Interaction Site Model Integral Equation Research Articles

Hydration free energy of hydrophobic solutes studied by a reference interaction site model with a repulsive bridge correction and a thermodynamic perturbation method

We modify the site–site as well as three-dimensional (3D) versions of the reference interaction site model (RISM) integral equations with the hypernetted chain (HNC) closures by adding a repulsive bridge correction (RBC). The RBC treats the overestimation of water ordering around a hydrophobic solute in the RISM/HNC approximation, and thus refines the entropic component in the hydration free energy. We build up the bridge functions on r−12 repulsive core potentials, and propose RBC expressions for both the site–site and 3D-RISM approaches. To provide fast calculation, we obtain the excess chemical potential of hydration by using the thermodynamic perturbation theory (TPT). The site–site RISM/HNC+RBC as well as 3D-RISM/HNC+RBC approaches are applied to calculate the structure and thermodynamics of hydration of rare gases and alkanes in ambient water. For both approaches, the RBC drastically improves the agreement of the hydration chemical potential with simulation data and provides its correct dependence on the solute size. For solutes of a nonspherical form, the 3D treatment yields the hydration structure in detail and better fits simulation results, whereas the site–site approach is essentially faster. The TPT approximation gives the hydration thermodynamics in good qualitative agreement with the exact results of the thermodynamic integration, and substantially reduces computational burden. The RBC–TPT approximation can improve the predictive capability of the hybrid algorithm of a generalized-ensemble Monte Carlo simulation combined with the site–site RISM theory, used to describe protein folding with due account for the water effect at the microscopic level. The RBC can be optimized for better fit to reference simulation data, and can be generalized for solute molecules with charged groups.

Potentials of mean force of simple ions in ambient aqueous solution. II. Solvation structure from the three-dimensional reference interaction site model approach, and comparison with simulations

We applied the three-dimensional reference interaction site model (3D-RISM) integral equation theory with the 3D hypernetted chain (3D-HNC) closure or its partial linearization (3D-PLHNC) to obtain the potentials of mean force (PMFs) and the solvation structure of sodium chloride in ambient water. The bulk solvent correlations are treated by the dielectrically consistent site–site RISM/HNC theory (DRISM/HNC) to provide a proper description of the dielectric properties of solution and to include the case of a finite salt concentration. The PMF is calculated as a difference in the solvation free energy of an ion pair and of individual ions. We obtained and analyzed in detail the PMFs and solvation structure for ion pairs of NaCl at infinite dilution and a concentration of 1 M. The results are in reasonably good agreement with molecular dynamics simulations for the same model of the solution species. Positions and orientations of water molecules in the first solvation shell around the ion pair are deduced. The short-range hydration structure of the ion pairs at infinite dilution and at moderate concentration is very similar. Ionic ordering and clustering is found in 1 M solution.

Liquid crystallinity in flexible and rigid rod polymers

We apply an anisotropic version of the polymer reference interaction site model (PRISM) integral equation description of flexible polymers to analyze athermal liquid crystallinity. The polymers are characterized by a statistical segment length, σo, and by a physical hard-core thickness, d, that prevents the overlap of monomers on different chains. At small segment densities, ρ, the microscopic length scale d is irrelevant (as it must be in the universal semidilute regime), but becomes important in concentrated solutions and melts. Under the influence of the excluded volume interactions alone, the chains undergo a lyotropic, first-order isotropic–nematic transition at a concentration dependent upon the dimensionless “aspect ratio,” σo/d. The transition becomes weaker as d→0, becoming second order, as has been previously shown. We extend the theory to describe the transition of rigid, thin rods, and discuss the evolution of the anisotropic liquid structure in the ordered phase.

Liquid structure at metal oxide–water interface: accuracy of a three-dimensional RISM methodology

We calculated the structure of water in contact with the MgO(100) surface by using the three-dimensional reference interaction site model (3D-RISM) integral equation theory. The spatial distributions of water oxygen and hydrogen over the surface unit cell are calculated and discussed. The water density profiles and the orientations obtained are in good agreement with computer simulations for the same model of the interface. The 3D-RISM approach shows considerable promise as a constituent of a self-consistent description of chemical processes at a metal oxide–water interface.

