Hard-sphere Chain Fluids Research Articles

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.

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.

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

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

The nematic–isotropic phase transition in linear fused hard-sphere chain fluids

We present a modification of the generalized Flory dimer theory to investigate the nematic (N) to isotropic (I) phase transition in chain fluids. We focus on rigid linear fused hard-sphere (LFHS) chain molecules in this study. A generalized density functional theory is developed, which involves an angular weighting of the dimer reference fluid as suggested by decoupling theory, to accommodate nematic ordering in the system. A key ingredient of this theory is the calculation of the exact excluded volume for a pair of molecules in an arbitrary relative orientation, which extends the recent work by Williamson and Jackson for linear tangent hard-sphere chain molecules to the case of linear fused hard-sphere chains with arbitrary intramolecular bondlength. The present results for the N–I transition are compared with previous theories and with computer simulations. In comparison with previous studies, the results show much better agreement with simulations for both the coexistence densities and the nematic order parameter at the transition.

Equations of state of freely jointed hard-sphere chain fluids: Theory

Using the analytical solution of a multidensity integral equation solved in our previous papers [J. Chem. Phys. 108, 6513, 6525 (1998)], we derive two compressibility and two virial equations of state (EOS) for freely jointed hard-sphere chain fluids on the basis of the approximations defined by the polymer Percus–Yevick (PPY) closure and of the PPY ideal-chain closure for the integral equations. We also extend a version of first-order thermodynamic perturbation theory to polymers, using a dimer fluid as the reference system, to treat mixtures of heteronuclear chain fluids and polymer solutions; the structural information of the dimer fluid is obtained from the PPY ideal-chain approximation in the complete-association limit. The attractive forces between monomers of chain molecules are treated using simple perturbation theory. We find that the compressibility EOS derived on the basis of the PPY approximation subject to the chain-connectivity condition reduces to the compressibility EOS based upon the PPY ideal-chain approximation in the complete-association limit, which is also equivalent to the EOS derived by Chiew [Mol. Phys. 70, 129 (1990)] and to the EOS derived by Kalyuzhnyi and Cummings [J. Chem. Phys. 105, 2011 (1996)]. On the other hand, the virial EOS derived on the basis of the PPY ideal-chain approximation coincides with Attard’s virial EOS [J. Chem. Phys. 102, 5411 (1995)] only in the zero-density limit. The advantages in numerical implementation of the EOS presented in this work are also discussed, but a full quantitative assessment of our results and a detailed numerical comparison among them are made in a companion paper, as is comparison with available simulation results.

Equations of state of freely jointed hard-sphere chain fluids: Numerical results

We continue our series of studies in which the equations of state (EOS) are derived based on the product-reactant Ornstein–Zernike approach (PROZA) and first-order thermodynamic perturbation theory (TPT1). These include two compressibility EOS, two virial EOS, and one TPT1 EOS (TPT1-D) that uses the structural information of the dimer fluid as input. In this study, we carry out the numerical implementation for these five EOS and compare their numerical results as well as those obtained from Attard’s EOS and GF-D (generalized Flory-dimer) EOS with computer simulation results for the corresponding chain models over a wide range of densities and chain length. The comparison shows that our compressibility EOS, GF-D, and TPT1-D are in quantitative agreement with simulation results, and TPT1-D is the best among various EOS according to its average absolute deviation (AAD). On the basis of a comparison of limited data, our virial EOS appears to be superior to the predictions of Attard’s approximate virial EOS and the approximate virial EOS derived by Schweizer and Curro in the context of the PRISM approach; all of them are only qualitatively accurate. The degree of accuracy predicted by our compressibility EOS is comparable to that of GF-D EOS, and both of them overestimate the compressibility factor at low densities and underestimate it at high densities. The compressibility factor of a polydisperse homonuclear chain system is also investigated in this work via our compressibility EOS; the numerical results are identical to those of a monodisperse system with the same chain length.

Equation of state for polymer systems

We developed an equation of state for systems containing chain-like molecules. The equation consists of three contributions. As a reference, we first adopted the equation for hard-sphere-chain fluids (HSCF) derived through the r-particle cavity-correlation function of sticky hard spheres. The HSCF equation has a simple form similar to the Carnahan–Starling equation and has been satisfactorily tested by computer-simulation results for chain fluids with a wide range of chain length. Secondly, for the perturbation, we used the square-well (SW) equation. Thirdly, for molecules with specific oriented interactions such as hydrogen bonding, we introduced a contribution from chemical association (ASS). The corresponding expression for association comes naturally from the sticky model used in deriving the reference equation. The ASS term was well tested with computer-simulation results for associated molecules. The equation of state was widely tested by fitting experimental vapor–liquid equilibrium data. For liquid–liquid equilibria, we adopted a lattice model developed previously as a high-pressure limit that virtually serves as a mixing rule.

The correlation functions of hard-sphere chains: Monodisperse chains as a complete association limit

The mixture of associating hard spheres with two random association sites is considered to model freely jointed tangent hard-sphere chains of fixed length. In the case of the complete association limit with infinite association strength, the associating fluid becomes the hard-sphere chain fluid. The multidensity Ornstein–Zernike equation is applied to this limiting case, and an analytical solution is obtained within the polymer Percus–Yevick (PPY) approximation. In doing so, we imposed connectivity constraints between bonded segments in order to avoid numerically inconvenient forms. Explicit expressions for the contact values of the correlation functions are obtained, and the correlation functions for the region beyond the hard core are calculated from a set of integral equations involving only finite quantities. Predictions of the theory for 4- and 8-mer fluid are compared to computer simulation results. For overall correlation functions accurate predictions are obtained over the whole density range. For the inter- and intramolecular correlation functions, a significant improvement is found at low density compared to our previous theory with the PPY ideal-chain approximation. As chain length increases, the theory overestimates the intermolecular correlation functions, and underestimates the intramolecular correlation functions. It is concluded that the good accuracy for the overall correlation functions is due to the cancellation of errors between the inter- and intramolecular correlation functions.

