An efficient method for computing excess free energy of liquid

We present a new theoretical method for efficient calculation of free energy of liquid. This interaction entropy method allows one to compute entropy and free energy of liquid from standard single step MD (molecular dynamics) simulation directly in liquid state without the need to perform MD simulations at many intermediate states as required in thermodynamic integration or free energy perturbation methods. In this new approach, one only needs to evaluate the interaction energy of a single (fixed) liquid molecule with the rest of liquid molecules as a function of time from a standard MD simulation of liquid and the fluctuation of distribution of this interaction energy is then used to calculate the interaction entropy of the liquid. Explicit theoretical derivation of this interaction entropy approach is provided and numerical calculations for the benchmark liquid water system were carried out using three different water models. Numerical analysis of the result was performed and comparison of the computational result with experimental data and other theoretical results were provided. Excellent agreement of calculated free energies with the experimental data using TIP4P model is obtained for liquid water.     

On the direct calculation of the free energy of quantization for molecular systems in the condensed phase

Using the path integral formalism or the Feynman-Hibbs approach, various expressions for the free energy of quantization for a molecular system in the condensed phase can be derived. These lead to alternative methods to directly compute quantization free energies from molecular dynamics computer simulations, which were investigated with an eye to their practical use. For a test system of liquid neon, two methods are shown to be most efficient for a direct evaluation of the excess free energy of quantization. One of them makes use of path integral simulations in combination with a single-step free energy perturbation approach and was previously reported in the literature. The other method employs a Feynman-Hibbs effective Hamiltonian together with the thermodynamic integration formalism. However, both methods are found to give less accurate results for the excess free energy of quantization than the estimate obtained from explicit path integral calculations on the excess free energy of the neon liquid in the classical and quantum mechanical limit. Suggestions are made to make both methods more accurate.     

Path integral calculation of free energies: quantum effects on the melting temperature of neon

The path integral formulation has been combined with several methods to determine free energies of quantum many-body systems, such as adiabatic switching and reversible scaling. These techniques are alternatives to the standard thermodynamic integration method. A quantum Einstein crystal is used as a model to demonstrate the accuracy and reliability of these free energy methods in quantum simulations. Our main interest focuses on the calculation of the melting temperature of Ne at ambient pressure, taking into account quantum effects in the atomic dynamics. The free energy of the solid was calculated by considering a quantum Einstein crystal as reference state, while for the liquid, the reference state was defined by the classical limit of the fluid. Our findings indicate that, while quantum effects in the melting temperature of this system are small, they still amount to about 6% of the melting temperature, and are therefore not negligible. The particle density as well as the melting enthalpy and entropy of the solid and liquid phases at coexistence is compared to results obtained in the classical limit and also to available experimental data.     

Melting temperature of Pb nanostructural materials from free energy calculation

The thermodynamic properties of lead, including the entropy, heat capacity, Gibbs free energy, and surface free energy have been studied. Based on bulk thermodynamic properties of lead, Gibbs free energy for nanostructural materials is obtained and used to calculate the size-dependent melting point depression for lead nanostructural materials. The studies indicate that the surface free energy difference between solid phase and liquid phase is a decisive factor for the size-dependent melting of nanostructural materials. The calculated results are in agreement with recent experimental values and the available molecular dynamics simulation data.     

Induced smectic phases in phase diagrams of binary nematic liquid crystal mixtures

To elucidate induced smectic A and smectic B phases in binary nematic liquid crystal mixtures, a generalized thermodynamic model has been developed in the framework of a combined Flory-Huggins free energy for isotropic mixing, Maier-Saupe free energy for orientational ordering, McMillan free energy for smectic ordering, Chandrasekhar-Clark free energy for hexagonal ordering, and phase field free energy for crystal solidification. Although nematic constituents have no smectic phase, the complexation between these constituent liquid crystal molecules in their mixture resulted in a more stable ordered phase such as smectic A or B phases. Various phase transitions of crystal-smectic, smectic-nematic, and nematic-isotropic phases have been determined by minimizing the above combined free energies with respect to each order parameter of these mesophases. By changing the strengths of anisotropic interaction and hexagonal interaction parameters, the present model captures the induced smectic A or smectic B phases of the binary nematic mixtures. Of particular importance is the fact that the calculated phase diagrams show remarkable agreement with the experimental phase diagrams of binary nematic liquid crystal mixtures involving induced smectic A or induced smectic B phase.     

