Hard-sphere Chain Research Articles

Primitive models of chemical association. III. Totally flexible sticky two-point model for multicomponent heteronuclear fixed-chain-length polymerization

A multidensity integral-equation theory for polymerization into freely jointed hard-sphere homonuclear chain fluids proposed earlier [J. Chem. Phys. 106, 1940 (1997)] is extended to the case of multicomponent heteronuclear chain polymerization. The theory is based on the analytical solution of the polymer Percus–Yevick (PPY) approximation for the totally flexible sticky two-point (S2P) model of associating fluids. The model consists of an n-component mixture of hard spheres of different sizes with species 2,…,n−1 bearing two sticky sites A and B, randomly distributed on its surface, and species 1 and n with only one B and A site per particle, respectively. Due to some specific restrictions imposed on the possibility of forming bonds between particles of various species, the present version of the S2P model represents an associating fluid that is able to polymerize into a mixture of heteronuclear chain macromolecules. The structural properties of such a model are studied in the complete-association limit and compared with computer-simulation results for homonuclear hard-sphere chain mixtures, symmetrical diblock copolymers, alternating copolymers, and homonuclear hard-sphere chains in a hard-sphere solvent. Some results for the case of partial association are also presented. The PPY theory represents a quantitatively successful theory for the mixtures of short homonuclear chains and the short copolymer systems studied here. We also expect that the theory will prove to be of the same order of accuracy in investigating the case of partial association.

Primitive models of chemical association. IV. Polymer Percus–Yevick ideal-chain approximation for heteronuclear hard-sphere chain fluids

We continue here our series of studies in which integral-equation theory is developed and used for the monomer-monomer correlation functions in a fluid of multicomponent freely jointed hard-sphere polymers. In this study our approach is based on Wertheim’s polymer Percus–Yevick (PPY) theory supplemented by the ideal-chain approximation; it can be regarded as a simplified version of Wertheim’s four-density PPY approximation for associating fluids considered in the complete-association limit. The numerical procedure of this simplified theory is much easier than that of the original Wertheim’s four-density PPY approximation, but the degree of accuracy is reduced. The theory can also be regarded as an extension of the PPY theory for the homonuclear polymer system proposed by Chang and Sandler [J. Chem. Phys. 102, 437 (1995)]. Their work is based upon a description of a system of hard-sphere monomers that associate into a polydisperse system of chains of prescribed mean length. Our theory instead directly describes a multicomponent system of associating monomers that form monodisperse chains of prescribed length upon complete association. An analytical solution of the PPY ideal-chain approximation for the general case of a multicomponent mixture of heteronuclear hard-sphere linear chain molecules is given. Its use is illustrated by numerical results for two models of copolymer fluids, a symmetrical diblock copolymer system, and an alternating copolymer system. The comparison with Monte Carlo simulations is given to gauge the accuracy of the theory. We find for the molecules we study here that predictions of our theory for heteronuclear chain systems have the same degree of accuracy as Chang and Sandler’s theory for homonuclear chain systems.

Isotropic to nematic transition in semiflexible polymer melts

Monte Carlo simulations are reported for the isotropic to nematic phase transition in semiflexible polymer melts. The molecules are modelled as tangent hard sphere chains with a bending energy that can be used to tune the stiffness of the chains. In the rod limit an isotropic to nematic transition occurs at high pressures. For freely-jointed chains no nematic phase is found, but a fairly sharp transition to a nematic phase is observed when the stiffness is increased at fixed density for sufficiently high densities. In semiflexible chains the isotropic to nematic transition is accompanied by a discontinuous change in the chain dimensions. The simulations show that existing theories are not quantitatively accurate for the isotropic to nematic transition in semiflexible chains.

On the linear hard sphere chain fluids

The thermodynamic and structural properties of linear hard sphere chain fluids are studied using geometrical properties of the single chain. This is an extension of the method recently proposed by the authors to study convex particle fluids. The validity of this application to non-convex particles is tested against other theories and the integral equations PY and HNC for which a new simpler solution method is proposed also. The cases of both tangent and fused hard sphere chains are examined. An accurate equation of state is proposed which is valid for both linear tangent hard spheres and fused hard spheres. A semi-analytical expression for the Ornstein–Zernike direct correlation function is proposed which compares well with PY and HNC numerical results. The orientational instabiilty of the isotropic fluid is examined for several cases. In particular the Flory conjecture for linear N -tangent hard sphere fluid which predicts the isotropic-nematic transition around N = 7 is confirmed by the integral equations results for the orientational instability criteria, whereas the present theory predicts an instability for smaller aspect ratio. The influence of chain flexibility on the equation of state is discussed, mainly in the context of very long chains. In particular, it is found that the expressions for the pressure proposed here can be valid also for fully flexible long chains.

