-Jones polymer while also demonstrating the distinction in the mechanism of osmolyte action between protein and hydrophobic polymer. This marks, to our knowledge, the first experimental observation of TMAO-induced hydrophobic collapse in a ternary aqueous system.

Vibration Modes and Characteristic Length Scales in Amorphous Materials

The numerical study of the mechanical responses of amorphous materials at the nanometer scale shows characteristic length scales that are larger than the intrinsic length of the microstructure. In this article, we review the different scales appearing upon athermal elastoplastic mechanical load and we relate it to a detailed study of the vibrational response. We compare different materials with different microstructures and different bond directionality (from Lennard–Jones model materials to amorphous silicon and silicate glasses). This work suggests experimental measurements that could help to understand and, if possible, to predict plastic deformation in glasses.

Revised Variable Soft Sphere and Lennard-Jones Model Parameters for Eight Common Gases up to 2200 K

Revised parameters for variable soft sphere (VSS) and Lennard-Jones (L-J) (6-12) models are presented for eight common gases in the temperature range 20–2200 K. The new parameters are based on a critical review of experimental measurements and ab-initio calculations of shear viscosity and self-diffusion coefficients. The optimized parameters are assessed relative to experimental measurements and ab-initio calculations of thermal conductivity. Results indicate most significant improvements for the VSS model occur when considering lighter gases where repulsive forces are more important. The agreement with transport property measurements for He gas extends to much higher temperatures using the revised parameters. The accuracy of the L-J model is more significantly improved when considering the heavier gases for which attractive forces are important over a wider temperature range.

On the determination of phase boundaries via thermodynamic integration across coexistence regions.

Specialized Monte Carlo methods are nowadays routinely employed, in combination with thermodynamic integration (TI), to locate phase boundaries of classical many-particle systems. This is especially useful for the fluid-solid transition, where a critical point does not exist and both phases may notoriously go deeply metastable. Using the Lennard-Jones model for demonstration, we hereby investigate on the alternate possibility of tracing reasonably accurate transition lines directly by integrating the pressure equation of state computed in a canonical-ensemble simulation with local moves. The recourse to this method would become a necessity when the stable crystal structure is not known. We show that, rather counterintuitively, metastability problems can be alleviated by reducing (rather than increasing) the size of the system. In particular, the location of liquid-vapor coexistence can exactly be predicted by just TI. On the contrary, TI badly fails in the solid-liquid region, where a better assessment (to within 10% accuracy) of the coexistence pressure can be made by following the expansion, until melting, of the defective solid which has previously emerged from the decay of the metastable liquid.

The ensemble switch method for computing interfacial tensions.

We present a systematic thermodynamic integration approach to compute interfacial tensions for solid-liquid interfaces, which is based on the ensemble switch method. Applying Monte Carlo simulations and finite-size scaling techniques, we obtain results for hard spheres, which are in agreement with previous computations. The case of solid-liquid interfaces in a variant of the effective Asakura-Oosawa model and of liquid-vapor interfaces in the Lennard-Jones model are discussed as well. We demonstrate that a thorough finite-size analysis of the simulation data is required to obtain precise results for the interfacial tension.

Freezing point depression in model Lennard-Jones solutions

Crystallisation of liquid solutions is of uttermost importance in a wide variety of processes in materials, atmospheric and food science. Depending on the type and concentration of solutes the freezing point shifts, thus allowing control on the thermodynamics of complex fluids. Here we investigate the basic principles of solute-induced freezing point depression by computing the melting temperature of a Lennard-Jones fluid with low concentrations of solutes, by means of equilibrium molecular dynamics simulations. The effect of solvophilic and weakly solvophobic solutes at low concentrations is analysed, scanning systematically the size and the concentration. We identify the range of parameters that produce deviations from the linear dependence of the freezing point on the molal concentration of solutes, expected for ideal solutions. Our simulations allow us also to link the shifts in coexistence temperature to the microscopic structure of the solutions.

