Calculating thermodynamic properties from perturbation theory: Part III. The mixtures of square-well chain fluid

A completely analytic perturbation theory has been developed to calculate the Helmholtz energy, compressibility factor, internal energy and constant-volume heat capacity for square-well chain fluid mixtures. This theory is based on the improved Barker–Henderson macroscopic compressibility (mc) approximation proposed by Zhang, the first-order perturbation theory of Wertheim in which Zhang’s analytic monomer radial distribution function as the function of temperature and monomer density is used, and a simple mixing rule similar to that of Hino–Prausnitz. The validity of the perturbation theory is evaluated by comparing the calculated compressibility factor, internal energy and constant-volume heat capacity for the freely jointed square-well chain mixtures from the theory to MC simulation data. The results show that the theory predicts results in good agreement with simulation results.     

Non-hard sphere thermodynamic perturbation theory

A non-hard sphere (HS) perturbation scheme, recently advanced by the present author, is elaborated for several technical matters, which are key mathematical details for implementation of the non-HS perturbation scheme in a coupling parameter expansion (CPE) thermodynamic perturbation framework. NVT-Monte Carlo simulation is carried out for a generalized Lennard-Jones (LJ) 2n-n potential to obtain routine thermodynamic quantities such as excess internal energy, pressure, excess chemical potential, excess Helmholtz free energy, and excess constant volume heat capacity. Then, these new simulation data, and available simulation data in literatures about a hard core attractive Yukawa fluid and a Sutherland fluid, are used to test the non-HS CPE 3rd-order thermodynamic perturbation theory (TPT) and give a comparison between the non-HS CPE 3rd-order TPT and other theoretical approaches. It is indicated that the non-HS CPE 3rd-order TPT is superior to other traditional TPT such as van der Waals/HS (vdW/HS), perturbation theory 2 (PT2)/HS, and vdW/Yukawa (vdW/Y) theory or analytical equation of state such as mean spherical approximation (MSA)-equation of state and is at least comparable to several currently the most accurate Ornstein-Zernike integral equation theories. It is discovered that three technical issues, i.e., opening up new bridge function approximation for the reference potential, choosing proper reference potential, and/or using proper thermodynamic route for calculation of f(ex-ref), chiefly decide the quality of the non-HS CPE TPT. Considering that the non-HS perturbation scheme applies for a wide variety of model fluids, and its implementation in the CPE thermodynamic perturbation framework is amenable to high-order truncation, the non-HS CPE 3rd-order or higher order TPT will be more promising once the above-mentioned three technological advances are established.     

A Thermodynamic Model for Square-well Chain Fluid: Theory and Monte Carlo Simulation

A thermodynamic model for the freely jointed square-well chain fluids was developed based on the thermodynamic perturbation theory of Barker-Henderson, Zhang and Wertheim. In this derivation Zhang's expressions for square-well monomers improved from Barker-Henderson compressibility approximation were adopted as the reference fluid, and Wertheim's polymerization method was used to obtain the free energy term due to the bond connectivity. An analytic expression for the Helmholtz free energy of the square-well chain fluids was obtained. The expression without adjustable parameters leads to the thermodynamic consistent predictions of the compressibility factors, residual internal energy and constant-volume heat capacity for dimer, 4-mer, 8-mer and 16-mer square-well fluids. The results are in good agreement with the Monte Carlo simulation. To obtain the MC data of residual internal energy and the constant-volume heat capacity needed, NVT MC simulations were performed for these square-well chain fluids.     

Calculating thermodynamic properties from perturbation theory: II. An analytic representation for the square-well chain fluid

An analytic representation of thermodynamic properties of the freely jointed square-well chain fluid is developed based on the thermodynamic perturbation theory of Barker–Henderson, Zhang and Weitheim. By using a real function expression for the radial distribution function and incorporating structural information for square-well monomer of TPT1 model, an analytic expression for the Helmholtz energy of square-well chain fluid is expanded from Zhang’s analytic expressions for thermodynamic properties of square-well monomer. The expression leads to good predictions of the compressibility factor, residual internal energy and constant-volume heat capacity for 4-mer, 8-mer and 16-mer square-well fluids when compared with the Monte Carlo (MC) simulation results. The incorporating structural information for square-well dimer of TPT-D model is also calculated. To obtain the constant-volume heat capacity needed, NVT MC simulations were performed.     