On the connection between the reference interaction site model integral equation theory and the partial wave expansion of the molecular Ornstein–Zernike equation

We develop an integral equation theory based on the partial wave expansion of the molecular Ornstein–Zernike (OZ) equation. The theory provides a rigorous and transparent framework for multiple site treatments of molecular fluids. We examine free-energy functional and closure expressions with pilot calculations of homonuclear diatomic Lennard-Jones liquids.

Comparison of the potentials of mean force for alanine tetrapeptide between integral equation theory and simulation

The dielectrically consistent reference interaction site model (DRISM) integral equation theory is applied to determine the potential of mean force (PMF) for an alanine tetramer. A stochastic dynamics simulation of the alanine tetramer using this PMF is then compared with an explicit water molecular dynamics simulation. In addition, comparison is also done with simulations using other solvent models like the extended reference interaction site model (XRISM) theory, constant dielectric and linear distance-dependent dielectric models. The results show that the DRISM method offers a fairly accurate and computationally inexpensive alternative to explicit water simulations for studies on small peptides.

Density functional theory of polymers: A Curtin-Ashcroft type weighted density approximation

A density functional theory is presented that combines an exact expression for the ideal gas free energy functional with a weighted density approximation for the excess free energy functional. The weighting function required in the theory is obtained from the Curtin-Ashcroft recipe, with a bulk fluid direct correlation function from the polymer reference interaction site model integral equation theory. The theory is in quantitative agreement with computer simulations for the density profiles of freely jointed tangent sphere hard chains at a hard wall, about as accurate as the Curtin-Ashcroft theory is for hard spheres at a hard wall. For a more realistic fused-sphere chain model with fixed bond angles and bond lengths, the theory is in excellent agreement with simulations at low and intermediate densities but overestimates the magnitude of layering at high densities for short chains. The theory becomes more accurate as the chain length is increased.

Efficient recursive implementation of the modified Broyden method and the direct inversion in the iterative subspace method: Acceleration of self-consistent calculations

An efficient recursive procedure to solve N-dimensional nonlinear equations using the modified Broyden method is described. This procedure is extended to include the direct inversion in the iterative subspace (DIIS) method to further improve the rate of convergence in iterative calculations. In the recursive procedures. the approximate solutions are constructed as linear combinations of n vectors of length N. The calculations are reduced to determine the appropriate coefficients of the linear combinations. The coefficients are evaluated through small matrix operations, the size of which are at most (n+1)×(n+1) except for the generation of a n×n matrix, where n is the iteration number. Storage is required only for the n vectors and the small matrices. The procedures described below can be applied to large systems. To examine the efficiency of the methods, some numerical results are presented in the context of self-consistent calculations of liquid structure using the reference interaction site model (RISM) integral equation and a molecule-site form of the Ornstein–Zernike integral equation. The results indicate that significant acceleration with respect to the Picard iteration method has been achieved by the recursive procedures: The converged solution is obtained in a very small number of iterations and in a fraction of the CPU time. Moreover, the extended method which includes the DIIS approach has further improved the rate of convergence. It also enables solutions to be obtained in an otherwise divergent case.