Revisiting the Frenkel-Ladd method to compute the free energy of solids: the Einstein molecule approach

In this paper a new method to evaluate the free energy of solids is proposed. The method can be regarded as a variant of the method proposed by Frenkel and Ladd [J. Chem. Phys. 81, 3188 (1984)]. The main equations of the method can be derived in a simple way. The method can be easily implemented within a Monte Carlo program. We have applied the method to determine the free energy of hard spheres in the solid phase for several system sizes. The obtained free energies agree within the numerical uncertainty with those obtained by Polson et al. [J. Chem. Phys. 112, 5339 (2000)]. The fluid-solid equilibria has been determined for several system sizes and compared to the values published previously by Wilding and Bruce [Phys. Rev. Lett. 85, 5138 (2000)] using the phase switch methodology. It is shown that both the free energies and the coexistence pressures present a strong size dependence and that the results obtained from free energy calculations agree with those obtained using the phase switch method, which constitutes a cross-check of both methodologies. From the results of this work we estimate the coexistence pressure of the fluid-solid transition of hard spheres in the thermodynamic limit to be p*=11.54(4), which is slightly lower than the classical value of Hoover and Ree (p*=11.70) [J. Chem. Phys. 49, 3609 (1968)]. Taking into account the strong size dependence of the free energy of the solid phase, we propose to introduce finite size corrections, which allow us to estimate approximately the free energy of the solid phase in the thermodynamic limit from the known value of the free energy of the solid phase with N molecules. We have also determined the free energy of a Lennard-Jones solid by using both the methodology of this work and the finite size correction. It is shown how a relatively good estimate of the free energy of the system in the thermodynamic limit is obtained even from the free energy of a relatively small system.     

Metastable extension of the liquid-vapor phase equilibrium curve and surface tension

The method of molecular dynamics has been used to calculate the parameters of liquid-vapor phase equilibrium and the surface tension in a two-phase system of 4096 Lennard-Jones particles. Calculations have been made in a range from the triple point to near-critical temperature and also at temperatures below the triple point corresponding to the metastable equilibrium of a supercooled liquid and supersaturated vapor. To determine the surface tension, along with a mechanical approach a thermodynamic one has been used as well. The latter was based on calculation of the excess internal energy of an interfacial layer. It has been shown that in accuracy the thermodynamic approach is as good as the more sophisticated mechanical one. Low-temperature asymptotics of the phase-equilibrium curve and also of liquid and vapor spinodals have been considered in the Lennard-Jones and the van der Waals models. The behavior of the surface tension and the excess internal energy of an interfacial layer at T-->0 is discussed.     

Surface tension of water droplets upon homogeneous droplet nucleation in water vapor

A method has been proposed for determining interfacial free energy from the data of molecular dynamics simulation. The method is based on the thermodynamic integration procedure and is distinguished by applicability to both planar interfaces and those characterized by a high curvature. The workability of the method has been demonstrated by the example of determining the surface tension for critical nuclei of water droplets upon condensation of water vapor. The calculation has been performed at temperatures of 273–373 K and a pressure of 1 atm, thus making it possible to determine the temperature dependence of the surface tension for water droplets and compare the results obtained with experimental data and the simulation results for a “planar” vapor–liquid interface.     

Computer simulation of molecular organization in a nematogen. The role of thermodynamic parameters

Computer simulations of molecular organization in a nematogen, alkenyl bicyclohexylnitrile (ALBCHNL), have been carried out with respect to translational and orientational motions. The atomic net charge and dipole moment at each atomic center has been evaluated using the complete neglect of differential overlap (CNDO/2) method. The modified Rayleigh-Schroedinger perturbation theory, along with multicentered-multipole expansion method, has been employed to evaluate long-range intermolecular interactions, while a “6-exp” potential function has been assumed for short-range interactions. The total interaction energy values obtained through these computations have been used as input to calculate the thermodynamical parameters such as entropy and Helmholtz free energy of each configuration at room temperature (300 K), nematic-isotropic transition (364.7 K) and above transition temperature (450 K). An attempt has been made to understand the molecular organization, to develop a new and interesting model for nematogen based on the thermodynamic parameters introduced in this article.     

Thermodynamics of the extraction of scandium(III) by the liquid cation exchangers dinonylnaphthalenesulfonic acid and bis(2-ethylhexyl) phosphoric acid

The thermodynamic functions for the extraction of Sc³⁺ by liquid cation exchangers HD and HDEHP are determined radiometrically by the temperature coefficient method. The role of diluent dielectric constant on the extraction of Sc³⁺ by HD is also studied. The thermodynamic parameters determined indicated that the free energy variation for the extraction of Sc³⁺ by HD is mainly determined by the entropic terms arising from the hydration—dehydration process of the exchanged ions. In the case of HDEHP as extractant, the free energy variations are determined mainly by the entalpic terms of the system.     