1D momentum-conserving systems: the conundrum of anomalous versus normal heat transport

Transport and the spread of heat in Hamiltonian one dimensional momentum conserving nonlinear systems is commonly thought to proceed anomalously. Notable exceptions, however, do exist of which the coupled rotator model is a prominent case. Therefore, the quest arises to identify the origin of manifest anomalous energy and momentum transport in those low dimensional systems. We develop the theory for both, the statistical densities for momentum- and energy-spread and particularly its momentum-/heat-diffusion behavior, as well as its corresponding momentum/heat transport features. We demonstrate that the second temporal derivative of the mean squared deviation of the momentum spread is proportional to the equilibrium correlation of the total momentum flux. Subtracting the part which corresponds to a ballistic momentum spread relates (via this integrated, subleading momentum flux correlation) to an effective viscosity, or equivalently, to the underlying momentum diffusivity. We next put forward the intriguing hypothesis: normal spread of this so adjusted excess momentum density causes normal energy spread and alike normal heat transport (Fourier Law). Its corollary being that an anomalous, superdiffusive broadening of this adjusted excess momentum density in turn implies an anomalous energy spread and correspondingly anomalous, superdiffusive heat transport. This hypothesis is successfully corroborated within extensive molecular dynamics simulations over large extended time scales. Our numerical validation of the hypothesis involves four distinct archetype classes of nonlinear pair-interaction potentials: (i) a globally bounded pair interaction (the noted coupled rotator model), (ii) unbounded interactions acting at large distances (the coupled rotator model amended with harmonic pair interactions), (iii) the case of a hard point gas with unbounded square-well interactions and (iv) a pair interaction potential being unbounded at short distances while displaying an asymptotic free part (Lennard–Jones model). We compare our findings with recent predictions obtained from nonlinear fluctuating hydrodynamics theory.

Thermodynamic correlation of molecular simulation data

Strategies to fit molecular simulation data-sets to low parameter fundamental equation of state correlations are reported. The Lennard-Jones model system truncated and shifted at interatomic distance rc/σ = 2.5 is used as an example. Homogeneous fluid states, vapour–liquid equilibrium including estimation of the critical point, and the Joule–Thomson inversion curve are investigated. The results suggest that small molecular simulation data-sets at homogeneous states are sufficient to provide consistent thermodynamic information over large portions of the fluid state.

Arrested Spinodal Decomposition in Polymer Brush Collapsing in Poor Solvent

We study the Brownian dynamics of flexible and semiflexible polymer chains densely grafted on a flat substrate, upon rapid quenching of the system when the quality of solvent becomes poor and chains attempt collapse into a globular state. The collapse process of such a polymer brush differs from individual chains, both in its kinetics and its structural morphology. We find that the resulting collapsed brush does not form a homogeneous dense layer, in spite of all chain monomers equally attracting each other via a model Lennard-Jones potential. Instead, a very distinct inhomogeneous density distribution in the plane forms, with a characteristic length scale dependent on the quenching depth (or equivalently, the strength of monomer attraction) and the geometric parameters of the brush. This structure is identical to the spinodal-decomposition structure, however, due to the grafting constraint we find no subsequent coarsening: the established random bundling with characteristic periodicity remains as the apparently equilibrium structure. We compare this finding with a recent field-theoretical model of bundling in a semiflexible polymer brush.

Ultrafast laser irradiation of spherical nanoparticles: molecular-dynamics results on fragmentation and small-angle scattering

Using molecular dynamics simulation we study the response of a spherical nanoparticle to a sudden homogeneous energization, such as effected by ultrashort pulse laser irradiation. We consider a Lennard-Jones model system and two different values of the energization. For the smaller one, the sphere expands while a multitude of voids are created inside; the sphere develops finally into an external shell filled with gas and small clusters. For the higher energization, the sphere expands uniformly and no shell structure is formed. An analysis of the pressure generated confirms that in the latter case the pressure is compressive throughout the sphere expansion, while it is temporarily tensile for the lower energization leading to void formation. The final state of both systems shows the fragmentation of the sphere into a multitude of clusters. With increasing fragmentation the cluster distribution becomes shifted to smaller sizes. Simulated small-angle scattering functions of the exploding NP are presented. The distribution of minima allows for an easy determination of the particle size during expansion.

