Free Energies in the Landau and Molecular Field Approaches

Abstract

An expression for the entropy as a power series in the order parameter is derived in the context of molecular field theory. The expression is valid both at and away from equilibrium. It is a unique generalization of molecular field theory to non-equilibrium situations. Discrepancies with certain expressions which have appeared in the literature are resolved. Analysis of the radius of convergence of the power series indicates that in certain cases, including the Maier-Saupe model of liquid crystals, molecular field theory and Landau theory cannot be made to agree over the entire range of possible values of the order parameter.     

Derivation of Martin-Hou Equation of State from Hard-particle Perturbation Theory

IntroductionIn the mid 1950's, an empirical equation of state was proposed by Martin and HouLI] abbreviated to the M--H equation. It is a more accurate equation of state applicable to real gasesincluding polar compounds, such as ammonia and water over a wide range of density and temperature. In 1981, the M--H equation was extended further to liquid domain by Hou et al. [2),and later, it was used to calculate the vapor--liquid equilibrium, liquid--liquid equilibrium andother thermodynamic p…     

Interfacial properties of Lennard-Jones chains by direct simulation and density gradient theory

We perform a series of molecular dynamics simulations of Lennard-Jones chains systems, up to tetramers, in order to investigate the influence of temperature and chain length on their phase separation and interfacial properties. Simulation results serve as a test to check the accuracy of a statistical associated fluid theory (soft-SAFT) coupled with the density gradient theory. We focus on surface tension and density profiles. The simulations allow us to discuss the success and limitations of the theory and how to estimate the only adjustable parameter, the influence parameter. This parameter is obtained by fitting the surface tension, and then used to obtain the density profiles in a predictive manner. A good agreement is found if the temperature dependence of this parameter is neglected.(c) 2004 American Institute of Physics.     

Relationships among coarse-grained field theories of fluctuations in polymer liquids

Two closely related field-theoretic approaches have been used in previous work to construct coarse-grained theories of corrections to the random phase approximation for correlations in block copolymer melts and miscible polymer blends. The "auxiliary field" (AF) approach is based on a rigorous expression for the partition function Z of a coarse-grained model as a functional integral of an auxiliary chemical potential field. The "effective Hamiltonian" (EH) approach is instead based on an expression for Z as a functional integral of an observable order parameter field. The exact effective Hamiltonian H(eff) in the EH approach is defined as the free energy of a system with a constrained order parameter field. In practice, however, H(eff) has often been approximated by a mean-field free energy functional, yielding what we call a mean-field effective Hamiltonian (MFEH) approximation. This approximation was the starting point of both the Fredrickson-Helfand analysis of fluctuation effects in diblock copolymers and earlier work on the Ginzburg criterion in polymer blends. A more rigorous EH approach by Holyst and Vilgis used an auxiliary field representation of the exact H(eff) and allowed for Gaussian fluctuations of this field. All applications of both AF and EH approaches have thus far relied upon some form of Gaussian, or "one-loop" approximation for fluctuations of a chemical potential and/or order parameter field about a mean-field saddle-point. The one-loop EH approximation of Holyst and Vilgis and the one-loop AF theory are equivalent to one another, but not to the one-loop MFEH theory. The one-loop AF and MFEH theories are shown to yield predictions for the inverse structure factor S(-1)(q) that (in the absence of further approximations to either theory) differ by a function that is independent of the Flory-Huggins interaction parameter χ. As a result, these theories yield predictions for the peak scattering intensity that exhibit a similar χ-dependence near a spinodal. The Fredrickson-Helfand theory for the structure factor in disordered diblock copolymer melts is an asymptotic approximation to the MFEH one-loop theory that captures the dominant asymptotic behavior of very long, symmetric copolymers very near the order-disorder transition.     

Analytical solutions for rotational diffusion in the mean field potential: application to the theory of dielectric relaxation in nematic liquid crystals

A theory of dielectric relaxation in nematics is developed for a molecular dipole moment directed at an arbitrary angle to the molecular long axis. Both exact and simple approximate analytical formulae for the longitudinal and transverse components of the complex dielectric permittivity tensor are obtained for the non-inertial rotational Brownian motion of a molecule in the mean field potential of Maier and Saupe. It appears that both longitudinal and transverse relaxation processes are effectively described by two Debye type mechanisms with corresponding relaxation times and dielectric strengths expressed in terms of the order parameter. The generalization of the theory for an arbitrary axially symmetric mean field potential is given.     

N.M.R. investigations and the molecular field theory of nematic mixtures

N.M.R. measurements have shown that the mixture E5 can be described by a single order parameter. Based on molecular field theory the nematic-isotropic transition and the temperature dependence of the order parameter in the nematic phase are considered for a binary mixture of nematogens. Guided by the results of the N.M.R. measurements the binary mixture is treated as an effective medium characterized by a single order parameter. Soft attractive forces are taken into account as well as the excluded volume. The coexistence of nematic and isotropic phases in the phase transition region is discussed in detail.     

