A general relation between the largest nucleus and all nuclei distributions for free energy calculations.

Prediction of nucleation rates in first order phase transitions requires the knowledge of the barrier associated with the free energy profile W. Molecular simulations offer a direct route through W = -kT ln pa, where k is Boltzmann's constant, T is temperature, and pa is the probability distribution of the size of any nucleus. However, in practice, the extremely scarce spontaneous occurrence of large nuclei impedes the full determination of pa, and a numerical bias must be introduced, which is generally done on the size of the largest nucleus in the system, leading to the probability size distribution of the largest nucleus pl. Although pl is known to be system size dependent, unlike pa, it has extensively been used as an approximation for pa. This communication demonstrates an improved relation between pa and pl, which cures this approximation and allows an accurate calculation of free energy barriers from biased simulations.

Silica-induced electron loss of silver nanoparticles.

Despite the frequent use of silver nanoparticles (Ag NPs) embedded in materials for medical or optical applications, the effect of the matrix on the nanoparticle properties remains largely unknown. This study aims to shed light on the effect of an amorphous silica matrix on the structure and charge distribution of 55- and 147-atom silver nanoparticles by means of dispersion-corrected DFT calculations. Particular attention is paid to nanoparticle size and concentration effects and to the impact of the presence of native defects in the matrix. Covalent bonding between the silver nanoparticles and the matrix is found to occur at the interface. Such interface reconstruction involves the breaking of Si-O bonds, which systematically leads to the formation of Ag-Si bonds, and in some cases, to the formation of Ag-O ones. Interestingly, these interface reconstructions are accompanied by electron depletion of the nanoparticles, a substantial number of electrons being transferred from the two outer shells of the Ag NPs to the surrounding silica medium. The electrons lost by the nanoparticles are captured by the Si atoms involved in the interface bonds, but also, unexpectedly, by the undercoordinated silica defects that act as electron pumps and by the atoms of the silica network inside a few angströms spherical shell around the silver nanoparticle. The numbers of interface bonds and electrons transferred to the surrounding silica shell appear to be proportional to the surface area of the Ag NP. The electronic extension within silica goes beyond that attributable to the Ag NP spill-out. The presence of additional electrons in the matrix, especially on defects, is consistent with the experimental literature.

Finite-size corrections in simulation of dipolar fluids.

Monte Carlo simulations of dipolar fluids are performed at different numbers of particles N = 100-4000. For each size of the cubic cell, the non-spherically symmetric pair distribution function g(r,Ω) is accumulated in terms of projections gmnl(r) onto rotational invariants. The observed N dependence is in very good agreement with the theoretical predictions for the finite-size corrections of different origins: the explicit corrections due to the absence of fluctuations in the number of particles within the canonical simulation and the implicit corrections due to the coupling between the environment around a given particle and that around its images in the neighboring cells. The latter dominates in fluids of strong dipolar coupling characterized by low compressibility and high dielectric constant. The ability to clean with great precision the simulation data from these corrections combined with the use of very powerful anisotropic integral equation techniques means that exact correlation functions both in real and Fourier spaces, Kirkwood-Buff integrals, and bridge functions can be derived from box sizes as small as N ≈ 100, even with existing long-range tails. In the presence of dielectric discontinuity with the external medium surrounding the central box and its replica within the Ewald treatment of the Coulombic interactions, the 1/N dependence of the gmnl(r) is shown to disagree with the, yet well-accepted, prediction of the literature.

On the Thermodynamics and Experimental Control of Twinning in Metal Nanocrystals.

This work demonstrates a new strategy for controlling the evolution of twin defects in metal nanocrystals by simply following thermodynamic principles. With Ag nanocrystals supported on amorphous SiO2 as a typical example, we establish that twin defects can be rationally generated by equilibrating nanoparticles of different sizes through heating and then cooling. We validate that Ag nanocrystals with icosahedral, decahedral, and single-crystal structures are favored at sizes below 7 nm, between 7 and 11 nm, and greater than 11 nm, respectively. This trend is then rationalized by computational studies based on density functional theory and molecular dynamics, which show that the excess free energy for the three equilibrium structures correlate strongly with particle size. This work not only highlights the importance of thermodynamic control but also adds another synthetic method to the ever-expanding toolbox used for generating metal nanocrystals with desired properties.

Influence of system size on the properties of a fluid adsorbed in a nanopore: Physical manifestations and methodological consequences.

