Wertheim's First-order Perturbation Theory Research Articles

Phase behavior and bulk structural properties of a microphase former with anisotropic competing interactions: A density functional theory study.

Using classical density functional theory, we investigate systems exhibiting interactions where a short-range anisotropic attractive force competes with a long-range spherically symmetric repulsive force. The former is modelled within Wertheim's first-order perturbation theory for patchy particles, and the repulsive part is assumed to be a Yukawa potential which is taken into account via a mean-field approximation. From previous studies of systems with spherically symmetric competing interactions, it is well known that such systems can exhibit stable bulk cluster phases (microphase separation) provided that the repulsion is sufficiently weak compared to the attraction. For the present model system, we find rich phase diagrams including both reentrant clustering and liquid-gas binodals. In particular, the model predicts inhomogeneous bulk phases at extremely low packing fractions, which cannot be observed in systems with isotropic competing interactions.

Very low surface tension liquid–vapour interfaces of patchy colloids

The liquid–vapour interface of a model of one-component patchy colloids is re-visited. The model consists of hard spheres decorated with short-ranged attractive sites (‘patches’) of different types on their surfaces. In an earlier paper [A. Oleksy and P.I.C. Teixeira, Phys. Rev. E 91 (2015) 012301], the density-functional theory (DFT) form of Wertheim's first-order perturbation theory (TPT1) was used to study a realisation of this model with two patches of type A and nine patches of type B (2A9B colloids), which exhibit re-entrant liquid–vapour coexistence curves and very low-density liquid phases. Here, it is shown that the non-monotonic temperature dependence of the surface tension and interface thickness in this model can be qualitatively reproduced by a local DFT, where the spatial non-uniformity is introduced through the square gradient of the density multiplied by a density-dependent prefactor. This simpler theory is then applied to colloids decorated with two patches of type A and ten patches of type B (2A10B colloids), and interactions chosen so as to exhibit a closed-loop phase diagram when only AA and AB bonds are present [N.G. Almarza et al., J. Chem. Phys. 137 (2012) 244902]. In this case, the surface tension goes to zero at both upper and lower critical points, with a maximum at some intermediate temperature. As BB attractions are gradually switched on, the lower critical point disappears and the surface tension vs temperature curve acquires a very deep minimum at a temperature lower than that of the maximum. On further increasing the strength of the BB attractions, both the minimum and the maximum disappear and the surface tension becomes a monotonically decreasing function of the temperature, as in atomic fluids. The very low surface tension at the minimum for weak BB attractions, combined with the ability to form physical gels, suggests that 2AnB colloids might be exploited as temperature-controlled surfactants and foam stabilizers.

Percolation in binary and ternary mixtures of patchy colloids.

We investigate percolation in binary and ternary mixtures of patchy colloidal particles theoretically and using Monte Carlo simulations. Each particle has three identical patches, with distinct species having different types of patch. Theoretically we assume tree-like clusters and calculate the bonding probabilities using Wertheim's first-order perturbation theory for association. For ternary mixtures, we find up to eight fundamentally different percolated states. The states differ in terms of the species and pairs of species that have percolated. The strongest gel is a trigel or tricontinuous gel, in which each of the three species has percolated. The weakest gel is a mixed gel in which all of the particles have percolated, but none of the species percolates by itself. The competition between entropy of mixing and internal energy of bonding determines the stability of each state. Theoretical and simulation results are in very good agreement. The only significant difference is the temperature at the percolation threshold, which is overestimated by the theory due to the absence of correlations between bonds in the theoretical description.

Analytic perturbative FMSA equation of state and thermodynamic properties from Monte Carlo simulation of the Kihara potential with a spherical core

A new equation of state is found to reproduce analytically the thermodynamic properties of fluids interacting with a spherical Kihara potential formulated for arbitrary values of the inner hard core. The results from the expressions developed are satisfactorily compared with the Monte Carlo simulation data for a wide range of densities and temperatures. The foundation of the new equation is strictly theoretical, based on the first-order mean spherical approximation perturbation approach of Tang and Lu, and provides a good starting point for the description of polyatomic fluids within the benchmark of Wertheim's first-order perturbation theory of associating fluids.

Liquid-vapor interfaces of patchy colloids.

We investigate the liquid-vapor interface of a model of patchy colloids. This model consists of hard spheres decorated with short-ranged attractive sites ("patches") of different types on their surfaces. We focus on a one-component fluid with two patches of type A and nine patches of type B (2A9B colloids), which has been found to exhibit reentrant liquid-vapor coexistence curves and very low-density liquid phases. We have used the density-functional theory form of Wertheim's first-order perturbation theory of association, as implemented by Yu and Wu [J. Chem. Phys. 116, 7094 (2002)], to calculate the surface tension, and the density and degree of association profiles, at the liquid-vapor interface of our model. In reentrant systems, where AB bonds dominate, an unusual thickening of the interface is observed at low temperatures. Furthermore, the surface tension versus temperature curve reaches a maximum, in agreement with Bernardino and Telo da Gama's mesoscopic Landau-Safran theory [Phys. Rev. Lett. 109, 116103 (2012)]. If BB attractions are also present, competition between AB and BB bonds gradually restores the monotonic temperature dependence of the surface tension. Lastly, the interface is "hairy," i.e., it contains a region where the average chain length is close to that in the bulk liquid, but where the density is that of the vapor. Sufficiently strong BB attractions remove these features, and the system reverts to the behavior seen in atomic fluids.