The convergence of energy expansions for molecules in electrostatic fields: A linear‐response coupled‐cluster study

The radii of convergence of power series expansions describing energy of a molecule in external electrostatic field are investigated usingD’Alembert ratio test, standard and generalized Cauchy–Hadamardcriteria, and Padé approximants. The corresponding coefficients at various field and field‐gradient components, representing multipole moments and (hyper)polarizabilities and including terms of tenth or even twentieth order, are determined using an ab initio linear responsecoupled‐cluster theory. Most calculations are performed for the HF molecule described by the basis set of double zeta quality, while the role of basis set is discussed by comparing the results with estimates of the radii of convergence obtained with the basis set of [5s3p2d/3s2p] quality. Emphasis is placed on the dependence of the interval of convergence of power series expansion describing energy of a molecule in applied electrostatic field on the nuclear geometry. The results might have important implications for various numerical methods used to calculate electrostatic molecular properties as functions of the internuclear geometry, including the finite‐field andfixed‐point‐charge approaches. This revised version was published online in July 2006 with corrections to the Cover Date.     

Preliminary communication Photopyroelectric investigation of the thermal conductivity anisotropy in oriented liquid crystals

Photopyroelectric measurements of heat transport parallel or perpendicular to the director have been carried out for compounds of the 4,4-dialkylazoxybenzene homologous series. A theoretical expression based on molecular length-width ratios predicts a thermal conductivity anisotropy ratio which is between 1.5 and 2 times larger than experimentally observed. However, the predicted order parameter temperature dependence seems to be consistent with the experimental data.     

Intermolecular interactions in mixtures of dichroic dyes with n-alkyl-4-(4-isothiocyanatophenyl)bicyclo[2,2,2]octanes

On the basis of the polarized absorption spectra, the guest order parameter as a function of temperature for five dichroic azo dyes dissolved in four members (pentyl to octyl) of the homologous series of n-alkyl-4-(4-isothiocyanatophenyl)bicyclo[2,2,2]octanes has been evaluated. Moreover, the influence of the dichroic dye addition on the nematic-isotropic phase transition temperature has been examined. The relation between the molecular geometry of the guest, its orientation efficiency in the nematic host and the clearing temperature of the azo dye-liquid crystal mixtures has been determined. The experimental results have been compared in some detail with calculations performed on the basis of the mean field theory for binary mixtures.     

Pressure calculation in hybrid particle-field simulations

In the framework of a recently developed scheme for a hybrid particle-field simulation techniques where self-consistent field (SCF) theory and particle models (molecular dynamics) are combined [J. Chem. Phys. 130, 214106 (2009)], we developed a general formulation for the calculation of instantaneous pressure and stress tensor. The expressions have been derived from statistical mechanical definition of the pressure starting from the expression for the free energy functional in the SCF theory. An implementation of the derived formulation suitable for hybrid particle-field molecular dynamics-self-consistent field simulations is described. A series of test simulations on model systems are reported comparing the calculated pressure with those obtained from standard molecular dynamics simulations based on pair potentials.     

Magnetic field effects on atomic and molecular collisions

The full quantal theory of heavy-particle collisions in the presence of a static magnetic field, B, has been formulated. A new mechanism of magnetic field effect was found which depends on the velocity of the molecular center of mass and plays an important role at high-speed collisions. A power series expansion of the cross section as a function of B was obtained, which can be applied to collisions at weak field. It was also found that the differential cross section can become zero due to the magnetic field when a rovibronic energy level of a molecule is in resonance with another energy level.     

MOLECULAR FIELD THEORY FOR NEMATIC LIQUID CRYSTAL POLYMER COMPRISING FLEXIBLE SPACER

Basea on the new model and concept of mtramolecular orientational order parameter, a molecular field theory was built up for main chain liquid crystalline polymer (MC-LCPs) with flexible spacers. The theory takes account of orientational correlation among all mesogens in a polymer chain and the relationship between the intramolecular orientation and spatial orientation of the mesogens. The free energy, temperature and entropy of the nematic-isotropic transition were determined with the theory and compared with experiments in current work. It was found that many unique transition properties of the MC-LCPs comprising flexible spacer are correctly predicted by the theory and the agreement of the theory with the experiments is impressive.     

