Molecular dynamics simulation in the grand canonical ensemble

An extended system Hamiltonian is proposed to perform molecular dynamics (MD) simulation in the grand canonical ensemble. The Hamiltonian is similar to the one proposed by Lynch and Pettitt (Lynch and Pettitt, J Chem Phys 1997, 107, 8594), which consists of the kinetic and potential energies for real and fractional particles as well as the kinetic and potential energy terms for material and heat reservoirs interacting with the system. We perform a nonlinear scaling of the potential energy parameters of the fractional particle, as well as its mass to vary the number of particles dynamically. On the basis of the equations of motion derived from this Hamiltonian, an algorithm has been proposed for MD simulation at constant chemical potential. The algorithm has been tested for the ideal gas, for the Lennard-Jones fluid over a wide range of temperatures and densities, and for water. The results for the low-density Lennard-Jones fluid are compared with the predictions from a truncated virial equation of state. In the case of the dense Lennard-Jones fluid and water our predicted results are compared with the results reported using other available methods for the calculation of the chemical potential. The method is also applied to the case of vapor-liquid coexistence point predictions.     

Size-effects on simulations in the grand canonical ensemble

Results are presented for the effect of periodic boundary conditions on predictions made using the grand canonical ensemble for systems of limited size. A Monte Carlo study of a realistic gas-solid adsorption model and an exact study of the Tonks gas both show that it is necessary for the minor dimension of the replicated system to be 5 σ or greater if errors in the partition function equivalent to δμ > 0.2 kT are to be avoided. The heat capacity C_μ has similar requirements. However, for quantities such as the isosteric heat of adsorption, <N > and <U > even a dimension as small as 3σ does not lead to serious errors. An examination is presented of possible implications for studies of phase changes.     

Isotropic-nematic interfacial tension of hard and soft rods: application of advanced grand canonical biased-sampling techniques

Coexistence between the isotropic and the nematic phase in suspensions of rods is studied using grand canonical Monte Carlo simulations with a bias on the nematic order parameter. The biasing scheme makes it possible to estimate the interfacial tension gamma(IN) in systems of hard and soft rods. For hard rods with LD=15, we obtain gammaIN approximately 1.4kBT/L2, with L the rod length, D the rod diameter, T the temperature, and kB the Boltzmann constant. This estimate is in good agreement with theoretical predictions, and the order of magnitude is consistent with experiments.     

Simulating prescribed particle densities in the grand canonical ensemble using iterative algorithms

We present two efficient iterative Monte Carlo algorithms in the grand canonical ensemble with which the chemical potentials corresponding to prescribed (targeted) partial densities can be determined. The first algorithm works by always using the targeted densities in the kT log(rho(i)) (ideal gas) terms and updating the excess chemical potentials from the previous iteration. The second algorithm extrapolates the chemical potentials in the next iteration from the results of the previous iteration using a first order series expansion of the densities. The coefficients of the series, the derivatives of the densities with respect to the chemical potentials, are obtained from the simulations by fluctuation formulas. The convergence of this procedure is shown for the examples of a homogeneous Lennard-Jones mixture and a NaCl-CaCl(2) electrolyte mixture in the primitive model. The methods are quite robust under the conditions investigated. The first algorithm is less sensitive to initial conditions.     

Acceleration of Monte Carlo simulations through spatial updating in the grand canonical ensemble

A new grand canonical Monte Carlo algorithm for continuum fluid models is proposed. The method is based on a generalization of sequential Monte Carlo algorithms for lattice gas systems. The elementary moves, particle insertions and removals, are constructed by analogy with those of a lattice gas. The updating is implemented by selecting points in space (spatial updating) either at random or in a definitive order (sequential). The type of move, insertion or removal, is deduced based on the local environment of the selected points. Results on two-dimensional square-well fluids indicate that the sequential version of the proposed algorithm converges faster than standard grand canonical algorithms for continuum fluids. Due to the nature of the updating, additional reduction of simulation time may be achieved by parallel implementation through domain decomposition.     

A new grand canonical ensemble method to calculate first-order phase transitions

A theory about first-order phase transition of pure fluids is proposed. The theory is developed by combining grand canonical ensemble with density functional for homogeneous fluids. It is based on the fact that the grand partition function of one macroscopic volume is the product of the grand partition functions of its subvolumes. Density fluctuations of molecules determine the relation between the grand partition function and the free energy density. By combining pairs of subvolumes successively, the free energy density is transformed and rapidly becomes stationary. The stationary curve versus molecule density is convex and its linear segments represent phase transitions. The transform leads to the new grand canonical method to calculate phase equilibrium, which is more robust than classic ones. The transform suggests that classical van der Waals loop is physical and essential to phase transition.     

Dissipative particle dynamics simulations in the grand canonical ensemble: applications to polymer brushes.

