Hard-sphere Chain Research Articles

An explicit molecular thermodynamic model for polyelectrolyte solutions

Polyelectrolyte solutions are modeled as linear tangent jointed and charged hard-sphere chains and counterions. Besides coulombic interaction, stickiness between polyions and counterions is taken into account as a short-range non-coulombic perturbation. The solvent is considered as a continuum medium with a certain permittivity. Expressions of thermodynamic properties derived in this work consist of two contributions. The first contribution responsible for the formation of polyion chains from monomers is derived by the statistical association theory that is the same as our previous work. The second contribution accounting for the additional stickiness is obtained by the same method. A molecular thermodynamic model with explicit expressions is then obtained. Thermodynamic properties for polyelectrolyte solutions can be described satisfactorily.

An application of cell theory to molecular models of n-alkane solids

Solid phase properties for hard sphere chain molecular models of n-alkanes are calculated using the cell theory, and a numerical method for implementation of cell theory for chain molecules is described. Good agreement with Monte Carlo simulations for solid phase properties is obtained from the theory. By using cell theory for the solid phase and an equation of state for the fluid phase, solid-phase equilibrium can be calculated. The predictions are in quite good agreement with Monte Carlo simulation results. Cell theory is used to assess the impact of an approximate treatment used in earlier work for the effect of the temperature dependence of the molecular flexibility upon the solid phase properties of a hard chain model with a realistic torsional potential.

Structure of polydisperse dipolar hard-sphere fluids

We report Monte Carlo simulations of dipolar hard-sphere fluids with a nonuniform distribution of the dipole strengths or different hard-sphere diameters. Dipolar polydispersity significantly reduces the spontaneous orientational order observed in the ferroelectric phase of monodisperse dipolar hard-sphere fluids at high densities and low temperatures. Equimolar binary mixtures of dipolar hard spheres of different diameters σa and σb also exhibit ferroelectric order. However, the order parameters are very dependent on the ratio σa/σb. Chains of dipolar hard spheres, typical of strongly interacting dipolar hard spheres at low densities and low temperatures, were observed in binary fluids of dipolar and neutral hard spheres of different sizes.

Equation of state for polymer chains in good solvents

A free-energy model is developed for polymer chains in good solvents. This model, which combines the ideas of polymer field theory with liquid-state theory, is valid in the dilute, semidilute, and concentrated regimes. The model is compared against computer simulation data for the equation of state of tangent hard-sphere chain fluids. At low concentrations, it accurately describes the thermodynamics of tangent hard-sphere chains, reproducing the universal scaling behavior of long chains. At high concentrations, the model has an accuracy comparable to those of previous equations of state for tangent hard-sphere chains.

Structural and thermodynamic properties of freely-jointed hard-sphere rings and chains

In this paper, we employ the product-reactant Ornstein–Zernike approach (PROZA) developed by the authors to investigate the structural and thermodynamic properties of freely-jointed hard-sphere ring fluids. Using an M×m component sticky two-point (S2P) model and specifying an appropriate association rule between various species, the associating monomers will form M rings with each ring composed of m beads in the complete-association limit. Applying the PROZA to such a Hamiltonian and considering the limit of complete association, we are able to derive analytical expressions for the average monomer–monomer radial distribution function (RDF) as well as its intermolecular and intramolecular contributions and a closed form of the compressibility pressure. To test the theory, we also perform Monte Carlo simulations for the freely-jointed hard-sphere ring model over a wide range of densities and ring sizes. Compared to the simulation results, we find that the predictions of the PROZA for the compressibility factor of flexible ring melts are quantitatively accurate and the average monomer–monomer RDF g(r) is in excellent agreement with the simulation data over a wide range of densities that includes the polymer-melt regime. Based upon such a comparison as well as theoretical considerations, we conclude that ring-size independence of g(r) is a quantitatively accurate approximation and also that the g(r) of rings will be a good approximation for melts of long chains. Finally, we find that we must go beyond our PROZA framework in order to accurately obtain the separate intramolecular and intermolecular parts of g(r), for which we give a quantitatively satisfactory recipe.

The phase behavior of a hard sphere chain model of a binary n-alkane mixture

Monte Carlo computer simulations have been used to study the solid and fluid phase properties as well as phase equilibrium in a flexible, united atom, hard sphere chain model of n-heptane/n-octane mixtures. We describe a methodology for calculating the chemical potentials for the components in the mixture based on a technique used previously for atomic mixtures. The mixture was found to conform accurately to ideal solution behavior in the fluid phase. However, much greater nonidealities were seen in the solid phase. Phase equilibrium calculations indicate a phase diagram with solid–fluid phase equilibrium and a eutectic point. The components are only miscible in the solid phase for dilute solutions of the shorter chains in the longer chains.