Surface induced segregation of hard-sphere chain mixtures

Structure and segregation properties of a hard-sphere chain fluid dissolved in a polymeric solvent near a hard wall are investigated using the Percus–Yevick (PY) approximation for the associative Henderson-Abraham-Barker (HAB) equation. Both the polymer and the solvent are modeled as freely jointed hard-sphere chains of different lengths mp and ms and with different bead diameters dp and ds. It is shown that there are two segregation factors: the bead diameter and the chain length difference, which cause structural features of the interface. It is determined that at low densities the segregation is mainly due to the chain-chain length difference, while the difference in the segment diameters is dominant at high densities. The structure is analyzed for various overall densities ζ3 and polymer concentrations xp. The partial density profiles as well as the segregation profiles are compared with Monte Carlo data to test the accuracy of the PY theory.

Equation of State for Systems Containing Chainlike Molecules

An equation of state for systems containing chainlike molecules is developed. The equation consists of three contributions. First, as a reference, we adopt the equation for hard-sphere-chain fluids (HSCF) derived through the r-particle cavity-correlation function of sticky hard spheres. The HSCF equation has a simple form similar to the Carnahan−Starling equation and has been satisfactorily tested by computer-simulation results for chain fluids covering a wide range of chain length. A new mixing rule is derived when the HSCF equation is used for mixtures. Second, as a perturbation, we use directly the square-well potential (SW). Finally, for molecules with specific oriented interactions such as hydrogen bonding, we introduce contribution from chemical association (ASS) that comes naturally from the sticky model used in deriving the reference equation. The equation of state is widely tested by fitting experimental vapor pressures and pVT relations for pure substances, as well as vapor−liquid equilibria, ex...

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.

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.

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...

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.

Interfacial structure of dimerizing and hard sphere chain fluids in the vicinity of a crystalline surface: effects of a cooperative adsorption

Density profiles have been studied of a hard sphere chain fluid and of dimerizing hard spheres near a crystalline wall represented as a lattice of sticky sites. The associative Henderson–Abraham–Barker equation in the PY approximation is used for the calculation of a structure with the adsorption isotherm assumed to be of the mean-field form. It is shown that the density profiles exhibit a sudden change corresponding to critical behaviour of the adsorption isotherm. The influence of the low-density critical point, caused by the intramolecular correlation in a chain, on the interfacial structure is determined. It is also shown that the associative interparticle interaction forms a sharp edge of the second adlayer. The density of this adlayer is analytically calculated.

Interfacial structure of dimerizing and hard sphere chain fluids in the vicinity of a crystalline surface: effects of a cooperative adsorption

Density profiles have been studied of a hard sphere chain fluid and of dimerizing hard spheres near a crystalline wall represented as a lattice of sticky sites. The associative Henderson–Abraham–Barker equation in the PY approximation is used for the calculation of a structure with the adsorption isotherm assumed to be of the mean-field form. It is shown that the density profiles exhibit a sudden change corresponding to critical behaviour of the adsorption isotherm. The influence of the low-density critical point, caused by the intramolecular correlation in a chain, on the interfacial structure is determined. It is also shown that the associative interparticle interaction forms a sharp edge of the second adlayer. The density of this adlayer is analytically calculated.

Molecular thermodynamic theory for polymer systems part II. Equation of state for chain fluids

An equation of state for fluids containing chain-like molecules has been established by combining a reference equation for hard-sphere chain fluids (HSCF) with Alder et al.'s square-well attractive potential served as a perturbation term. The equation for HSCFs has been derived through the r-particle cavity-correlation function (CCF) based on a sticky-point model for chemical association, where r is the number of segments of a chain molecule. The r-particle CCF is approximated by a product of effective two-particle CCFs accounting for correlations in nearest-neighbour and next-to-nearest-neighbour segment pairs. Their density dependence is determined by using the Tildesley-Streett equation and computer-simulation results for hard-sphere trimers. The obtained HSCF equation can excellently predict compressibility factors and second virial coefficients for homonuclear HSCFs covering a wide range of chain length as well as for homonuclear HSCF mixtures. The final equation of state has been satisfactory tested by fitting vapor pressures and volumetric properties for various pure fluids and polymers.

Density profiles of a hard sphere chain fluid near a hard wall: applications to adsorption onto a crystalline surface

Density profiles and adsorption coefficient of a hard sphere chain fluid near a hard wall are investigated using the Percus-Yevick approximation for the associative Henderson-Abraham-Barker equation. The chain molecule is represented as a hard sphere with two independent attractive sites. The wall-particle correlation functions are given both numerically and analytically (as the Laplace transform) as functions of the bulk number density η and mean chain length m. An analytical expression for the adsorption coefficient is obtained. The density profile prediction for the Wertheim totally dimerized hard sphere case and the case under consideration (with m = 2) are compared. The results for the density profiles (m = 3, m = 4, m = 20) are compared with Monte Carlo data and with the results of density functional theory. The contact values of the correlation functions are used to calculate the adsorption isotherm of polymer chains onto a crystalline wall. It is shown that the intramolecular correlations lead to a new kind of critical behaviour at low densities, which could be interpreted as the adsorption of distinct chains.

Density profiles of a hard sphere chain fluid near a hard wall: application to adsorption onto a crystalline surface

Density profiles of a hard sphere chain fluid near a hard wall: application to adsorption onto a crystalline surface