We have used the dissipative particle dynamics (DPD) method in the grand canonical ensemble to study the compression of grafted polymer brushes in good solvent conditions. The force-distance profiles calculated from DPD simulations in the grand canonical ensemble are in very good agreement with the self-consistent field (SCF) theoretical models and with experimental results for two polystyrene brush layers grafted onto mica surfaces in toluene.     

Computation of interfacial properties via grand canonical transition matrix Monte Carlo simulation

We examine two free-energy-based methods for studying the wetting properties of a fluid in contact with a solid substrate. Application of the first approach involves examination of the adsorption behavior of a fluid at a single substrate, while the second technique requires investigation of the properties of a system confined between two parallel substrates. Both of the techniques rely upon computation and analysis of the density dependence of a system's surface free energy and provide the contact angle and solid-vapor and solid-liquid interfacial tensions for substrate-fluid combinations within the partial wetting regime. Grand canonical transition matrix Monte Carlo simulation is used to obtain the required free-energy curves. The methods examined within this work are general and are applicable to a wide range of molecular systems. We probe the performance of the methods by computing the interfacial properties for two systems in which an atomistic fluid interacts with a fcc crystal. For both of the systems studied we find good agreement between our results and those obtained via the mechanical definition of the interfacial tension.     

On a new self-consistent-field theory for the canonical ensemble

A significant amount of many-body problems of quantum or classical equilibrium statistical mechanics are conveniently treated at fixed temperature and system size. In this paper, we present a new functional integral approach for solving canonical ensemble problems over the entire coupling range, relying on the method of Gaussian equivalent representation of Efimov and Ganbold. We demonstrate its suitability and competitiveness for performing approximate calculations of thermodynamic and structural quantities on the example of a repulsive potential model, widely used in soft matter theory.     

Calculation of critical temperature Tc in high temperature superconductivity using grand canonical ensemble

In this paper we use the grand canonical ensemble to calculate the T_c of superconductivity for the compound YBa₂Cu₃O₇ under one dimensional Cu–O chain model. The results are consistent with experiments.     

Solvation of sodium-chloride ion pair in water cluster at atmospheric conditions: grand canonical ensemble Monte Carlo simulation

Open statistical ensemble simulations are used to study the mechanism of nucleation of atmospheric water on sodium-chloride ion pair in a wide range of temperature and relative humidity values. The extended simple point-charge model is used for water molecules. Ions-water nonadditive interactions are taken into account by introducing the mutual polarization of ions and water in the field of each other. Gibbs free-energy variations are calculated from Na+-Cl- pair-correlation function and used as a criterion for determining the possible stable states of the cluster. In this relation, it was found that the dissociation of ion pairs in water clusters occurs even at vapor pressures of only a few millibars. In the conditions under consideration solvent-separated ion-pair states are found to be more probable than contact ion-pair configurations. The susceptibilities of water and ions are found to play an essential role in the stabilization of ions at large separations. The structure of ion-induced clusters is analyzed in terms of binary correlation functions. The non-pair interactions influence essentially the structure of ion solvation shells. The results of simulation show that the separation of the charges in water clusters containing simple ions can take place under atmospheric conditions.     

A comparison between two massively parallel algorithms for Monte Carlo computer simulation: An investigation in the grand canonical ensemble

We present a comparison between two different approaches to parallelizing the grand canonical Monte Carlo simulation technique (GCMC) for classical fluids: a spatial decomposition and a time decomposition. The spatial decomposition relies on the fact that for short-ranged fluids, such as the cut and shifted Lennard-Jones potential used in this work, atoms separated by a greater distance than the reach of the potential act independently, and thus different processors can work concurrently in regions of the same system which are sufficiently far apart. The time decomposition is an exactly parallel approach which employs simultaneous (GCMC) simulations, one per processor, identical in every respect except the initial random number seed, with the thermodynamic output variables averaged across all processors. While scaling characteristics for the spatial decomposition are presented for 8–1024 processor systems, the comparison between the two decompositions is limited to the 8–128 processor range due to the warm-up time and memory imitations of the time decomposition. Using a combination of speed and statistical efficiency, the two algorithms are compared at two different state points. While the time decomposition reaches a given value of standard error in the system's potential energy more quickly than the spatial decomposition for both densities, the warm-up time demands of the time decomposition quickly become insurmountable as the system size increases. © 1996 by John Wiley & Sons, Inc.     

