Statistical mechanics of linear molecules I: Percus-Yevick and convolution-hypernetted-chain approximations

It is shown how the Percus-Yevick and convolution-hypernetted-chain approximations for dense fluids of linear molecules can be brought into practically applicable forms by using cartesian tensor formalism and pertubation theory. The possibility of expanding the intermolecular potential and the correlation functions is discussed in detail. Also, exact reductions for the Ornstein-Zernike relation in real space and for the Percus-Yevick equation are given. Some other approaches to the statistical mechanics of linear molecules are critically discussed. It is shown in particular, that the mean spherical model is an inconsistent type of perturbation theory. The necessary cartesian tensor formulae are given in appendices as well as formally exact expressions for the second virial coefficient and for the Fourier-transformed Ornstein-Zernike relation.     

Orientational correlations in liquid and supercritical CO2: neutron diffraction experiments and molecular dynamics simulations

Neutron diffraction data from sub- and supercritical CO₂, collected over three different experimental runs, are analysed together, in order to investigate the evolution of the microscopic structure with the thermodynamic parameters. The relative orientations of neighbouring molecules are analysed also in terms of the angular radial correlation function, obtained by molecular dynamics simulation runs compatible with the total neutron weighted radial distribution function. Orientational correlations between neighbouring molecules are seen to persist across the whole thermodynamic range investigated. However, these are limited to the first neighbouring shell, and mainly ascribable to the quadrupolar interaction.     

TPT2 and SAFTD equations of state for mixtures of hard chain copolymers

We present the second-order thermodynamic perturbation theory (TPT2) and the dimer statistical associating fluid theory (SAFTD) equations of state for mixtures consisting of hetero-nuclear hard chain molecules based on extensions of Wertheim's theory for associating fluids. The second-order perturbation theory, TPT2, is based on the hard sphere mixture reference fluid. SAFTD is an extension of TPT1 (= SAFT) and is based on the non-spherical (hard disphere mixture) reference fluid. The TPT2 equation of state requires only the contact values of the hard sphere mixture site-site correlation functions, while the SAFTD equation of state requires the contact values of site-site correlation functions of both hard sphere and hard disphere mixtures. We test several approximations for site-site correlation functions of hard disphere mixtures and use these in the SAFTD equation of state to predict the compressibility factor of copolymers. Since simulation data are available only for a few pure copolymer systems, theoretical predictions are compared with molecular simulation results for the compressibility factor of pure hard chain copolymer systems. Our comparisons show a very good performance of TPT2, which is found to be more accurate than TPT1 (= SAFT). Using a modified Percus-Yevick site-site correlation function SAFTD is found to represent a significant improvement over SAFT and is slightly more accurate than TPT2. Comparison of SAFTD with generalized Flory dimer (GFD) theory shows that both are equivalent at intermediate to high densities for the compressibility factor of copolymer systems investigated here.     

Analytical expressions for the correlation function of a hard sphere dimer fluid

A closed form expression is given for the correlation function of a hard sphere dimer fluid. A set of integral equations is obtained from Wertheim's multidensity Ornstein-Zernike integral equation theory with Percus-Yevick approximation. Applying the Laplace transformation method to the integral equations and then solving the resulting equations algebraically, the Laplace transforms of the individual correlation functions are obtained. By the inverse Laplace transformation, the radial distribution function (RDF) is obtained in closed form out to 3D (D is the segment diameter). The analytical expression for the RDF of the hard dimer should be useful in developing the perturbation theory of dimer fluids.     

Evaluation of the SSC/LHNC,SSCF and PY approximations for short ranged,anisotropic potentials

Three approximations for the orientational correlation function of molecular fluids—the single super chain/linearized hypernetted chain (SSC/LHNC), single super chain f-expansion (SSCF) and Percus-Yevick (PY) approximation—are evaluated in the dense liquid state for a fluid inter-acting with short ranged anisotropic interactions. These interactions are prototypical of the forces that determine the non-spherical shape of symmetric linear molecules. We find that SSC/LHNC is very poor, failing to predict many of the structural features present in the Monte Carlo (MC) simulation results. The PY and SSCF approximations produce much better results; however, neither approximation is completely satisfactory.     