Hysteresis and discontinuities in the isotherms of a fluid adsorbed in a nanopore in general hamper the determination of equilibrium thermodynamic properties, even in computer simulations. A way around this has been to consider both a reservoir of small size and a pore of small extent in order to restrict the fluctuations of density and approach a classical van der Waals loop. We assess this suggestion by thoroughly studying through Monte Carlo simulations and density functional theory the influence of system size on the equilibrium configurations of the adsorbed fluid and on the resulting isotherms. We stress the importance of pore-symmetry-breaking states that even for modest pore sizes lead to discontinuous isotherms and we discuss the physical relevance of these states and the methodological consequences for computing thermodynamic quantities.

Bridge function for the dipolar fluid from simulation.

The exact bridge function of the Lennard-Jones dipolar (Stockmayer) fluid is extracted from Monte Carlo simulation data. The projections g(mnl)(r) onto rotational invariants of the non-spherically symmetric pair distribution function g(r, Ω) are accumulated during simulation. Making intensive use of anisotropic integral equation techniques, the molecular Ornstein-Zernike equation is then inverted in order to derive the direct correlation function c(mnl)(r), the cavity function y(mnl)(r), the negative excess potential of mean force lny|(mnl)(r), and the bridge function b(mnl)(r) projections. b(r, Ω) presents strong, non-universal anisotropies at high dipolar coupling. This simulation data analysis may serve as reference and guide for approximated bridge function theories of dipolar fluids and is a valuable step towards the case of more refined, nonlinear water-like geometries.

Numerical characterization of the density of metastable states within the hysteresis loop in disordered systems.

An improved approach is proposed to analyze the density of metastable states within any hysteresis loop, such as those observed in magnetic materials or for adsorption in porous materials. Except for a few analytically tractable models, most calculations have to be performed numerically on finite systems. The main points to be addressed thus concern the average over various material samples (the so-called realizations of the disorder), and the finite size analysis to estimate the thermodynamic limit. As an improvement of previously existing methods, it is proposed to introduce the Fourier transform of the density of metastable states (characteristic function). Its logarithm is shown to be additive and can straightforwardly be averaged over disorder. This procedure leads to a new definition of the complexity in finite size, giving the usual quenched complexity in the thermodynamic limit, while being better suited to performing finite size analysis. The calculations are illustrated on a molecular simulation based model for a simple fluid adsorbed in heterogeneous siliceous tubular pores mimicking mesoporous materials like MCM-41 or porous silicon. This approach is expected to be of general interest for hysteresis phenomena, including magnetic materials.

Stability intervals of metastable states in hysteretic systems.

Hysteresis in disordered systems originates in a plethora of metastable states. Previous works focused on their distribution inside the hysteresis. In contrast, an analysis of their range of metastability is proposed. This model, designed to catch the main features of fluid adsorption in porous materials, shows strong evidence that, in the thermodynamic limit, despite that metastable states of finite range can be found, they are exponentially dominated by those infinitely localized states.

Counting metastable states within the adsorption/desorption hysteresis loop: A molecular simulation study of confinement in heterogeneous pores.

A molecular simulation approach has been used to model simple fluid adsorption in heterogeneous tubular pores mimicking mesoporous materials such as MCM-41 or porous silicon, allowing to determine the amount adsorbed ρ as a function of the chemical potential µ. A hysteresis loop is observed in adsorption/desorption cycles, which is closely connected to the appearance of many metastable states. The density of these metastable states is studied in the µ-ρ plane. Experimentally, the accessible metastable states are those that can be attained by the µ-path, i.e., a series of increasing or decreasing µ steps. One could also imagine using a quench from high temperature. Although the total density of metastable states is not directly accessible to experiments, it is of primary theoretical importance to understand the structure of metastable states in the hysteresis as determined experimentally. The disorder associated with the porous material realizations is accurately taken into account, and a systematic system size analysis is also performed in order to study the thermodynamic limit. It is shown that the quenched complexity is the relevant quantity to understand the hysteresis structure in the thermodynamic limit. It clearly exhibits a distinctive behavior depending on the distribution of heterogeneities characterizing the disorder in the pore. Some analogies can be found with the situation where an out-of-equilibrium transition appears, but careful examination of the data suggests another interpretation.

Influence of reservoir size on the adsorption path in an ideal pore.