On the curvature dependence of the interfacial tension in a symmetric three-component interface

We consider a symmetric interface between two polymers A(N) and B(N) in a common monomeric solvent S using the mean-field Scheutjens-Fleer self-consistent field theory and focus on the curvature dependence of the interfacial tension. In multi-component systems there is not one unique scenario to curve such an interface. We elaborate on this by keeping either the chemical potential of the solvent or the bulk concentration of the solvent fixed, that is we focus on the semi-grand canonical ensemble case. Following Helfrich, we expand the surface tension as a Taylor series in the curvature parameters and find that there is a non-zero linear dependence of the interfacial tension on the mean curvature in both cases. This implies a finite Tolman length. In a thermodynamic analysis we prove that the non-zero Tolman length is related to the adsorption of solvent at the interface. Similar, but not the same, correlations between the solvent adsorption and the Tolman length are found in the two scenarios. This result indicates that one should be careful with symmetry arguments in a Helfrich analysis, in particular for systems that have a finite interfacial tension: one not only should consider the structural symmetry of the interface, but also consider the constraints that are enforced upon imposing the curvature. The volume fraction of solvent, the chain length N as well as the interaction parameter chi(AB) in the system can be used to take the system in the direction of the critical point. The usual critical behavior is found. Both the width of the interface and the Tolman length diverge, whereas the density difference between the two phases, adsorbed amount of solvent at the interface, interfacial tension, spontaneous curvature, mean bending modulus as well as the Gaussian bending modulus vanish upon approach of the critical 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.     

Storage of hydrogen at 303 K in graphite slitlike pores from grand canonical Monte Carlo simulation

Grand canonical Monte Carlo (GCMC) simulations were used for the modeling of the hydrogen adsorption in idealized graphite slitlike pores. In all simulations, quantum effects were included through the Feynman and Hibbs second-order effective potential. The simulated surface excess isotherms of hydrogen were used for the determination of the total hydrogen storage, density of hydrogen in graphite slitlike pores, distribution of pore sizes and volumes, enthalpy of adsorption per mole, total surface area, total pore volume, and average pore size of pitch-based activated carbon fibers. Combining experimental results with simulations reveals that the density of hydrogen in graphite slitlike pores at 303 K does not exceed 0.014 g/cm(3), that is, 21% of the liquid-hydrogen density at the triple point. The optimal pore size for the storage of hydrogen at 303 K in the considered pore geometry depends on the pressure of storage. For lower storage pressures, p < 30MPa, the optimal pore width is equal to a 2.2 collision diameter of hydrogen (i.e., 0.65 nm), whereas, for p congruent with 50MPa, the pore width is equal to an approximately 7.2 collision diameter of hydrogen (i.e., 2.13 nm). For the wider pores, that is, the pore width exceeds a 7.2 collision diameter of hydrogen, the surface excess of hydrogen adsorption is constant. The importance of quantum effects is recognized in narrow graphite slitlike pores in the whole range of the hydrogen pressure as well as in wider ones at high pressures of bulk hydrogen. The enthalpies of adsorption per mole for the considered carbonaceous materials are practically constant with hydrogen loading and vary within the narrow range q(st) congruent with 7.28-7.85 kJ/mol. Our systematic study of hydrogen adsorption at 303 K in graphite slitlike pores gives deep insight into the timely problem of hydrogen storage as the most promising source of clean energy. The calculated maximum storage of hydrogen is equal to approximately 1.4 wt %, which is far from the United States Department of Energy (DOE) target (i.e., 6.5 wt %), thus concluding that the total storage amount of hydrogen obtained at 303 K in graphite slitlike pores of carbon fibers is not sufficient yet.     

Comparison of the capillary wave method and pressure tensor route for calculation of interfacial tension in molecular dynamics simulations

We have studied the calculation of surface and interfacial tension for a variety of liquid–vapor and liquid–liquid interfaces using molecular dynamics (MD) simulations. Because of the inherently small scale of MD systems, large pressure fluctuations can cause imprecise calculations of surface tension using the pressure tensor route. The capillary wave method exhibited improved precision and stability throughout all of the simulated systems in this study. In order to implement this method, the interface was defined by fitting an error function to the density profile. However, full mapping of the interface from coordinate files produced enhanced accuracy. Upon increasing the system size, both methods exhibited higher precision, although the capillary wave method was still more reliable. © 2013 Wiley Periodicals, Inc.     

On phase equilibria,interfacial tension and phase growth in ternary polymer blends

A gradient squared free energy functional of the Landau-Ginzburg type is combined with Flory-Huggins theory to calculate minimum domain sizes, concentration profiles and interfacial tensions in ternary polymer blends. The dynamic equations governing spinodal decomposition are linearized to show that the minimum size for growth is identical to the thermodynamic minimum on phase volume. It is shown that unseparated, third components are enriched at the interface, reduce interfacial tension, increase stability and increase the minimum domain sizes. Enrichment of the third component at the interface causes concentrations at the major components to lie outside their binodal limits at a distance from the interface. Although the effects are most pronounced when the third component is a compatibilizer, the general phenomena remain true even when the third component is relatively incompatible. Generalizations to blends of N components are presented, and a robust method for calculating multicomponent phase diagrams is described.