LOCAL DENSITY DISTRIBUTION OF NONUNIFORM POLYMER MELTS AT SOLID-FLUID INTERFACES

We have developed the density functional theory (DFT) and density functional perturbation theory (DFPT) which are based both on the hybrid weighted-density approximation and on the higher-order weighted-density approximation, respectively, to study the structural and thermodynamic properties of polymer melts at interfaces. They were applied to predict the local density distributions, adsorption isotherms, surface excesses, and solvation forces of a freely jointed tangent hard-sphere chain in hard slit pores. The Wertheim's first-order perturbation theory extended by Yu and Wu [J. Chem. Phys.117, 2368 (2002)] was used to calculate the excess free energy and second-order direct correlation function due to the chain connectivity. The weight functions were calculated from the second-order direct correlation functions. The calculated results show that the DFT is in excellent agreement with the computer simulations and slightly better than the DFPT.

Structure of a freely jointed tangent hard-sphere chain in hard slit pores

A density functional theory (DFT), which is based both on the density functional expansion of the first-order direct correlational functional with respect to the bulk molecular density and on the contact-value theorem at the hard wall, has been developed for studying the structure of a freely jointed tangent hard-sphere chain in hard slit pores. Two different second-order direct correlation functions (DCFs) based on the Polymer Reference Interaction Site model (PRISM) coupled with the Percus–Yevick closure relation and the Wertheim's first-order perturbation theory modified by Yu and Wu (modified TPT1) were used as inputs. The calculated results show that (i) the PRISM theory is in excellent agreement with the computer simulations even for high densities, and that (ii) the calculated local density distribution strongly depends on the second-order DCFs as well as the DFT version employed.

Chemical potential of the two dimensional fluid and solid phases

The statistical associating fluid theory (SAFT) for hard sphere chain fluids has been extended to the two-dimensional hard disk chain in fluid and solid phases. For fluids, SAFT dimer has been employed to determine the equation of state (EOS) and chemical potential of the hard disk chain fluids for various chain lengths. For solids, the SAFT also known as Wertheim's first-order perturbation theory has been employed to determine the equation of state and chemical potential of the two dimensional hard homonuclear the dumbbell-shaped Herringbone (HB) crystal with a high anisotropy parameter (ℓ*=0.924). The fluid–solid transition for dumbbell shape (ℓ*=0.924) molecule has also been determined.

Extending Wertheim's perturbation theory to high‐density molecular solids

AbstractWe report a theoretical study of a two‐dimensional hard homonuclear dumbbell‐shaped molecular system with a high anisotropy parameter d* = 0.924 employing Wertheim's first‐order perturbation theory. The thermodynamic state of the molecular solid known as HB‐crystalline phase is discussed. The equation of state, free energy and fluid‐solid phase transition are reported. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Structure of nonuniform polymer melts: density functional perturbation approach.

A density functional perturbation approximation for polyatomic molecules, which is based on the fundamental-measure theory for hard-core repulsion and the hybrid weighted-density approximation for chain connectivity, was proposed to clarify the structure of polymer melts at interfaces. It was applied to predict the local density distributions, adsorption isotherms, and surface excess of a freely jointed tangent hard-sphere chain in hard slit pores. Wertheim's first-order perturbation theory extended by J. Chem. Phys. 112, 2368 (2002)] was used to calculate the weight function and second-order direct correlation function due to the chain connectivity. The theoretical results are in excellent agreement with the computer simulations.

A computer simulation study of racemic mixtures

A simple model for a chiral molecule is proposed. The model consists of a central atom bonded to four different atoms in tetrahedral coordination. Two different potentials were used to describe the pair potentials between atoms: the hard sphere potential and the Lennard-Jones potential. For both the hard sphere and the Lennard-Jones chiral models, computer simulations have been performed for the pure enantiomers and also for the racemic mixture. The racemic mixture consisted of an equimolar mixture of the two optically active enantiomers. It is found that the equations of state are the same, within statistical uncertainty, for the pure enantiomer fluid and for the racemic mixture. Only at high pressures does the racemic mixture seem to have a higher density, for a given pressure, than the pure enantiomer. Concering the structure, no difference is found in the site-site correlation functions between like and unlike molecules in the racemic mixture either at low or at high densities. However, small differences are found for the site-site correlations of the pure enantiomer and those of the racemic mixtures. In the Lennard-Jones model, similar conclusions are drawn. The extension of Wertheim's first-order perturbation theory, denoted bonded hard sphere theory (ARCHER, A. L., and JACKSON, G., 1991, Molec. Phys., 73, 881; AMOS, M. D., and JACKSON, G., 1992, J. chem. Phys., 96, 4604), successfully reproduces the simulation results for the hard chiral model. Virial coefficients of the hard chiral model up to the fourth have also been evaluated. Again, no differences are found between virial coefficients of the pure fluid and of the racemic mixture. All the results of this work illustrate the quasi-ideal behaviour of racemic mixtures in the fluid phase.