Aggregation of colloidal particles induced by polymer chains: The RISM integral equation theory

Using the reference interaction site model (RISM) integral equation technique we study aggregation of small colloidal particles in the environment of polymer chains. In particular, we are interested in the following two questions: (i) what is the effect of density of the polymer matrix, ϱ p , and the chain length, N p , on the process of aggregation and (ii) how does the average cluster size depend on the polymer-particle interaction? The central object in our studies is the connectedness pair correlation function, h αβ +( r,r ′) , which gives the probability density that colloidal particle α has a position r and particle β in the same cluster has a position r′. The mean cluster (aggregate) size s is given by s = 1 + ϱ c lim q→0 h ̂ αβ +(q) , where h ̂ αβ +(q) denotes the spatial Fourier transform of h αβ +( r,r ′) and ϱ c is the number density of particles. It is shown that at higher density of the polymer medium aggregation of the colloidal particles is more intensive. With the chain length N p increasing, the effect of the polymer environment on the aggregation of the particles is diminished; however, at sufficiently large polymer densities and N p ∼ 10 3 convergence to the asymptotic limit is observed. In the case when polymer segments are adsorbed on the particles, the temperature dependence of the average aggregate size exhibits a nontrivial behavior: with temperature decreasing, the value of s first decreases and then begins to grow. The effect of varying colloid particle size and colloid density on the mean cluster size is also studied.

Solvent-induced interactions between hydrophobic and hydrophilic polyatomic sheets in water and hypothetical nonpolar water

Hydrophobic and hydrophilic interactions are two major intermolecular forces between hydrophobic nonpolar and hydrophilic polar sites of macromolecules or materials surfaces in solvents. To further understand these two interactions at the microscopic level, an idealized polyatomic model is devised, which includes hydrophobic, hydrophilic, and partially hydrophilic polyatomic planar square molecular sheets. The hydrophobic molecular sheet is composed of the Lennard-Jones particles while the hydrophilic molecular sheet consists of positive and negative charge sites. In the framework of the extended reference interaction site model integral equation theory the solvent-induced interactions (or the potential of mean forces) between two parallel molecular sheets in water and in the hypothetical nonpolar water are investigated in a systematic fashion. Such a highly idealized model allows us to isolate and to explore the important effects of molecular size, relative intermolecular position (e.g., in- or out-of-registry configuration), and hydrophilic site distribution on the hydrophobic and hydrophilic interactions in both water and the hypothetical nonpolar water. Significant insight into these effects at the molecular level is obtained. For the hydrophobic planar molecules in water we find solvent separated hydrophobic interaction becomes less favored as sheet size increases. Moreover, the contact hydrophobic interaction between two molecular sheets in the out-of-registry configuration is always most favorable. For the latter case we find it is the van der Waals attraction, rather than the hydrophobic attraction, that dominates the total interaction. We also find that in both water and the hypothetical nonpolar water the solvent-induced interaction between two hydrophobic sheets behaves similarly. One possible explanation is that the hydrophobic hydration originating from the hydrogen bonding network in water plays an insignificant role in the solvent-induced interaction, at least in the infinitely dilute aqueous solution. For hydrophilic planar molecular sheets in water, we find water-induced hydrophilic interaction is much more substantial compared with the hydrophobic one. In many cases, the hydrophilic interaction is found directly against the intermolecular force between two parallel molecular sheets in vacuum. Finally, for the partially hydrophilic planar molecules in water, a newly discovered feature is that a disperse hydrophilic site distribution gives rise to stronger solvent-induced interaction compared with the clustered hydrophilic site distribution.

Study of polar dumbbell fluids from the gaseous to the liquid densities by the reference interaction site model-1 and -2 integral equations

The structures and the thermodynamic properties of a fluid composed of a nonpolar or a polar dumbbell molecule have been studied from the gaslike to the liquidlike density regions based on two types of the reference interaction site model (RISM) integral equation theories (RISM-1 and RISM-2). This is the first application of the RISM-2 theory to a polar dumbbell fluid. We have proven that the RISM-2 theory with the hypernetted-chain (HNC) approximation has an inconsistency with respect to the zeroth-order relation between site–site total correlation functions and site–site direct correlation functions in the Fourier space. An hypothetical bridge function is introduced to remedy this inconsistency, which works well to give good information on the structure and the dielectric constant in the lower-density region. On the other hand, the RISM-1 theory works well in the higher-density region, but not well in the lower-density region. Complemental application of these theories, that is, the RISM-1 theory for the high-density region and the RISM-2 theory for the low-density region, allows us to understand the properties of fluid over wide density regions.