Kinetic equation for a weakly interacting classical electron gas

A kinetic equation is derived for the two-time phase space correlation function in a dilute classical electron gas in equilibrium. The derivation is based on a density expansion of the correlation function and the resummation of the most divergent terms in each order in the density. It is formally analogous to the ring summation used in the kinetic theory of neutral fluids. The kinetic equation obtained is consistent to first order in the plasma parameter and is the generalization of the linearized Balescu-Guersey-Lenard operator to describe spatially inhomogeneous equilibrium fluctuations. The importance of consistently treating static correlations when deriving a kinetic equation for an electron gas is stressed. A systematic derivation as described here is needed for a further generalization to a kinetic equation that includes mode-coupling effects. This will be presented in a future paper.     

An equation of state for the isotropic phase of linear,partially flexible and fully flexible tangent hard-sphere chain fluids

A new equation of state is developed that accurately describes the isotropic phase behaviour of linear, partially flexible and fully flexible tangent hard-sphere chain fluids and their mixtures. The equation of state is based on the equation of state of Liu and Hu [H. Liu and Y. Hu, Fluid Phase Equilibr. 122, 75 (1996)] for fully flexible chain fluids. The effect of molecular flexibility is described by a pure-component parameter that is introduced in the theory at the level of the cavity correlation function of next-to-nearest neighbour segments in a chain molecule. The equation of state contains a total of three adjustable model constants. The extension to partially flexible- and linear chain fluids is based on a refitting of the first model constant to numerical data of the second virial coefficient of partially flexible and linear tangent hard-sphere chain fluids. The numerical data were obtained from an analytical approximation for the pair-excluded volume. The other two parameters were adjusted to molecular simulation data for the pressure of linear tangent hard-sphere chain fluids. For both, pure component systems and mixtures of chains of variable flexibility, the pressure and second virial coefficient obtained from the equation of state, are in excellent agreement with the results from Monte Carlo simulations. A significant improvement to TPT1, TPT2, generalised Flory-dimer theory and scaled particle theory is observed.     

Many-body correlations in the problems of electrodynamics of fluids

This paper presents a review of the theory of the many-body correlation phenomena in fluids, in which the collective character of the fluctuations is caused by the long-range interaction of electrodynamic origin. The procedure of the statistical mechanical averaging of the microscopic electrodynamic equations is developed for a classical system of the interacting polarizable molecules with a subsequent account of the molecular correlations. As a result an effective expansion for the refractive index is obtained. The dependence of the refractive index on the thermodynamic parameters near the critical point is investigated using the scaling-law asymptotics for the many-body correlation functions. A molecular theory of the multiple light scattering is suggested. A method of evaluation of the many-body any-order scattering intensity in the critical region is described. Using a resummation procedure, similar to that developed for the light propagation problem, the correlation function expansion is obtained for a classical system of charged and neutral particles, correlations due to the short-range forces are taken into account. The expansion gives series in terms of the effective “dressed” electrostatic potential containing no long-range Coulomb divergencies, nor short-range ones.     

Representation of pair correlations in nematics

A complete expansion of equilibrium pair correlation functions for a uniaxial nematic phase composed of axially symmetric, non-polar and chiral molecules is proposed. Full advantage is taken of the symmetry of the nematic state and of the molecules. The explicit analysis and classification of the terms involving spherical harmonics with indices not exceeding 4 is given and illustrated using computer simulations for the nematic phase of a Gay-Berne mesogen. The theory is contrasted with the commonly used approach employing invariants that describe orientational correlations in molecular fluids. The role of the new representation in obtaining a correct understanding of a variety of observables, like the elastic constants, is demonstrated. In particular, the long-standing puzzle concerning the splay-bend degeneracy is resolved. Received 11 April 2000 and Received in final form 19 July 2000     

Binary fluid mixture of hard ellipses: Integral equation and weighted density functional theory

We study a two-dimensional (2D) classical fluid mixture of hard convex shapes. The components of the mixture are two kinds of hard ellipses with different aspect ratios. Two different approaches are used to calculate the direct, pair and total correlation functions of this fluid and results are compared. We first use a formalism based on the weighted density functional theory (WDFT), introduced by Chamoux and Perera [Phys. Rev. E 58 (1998) 1933]. Second, in general the Percus-Yevick (PY) and the hypernetted chain (HNC) integral equations are solved numerically for the 2D fluid mixtures of hard noncircular particles. Explicit results are obtained for the fluid mixtures of hard ellipses and comparisons are made by the two approaches. Also, the results are compared with the recent Monte Carlo simulation for the one-component fluids of hard ellipses. Finally we obtained the equation of state of hard ellipses for the aspect ratio sufficiently close to 1 and compared our results with the simulations of the fluid mixtures of hard disks.     

