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.     

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.     

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.     

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.     

Calculating thermodynamic properties from perturbation theory: I. An analytic representation of square-well potential hard-sphere perturbation theory

The Barker–Henderson macroscopic compressibility approximation of the second-order perturbation term is improved by assuming that the numbers of molecules in every two neighbour shells are correlated, based upon the original assumptions. The results are better than those for the original macroscopic compressibility and local compressibility approximation, especially at high densities. A simple analytic representation of square-well potential hard-sphere perturbation theory is derived based upon this improvement. The method is tested by calculating thermodynamic properties with the four-term truncated form, and the results are in good agreement with those of Monte Carlo and Molecular Dynamics simulation.     

The phase behavior of polyethylene ring chains

The equilibrium properties of an isolated polyethylene ring chain are studied by using molecular dynamics (MD) simulations. The results of an 80-bond linear chain are also presented, which are in agreement with previous studies of square-well chains and Lennard-Jones (LJ) homopolymers. Mainly, we focus on the collapse of polyethylene ring chains. At high temperatures, a fully oblate structure is observed for the ring chains with different chain lengths. For such an oblate structure, a shape factor of delta(*)=0.25 and a rodlike scaling relation between the radius of gyration and chain lengths could be deduced easily in theory, and the same results are obtained by our MD simulations. Such an oblate structure can be obtained by Monte Carlo simulation only for sufficient stiff ring chains. When the temperature decreases, an internal energy barrier is observed. This induces a strong peak in the heat capacity, denoting a gas-liquid-like transition. This energy barrier comes mainly from the local monomer-monomer interactions, i.e., the bond-stretching, the bond-bending, and the torsion potentials. A low temperature peak is also observed in the same heat capacity curve, representing a liquid-solid-like transition. These numerical simulation results support a two-stage collapse of polyethylene ring chains; however, the nature should be different from the square-well and LJ ring chains.     

A new development of equation of state for square-well chain-like molecules with variable width 1.1 ≤ λ ≤ 3

A new equation of state (EOS) for square-well chain molecules and their mixtures with variable well-width range (SWCF-VR-EOS) has been developed based on the sticky-point model for chemical association. Two important modifications have been made. Firstly, a new dispersion contribution to the Helmholtz function of monomers due to square-well potential with variable well-width range of 1.1 ≤ λ ≤ 3 was established by combining the second-order perturbation theory and Chiew's PY2 approximation of the integral equation. Secondly, the contribution of chain formation to the Helmholtz function is divided into two parts: One is from the hard sphere, and the other is from the effect of square-well potential described via the nearest-neighbor and next-to-nearest-neighbor residual cavity correlation functions (CCFs). The predicted compressibility factors and vapor–liquid coexistence curves for square-well fluids as well as for their mixtures are in good agreement with simulations. The new EOS has been applied to real non-associating fluids and the corresponding mixtures by adopting one-fluid mixing rule. The pVT and vapor–liquid equilibria (VLE) can be correlated satisfactorily. The model parameters for some homologous compounds are found to be linear with the molar mass indicating that the pVT and VLE of those homologous compounds can be predicted even if no accurate data are available.     

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。     

A cubic perturbed hard-chain equation of state

A cubic equation of state is developed on the basis of perturbation theory. The equation is an association of three segments: the hard-sphere, the hard-chain, and the attraction. The expression for each segment was invoked from approximations of computer simulations of rigorous molecular theories of fluids, but compromised to some extent accuracy and theory for simplicity. This model equation is shown to be potentially capable of describing the PVT behavior of real fluids. As limiting cases, the new equation is reduced to expressions for the hard-sphere and the hard-body fluids. It also represents square-well fluids when the hard-chain contribution is eliminated. The square-well equation was found satisfactory in conforming with the molecular simulation results for square-well fluids and their mixtures.     