Equations of state for hard spheres and hard-sphere chains

Several well-known equations of state developed for hard spheres are reviewed and applied to describe the properties of hard-sphere chains. Two different types of bonding terms that account for chain connectivity are adapted for hard-sphere-chain equations. Each model is then compared with computer simulation data for hard spheres and hard-sphere chains. For hard spheres, the Carnahan–Starling and Kolafa models best describe the computer simulation data. For hard-sphere chains, the accuracy of predictions from each model strongly depends on the choice of bonding terms.

A mapping equation of state: development for the dipolar hard sphere fluid

A new generalised equation of state is proposed and applied to a dipolar hard sphere fluid. This general equation of state has been successfully employed earlier for modelling a hard sphere chain fluid. It is shown that it is possible to map complicated equations which are derived by perturbation approaches on this simple equation. In addition different applications of the c-factor approach are discussed and implemented in the proposed equation of state. Furthermore, a simplified version of a molecular based dispersion term is proposed. The result is a simple fourth-order equation of state for molecular fluids.

Lennard–Jones Chain Model for Self-Diffusion of n-Alkanes

The Lennard–Jones chain model, which was developed from the equation for the self-diffusion coefficient in a Lennard–Jones fluid and the molecular dynamics simulation data of a tangent hard-sphere chain fluid, is used to calculate the self-diffusion coefficient of n-alkanes. n-Alkanes are characterized by a Lennard–Jones segment diameter, a segment–segment interaction energy, and a chain length expressed as the number of segments. The equation represents the experimental self-diffusion coefficients with an average absolute deviation of 3.93% for 16 n-alkanes covering wide ranges of temperature and pressure. The correlated results are compared with those of the rough Lennard–Jones model. A generalized version of the Lennard–Jones chain model is presented which requires only the carbon number in order to predict n-alkane self-diffusivity.

A novel approach to thermophysical properties prediction for chloro-fluoro-hydrocarbons

In this paper we present a new procedure, based on quantum/molecular (QM/MM) mechanics and molecular dynamics (MD) computer simulations, for estimating the Perturbed Hard Sphere Chain Theory (PHSCT) equation of state (EOS) parameters of a set of 14 alternative chloro-fluoro-hydrocarbons (CFH) of industrial relevance and to predict their thermophysical properties and PVT behavior. Force field and quantum-mechanical techniques were employed in molecular modeling and for the calculation of geometrical and chemico-physical parameters. The Connolly surface algorithm, corrected for quantum-mechanical effects, was used in the evaluation of molecular surfaces and volumes. From these data, the parameters of the PHSCT EOS, V* and A*, were obtained. The third parameter, E*, was calculated from extensive MD simulations under NPT conditions. The new, original method proposed in this work gives good results, is relatively inexpensive, is absolutely general and can be applied in principle to any EOS, provided the parameters have a physical meaning. The tuning of the energetic parameter to a generated data set accounts for the degree of empiricism introduced at a certain stage in the development of any EOS.

A Molecular-Based Equation of State for Simple and Chainlike Fluids

A perturbed Lennard-Jones chain (PLJC) equation of state is developed for simple chainlike fluids and polymers. The PLJC equation is formulated on the basis of a well-defined molecular model - the Lennard-Jones chains. Analytical expressions for the Helmholtz energy and compressibility factor were derived on the basis of a first-order variational perturbation theory using the hard-sphere chain as reference. The PLJC equation of state is characterized by three segment-based parameters, namely, the number of segments per molecule m, segment size a, and segment energy e/k. Unlike other perturbation theory-based models, the diameter of the hard-sphere chain segments depends not only on temperature kT/e but also on chain length m. Model parameters for nonassociating molecular fluids were obtained by regressing against vapor pressure and saturated liquid density data. The PLJC model was also applied to high molecular weight polymers; model parameters were obtained by fitting measured specific volumes.

SAFT-D theory for hard sphere and hard disc chain fluids

SAFT-D theory for hard sphere and hard disc chain fluids

SAFT-D theory for hard sphere and hard disc chain fluids

SAFT-dimer (SAFT-D) theory is reformulated to yield an improved equation of state for the hard sphere chain fluid. Two sets of the equation of state are proposed by employing Chiew's expressions for the contact values of the m hard sphere site-site correlation function g(σ). Comparison with molecular simulation data shows that the improved SAFT-D equation of state predicts the compressibility factor more accurately than Ghonasgi and Chapman's equation of state. It has been shown that SAFT-dimer theory can be applied readily to fused hard sphere chain fluids by considering the correct value of the effective chain length (m*). SAFT-dimer theory is also extended to the 2-dimensional tangent and fused hard disc chain fluids. For the fused hard disc dimer fluid, the SAFT equation of state is found to be more accurate than the Boublik hard disc dimer equation of state. For tangent hard disc chain fluids, the results obtained from SAFT-dimer theory are compared with Monte Carlo results for 5-mers and with GFD theory for 4-mers, 8-mers and 16-mers.