The effects of local stiffness disparity on the surface segregation from binary polymer blends

The surface segregation from free space polymer blends based on purely entropic effects is investigated using computer simulation and integral equation theory. Computer simulations are performed for tangent-hard-sphere chains of length ranging from short 10 bead chains to experimentally realistic 500 bead chains. The chain segments of one species experience a bending potential which is introduced between any two consecutive bonds and this serves to make this component stiffer than the other blend component. Computer simulations and numerical wall polymer reference interaction site model (wall-PRISM) integral equation calculations for finite hard core athermal chains demonstrate that at liquidlike densities the segments of the stiffer polymer always partition to a neutral surface, apparently independent of the length of the polymer chains in question. Although the primary factor affecting this segregation is the better local packing of the stiff chains at the surface, lattice mean-field calculations suggest that local conformational changes in the molecules also favor the stiff chains at the surface under these conditions. Further, nonlocal effects appear to be irrelevant in this context. Recently, field theoretic based models have suggested in the context of an incompressible approximation that stiffness disparity is the underlying cause for the experimentally observed surface segregation of branched molecules from blends of linear and branched hydrocarbon polymers (the branched molecules were considered more ‘‘flexible’’ or ‘‘conformationally smaller’’). The segregation observed in the simulations, however, is both much smaller in magnitude and of the opposite sign to that seen in the field theoretic calculations. Coupled with results of independent work on the bulk behavior of these athermal mixtures, which do not capture the experimentally observed phase separation, we suggest that hydrocarbon blends, at least over the chain lengths examined, cannot be modeled in terms of purely entropic effects, but rather through the incorporation of energetics. Analytic wall-PRISM results for a thread like model of the polymer molecules are also presented, and show that the various approximations made in deriving analytical theories critically affect the magnitude and the sign of the predicted athermal segregation. The connections of our analytical work to recent field theoretic analyses is also discussed.

Athermal stiffness blends: A comparison of Monte Carlo simulations and integral equation theory

Off-lattice Monte Carlo computer simulations and numerical polymer reference interaction site model (PRISM) integral equation calculations were performed to quantitatively probe the origins of entropic corrections to Flory–Huggins theory for athermal polymer blends with stiffness disparity. This model system is of interest since it has been recently proposed for describing commercially relevant hydrocarbon polymer mixtures. The novelty of the simulations is that the chemical potential changes on mixing for both components are evaluated. We have considered mixing under constant density conditions, and find surprisingly that the stiffer component is stabilized on blending, while the flexible component is characterized by a positive interaction or χ parameter. The net effective single χ parameter describing these blends, however, is close to zero suggesting that they are completely miscible over a wide range of stiffness disparities and chain lengths. PRISM theory is found to be in good agreement with the simulations for both structural and mixing thermodynamic properties. While purely entropic nonrandom mixing effects could be relevant in determining system thermodynamics, especially for large stiffness disparity, the dominant contribution to the chemical potential changes on mixing arise from equation-of-state (EOS) effects since the two pure components and the mixture are at different pressures when examined at the same density. The EOS contribution to the mixing free energy for small stiffness mismatch is shown to be quantitatively reproduced through an extension of the generalized Flory approach. Through the use of PRISM theory we find that athermal, nonlocal entropy-driven phase separation can occur for long enough chains and high enough stiffness disparity. However, since no phase separation is predicted for stiffness disparities relevant to experimental hydrocarbon systems, regardless of chain length, we suggest that enthalpic effects have to be evoked to explain the limited miscibility of these commercially important mixtures.

Microstructure and phase behaviour of a mixed-dimer amphiphile

Microphase ordering and phase behaviour of a mixed dimer composed of a hard sphere and a sphere with a square-well interaction energy are investigated using Monte Carlo simulation, the reference interaction site model (RISM) integral equation method, and the statistical associating fluid theory (SAFT) equation of state. The RISM model and the computer simulations both indicate strong segmental ordering in the fluid due to the asymmetry of the mixed dimer molecule. Monte Carlo results also show the appearance of different microphase morphologies in mixed dimer-hard dimer solutions. Equation of state modelling using the SAFT equations provides information on the bulk phase behaviour of the mixed dimer.