The extension of coordination number model by using the local composition theory: mixtures

The extension of a new coordination number model to mixture is presented in this work. Extended model agrees well with the Monte Carlo (MC) simulation results for square-well (SW) mixture fluids and shows better results compared with other models. To test our model, we compare the compressibility factors from various models for SW fluids at different λ values and for SW fluid mixtures at λ=1.5. Although our model is obtained by fitting simulation data at λ=1.5, it shows better results for the different λ values than other coordination number model. Compared with the compressibility factors of various binary mixtures of SW fluids calculated from other models, this model presents better results. Because our model considers the temperature dependency importantly by using the total site number, it predicts coordination number and compressibility factor well in the wide temperature range and enables one to derive an equation of state (EOS) through integration of the coordination number equation.     

Simulation and model development for the equation of state of self assembling non-additive hard chain–hard monomer mixture

Molecular dynamics simulations for a short hard chain composed of a head and three tail groups interacting with non-additive size interactions with a hard sphere solvent were performed. Different densities and non-additivities were used. The equation of state for this mixture was investigated and models based on the first-order thermodynamic perturbation theory (TPT1) and the polymeric analog of the Percus–Yevick approximation (PPY) were developed to predict the compressibility factor of the mixture. The models predicted the compressibilities of the mixtures accurately at zero and negative non-additivities. However, at positive non-additivities, the models overpredicted the compressibilities especially at high densities. The TPT1 model was generally more accurate in predicting the compressibility factor than the PPY model. Microphase separation was observed at high densities and positive non-additivities.     

Second order of perturbation theory correction to the radial distribution function of a liquid with the interaction potential of hard spheres plus a square-well

A liquid with the interaction potential of hard spheres plus a square-well is analyzed using the Monte-Carlo technique. Numerical results for the perturbation theory series over a square-well potential are obtained in the form of the Barker and Henderson discrete representation. Approximating expressions for the correction to a liquid radial distribution function in the second order of perturbation theory are presented. The obtained results allow us to define this correction with a root-mean-square deviation of about 0.007. It is shown that the given approach provides a complete calculation in the second order of perturbation theory, and also the determination of the third order correction to the free energy for a liquid interacting with the potential of the Lennard-Jones type.     

改进压缩性近似并导出新的硬球微扰理论表达式

提出在Barker与Henderson的压缩性近似推导中,相邻壳层之间的分子数应该相关,导出了相关系数和改进的二级微扰项,提高了高级微扰项在较高密度区的准确性。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,用一新的级数形式的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。其四项截断式的计算结果与MC、MD计算机模拟值符合较好。     

An analytical perturbed equation of state for hard chain fluids: Application to n-alkanes and n-perfluoroalkanes

A completely analytical equation of state for pure hard chain fluids, derived on the basis of perturbation theory and reported in our previous work, is applied for the calculation of pVT properties and the prediction of vapour–liquid equilibria of n-alkanes and n-perfluoroalkanes. The molecules are treated as a chain formed from freely joined spheres which interact via an extended site-site square-well potential. The molecular parameters of compounds are obtained from the experimental compressibility factor data above the critical temperature. These parameters are capable of relatively satisfactory prediction of the vapour–liquid equilibrium coexistence curves of compounds. Linear relationships have been found between the potential parameters of fluids and their molecular weight, which make it possible to predict the pVT data and vapour–liquid phase equilibria of heavier compounds.     

The generalized van der Waals partition function. III. Local composition models for a mixture of equal size square-well molecules

New local composition models for mixtures of equal size molecules with differing attractive forces are presented, and compared with our results of Monte Carlo computer simulations for mixtures of square-well molecules which are also reported here. Unlike most previous local composition models, these new models predict random mixing in the high density limit and Boltzmann factor nonrandomness at low density; both limiting behaviors are in agreement with statistical mechanical theory. The predictions of these new models as a function of composition, density and temperature are in good agreement with the Monte Carlo computer simulation results.     