Computing the crystal growth rate by the interface pinning method.

An essential parameter for crystal growth is the kinetic coefficient given by the proportionality between supercooling and average growth velocity. Here, we show that this coefficient can be computed in a single equilibrium simulation using the interface pinning method where two-phase configurations are stabilized by adding a spring-like bias field coupling to an order-parameter that discriminates between the two phases. Crystal growth is a Smoluchowski process and the crystal growth rate can, therefore, be computed from the terminal exponential relaxation of the order parameter. The approach is investigated in detail for the Lennard-Jones model. We find that the kinetic coefficient scales as the inverse square-root of temperature along the high temperature part of the melting line. The practical usability of the method is demonstrated by computing the kinetic coefficient of the elements Na and Si from first principles. A generalized version of the method may be used for computing the rates of crystal nucleation or other rare events.

Structural and Dynamical Properties of POPC Bilayers Supported on Nanoporous Substrates

The structure, dynamics and organization of lipid bilayers depend on a variety of environmental factors. Here we examine, using molecular dynamics (MD) simulations, the specific effects of nanoporous substrates on palmitoyl-oleyl phosphatidylcholine (POPC) bilayers. We expose POPC bilayers unilaterally and separately to various model Lennard-Jones (LJ) solid substrates differing in surface chemistry. In an earlier study, we had found that substrates with surface hydroxyl densities in the range 10-20% kept POPC bilayers juxtaposed to the substrates. While the bilayers did not interact directly with these hydroxylated substrates, they were separated from substrates by thin (< 0.6 nm) layers of water. Additionally, despite such buffered interactions, these supported bilayers exhibited physical properties notably different from unsupported bilayers. Recent experiments show that zwitterionic lipid bilayers supported on partially charged, nanoporous silica wafers also do not interact with silica directly. However, in such cases the bilayer is separated from the silica substrate by a relative thicker (∼1.5 nm) layer of solvent. We report here results from our MD simulation that reconcile these two seemingly disparate observations. We find that LJ nano-substrates whose porosity and surface hydroxyl density match those of nanoporous silica wafers do not increase the substrate-bilayer distance. Instead, the introduction of partial charge on the LJ substrate increases the substrate-bilayer distance to 1.5 nm. The negative charge on the substrate attracts hydrated ions, creating an electric double layer, which is a sufficient condition to increase the substrate-bilayer distance. While the properties of the bilayer supported on a charged substrate are also different from those of unsupported bilayers, the extent of the perturbation is not as large compared to that induced by uncharged hydroxylated substrates.

Molecular dynamics investigation of nanoscale cavitation dynamics.

We use molecular dynamics simulations to investigate the cavitation dynamics around intensely heated solid nanoparticles immersed in a model Lennard-Jones fluid. Specifically, we study the temporal evolution of vapor nanobubbles that form around the solid nanoparticles heated over ps time scale and provide a detail description of the following vapor formation and collapse. For 8 nm diameter nanoparticles we observe the formation of vapor bubbles when the liquid temperature 0.5-1 nm away from the nanoparticle surface reaches ∼90% of the critical temperature, which is consistent with the onset of spinodal decomposition. The peak heat flux from the hot solid to the surrounding liquid at the bubble formation threshold is ∼20 times higher than the corresponding steady state critical heat flux. Detailed analysis of the bubble dynamics indicates adiabatic formation followed by an isothermal final stage of growth and isothermal collapse.