A hard sphere fluid model for superionic conductors

We develop a theory for the mobile constituent of a superionic conductor using the Ornstein-Zernike integral equation for the pair correlation function of an inhomogeneous fluid. We solve this equation in the Percus-Yevick approximation using a simple decoupling procedure and hard core potentials. Comparison is made with molecular dynamics calculations on α-AgI.     

Statistical mechanics of linear molecules II: Density expansions,intermolecular potentials and virial coefficients

In the Percus-Yevick and convolution-hypernetted-chain equations obtained in the previous paper, a density expansion for the correlation functions is introduced. To first order in the density, the so-obtained equations are identical and exact. By solving these, the pair correlation functions for linear molecules are obtained explicitly to first order in the density and for arbitrary order in the potential perturbation expansion. From these, the second and third virial coefficients can be extracted for all orders. A generalized charge is defined and used to give generalized multipole expansions for the intermolecular potential. Explicit expressions for this potential model are given up to fourth order. It is shown how the correlation functions, and second and third virial coefficients can be obtained to fourth order for any intermolecular potential with the same perturbation structure.     

Dynamic Theory of Die Swell for Entangled Polymeric Liquids in Tube Extrusion: Correlations of Total and Ultimate Extrudate Swell Effects to Growth Time, Shear Stress and Aspect Ratio Under the Free States

利用挤出胀大动力学理论研究稳态剪切流下HDPE 和PBD液体的挤出胀大行为,建立了自由动态下线团回复和挤出胀大增长时间的回复机制和动力学. 结果表明在自由回复过程中自由线团回复和挤出物胀大增长可分为两个区域(瞬间和推迟区)、三个增长阶段(瞬间、推迟和最终阶段). 证明了自由线团回复和挤出胀大增长可表征为增长时间、剪切应力和长径比的函数。从而从动力学理论推倒出了三种挤出胀大效应(瞬间、推迟和最终)同分子结果参数和挤出操作条件间的相关性. 并建立了总合(TESE)和最终(UESE)两组挤出胀大效应的普适方程.     

The 13C chemical shift tensor principal values and orientations in dialkyl carbonates and trithiocarbonates

The 13C chemical shift tensor principal values for the trigonal carbonate and thiocarbonate carbon atoms in the dialkyl carbonates, dimethyl carbonate, ethylene carbonate, and diphenyl carbonate, and in the trithiocarbonates, ethylene trithiocarbonate and dimethyl trithiocarbonate, respectively, were measured in various solid-state one-dimensional and two-dimensional nuclear magnetic resonance experiments. Furthermore, the chemical shift tensor principal values and orientations were calculated for the corresponding isolated molecules with quantum mechanically fully optimized geometries. Proton-optimized X-ray geometries of ethylene carbonate, ethylene trithiocarbonate, and diphenyl carbonate were used in embedded ion method (EIM) calculations and in calculations on the isolated molecules to obtain the theoretical principal values and to assign the chemical shift tensor orientations in these three compounds. Considerable improvement in the correlation between the experimental and calculated principal values is obtained when the electrostatic crystal potentials are included in EIM calculations. The chemical shift tensor orientations and principal values obtained for the dialkyl compounds in this study complement the previous data on a series of ionic potassium carbonates and thiocarbonates.     