A molecular dynamics study of the lateral free energy profile of a pair of cholesterol molecules as a function of their distance in phospholipid bilayers

Free energy profile of a pair of cholesterol molecules in a leaflet of 1-palmitoyl-2-oleoyl-phosphatidylcholine (POPC) bilayers in the liquid-crystalline phase has been calculated as a function of their lateral distance using a combination of NPT-constant atomistic molecular dynamics calculations (P = 1 atm and T = 310.15 K) and the thermodynamic integration method. The calculated free energy clearly shows that the two cholesterol molecules form a dimer separated by a distance of 1.0-1.5 nm in POPC bilayers. Well depth of the free energy profile is about 3.5 kJ/mol, which is comparable to the thermal energy k(B)T at 310.15 K. This indicates that the aggregation of cholesterol molecules in the bilayers depends on the temperature as well as the concentration of the system. The free energy function obtained here may be used as a reference when coarse grained potential model is investigated for this two-component system. Local structure of POPC molecules around two cholesterol molecules has also been investigated.     

Thermodynamic model of quasiliquid formation on H2O ice: comparison with experiment

We have developed a new thermodynamic theory of the quasiliquid layer, which has been shown to be effective in modeling the phenomenon in a number of molecular systems. Here we extend our analysis to H(2)O ice, which has obvious implications for environmental and atmospheric chemistry. In the model, the liquid layer exists in contact with an ice defined as a two-dimensional lattice of sites. The system free energy is defined by the bulk free energies of ice I(h) and liquid water and is minimized in the grand canonical ensemble. An additional configurational entropy term arises from the occupation of the lattice sites. Furthermore, the theory predicts that the layer thickness as a function of temperature depends only on the liquid activity. Two additional models are derived, where slightly different approximations are used to define the free energy. With these two models, we illustrate the connection between the quasiliquid phenomenon and multilayer adsorption and the possibility of a two-dimensional phase transition connecting a dilute low coverage phase of adsorbed H(2)O and the quasiliquid phase. The model predictions are in agreement with a subset of the total suite of experimental measurements of the liquid thickness on H(2)O ice as a function of temperature. The theory indicates that the quasiliquid layer is actually equivalent to normal liquid water, and we discuss the impact of such an identification. In particular, observations of the liquid layer to temperatures as low as 200 K indicate the possibility that the quasiliquid is, in fact, an example of deeply supercooled normal water. Finally, we briefly discuss the obvious extension of the pure liquid theory to a thermodynamic theory of interfacial solutions on ice in the environment.     

Interlamellar forces and the thermodynamic characterization of lamellar phospholipid systems

In this review, we summarize a series of experimental studies of the swelling of zwitterionic lamellar phospholipid and phospholipid-cholesterol systems using a novel double twin calorimeter. With this method, one can obtain simultaneous measures of the partial molar free energy and the partial molar enthalpy, and the experimental studies thus provide a complete thermodynamic characterization of the isothermal swelling process. A major finding is that the swelling of lamellar zwitterionic phospholipid systems at higher water contents (> 4 water molecules per lipid) is endothermic. The enthalpy has the opposite sign relative to the free energy, thus demonstrating that the swelling process is entropy driven. The water uptake also triggers a transition from a gel to a liquid crystalline state showing that, at given water content, the swelling pressure is much higher in the liquid crystal than in the gel. When cholesterol is added to the system the liquid ordered phase is formed at all available water contents. In this phase the swelling pressure varies smoothly and takes relatively low values at water contents below two per phospholipid, while it is substantially higher than in the gel phase at higher water contents. Together, these data demonstrate that the swelling pressure is sensitive to the phase state of the lipids. We also describe a series of studies that demonstrate that the addition of a second polar solute to the phospholipid–water system has a remarkably small effect on the swelling behavior when analyzed with respect to solvent volume. The reviewed experimental studies provide a thermodynamic characterization of the swelling of lamellar zwitterionic phospholipid systems that should be encompassed in the mechanistic molecular interpretation of the “hydration force.”     