Studies on fluids and their mixtures using the Born-Green-Yvon integral equation technique

We have developed the Bom-Green-Yvon (BGY) integral equation theory for investigating the equilibrium properties of fluids and their mixtures both on the lattice and in the continuum. Using the continuum theory we have studied hard sphere fluids over a range in density having chain lengths between one and fifty sites. We have also investigated the collapse transition of a square well chain and a square well ring, each having up to four hundred sites, and have predicted the theta temperature for these systems. Turning to the case of a dilute (hard-sphere) solution we have been able to show the effect of solvation on a hard sphere chain, and captured the dependence of this effect on the ratio of hard sphere diameters of the solvent and chain segments. In all the continuum studies we have found good to excellent agreement with simulation results. We have also derived a lattice BGY theory which, while less sophisticated than the continuum version, has the advantage of producing simple closed-form expressions for thermodynamic properties of interest. This theory is capable of exhibiting the full range of miscibility behaviour observed experimentally, including upper and lower critical solution temperatures and closed-loop phase diagrams. We find that the theory does an excellent job of fitting to different kinds of experimental data and, making use of the parameters derived from fits to pure component data alone, we have been able to predict properties ranging from pure fluid vapour pressures and critical temperatures to changes in the volume and enthalpy on mixing as well as coexistence curves for solutions.

On the linear hard sphere chain fluids

The thermodynamic and structural properties of linear hard sphere chain fluids are studied using geometrical properties of the single chain. This is an extension of the method recently proposed by the authors to study convex particle fluids. The validity of this application to non-convex particles is tested against other theories and the integral equations PY and HNC for which a new simpler solution method is proposed also. The cases of both tangent and fused hard sphere chains are examined. An accurate equation of state is proposed which is valid for both linear tangent hard spheres and fused hard spheres. A semi-analytical expression for the Ornstein–Zernike direct correlation function is proposed which compares well with PY and HNC numerical results. The orientational instabiilty of the isotropic fluid is examined for several cases. In particular the Flory conjecture for linear N -tangent hard sphere fluid which predicts the isotropic-nematic transition around N = 7 is confirmed by the integral equatio...

Phase behavior of semiflexible tangent hard sphere chains

The isotropic–nematic phase transition in semiflexible hard chain fluids is investigated via an Onsager type density functional theory. The angle-dependent excluded volume of two chains required in the theory is obtained via Monte Carlo simulations. The theory predicts an isotropic to nematic phase transition at lower densities than those predicted by previous theories. These results compare favorably with available simulation data.

A molecular-thermodynamic model for polyelectrolyte solutions

Polyelectrolyte solutions are modeled as freely tangent-jointed, charged hard-sphere chains and corresponding counterions in a continuum medium with permitivity ε. By adopting the sticky-point model, the Helmholtz function for polyelectrolyte solutions is derived through the r-particle cavity-correlation function (CCF) for chains of sticky, charged hard spheres. The r-CCF is approximated by a product of effective nearest-neighbor two-particle CCFs; these are determined from the hypernetted-chain and mean-spherical closures (HNC/MSA) inside and outside the hard core, respectively, for the integral equation theory for electrolytes. The colligative properties are given as explicit functions of a scaling parameter Γ that can be estimated by a simple iteration procedure. Osmotic pressures, osmotic coefficients, and activity coefficients are calculated for model solutions with various chain lengths. They are in good agreement with molecular simulation and experimental results.