Density and chain conformation profiles of square-well chains confined in a slit by density-functional theory

Density and chain conformation profiles of square-well chains between two parallel walls were studied by using density-functional theory. The free energy of square-well chains is separated into two contributions: the hard-sphere repulsion and the attraction. The Heaviside function is used as the weighting function for both of the two parts. The equation of state of Hu et al. is used to calculate the excess free energy of the repulsive part. The equation of state of statistical associating fluid theory for chain molecules with attractive potentials of variable range [A. Gil-Villegas et al. J. Chem. Phys. 106, 4168 (1997)] is used to calculate the excess free energy of the attractive part. Because the wall is inaccessible to a mass center of a longer chain, there exists a sharp fall in the distribution of end-to-end distance near the wall as the chain length increases. When the average density of the system is not too low, the prediction of this work is in good agreement with computer simulation results for the density profiles and the chain conformation over a wide range of chain length, temperature, and attraction strength of the walls. However, when the average density and the temperature are very low, the prediction deviates to a certain degree from the computer simulation results for molecules with long chain length. A more accurate functional approximation is needed.     

Monte Carlo Simulation of Coil-to-Globule Transition of Compact Polymer Chains: Role of Monomer Interacting

Coil-to-globule transitions are fundamental problems existing in polymer science for several decades; however, some features are still unclear, such as the effect of chain monomer interaction. Herein, we use Monte Carlo simulation to study the coil-to-globule transition of simple compact polymer chains. We first consider the finite-size effects for a given monomer interaction, where the short chain exhibits a one-step collapse while long chains demonstrate a two-step collapse, indicated by the specific heat. More interestingly, with the decrease of chain monomer interaction, the critical temperatures marked by the peaks of heat capacity shift to low values. A closer examination from the energy, mean-squared radius of gyration and shape factor also suggests the lower temperature of coil-to-globule transition.     

基于化学缔合统计理论的链状流体状态方程

基于化学缔合统计理论的链状流体状态方程(EOS)能够反映实际分子的形状、链节成链、缔合等具体信息,在实际流体热力学性质计算中有着广泛应用.一般的链状流体EOS仅考虑相邻链节间的相关性,我们则借助统计力学和计算机模拟结果在模型中纳入了相间链节间的相关性,获得的硬球链流体(HSCF)模型能够更好地预测模型流体的压缩因子和第二维里系数.以HSCF为参考,引入方阱色散微扰项获得了实际方阱链流体(SWCF)EOS;结合根据黏滞球模型导得的缔合项,进一步构建了缔合流体EOS.最近,我们根据微扰理论和积分方程方法又开发了一新的变阱宽方阱链流体(SWCF-VR)模型.SWCF和SWCF-VREOSs可很好地用于计算小分子、聚合物、离子液体等纯流体及混合物的相行为、热焓、表面张力、黏度等热力学及传递性质,显示了模型良好的工程应用价值.本文就本课题组多年来在自由空间范畴内基于化学缔合统计理论开发链状流体EOS及其实际应用作系统的总结.     

Thermodynamic and structural properties of mixed colloids represented by a hard-core two-Yukawa mixture model fluid: Monte Carlo simulations and an analytical theory

The interaction between colloidal particles is well represented by a hard-core two-Yukawa potential. In order to assess the accuracy of theoretical predictions for the thermodynamic and structural properties of mixed colloids, standard Monte Carlo simulations are carried out for the hard-core two-Yukawa mixtures. In the simulations, one range parameter in the two-Yukawa potential is taken as 1.8 or 2.8647, and another is taken as 4, 8, or 13.5485. Both attractive and repulsive dominant cases of the potential outside the hard core are considered. The effects of temperature, density, composition, size and energy parameter ratios on internal energy, compressibility factor, and radial distribution function are investigated extensively. Theoretical calculations are performed in the framework of analytical solution for the Ornstein-Zernike equation with the first-order mean spherical approximation (FMSA). Our analysis shows that the FMSA is very accurate for the prediction of the compressibility factor of the hard-core two-Yukawa mixtures at all conditions studied. The FMSA generally predicts accurate internal energy, but overestimates the internal energy of the systems at lower temperatures. Furthermore, we found that a simplified exponential version of the FMSA predicts fairly good radial distribution function at contact for the mixed two-Yukawa fluids. The comparison of the theoretical compressibility factor with that from the Monte Carlo simulations suggests that the FMSA can be used to investigate the fluid-fluid equilibria of hard-core two-Yukawa mixtures.