Monte Carlo Simulation Study of the Induced Deformation of Polymer Chains Dissolved in Stretched Networks

We examine the preferential alignment of flexible hard-sphere polymer chains along a direction defined by a network of parallel hard rods. For relatively short chains (of 50 hard-sphere segments) we are able to quantify the alignment using a function, φ, which we have called the “nonsphericity” and which is analogous to the nematic order parameter P2. Using this function, we show the alignment to be a weakly increasing function of the density of the chains and a much more strongly decreasing function of the separation of the rods comprising the network. In particular, for shorter chains, we demonstrate the presence of alignment even in the dilute regime, which is unexpected. Using three-dimensional distribution and scattering functions, we demonstrate preferential alignment for longer chains (of 2000 hard-sphere segments) at a density corresponding to the crossover between the dilute and semidilute regimes, although we see no evidence of alignment in the dilute region in this case.

Analytical integral equation theory for a restricted primitive model of polyelectrolytes and counterions within the mean spherical approximation. I. Thermodynamic properties

We present an analytical integral equation theory for polyelectrolyte solutions modeled as linear freely-jointed tangent hard-sphere polyanionic chains and cationic hard-sphere monomeric counterions embedded in a continuum dielectric medium. Each hard-sphere segment on the polyelectrolyte chain and hard-sphere counterion are univalent with unit diameters. The model was formulated in the context of the multi density Ornstein–Zernike integral equation theory within the mean spherical approximation. Analytical solutions for the model were obtained using the ideal chain approximation. The contact values of the radial distribution functions, internal energy, Helmholtz energy, osmotic pressure, and activity coefficient of the system were derived as a function of chain length, density, and Bjerrum length via the energy route. Predictions from the theory were compared with computer simulation data reported in the literature, and very good agreement was found.

Application of Wertheim's thermodynamic perturbation theory to dipolar hard sphere chains

We present results from molecular simulation and statistical mechanics based theory for dipolar hard sphere chains. We consider tangent hard sphere chains with dipoles on alternate segments. The equation of state is obtained by applying Wertheim's associating fluid theory in the total bonding limit to a mixture of non-polar and dipolar hard spheres. The equation of state of the chain requires the compressibility factor and pair correlation functions of this mixture which is the reference fluid. Computer simulation shows that the distribution function between the non-polar and polar hard spheres perpendicular to the dipole can be closely approximated by the hard sphere distribution function. This result is consistent with results from the mean spherical approximation and from reference linear hypernetted chain theory. With this approximation we propose an equation of state for dipolar hard sphere chains.

Application of Wertheim's thermodynamic perturbation theory to dipolar hard sphere chains

Application of Wertheim's thermodynamic perturbation theory to dipolar hard sphere chains

An investigation of the shape and crossover scaling of flexible tangent hard-sphere polymer chains by Monte Carlo simulation

Results are presented from our simulation study of flexible, tangent hard-sphere polymer chains, using a Monte Carlo technique with a reptation algorithm. We examine the crossover region from the swollen (dilute) region to the semidilute regime using chains of degree up to N=2000. The (average) chain end-to-end distance is analyzed as a function of the number of chain links, as is a correlation function for the internal separation of segments, expressed in terms of their connectivity. We study the structure of the polymer chains using scattering functions, which are presented in addition to the segment–segment distribution functions from which they are calculated. Not surprisingly, the Flory exponent ν changes gradually from the dilute value of ν≈0.59 to ≈12 at high density. We find that for chains of 2000 segments, this transition begins at the very low molecular volume fraction of about 0.1%. An analysis of the correlation function for internal segments provides a similar but more detailed picture, in that the role of the screening length ξ becomes explicit. The pair distribution and scattering functions in the swollen, crossover, and melt regions are determined and the scaling behavior is reexamined in the context of these functions. The system considered will form a reference system for subsequent studies in which the model will be broadened to include a network and/or attractive interactions. In this paper we seek first of all to establish that reference, and to relate our work to previous studies in this area of interest. As an important technical point, we also note that the use of an inappropriate random number generator can cause the complete failure of simulations of this type.

Analytical expression for the correlation function of a hard sphere chain fluid

Analytical expression for the correlation function of a hard sphere chain fluid

Analytical expression for the correlation function of a hard sphere chain fluid

A closed form expression is given for the correlation function of flexible hard sphere chain fluid. A set of integral equations obtained from Wertheim's multidensity Ornstein—Zernike integral equation theory with the polymer Percus—Yevick ideal chain approximation is considered. Applying the Laplace transformation method to the integral equations and then solving the resulting equations algebraically, the Laplace transforms of individual correlation functions are obtained. By inverse Laplace transformation the inter- and intramolecular radial distribution functions (RDFs) are obtained in closed forms up to 3D (D is segment diameter). These analytical expressions for the RDFs would be useful in developing the perturbation theory of chain fluids.