Predicting the high-pressure phase equilibria of binary aqueous solutions of 1-butanol, n-butoxyethanol and n-decylpentaoxyethylene ether (C10E5) using the SAFT-HS approach

The phase equilibria of water + n-butan-1-ol is characterized by the presence of so-called closed-loop curves which represent regions of liquid-liquid immiscibility in the phase diagram. The system exhibits lower critical solution temperatures (LCSTs), which denote the lower limit of immiscibility together with upper critical solution temperatures (UCSTs). This behaviour can be explained in terms of the competition between the incompatibility of the water with the alkyl chain and the hydrogen bonding between water and the hydroxyl group. To study the phase equilibria we have used a simplified version of the statistical associating fluid theory (SAFT), which is based on the thermodynamic perturbation theory of Wertheim for associating fluids: the original SAFT-LJ equation of state treats the molecules as chains of Lennard-Jones segments, while the simplified SAFT-HS equation treats molecules as chains of hardsphere repulsive segments with van der Waals interactions. The water molecules are modelled as hard spheres with four association sites to treat the hydrogen bonding; the dispersion forces are treated at the van der Waals mean-field level. The 1-butanol molecules are modelled as chains of hard-sphere segments with two/three bonding sites to treat the terminal hydroxyl group; the dispersion forces also are treated at the mean-field level. The work has been extended to examine water + n-butoxyethanol (C4E1) and water + n-decylpentaoxyethylene ether (C10E5), which are n-alkyl polyoxyethylene ethers (C i E j ). Both of these systems exhibit closed-loop behaviour. These longer chain molecules also are modelled as chains of hard spheres with bonding sites to treat the terminal hydroxyl group, but with an additional three sites per oxyethylene group. For appropriate choices of the intermolecular parameters, the SAFT-HS approach predicts cloud curves with both a UCST and an LCST. The critical temperatures and the extent of immiscibility are in very good agreement with experimental data. The high-pressure critical behaviour is also well described by the theory, including the critical dome for H2O + C4E1.

Polymers near a solid surface. Fused hard sphere chain model

Density profiles and adsorption coefficient of a fused hard sphere chain fluid near a hard wall are studied using the polymer PY approximation for the associative Henderson-Abraham-Barker equation. Adsorption of polymer chains onto an adhesive wall is considered and discussed. The chain molecule is represented to be formed by hard sphere beads with two independent attractive sites displaced inside the hard core. The beads adjacent within a given chain penetrate each other at a distance L between their centres. The wall-particle correlation functions are given both numerically and analytically (as the Laplace transform) as functions of bulk number density η, mean chain length m and bond length L . Dependence of the interfacial structure on the bond length L is investigated. The results indicate that with increasing density and elongation L the paralel orientation of the chains adjacent to the wall is more probable. The appearence of the second adlayer with increasing adhesion is displayed. It is shown that increasing of the intramolecular correlations induces a high coverage of adhesive surface at low bulk densities.

Mixing rules for hardsphere chain mixtures and their extension to heteronuclear hardsphere polyatomic fluids and mixtures

Mixing rules are very important for the calculation of fluid properties using different equations of state. In order to find the theoretical lead of the mixing rule for the size parameter, a mixing rule [1] for hardsphere mixtures has been proposed on the basis of Carnahan-Starling equation and Boublik-Mansoori equation. As its extension, mixing rules for hardsphere chain mixtures are proposed in this work. A mixing rule for the segment number (or chain length) is derived on the limitation of the equality of segment diameters, from the first order thermodynamic perturbation theories (TPT1) for pure chain fluids and for chain mixtures. Meanwhile, the mixing rule for the segment diameter is the same as the mixing rule for hardsphere mixtures on the limitation of monomer mixtures. The two mixing rules are checked together over wide ranges of conditions for hardsphere chain mixtures and compared with the first order thermodynamic perturbation theory (TPT1) and also with simulation data available in literature. An another interesting usage of new mixing rules is to describe the heteronuclear hardsphere polyatomic pure fluids, which consist of hardspheres with different segment diameters as in methane and ethane in which carbon and hydrogen atoms are looked as bonded spheres, and heteronuclear hardsphere polyatomic mixtures. The comparison with simulation data shows the validity of the mixing rules.

The high density equation of state and solid-fluid equilibrium in systems of freely jointed chains of tangent hard spheres

We present a study of the solid-fluid phase equilibrium for systems of freely jointed chains of tangent hard spheres using Monte Carlo computer simulation. Extensive calculations of the fluid and solid phase equations of state have been made. Our results for the fluid properties are compared with various theoretical equations of state which have been proposed for these systems. A method for calculating the solid phase free energy of chain molecules has been developed based on finding a reversible path to an Einstein crystal. The stable solid phase structure of these systems approximates cubic close packing of the atoms without long range orientational ordering of the bond vectors. The contribution to the solid phase free energy from the configurational degeneracy has been approximated by the expression developed by Huggins. The dependence of the solid phase thermodynamics upon the chain conformation has been studied and solid-fluid phase diagrams for chains with lengths ranging from three through eight atoms have been determined.