We consider the influence of the relative size of the gas reservoir on the states visited by a simple fluid adsorbed in a nanopore of ideal geometry (a slit). We focus on the intermediate states that appear in between the main hysteresis branches comprising gaslike and liquidlike states and we study the adsorption and desorption paths actually followed by the system as one changes the reservoir size. We find that these paths may display discontinuous sections associated with transitions between different nonuniform states. We also discuss the stability of the states in such situations.

Monte-Carlo multiscale simulation study of argon adsorption/desorption hysteresis in mesoporous heterogeneous tubular pores like MCM-41 or oxidized porous silicon.

In a recent paper [J. Chem. Phys. 2007, 127, 154701] a multiscale approach was introduced which allowed calculation of adsorption/desorption hysteresis for fluid confined in a single mesoporous, heterogeneous tubular pore. The main interest in using such an approach is that it allows one to reconcile a molecular simulation approach generally limited to the nanometer scale (atomistic description of the confined fluid and pore roughness) with the much larger scale (micrometer) relevant to understand the complexity of adsorption/desorption hysteresis (the numerous metastable states in the hysteresis loop are a consequence of the large-scale disorder in the porous material). In this paper, this multiscale approach is used to study adsorption phenomena in mesoporous models made of a collection of disordered, noninterconnected tubular pores, as MCM-41 or porous silicon. A double distribution is introduced: one to characterize the disorder in a given pore, and the other to characterize the disorder between the pores. We consider two distribution shapes: Gaussian and uniform truncated and two cases of pores open at one or both ends. These models are expected to cover a wide variety of real materials made of independent pores, as MCM-41 and oxidized porous silicon. A large variety of hysteresis shapes is obtained, ranging from almost parallel adsorption/desorption branches typical of MCM-41 adsorption to triangular hysteresis typical of porous silicon. The structure of the metastable states inside the hysteresis (scanning adsorption/desorption curves) is also examined. The results are expected to be useful to experimentalists who want to infer pore structure and level of disorder from experimental adsorption/desorption experiments.

Pseudocritical or hysteresis temperature versus pore size for simple fluids confined in cylindrical nanopores.

The adsorption/desorption isotherms measured in nanoporous materials generally present a hysteresis. The hysteresis shrinks upon increasing the temperature (for a given pore size) or decreasing the pore size (for a given temperature), until it finally disappears at the so-called hysteresis (or pseudocritical) temperature T(h) or hysteresis (or pseudocritical) pore size R(h), not to be confused with a true critical point. In this paper, a Monte Carlo approach allowed calculating the surface free energy of confined fluid along the adsorption/desorption isotherms for various cylindrical pore sizes and temperatures. A simple phenomenological model then allowed exploiting these results to determine the relation between T(h) and R(h). The prediction is compared to various literature models and experimental data, showing agreement within uncertainties. On the other hand, the simulations cannot be used directly to predict T(h) and R(h) since they significantly overestimate the hysteresis width. The model predicts a nonlinear relation between the reduced hysteresis temperature and the inverse pore radius.

Grand canonical monte carlo simulation study of water adsorption in silicalite at 300 K.

This molecular simulation work focuses on the adsorption of water in a priori hydrophobic silicalite-1, a microporous ordered silica. The water-water interactions are described with the SPC model, while water-silica interactions are calculated in the framework of the PN-TrAZ model. The water adsorption isotherm at 300 K, the configurational energies, and the isosteric heat of adsorption are calculated by the grand canonical Monte Carlo (GCMC) simulation method. The thermodynamic integration scheme allows one to calculate the grand potential along the adsorption isotherm. The adsorption results are compared with experiments, showing only qualitative agreement. Indeed, the simulations do not reproduce the expected hydrophobicity of silicalite (Eroshenko, V.; Regis, R.-C.; Soulard, M.; Patarin, J. C. R. Phys. 2002, 3, 111). This indicates that common models used to describe confined polar molecules are far from being operative. In this work, it is shown, on the basis of periodic ab initio calculations, that confined water molecules in silicalite have a dipole value roughly 10% smaller than that in the bulk liquid phase, indicating that the environment felt by a confined water molecule in silicalite pores is not equivalent to that in the bulk liquid. This suggests that effective intermolecular potentials parametrized for bulk water are inefficient to describe ultraconfined water molecules. Reducing the SPC water dipole moment by 5% (i.e., decreasing water partial charges in magnitude) in GCMC calculations does allow reproducing the experimental water/silicalite isotherm at 300 K.