The C Chemical Shift Tensor Principal Values and Orientations in Dialkyl Carbonates and Trithiocarbonates

The ¹³C chemical shift tensor principal values for the trigonal carbonate and thiocarbonate carbon atoms in the dialkyl carbonates, dimethyl carbonate, ethylene carbonate, and diphenyl carbonate, and in the trithiocarbonates, ethylene trithiocarbonate and dimethyl trithiocarbonate, respectively, were measured in various solid-state one-dimensional and two-dimensional nuclear magnetic resonance experiments. Furthermore, the chemical shift tensor principal values and orientations were calculated for the corresponding isolated molecules with quantum mechanically fully optimized geometries. Proton-optimized X-ray geometries of ethylene carbonate, ethylene trithiocarbonate, and diphenyl carbonate were used in embedded ion method (EIM) calculations and in calculations on the isolated molecules to obtain the theoretical principal values and to assign the chemical shift tensor orientations in these three compounds. Considerable improvement in the correlation between the experimental and calculated principal values is obtained when the electrostatic crystal potentials are included in EIM calculations. The chemical shift tensor orientations and principal values obtained for the dialkyl compounds in this study complement the previous data on a series of ionic potassium carbonates and thiocarbonates.     

Direct and indirect correlations in low density supercritical Lennard-Jones fluids

We have carried out a detailed comparison of the direct and indirect correlation functions obtained from canonical molecular dynamics (NVT-MD) simulation of supercritical Lennard-Jones fluids to the results obtained by solving the Ornstein-Zernike equation with the approximate Duh-Henderson (DH) closure. The variations of equilibrium correlations are studied as functions of density at two supercritical isotherms near and away from the critical point. The direct and indirect correlations predicted using the DH closure provides a very good agreement with our simulation results at low densities. However, a marked deviation is observed at higher densities. These results are correlated to the discrepancies between the density and temperature dependence of the underlying bridge function. The implication of these results on the calculation of chemical potential and the Krichevskii parameter is also presented.     

广义Morse流体解析状态方程及对氮的应用

 根据Ross变分微扰理论以及硬球流体Percus-Yevick（PY） 径向分布函数表达式,建立了广义Morse势流体的解析状态方程。与模拟结果的比较一方面证实了广义Morse 势模型的合理性;另一方面表明了解析Ross变分微扰理论的精度相当或略好于非解析的Weeks-Chandler-Anderson （mWCA）理论,而优于复杂的优化超网络链积分方程理论（RHNC）。该解析状态方程被应用于拟合处于环境温度和压强小于1 GPa情形流体氮的实验数据,所得到的势能参数被用于预测高温高密度情形氮流体的压强,预测结果证实,该解析状态方程可以很好地适用于较宽的压强和温度范围。     

Low-frequency velocity correlation spectrum of fluid in a porous media by modulated gradient spin echo.

In addition to the fast correlation for local stochastic motion, the molecular velocity correlation function in a fluid enclosed within the pore boundaries features a slow long time-tail decay. Here we present its study by the NMR modulated gradient spin-echo method (MGSE) [1] on a system of water trapped in the space between the closely packed polystyrene beads. With MGSE pulse sequence, a repetitive train of RF pulses with interspersed gradient pulses periodically modulates the spin phase. It gives the spin echo attenuation proportional to a value of the molecular velocity correlation spectrum at the modulation frequency. Covering the frequency range between Hz and MHz, it is a complement to the quasi-elastic neutron scattering, and so a suitable technique for the investigation of low frequency molecular dynamics in fluids. In our experiment, it enables to extract the low frequency correlation spectrum of water molecules confined in porous media. The function exhibits a negative long time-tail characteristic (a low frequency decay of the spectrum), which can be interpreted as a molecular back scattering on boundaries. The results can be well fitted with the spectrum calculated from the solution of the Langevin equation for restricted diffusion (which exhibits an exponential decay) [2] as well as with the spectrum obtained when simulating the hydrodynamics of molecular motion constrained by capillary walls (which gives an algebraic decay) [3]. Despite much work on theories and simulation, which predict slow negative long time tail of molecular velocity correlation dynamics in confined fluids, the obtained velocity correlation spectrum is the first experimental evidence to confirm these effects. The obtained dependence of spin echo attenuation on time, gradient strength and modulation frequency is also the first experimental verification of the recently developed approach to the spin echo in porous media, that uses the spin phase average with the cumulant expansion to get the attenuation as a discord of spin spatial coherence [4].