Analytical treatment of the fused hard-sphere chain model. 0.5 < L < 1

Multidensity integral equation theory for a model of an associating fluid forming freely jointed fused hard-sphere chain, is presented. Our approach is based on the Wertheim's polymer Percus-Yevick (PPY) theory supplemented by the ideal chain approximation and can be regarded as an extension of the PPY theory for tangent hard-sphere fluids proposed by Chang and Sandler (J. Chem. Phys. 102, 1995, 437). The radial distribution function and the structure factor are calculated for the different model parameters. We compare the resulting predictions for the intermolecular distribution function with the Monte Carlo simulation results for the fused diatomic system. It is found that the accuracy of the prediction of the structure of such system is realiable in a rather wide range of density. It is shown that structure factor exhibits a peculiarity (so-called pre-peak) at small wave numbers, connected with the formation of relatively large molecular aggregates. The dependence of the pre-peak magnitude on the degree of penetrability is investigated and discussed.

Vapour-liquid equilibria for some binary and ternary polymer solutions

Vapour-liquid equilibrium (VLE) data for seven binary polymer + solvent systems and for one ternary polymer + mixed solvent system have been obtained using the gravimetric-sorption method at 50°C. The binary polymer + solvent mixtures studied include polystyrene (PS), poly(methyl methacrylate) (PMMA) and the random copolymer poly(styrene-co-50% methyl methacrylate) in acetone, methyl acetate and chloroform. Solvent absorption by the copolymer was inbetween those by PS and PMMA for chloroform, but it was less than those by either homopolymer for acetone and methyl acetate. A modification to the classic gravimetric-sorption technique was established for acquiring sorption data for polymer + mixed solvent systems. VLE data were obtained for polystyrene + chloroform + carbon tetrachloride. A perturbed hard-sphere chain (PHSC) equation of state was used to correlate binary and ternary VLE data.

A perturbed hard-sphere-chain equation of state for normal fluids and polymers using the square-well potential of variable width

A perturbed hard-sphere-chain (PHSC) equation of state is developed for normal fluids and polymers, including mixtures. The new PHSC equation of state for pure fluids uses a perturbation term based on the analytic solution by Chang and Sandler to the second-order perturbation theory of Barker and Henderson for the square-well fluid of variable width. The reference equation of state is the modified Chiew equation of state for athermal mixtures of hard-sphere chains as used in the original PHSC equation of state. The analytic solution by Chang and Sandler is simplified such that theory is readily extended to mixtures by conventional mixing rules. Using an optimum system-independent reduced well width, the new PHSC equation of state correlates the vapor pressures and liquid densities of saturated liquids with good accuracy. Combined with one-fluid type mixing rules for the perturbation term, theory is applied to liquid-liquid and vapor-liquid equilibria for binary mixtures where all components have the same reduced well width. Calculations were also made for a polymer solution where each component has a different reduced well width. In solvent/polymer systems, calculated liquid-liquid equilibria are sensitive to the reduced well width. The perturbation term, however, neglects chain connectivity.

A Computational Study of the Role of Solvation Effects in Reverse Turn Formation in the Tetrapeptides APGD and APGN

The tetrapeptides APGD and APGN are known by NMR analysis to adopt reverse turn conformations to a significant degree in aqueous solution. We have carried out a 7.7 ns molecular dynamics simulation of Ace-APGD-NHMe in explicit water, and have analyzed the energetics of snapshots from this simulation in terms of a molecular mechanics energy function, estimates of solvation free energy based on numerical solutions of the Poisson−Boltzmann equation (in which the solvent is treated as a high-dielectric continuum), and an estimate of chain entropy effects derived from a systematic search procedure. In the unconstrained trajectory, 17 transitions occur between turn and extended conformers, suggesting that the free energy profile is nearly flat and that the simulation is moderately-well-equilibrated with respect to this transition; the turn population found is within the experimental range. The potential of mean force, constructed as the sum of solute force-field energies, continuum solvation, and hard-sphere chain entropy, agrees with that computed directly from the simulation to within 2 kcal/mol across the entire range of configurations sampled. The study has been extended to the tetrapeptide APGN by repeating the energetic analysis with an Asn side chain replacing the Asp in the APGD snapshots. A comparison of energetics with Asp and Asn side chains shows a complex interplay among vacuum electrostatic terms, dielectric screening terms, and solvation free energy terms such that the net effect of side chain substitution on turn formation is very small. Prospects for application of this sort of analysis to other peptide and protein conformational problems are discussed.