Determination of micelle/water partition coefficients and associated thermodynamic data for dialkyl phthalate esters

Micelle/water partition coefficients for three dialkyl phthalate esters — dimethyl phthalate ester (DMP), diethyl phthalate ester (DEP) and dipropyl phthalate ester (DPP) were obtained by micellar liquid chromatography (MLC). Experiments were conducted over a temperature range which led to calculation of a Gibbs free energy, enthalpy and entropy of transfer for the phthalate esters. In addition, small angle neutron scattering (SANS) experiments were conducted with no substantial change observed in micelle size before and after phthalate ester incorporation. Overall, a novel method for obtaining thermodynamic information, based on partitioning data, has been developed for dialkyl phthalate esters using micellar liquid chromatography.     

Morphology development of main‐chain liquid crystalline polymer fibers during solvent evaporation

A nonequilibrium thermodynamic approach has been developed for describing the emergence of fiber morphologies from a liquid crystalline polymer solution undergoing solvent evaporation, including fibrillar structures, concentric rings, and spiral structures. We utilized Matsuyama–Kato free energy for main‐chain liquid crystalline polymer (MCLCP) solutions, which is an extension of Maier–Saupe theory for nematic ordering and incorporates a chain‐stiffening, combined with Flory‐Huggins free energy of mixing. Temporal evolution of the concentration and nematic order parameters pertaining to the above free energy density of liquid crystalline polymer solution was simulated in the context of time‐dependent Ginzburg–Landau theory coupled with the solvent evaporation rate equation under the quasi‐steady state assumption. The emerged morphological patterns are discussed in relation to the phase diagram of the MCLCP solution and the rate of solvent evaporation. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 429–435, 2007     

Tests of Monte Carlo perturbation theory for the free energy of liquid copper

Monte Carlo perturbation theory, in which terms in the thermodynamic perturbation series are evaluated by Monte Carlo averaging, has potentially large advantages in efficiency for calculating free energies of liquids from ab initio potential surfaces. In order to test the accuracy of perturbation theory for liquid metals, a series of calculations has been done on liquid copper, modeled by an embedded atom potential. A simple 1/r(12) pair potential is used as the reference system. The free energy is calculated to third order in perturbation theory, and the results are compared to an exact formula. It is found that for optimal reference potential parameters, second order perturbation theory is essentially exact. Second and third order theories give accurate results for significantly nonoptimal reference parameters. The relation between perturbation theory and reweighting is discussed, and an approximate formula is derived that shows an exponential dependence of the efficiency of reweighting on the second order free energy correction. Finally, techniques for application to ab initio potentials are discussed. It is shown that with samples of 100 configurations, it is possible to obtain accuracy and precision at the level of approximately 1 meV/atom.     

Pb-Al二元体系液-固界面自由能的热力学理论计算

以复杂的Warren二元及赝二元常规系统下的液-固界面自由能理论为基础, 借助Pb-Al二元体系为例对其进行简化, 获得了二元非混溶体系液-固界面自由能物理模型, 然后对其热力学公式进行推导, 得出只含两个参变量的理论公式, 并对几种温度下液-固界面自由能(γSL)计算值及用多相平衡(MPE)法得到的实验值作了对比. 结果表明, 改进的物理模型及理论公式易于理解、计算简便, γSL的计算值取决于温度及Al原子分数的两个参变量, 与实验值较好地吻合, 证明了该模型具有结构简单、精度较高的优点, 并可作为其它非混溶体系γSL的计算模型, 为其推广应用奠定基础.     

Wall-liquid and wall-crystal interfacial free energies via thermodynamic integration: A molecular dynamics simulation study

A method is proposed to compute the interfacial free energy of a Lennard-Jones system in contact with a structured wall by molecular dynamics simulation. Both the bulk liquid and bulk face-centered-cubic crystal phase along the (111) orientation are considered. Our approach is based on a thermodynamic integration scheme where first the bulk Lennard-Jones system is reversibly transformed to a state where it interacts with a structureless flat wall. In a second step, the flat structureless wall is reversibly transformed into an atomistic wall with crystalline structure. The dependence of the interfacial free energy on various parameters such as the wall potential, the density and orientation of the wall is investigated. The conditions are indicated under which a Lennard-Jones crystal partially wets a flat wall.     

Determination of the Molecular Area on a Liquid Surface from Thermodynamic Functions: Application to Alkanes

A method is proposed to calculate the molecular area on the surface of a liquid from thermodynamic parameters such as the molar internal energy, the surface free energy, and the surface entropy. When the method is applied to the series of normal alkanes, it allows calculation of the area of the molecules on these liquid surfaces and to deduce the orientation of these molecules. Moreover, the molecular areas of the first terms of the alkane series and of hydrogen are also obtained by extrapolation.