Thermodynamics and local structure of vinyl polymer melts

Monte Carlo simulation results are reported for the site-site pair correlations and equation of state of model vinyl polymer melts. The molecules are freely jointed hard chains with a hard sphere side-group attached to every other backbone bead. The local structure and pressure are investigated as a function of the diameter of the side group for melt-like densities. The intramolecular correlation functions are well represented by a single chain model where excluded volume interactions are included for beads separated by four bonds or less and neglected otherwise. The intermolecular correlation functions show interesting packing effects. The side group shields the backbone beads from approaching each other, to a degree that increases with increasing diameter of the side group. The polymer reference interaction site model integral equation theory is in good agreement with the simulation results for the pair correlation functions. At fixed volume fraction, the pressure is found to be a non-monotonic function of the size of the side group.

Self-consistent integral-equation theory of chain-molecular liquids: Structure and thermodynamics

Self-consistent integral equations for the pair intramolecular and intermolecular correlation functions are derived from a general hierarchy of integral equations for chain-molecular liquids. These coupled equations are obtained by using superposition approximations for the triplet correlation functions, an approximate translational symmetry for the site–site intramolecular correlation functions and the equivalence of sites for intermolecular correlation functions. In addition to this self-consistent set of integral equations, the polymer reference interaction site model (PRISM) integral equation is also made self-consistent by coupling this intermolecular equation to the equations for the intramolecular correlation functions derived in the present theory. The intra- and intermolecular correlation functions of the self-consistent schemes considered in this work obey integral equations, and they are different from the other self-consistent schemes proposed in the literature. Self-consistent solutions for the structural properties, such as intra- and intermolecular correlation functions and structure factor, and macroscopic properties, such as chain expansion factor and thermodynamic functions of athermal polymer melts, are compared with available Monte Carlo results and other theories. For the properties examined, self-consistent solutions yield better results than the non-self-consistent calculations with ad hoc, ideal Gaussian inputs for the intramolecular correlation functions.

Numerical solution of the hypernetted chain equation for a solute of arbitrary geometry in three dimensions

The average solvent distribution near complex solid substrates of arbitrary geometry is calculated by solving the hypernetted chain (HNC) integral equation on a three-dimensional discrete cubic grid. A numerical fast Fourier transform in three dimensions is used to calculate the spatial convolutions appearing in the HNC equation. The approach is illustrated by calculating the average solvent density in the neighborhood of small clusters of Lennard-Jones particles and inside a periodic array of cavities representing a simplified model of a porous material such as a zeolite. Molecular dynamics simulations are performed to test the results obtained from the integral equation. It is generally observed that the average solvent density is described accurately by the integral equation. The results are compared with those obtained from a superposition approximation in terms of radial pair correlation functions, and the reference interaction site model (RISM) integral equations. The superposition approximation significantly overestimates the amplitude of the density peaks in particular cases. Nevertheless, the number of the nearest neighbors around the clusters is well reproduced by all approaches. The present calculations demonstrate the feasibility of a numerical solution of HNC-type integral equations for arbitrarily complex geometries using a three-dimensional discrete grid.