Computer simulation studies of anisotropic systems. XI. Second- and fourth-rank molecular interactions

The Maier-Saupe theory for nematic liquid crystals provides a reasonable account of their orientational order and its temperature dependence. The theory is based on second-rank anisotropic interactions and its predictions can be improved by the introduction of higher-rank terms as in the Humphries-James-Luckhurst theory. However comparison with the properties of real nematogens does not allow an unambiguous test of the theory because the form of the pair potential is unknown. This is not the case for computer simulations where the intermolecular potential is defined. We have therefore undertaken a Monte Carlo study of the influence fourth-rank interactions on nematic behaviour and report the results of our simulations here. The model nematogen used as a reference is that developed by Lebwohl and Lasher in which the particles are confined to the sites of a simple cubic lattice and interact via a second-rank anisotropic potential. The simulation gives the internal energy, the heat capacity at constant volume and the second-rank order parameter as a function of temperature, as well as the nematic-isotropic transition temperature. These results are used to provide the first unambiguous test of the Humphries-James-Luckhurst theory. We also discuss the changes in the transition temperature which are caused by the introduction of the fourth-rank term in the pair potential using thermodynamic perturbation theory for the Helmoltz free energy.     

An improved thermodynamic perturbation theory for Mercedes-Benz water

We previously applied Wertheim's thermodynamic perturbation theory for associative fluids to the simple Mercedes-Benz model of water. We found that the theory reproduced well the physical properties of hot water, but was less successful in capturing the more structured hydrogen bonding that occurs in cold water. Here, we propose an improved version of the thermodynamic perturbation theory in which the effective density of the reference system is calculated self-consistently. The new theory is a significant improvement, giving good agreement with Monte Carlo simulations of the model, and predicting key anomalies of cold water, such as minima in the molar volume and large heat capacity, in addition to giving good agreement with the isothermal compressibility and thermal expansion coefficient.     

复合固体功能材料热导率测定的Monte Carlo模拟

本文首先把固体功能材料格点化,在无源、稳态情况下得到格点化了的热传导方程。利用格点模型,用Monte Carlo方法模拟了复合固体功能材料在无源、稳态下的温度场,以此来求其热导率。模拟的结果表明,在掺入的良导热体的体积分数固定的情况下,随着掺入物热导率的增加,复合材料的热导率呈对数规律增长;在掺入物的热导率固定不变的情况下,随着掺入物在复合材料中所占比例的增加,复合材料的热导率呈指数规律增长。这些模拟结果与实验结果符合得较好。     

链状分子状态方程的推导及热容的推算

在Barker Henderson, Zhang以及Wertheim 等微扰理论的基础上,以方阱势硬球流体为参考体系,将Zhang的解析表达方法与Wertheim 的链成键自由能的处理方法结合起来,推导出自由链接的链状分子流体的Helmholtz自由能的解析表达式,并得到了压缩因子、内能、恒容热容等热力学性质的计算式.计算结果与MC(Monte Carlo)模拟结果吻合良好.对Zhang的解析表达式与“TPT D”(二阶Wertheim微扰理论)的结合也作了推导和计算.     

Coarse-grained molecular dynamics modeling of strongly associating fluids: thermodynamics, liquid structure, and dynamics of symmetric binary mixture fluids