The conformation of hard-sphere polymers in hard-sphere solution calculated by single-chain simulation in a many-body solvent influence functional

A fully many-body solvent influence functional for hard-sphere polymer chains in hard-sphere solvents is derived from scaled particle theory. Monte Carlo simulation of single chains in this “solvation potential” is used to calculate conformational properties of 3mers, 4mers and 10mers dissolved in 1mers. Comparison is made to the exact results of Monte Carlo simulation of the whole polymer+solvent fluid, and with the predictions of pairwise-additive integral-equation-based solvation potentials. The greater accuracy of the many-body potential suggests that at least in these systems solvent influence on polymer conformation is a strongly many-body phenomenon.

A perturbed hard-sphere-chain equation of state for nematic liquid crystals and their mixtures with polymers

A perturbed hard-sphere-chain (PHSC) equation of state is presented to compute nematicisotropic equilibria for thermotropic liquid crystals, including mixtures. The equation of state consists of an isotropic term and an anisotropic term given by the Maier-Saupe theory whose contribution disappears in the isotropic phase. The isotropic contribution is the recently presented PHSC equation of state for normal fluids and polymers which uses a reference equation of state for athermal hard-sphere chains and a perturbation theory for the squarewell fluid of variable well width. The PHSC equation of state gives excellent correlations of pure-component pressure-volume-temperature data in the isotropic region and, combined with the Maier-Saupe theory, correlates the dependence of nematic-isotropic transition temperature on the pressure. Theory also predicts a nematic-isotropic biphasic region and liquid-liquid phase separation in a temperature-composition diagram of binary mixtures containing a nematic liquid crystal and a normal fluid or polymer. Theory and experiment show good agreement for pure fluids as well as for mixtures.

Effect of pressure on solid-liquid equilibrium for the system carbon dioxide/n-decane/n-octacosane

Solid-liquid saturation lines (pressure vs temperature for a saturated solution of given composition) were determined for n-octacosane in mixtures of n-decane and carbon dioxide. The carbon dioxide content of the solutions ranged from 0 to 90 mol %, at pressures up to 25 MPa. The data were correlated by means of the perturbed hard sphere chain equation of state. The equation of state requires three pure component constants for each component plus three binary interaction parameters. Two of the interaction parameters were obtained from binary data. The third was adjusted to fit the ternary data. The correlation was satisfactory for carbon dioxide contents up to 80 mol %. For higher carbon dioxide levels the octacosane solubility predicted by the correlation was consistently low.

Primitive models of chemical association. II. Polymerization into flexible chain molecules of prescribed length

The structural properties of the totally flexible sticky two-point (S2P) model for polymerization into chain molecules of fixed length are studied. The model is represented by an n-component mixture of hard spheres of the same size with species 2,…,n−1 bearing two attractive sticky sites A and B, randomly distributed on the surface. The hard spheres of species 1 and n have only one site per particle, site B for species 1 and site A for species n. Due to the specific choice for the attractive interaction, which is present only between site B of the particles of species a and site A of the particles of species a+1, this version of the S2P model represents an associating fluid that polymerizes into freely jointed tangent hard-sphere chain molecules. The correlation functions of this model are studied at all degrees of association using a recently obtained general solution of the polymer Percus–Yevick (PPY) approximation [Yu. Kalyuzhnyi and P. Cummings, J. Chem. Phys. 103, 3265 (1995)]. Comparison of the results of the present theory in the complete association limit with corresponding computer-simulation results and results of other theories is presented and discussed. The complete-association results constitute a quantitatively successful theory of the mean monomer–monomer distribution functions for n⩽16 but for n=50 these functions are no longer quantitatively accurate.