Adsorption/desorption hysteresis of simple fluids confined in realistic heterogeneous silica mesopores of micrometric length: a new analysis exploiting a multiscale Monte Carlo approach.

Adsorption/desorption isotherms in porous materials are commonly used for characterization. In order to analyze the data, accurate calculations of fluid adsorption in various complex pore models are required. The reversible, low adsorption portion of the isotherm is generally well described by molecular simulation, since the relevant fluid/substrate interactions are described at molecular level. This molecular approach is, however, ineffective in the hysteresis region because the large scale spatial distribution of heterogeneities in the pore network is beyond the computer capabilities. On the other hand, coarse grained approaches are more suited to take into account this porous network complexity at large scale and discuss the hysteresis nature, but the molecular description is lost. In this paper, a multiscale approach is introduced which allows both a molecular description of fluid/substrate interactions, and taking into account the connectivity between the various domains in a porous material. The case of argon confined in heterogeneous tubular silica mesopores (MCM-41 or oxidized porous silicon) is considered. Comparison with the simple independent domain theory shows the strong influence of quenched disorder. It is also shown that the independent pore model significantly overestimates the hysteresis width. The effect of pore ends open at only one or at both ends is addressed.

Thermodynamic pressure of simple fluids confined in cylindrical nanopores by isothermal-isobaric Monte Carlo: influence of fluid/substrate interactions.

The thermodynamic pressure or grand potential density is calculated by isobaric-isothermal Monte Carlo algorithm for simple Lennard-Jones fluid confined in cylindrical pores presenting chemical heterogeneities along their axis. Heuristic arguments and simulation results show that the thermodynamic pressure of the confined fluid contains two contributions. The first term is the usual pressure of the bulk fluid for a density equal to the confined fluid density defined as the total number of confined particles divided by the accessible volume due to thermal agitation. A second term has to be added, which is empirically shown to be proportional to the fluid/wall interface area and almost constant along the adsorption and desorption branches. This interfacial contribution, calculated for various pore models, has small variations reminiscent of the fluid adsorption/desorption properties calculated in the various pores. In particular, it is shown that this interfacial quantity is maximum for a fluid/substrate interaction intensity of the same order as the fluid/fluid one, while the thermodynamic pressure at which rapid desorption occurs presents a minimum. Stronger or weaker fluid/wall affinity favors gas state nucleation on the desorption of confined fluids.

Surface excess free energy of simple fluids confined in cylindrical pores by isothermal-isobaric Monte Carlo: influence of pore size.

Confined fluid properties are mainly determined by interfacial phenomena characterized by surface quantities. Based on a simple model of Lennard-Jones particles confined in a cylindrical pore, this study introduces a grand potential surface quantity to quantify the difference in the thermodynamic pressure between the bulk and the confined fluids. The usual surface tension gamma defined as this grand potential difference for the same chemical potential in both confined and bulk states is generally strongly dependent on both the chemical potential and temperature. It is proposed here to introduce another surface quantity zeta which measures the thermodynamic pressure difference between confined and bulk states for identical densities. It is shown that this quantity is much less dependent on confined fluid density or chemical potential. It is actually constant along the gas-like and liquid-like adsorption/desorption branches for an irreversible isotherm (hysteresis), with a different value for each branch. For reversible supercritical isotherms, zeta is shown to remain constant in the low and high density parts of the isotherm. This independence on chemical potential (or equivalently fluid density) is believed to be of great interest for practical applications when one desires to calculate thermodynamic quantities such as the usual surface tension gamma or the thermodynamic pressure of a confined fluid for any given chemical potential and temperature. Such calculations are required to determine fundamental properties such as metastability or coexistence. The effects of temperature, fluid/substrate interaction strength, and pore size are studied.

Influence of surface chemical heterogeneities on adsorption/desorption hysteresis and coexistence diagram of metastable states within cylindrical pores.

Grand canonical Monte Carlo simulations are performed to determine the adsorption/desorption isotherms at different temperatures of a Lennard-Jones fluid confined within a simple model of cylindrical pores presenting chemical heterogeneities. A complex hysteresis loop is observed, showing hysteresis subloops (scanning curves). This is shown to be consistent with the existence of several metastable states (local minima in the system free energy). A recent extension to the Gibbs ensemble technique is then used to calculate the complete coexistence diagram of these local minima.

Phase coexistence in heterogeneous porous media: a new extension to Gibbs ensemble Monte Carlo simulation method.