Symmetric binary mixtures capable of strong association via a highly directional and saturable specific interaction between unlike molecules are investigated by canonical molecular dynamics simulations. The specific interaction of the molecules is defined in a new coarse-grained pair potential that can be applied in continuous molecular dynamics as well as in Monte Carlo simulations. The thermodynamic, structural, and dynamic properties of the associating mixture fluids are investigated as a function of density, temperature, and association strength of the specific interaction. Detailed analysis of the simulation data confirms a two-stage mechanism in the formation of specific bonds with increasing interaction strength, including a fast dimerization process and a subsequent stage of perfecting the bonds. A large heat capacity peak is found during the formation or breaking of the bonds, reflecting the large energy fluctuation introduced by the strong association. The fractions of nonbonded molecules obtained from the simulations as a function of density, temperature, and interaction strength are in excellent agreement with the predictions of Wertheim's thermodynamic perturbation theory. The translational and rotational dynamics of the Tmer mixture are effectively retarded with increasing association strength and are analyzed in terms of autocorrelation functions and a non-Gaussian parameter for the translational dynamics. The lifetimes of molecules in bonded and nonbonded states provide detailed information about the transformation of molecules between the bonded state and the nonbonded state. Finally, simulation sampling problems inherent to strongly interacting systems are easily overcome using the parallel tempering simulation technique. This latter result confirms that with the new continuous coarse-grained simulation potential we have a versatile and flexible interaction potential that can be used with many available molecular dynamics and Monte Carlo algorithms under various ensembles.     

A completely analytic equation of state for the square-well chain fluid of variable well width

A completely analytic perturbation theory equation of state for the freely-jointed square-well chain fluid of variable well width (1 ≤ λ ≤ 2) is developed and tested against Monte Carlo simulation data. The equation of state is based on second-order Barker and Henderson perturbation theory to calculate the thermodynamic properties of the reference monomer fluid, and on first-order Wertheim thermodynamic perturbation theory to account for the connectivity of monomers to form chains. By using a recently developed real function expression for the radial distribution function of hard spheres in perturbation theory, we obtain analytic, closed form expressions for the Helmholtz free energy and the radial distribution function of square-well monomers of any well width. This information is used as the reference fluid in the perturbation theory of Wertheim to obtain an analytic equation of state, without adjustable parameters, that leads to good predictions of the compressibility factors and residual internal energies for 4-mer, 8-mer and 16-mer square-well fluids when compared with the simulation results. Further, very good results are obtained when this equation of state with temperature-independent parameters is used to correlate the vapor pressures and critical points of the linear alkanes from methane to n-decane.     

从统计力学推导流体状态方程——方阱势硬球微扰理论的解析表达式

本文在Zwanzlg微扰理论的基础上, 对二级以上的高级微扰项采用Barker与Henderson的近似方法, 得到一个简单的微扰理论表达式。以硬球势为参考势, 方阱势为微扰势,用一新的级数表达式g(R)=1/ηgj(η/(1-η))~j为径向分布函数, 导出了自由能。内能、比热、压缩因子的级数表达式。为了检验理论的正确性, 取g(R)级数的前四项代入各热力学性质的表达式, 与Monte-Carlo(MC)及分子动力学(MD)计算机模拟数据作了比较, 结果符合较好。     

Monte carlo simulation and thermodynamic perturbation theory for mixtures of diatomic molecules

The thermodynamic properties and site—site distribution functions of mixtures of non-spherical molecules are obtained by Monte Carlo simulation. A non-spherical reference-system perturbation theory based on the RISM equation is developed to predict these results. The agreement between theory and simulation for the thermodynamic properties is encouraging. Important differences in the relative peak heights of the site—site distribution functions from theory and simulation are attributed to the role of attractive forces in determining local structure in the fluid mixtures, where the volumes of the components are similar but the well depths differ.     

Hard sphere perturbation theory for fluids with soft-repulsive-core potentials

The thermodynamic properties of fluids with very soft repulsive-core potentials, resembling those of some liquid metals, are predicted with unprecedented accuracy using a new first-order thermodynamic perturbation theory. This theory is an extension of Mansoori-Canfield/Rasaiah-Stell (MCRS) perturbation theory, obtained by including a configuration integral correction recently identified by Mon, who evaluated it by computer simulation. In this work we derive an analytic expression for Mon's correction in terms of the radial distribution function of the soft-core fluid, g(0)(r), approximated using Lado's self-consistent extension of Weeks-Chandler-Andersen (WCA) theory. Comparisons with WCA and MCRS predictions show that our new extended-MCRS theory outperforms other first-order theories when applied to fluids with very soft inverse-power potentials (n< or =6), and predicts free energies that are within 0.3 kT of simulation results up to the fluid freezing point.     