A SAFT model for associating Lennard-Jones chain mixtures

The recently proposed equation of state of statistical associated fluid theory (SAFT) is extended to associating Lennard-Jones (LJ) chain mixtures. In this extension, a new radial distribution function (RDF) for LJ mixtures is derived around the LJ potential size (σ ij ). The RDF expression is completely analytical and real. Comparisons with computer simulation data under various conditions indicate that the RDF is very accurate up to its first peak. The new RDF, together with a previously established equation of state for LJ mixtures, is employed to study LJ chain mixtures by combining with Wertheim's first-order perturbation theory. The resulting equation of state is tested satisfactorily against computer simulation data for both non-associating and associating LJ chain mixtures, with a performance similar to its predecessors for pure LJ chains and LJ mixtures. The SAFT model is uniquely featured by being totally mixing-rule free and by being adjustable at both chain bonding and association sites. Moreover, the SAFT model is formulated very generally, so that it is applicable to both homonuclear and heteronuclear chain mixtures.

A study of associating Lennard–Jones chains by a new reference radial distribution function

A new radial distribution function (RDF) for the Lennard–Jones (LJ) fluid is derived around the LJ potential size ( σ). The theoretically based RDF is completely analytical and real. Comparisons with computer simulation data at various conditions indicate that the RDF is very accurate at r<1.2 σ, and the accuracy is consistent from low to high densities. The new RDF is utilized to study associating LJ chains by combining with Wertheim's first-order perturbation theory. The resulting equation of state is tested against computer simulation data for second virial coefficient and pressure of LJ chains, and for pressure and the fraction of non-associating molecules in associating LJ chains. The calculated results indicate that the new equation of state is very satisfactory and its performance is comparable to several empirical equations reported in the literature. Another feature of the new equation of state is that both chain bonding and association sites are adjustable for practical purposes.

A two-fluid theory for chain fluid mixtures from thermodynamic perturbation theory

A two reference fluid equation of state is derived for mixtures consisting of chain molecules, using Wertheim's first-order thermodynamic perturbation theory (TPT1). This equation of state is based on the compressibility factors of two reference fluids of different chain lengths, and avoids the need to estimate the correlation functions of reference fluids. It is shown that the recently proposed pure fluid equation of state based on observations of Wertheim's firstorder perturbation theory and generalized Flory theory due to Escobedo, F. A., and de Pablo, J. J. (1995, J. chem. Phys., 103, 1946) and Sheng, Y.-J., Panagiotopoulos, A. Z., and Kumar, S. K. (1995, J. chem. Phys., 103, 10315) is a direct prediction of this approach. The present study involves a rigorous derivation of the equation of state, and provides the basis to extend the equation of state to mixtures. For particular chain lengths of the two reference fluids, the equivalence of the SAFT and SAFTD equations of state are recovered. The equation of state is shown to be in good agreement with recent molecular simulation results for pure homonuclear and alternating copolymer hard chain fluids.

SAFT equation of state for fluid mixtures of hard chain copolymers

We present the SAFT (statistical associating fluid theory) equation of state for mixtures consisting of heteronuclear hard chain molecules using Wertheim's first-order perturbation theory extended to monodisperse mixtures. The reference fluid is a hard sphere mixture. Since only pure fluid simulation results are available, comparisons of theoretical results with molecular dynamics simulation data are made for the compressibility factor of pure hard chain copolymer systems, namely, block, alternate and random copolymers. Our comparisons show a very good performance of the equation of state. The comparisons of our theory with other equations of state for heteronuclear hard chain fluids show that the SAFT equation of state is almost equivalent to the generalized Flory dimer theory, and it represents a significant improvement over the generalized Flory, Chiew-Percus-Yevick and the modified form of Chiew-Percus-Yevick theories.

SAFT equation of state for fluid mixtures of hard chain copolymers

We present the SAFT (statistical associating fluid theory) equation of state for mixtures consisting of heteronuclear hard chain molecules using Wertheim's first-order perturbation theory extended to monodisperse mixtures. The reference fluid is a hard sphere mixture. Since only pure fluid simulation results are available, comparisons of theoretical results with molecular dynamics simulation data are made for the compressibility factor of pure hard chain copolymer systems, namely, block, alternate and random copolymers. Our comparisons show a very good performance of the equation of state. The comparisons of our theory with other equations of state for heteronuclear hard chain fluids show that the SAFT equation of state is almost equivalent to the generalized Flory dimer theory, and it represents a significant improvement over the generalized Flory, Chiew-Percus-Yevick and the modified form of Chiew-Percus-Yevick theories.