Influence of Conformational Asymmetries on Local Packing and Small-Angle Scattering in Athermal Diblock Copolymer Melts

Local structural correlations and small-angle scattering profiles arestudied using the polymer reference interaction site model (PRISM) integral equation theory for a series of structurally asymmetric diblock copolymer melts in the athermal limit. Motivated by recent experimental work on polyolefin and polydiene copolymers, we have chosen the block aspect ratios and the overall degree of copolymer asymmetry to be representative of those systems. The extent of nonrandom mixing in the fluid is quantified by calculating length scale dependent «effective compositions». These deviations from the bulk composition in the athermal fluid are compared to those of a fully interacting thermal system. The athermal contribution to nonrandom mixing, resulting from pure stiffness asymmetry, is quite small relative to the effects brought on by unfavorable enthalpic AB interactions. Nonetheless, deviations from random mixing in the athermal limit can lead to appreciable unfavorable enthalpic interactions between components within a thermodynamic perturbative scheme. Large departures from the widely assumed equality of the three partial collective structure factors are also found, suggesting that the extraction of a single χ parameter by fitting to a form which assumes incompressibility may in some cases be unreliable

Microscopic Solubility-Parameter Theory of Polymer Blends: General Predictions

A microscopic solubility parameter theory for polymer blends is developed based on liquid-state polymer reference interaction site model (PRISM) integral equation methods. The well-defined approximations to obtain such a description are identified, and tractable schemes to go beyond them are discussed. Analytical predictions for the purely enthalpic χ-parameter are derived using the Gaussian thread model. Many novel, non-Flory-Huggins effects are predicted including the failure of mean-field theory for random copolymer alloys, strong deuteration swap effects, nonadditivity of chemical and conformational asymmetry contributions to the χ-parameter, and unusual and subtle temperature dependences of χ due to thermally-induced density and chain dimension changes. Both general model calculations and applications to specific polyolefins are presented. The simple enthalpy-based theory accounts for the absolute magnitude of olefinic χ-parameters and a wide range of «anomalous» non-Flory-Huggins phenomena observed in recent small-angle neutron scattering and cloud-point experiments. New, experimentally testable predictions are also made. The fundamental origin of the non-mean-field behavior is the dependence of local intermolecular packing, and hence cohesive energy density, on the chain aspect ratio or effective stiffness. Numerical studies using more realistic semiflexible chain models are also presented and are in qualitative agreement with the analytic predictions. The influence of nonlocal, excess entropy of mixing processes is investigated and found to be a relatively small effect for chain aspect ratios characteristic of flexible polymers. Moreover, for blends which can be studied in the miscible phase at experimentally relevant temperatures, the enthalpic contribution to χ arising from conformational asymmetry is generally orders of magnitude larger than the excess entropic contribution. These conclusions provide support for a fundamental assumption of regular solution theory that spatially local enthalpic effects make the dominant contribution to the excess mixture free energy

Structure-Property Correlations of Atomistic and Coarse-Grained Models of Polymer Melts

Polymer reference interaction site model (PRISM) integral equation theory of polymer melts is applied to investigate the role of local chemical structure on interchain packing and thermodynamic properties such as the isothermal compressibility, cohesive energy, and solubility parameters. Chemically realistic rotational isomeric state model (RIS) level calculations are carried out for polyethylene and isotactic and syndiotactic polypropylene. A strong correlation between the thermodynamic properties and the local depletion regime of interchain packing is found. A series of coarse-grained models of decreasing local realism are then investigated, and various schemes are constructed to estimate the coarse-grained model parameters in terms of polymer structure. For the discrete semiflexible chain and the Gaussian thread models the effects of backbone characteristic ratio and chain branching are incorporated by employing an effective stiffness, or aspect ratio. A strategy for carrying out the mapping from the atomistic to coarse-grained level based on single-chain conformational and interchain packing considerations is studied. For polyethylene and polypropylene, by enforcing an equality of the zero-angle scattering of the atomistic and coarse-grained models, good agreement is found between the local radial distribution functions and reduced solubility parameters predicted by the various models. Numerical predictions of the semiflexible chain model and analytical results for the Gaussian thread liquid are obtained and compared against RIS-level calculations and experimental estimates of the melt solubility parameters. At least within a homologous series of materials, it appears that appropriately calibrated coarse-grained descriptions can reproduce various structural and thermodynamic properties. This suggests that polymer alloy miscibility can be studied within a homologous series of materials by computationally convenient, coarse-grained models.