The effect of confinement on phase behavior of simple fluids is still an area of intensive research. In between experiment and theory, molecular simulation is a powerful tool to study the effect of confinement in realistic porous materials, containing some disorder. Previous simulation works aiming at establishing the phase diagram of a confined Lennard-Jones-type fluid, concentrated on simple pore geometries (slits or cylinders). The development of the Gibbs ensemble Monte Carlo technique by Panagiotopoulos [Mol. Phys. 61, 813 (1987)], greatly favored the study of such simple geometries for two reasons. First, the technique is very efficient to calculate the phase diagram, since each run (at a given temperature) converges directly to an equilibrium between a gaslike and a liquidlike phase. Second, due to volume exchange procedure between the two phases, at least one invariant direction of space is required for applicability of this method, which is the case for slits or cylinders. Generally, the introduction of some disorder in such simple pores breaks the initial invariance in one of the space directions and prevents to work in the Gibbs ensemble. The simulation techniques for such disordered systems are numerous (grand canonical Monte Carlo, molecular dynamics, histogram reweighting, N-P-T+test method, Gibbs-Duhem integration procedure, etc.). However, the Gibbs ensemble technique, which gives directly the coexistence between phases, was never generalized to such systems. In this work, we focus on two weakly disordered pores for which a modified Gibbs ensemble Monte Carlo technique can be applied. One of the pores is geometrically undulated, whereas the second is cylindrical but presents a chemical variation which gives rise to a modulation of the wall potential. In the first case almost no change in the phase diagram is observed, whereas in the second strong modifications are reported.

Water adsorption in disordered mesoporous silica (Vycor) at 300 K and 650 K: a Grand Canonical Monte Carlo simulation study of hysteresis.

This numerical simulation paper focuses on the adsorption/desorption of water in disordered mesoporous silica glasses (Vycor-like). The numerical adsorbent was previously obtained by off lattice method, and was shown to reproduce quite well the micro- and mesotextural properties of real Vycor, as well as morphological (pore size distribution) and topological (pore interconnections) disorder. The water-water interactions are described by the SPC model while water-silica interactions are calculated in the framework of the PN-TrAZ model. The water adsorption/desorption isotherms and the configurational energies are calculated by the Grand Canonical Monte Carlo simulation method. The low pressure results compare well with experiments, showing the good transferability of the intermolecular potential. It is shown that if the hysteresis loop observed in the adsorption/desorption isotherm is considered as a true phase transition (which is actually still an open question in the case of disordered porous materials), then it is possible to calculate the grand potential by applying the thermodynamic integration scheme. The grand potential is shown to be multivalued for low (subcritical) temperature, and continuous for high (supercritical) temperature. A coexistence point is found within the hysteresis loop, actually close to the vertical desorption line. Below the equilibrium chemical potential, the gaslike branch is stable whereas the liquidlike branch is metastable. The situation is reversed above the coexistence point.

Grand potential, helmholtz free energy, and entropy calculation in heterogeneous cylindrical pores by the grand canonical Monte Carlo simulation method.

The adsorption of fluids in porous media is still an open area of research, since no model is able to explain all experimental features. The difficulties rise from the complexity of the real porous materials which present surface heterogeneities, large pore size distributions, and complex networks of interconnected pores. In parallel to experimental efforts trying to produce more ordered porous materials, theoreticians try to introduce more disorder in their models, with the help of molecular simulation for instance. This grand canonical Monte Carlo simulation study concentrates on the adsorption of a simple Lennard-Jones fluid in three porous substrates, to compare the effect of purely geometric heterogeneity (spatial deformation of the external potential) as opposed to purely chemical heterogeneity (amplitude variations of the external potential). This separation is unrealistic, since geometric fluctuations of a real pore diameter along its axis generally induce variations in the amplitude of the external potential created by the pore. However it enables one to compare both effects. In this paper, a thermodynamic integration scheme is applied to a complete set of adsorption/desorption isotherms. The grand potential, free energy, and entropy are calculated, which allows one to discuss the features of the phase diagrams. It is shown that a purely geometric deformation (undulation) of the external potential does not affect the thermodynamic characteristics of the confined fluid. On the other hand, amplitude modulation of the external potential (chemical heterogeneity) strongly distorts the phase diagram. This heterogeneity is actually able to stabilize a "bridgelike" phase which corresponds to an accumulation of molecules in the most attractive region of the pore.