Probabilistic evolution theory for the solution of explicit autonomous ordinary differential equations: squarified telescope matrices

Probabilistic evolution theory (PREVTH) is used for the solution of initial value problems of first order explicit autonomous ordinary differential equation sets with second degree multinomial right hand side functions. It is an approximation method based on Kronecker power series: a rewriting of multivariate Taylor series using matrices having certain flexible parameters. Kronecker power series have matrices which are called telescope matrices: \(n \times n^{j+1}\) matrices where j is the index of summation. The additive terms of each telescope matrix is formed through Kronecker product from both sides by Kronecker powers of identity matrices. Recently, squarification is proposed in order to avoid the growing of the matrices in size at each additive term of the series. This paper explains the squarification procedure: the procedure used in order to avoid Kronecker multiplications within PREVTH so that the sizes of the matrices do not grow and so that the amount of necessary computation is reduced. The recursion between squarified matrices is also given. As a numerical application, the solution of a Hénon–Heiles system is provided.     

Orientation of semiflexible biopolymers in a magnetic field

A theory is presented to explain the stretching and orientation of a semiflexible chain such as the biopolymeric filamentous actin and deoxyribonucleic acid in a magnetic field. Both dilute and concentrated solutions are considered. The theory predicts the requirement of a high magnetic field to achieve a high degree of orientation in a dilute solution. The experimental value of the order parameter for a dilute, polydisperse F‐actin solution has been compared with the theoretical value. On the other hand, a molecular field (nematic liquid crystal) which exists in a concentrated solution can achieve the same effects of stretching and orientation and can be assisted by a magnetic field. Sources of diamagnetic anisotropy in biopolymers have been cited. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 3256–3270, 2000     

Theoretical analysis of resonant raman scattering: Simulations of lineshapes and excitation profiles

The steady-state response of a three-level system in the presence of a strong laser field is described with the density operator formalism, in order to derive an analytical expression for the intensity of resonance Raman scattering. The Liouville equation for the density operator is written in quantum form for both the molecular system and the radiation field, making use of the dipole and rotating-wave approximations for the matter-radiation interaction, and of the Markov approximation for the molecular decays. The equation is solved exactly for the laser field, and in perturbative way for the generated field. The results account for power broadening of the generated line. Rabi intensity-dip of the generated signal, and for the excitation profiles in the Franck-Condom approximation.     

Study of the molecular tilt angle and the order parameter of a ferroelectric liquid crystal mixture using IR spectroscopy

We report measurements of the molecular tilt angle of a ferroelectric liquid crystal mixture (ZLI 3654), using infra-red spectroscopy, and of the optical tilt angle using polarizing microscopy. The molecular tilt angle is found to be approximately the same for different molecular vibrations. The second rank orientational order parameter for the axis of molecular vibrations of different groups of the molecule for homogeneously aligned samples have been determined as a function of temperature. The order parameter depends slightly on the sample thickness and is about 8 per cent higher for a lower thickness (∼ 7 μm) compared to a higher thickness (∼ 12 μm). The order parameter does not change significantly at the S_C*-S_A transition. The order parameter of S_C* is found to be unaffected by the application of a constant electric field. The order parameter for the alkyl chain is found to be about three times lower than for the molecular core.     

A study on the relaxation phenomena of nematic polymers after cessation of shear flow

A viscoelastic model, composed of the Ericksen and Landau-de Gennes nematic continuum theories, is used to study numerically the relaxation phenomena after cessation of simple shear flow for a model rigid rod uniaxial nematic polymer. This model predicts that under certain conditions the relaxation of stored molecular and coupling elastic free energies due to periodic fluctuations in the scalar order parameter results in a transient periodic distortion of the director field. These conditions are that: (1) the ratio of the wavelength scales of the initial periodic spatial variation in the scalar order parameter k_s to the initial periodic planar director orientation fluctuation kφ (i.e. k_S/kφ) and the amplitude of the initial S spatial variation exceed certain minimum values, and (2) kφ is not zero. It is shown that the wavelength selection mechanism is controlled by the director reorientation-induced backflows. The digitized optical patterns of the transient periodic director field show transient periodic optical patterns similar to the transient banded texture nematic polymers exhibit after cessation of shear flow when observed between crossed polars. The numerical results and digitized optical patterns replicate frequently reported experimental observations.     

Probabilistic evolution approach for the solution of explicit autonomous ordinary differential equations. Part 1: Arbitrariness and equipartition theorem in Kronecker power series

Probabilistic evolution approach is a newly developed theory which may be utilized for the solution of ordinary differential equations. The approach may directly be applied for initial value problems of explicit first order autonomous ordinary differential equation sets with analytic right hand side functions. Analyticity plays an important role since it facilitates the expansion into direct power series which is the key element of the approach. Direct power series appear not only in all applications of probabilistic evolution but also show themselves as a promising tool for novel approximation methods. In this work, similarities and differences between Taylor series and direct power series are rigorously studied. Arbitrariness in transposed vector coefficients of direct power series is detailed. Equipartition theorem of direct power series is conjectured and proven in order to obtain unique transposed vector coefficients.