Thermodynamic and structural properties of repulsive hard-core Yukawa fluid: integral equation theory, perturbation theory and Monte Carlo simulations

The thermodynamic and structural properties of purely repulsive hard-core Yukawa particles in the fluid state are determined through Monte Carlo simulation and modeled using perturbation theory and integral equation theory in the mean spherical approximation (MSA). Systems of particles with Yukawa screening lengths of 1.8, 3.0, and 5.0 are examined with results compared to variations of MSA and perturbation theory. Thermodynamic properties were predicted well by both theories in the fluid region up to the fluid-solid phase boundary. Further, we found that a simplified exponential version of the MSA is the most accurate at predicting radial distribution function at contact. Radial distribution function of repulsive hard-core Yukawa particles are also reported. The results show that methods based on MSA and perturbation theory that are typically applied to the attractive hard-core Yukawa potential can also be extended to the purely repulsive hard-core Yukawa potential.     

Lattice dynamics of single-bonded cubic nitrogen

The first principle calculations of the lattice dynamical properties of the single-bonded cubic nitrogen were performed using the density-functional perturbation theory together with plane-wave expansion and nonlocal pseudopotentials. The equilibrium structure of the single-bonded cubic nitrogen was first evaluated via the minimization of the total energy. Then, the harmonic phonon dispersion curves and the density of phonon states of the single-bonded cubic nitrogen have been evaluated within the linear-response framework. Furthermore, the heat capacity, enthalpy, free energy, entropy and velocity of sound of the single-bonded cubic nitrogen were calculated.     

氯化钠水溶液的monte Carlo分子模拟研究

采用MonteCarlo计算机分子模拟方法,研究了氯化钠水溶液在常温和高温情况下粒子的径向分布函数和粒子间各种作用位能的热力学性质。模拟过程采用了NTV正则系综,粒子间的作用能包括离子库仑静电作用,偶极子作用以及色散作用,这些作用构成了电解质溶液的基本框架,模拟结果与微扰理论和平均球近似积分方程理论的预测值进行了比较。     

Dynamic behaviors of Monod type chemostat model with impulsive perturbation on the nutrient concentration

In this paper, the dynamic behaviors of a Monod type chemostat model with impulsive perturbation are investigated. Using Floquet theory and small amplitude perturbation method, we prove that the microorganism-eradication periodic solution is asymptotically stable if the impulsive period satisfies some conditions. Moreover, the permanence of the system is discussed in detail. Finally, we verify the main results by numerical simulation.      

Reaction rate calculation with time-dependent invariant manifolds

The identification of trajectories that contribute to the reaction rate is the crucial dynamical ingredient in any classical chemical reactivity calculation. This problem often requires a full scale numerical simulation of the dynamics, in particular if the reactive system is exposed to the influence of a heat bath. As an efficient alternative, we propose here to compute invariant surfaces in the phase space of the reactive system that separate reactive from nonreactive trajectories. The location of these invariant manifolds depends both on time and on the realization of the driving force exerted by the bath. These manifolds allow the identification of reactive trajectories simply from their initial conditions, without the need of any further simulation. In this paper, we show how these invariant manifolds can be calculated, and used in a formally exact reaction rate calculation based on perturbation theory for any multidimensional potential coupled to a noisy environment.     

Toward a statistical mechanical theory for water: analytical theory for a short-ranged reference system

Starting from a realistic Hamiltonian and making use of recent findings that the properties of associating fluids are determined primarily by short-ranged interactions, this methodology has been implemented using statistical mechanical approaches and thermodynamic perturbation theory for the TIP4P model of water. We focus on the short-range reference system for which an analytic expression for the Helmholtz free energy is derived. It is found that the model (reference system) exhibits, in addition to a faithful representation of the structure of water, the same features that are characteristic for real water, namely, (i) the temperature of the density maximum and its pressure dependence, including the inflection point at high pressures and (ii) the temperature minima of the constant pressure heat capacity and the coefficient of isothermal compressibility.