Wertheim's Perturbation Theory Research Articles

Coarse-grained polarizable soft solvent models, with applications in dissipative particle dynamics.

We critically examine a broad class of explicitly polarizable soft solvent models aimed at applications in dissipative particle dynamics. We obtain the dielectric permittivity using the fluctuating box dipole method in linear response theory and verify the models in relation to several test cases, including demonstrating ion desorption from an oil-water interface due to image charge effects. We additionally compute the Kirkwood factor and find that it uniformly lies in the range gK≃0.7-0.8, indicating that dipole-dipole correlations are not negligible in these models. This is supported by the measurements of dipole-dipole correlation functions. As a consequence, Onsager theory over-predicts the dielectric permittivity by 20%-30%. The mean square molecular dipole moment can be accurately estimated with a first-order Wertheim perturbation theory.

Thermodynamic modelling of industrially important associating mixtures with a group-contribution equation of state

The accurate prediction of phase equilibria of mixtures containing hydrocarbons and associating components such as water, alcohols and glycols is very important from an industrial perspective, since it ensures product quality and process safety, leading to a more effective design. To this purpose, a new model, that combines the Peng-Robinson equation of state coupled with the Universal Mixing Rules (UMR) with the association term based on Wertheim's perturbation theory is developed. The new model, referred as UMR-CPAPRU, is a predictive group contribution equation of state since it utilizes the UNIFAC activity coefficient model. UMR-CPAPRU leads to improved results in binary mixtures with a 5.4% average absolute relative deviation in bubble point pressure as compared to the 8.8% obtained by UMR-PRU, while it yields improved predictions over UMR-PRU for liquid-liquid equilibrium in alkane/water/methanol mixtures and dew points of wet natural gasses.

Phase behavior of a binary mixture of patchy colloids: Effect of particle size and gravity.

We study theoretically the effect of size difference and that of gravity in the phase behavior of a binary mixture of patchy particles. The species, 2A and 3B, have two A and three B patches, respectively, and only bonds between patches A and B (AB bonds) are allowed. This model describes colloidal systems where the aggregation of particles (3B) is mediated and controlled by a second species, the linkers (2A) to which they bind strongly. Thermodynamic calculations are performed using Wertheim's perturbation theory with a hard sphere reference term that accounts for the difference in the size of the two species. Percolation lines are determined employing a generalized Flory-Stockmayer theory, and the effects of gravity are included through a local density approximation. The bulk phase diagrams are calculated, and all the stacking sequences generated in the presence of gravity are determined and classified in a stacking diagram. The relative size of the particles can be used to control the phase behavior of the mixture. An increase in the size of particles 3B, relative to the size of the linkers 2A, is found to promote mixing while keeping the percolating structures and, in certain cases, leads to changes in the stacking sequence under gravity.

Effect of Buffer on Protein Stability in Aqueous Solutions: A Simple Protein Aggregation Model.

Liquid–liquid phase separation (LLPS) of proteins has recently been associated with the onset of numerous diseases. Despite several studies in this area of protein aggregation, buffer-specific effects always seem to be overlooked. In this study we investigated the influence of buffers on the phase stability of hen egg-white lysozyme (HEWL) and its respective protein–protein interactions by measuring the cloud point temperature, second virial coefficient, and interaction diffusion coefficient of several HEWL–buffer solutions (MOPS, phosphate, HEPES, cacodylate) at pH 7.0. The results indicate that the buffer molecules, depending on their hydration, adsorb on the protein surface, and modulate their electrostatic stability. The obtained information was used to extend the recently developed coarse-grained protein model to incorporate buffer-specific effects. Treated by Wertheim’s perturbation theory the model qualitatively correctly predicted the experimentally observed phase separation of all investigated HEWL–buffer solutions, and further allowed us to predict the phase stability of protein formulations even in experimentally unattainable conditions. Since the theory can be straightforwardly extended to include multiple components it presents a useful tool to study protein aggregation in crowded cell-like systems.

Nonrandom two‐liquid activity coefficient model with association theory

AbstractConsidered one of the most versatile and widely used classical thermodynamic models to correlate phase equilibria behavior of nonideal systems, the nonrandom two‐liquid (NRTL) theory does not explicitly account for specific chemical associations such as hydrogen bonding. This deficiency has been recognized as the cause for unsatisfactory representation of association systems like methanol–alkanes binaries. This work presents a practical approach to integrate Wertheim's perturbation theory for association contribution with the classical NRTL model. Specifically, the association contribution is calculated from pre‐determined molecule‐specific association strengths while the physical interaction contribution is captured with NRTL binary interaction parameters. The resulting association NRTL model correlates fluid phase equilibria of association systems with few adjustable parameters and offers improved predictive capability for higher order systems.

A new study of associating inhomogeneous fluids with classical density functional theory

ABSTRACT A new density functional for the study of associating inhomogeneous fluids based on Wertheim's first-order thermodynamic perturbation theory is presented and compared to the most currently used associating density functionals. This functional is developed using the weighted density approximation in the range of association of hard spheres. We implement this functional within the framework of classical density functional theory together with modified fundamental measure theory to account for volume exclusion of hard spheres. This approach is tested against molecular simulations from literature of pure associating hard spheres and mixtures of non-associationg and associating hard spheres with different number of bonding sites close to a hard uniform wall. Furthermore, we compare and review our results with the performance of associating functionals from literature, one based on fundamental measure theory and the inhomogeneous version of Wertheim's perturbation theory. Results obtained with classical DFT and the three functionals show excellent agreement with molecular simulations in systems with one hard wall. For the cases of small pores where only one or two layers of fluid are allowed discrepancies between results with classical DFT and molecular simulations were found.

Relation of Wertheim association constants to concentration-based equilibrium constants for mixtures with chain-forming components

Several association modeling approaches have been developed to accurately describe the properties of polar solutions. Chemical theory and Wertheim's perturbation theory are among the most popular of these and they have been shown to yield similar functional forms for the contributions of association to Helmholtz energy and activity coefficients. In this paper, we study Flory polymerization theory through the work of Campbell and elucidate its correlation to Wertheim's theory. A simple key relationship between the concentration-based equilibrium constant and Wertheim's association constant is developed for systems in which all associating components have one acceptor and one donor site. Algebraic and numerical proofs are given for the equivalence of Flory's polymerization theory and Wertheim's perturbation theory for pure fluids and mixtures. Additionally, a new generalized activity expression is developed for Wertheim's theory.

Wertheim perturbation theory: thermodynamics and structure of patchy colloids

We critically discuss the application of the Wertheim’s theory to classes of complex associating fluids that can be today engineered in the laboratory as patchy colloids and to the prediction of their peculiar gas–liquid phase diagrams. Our systematic study, stemming from perturbative version of the theory, allows us to show that, even at the simplest level of approximation for the inter-cluster correlations, the theory is still able to provide a consistent and stable picture of the behaviour of interesting models of self-assembling colloidal suspension. We extend the analysis of a few cases of patchy systems recently introduced in the literature. In particular, we discuss for the first time in detail the consistency of the structural description underlying the perturbative approach and we are able to prove a consistency relationship between the valence as obtained from thermodynamics and from the structure for the one-site case. A simple analytical expression for the structure factor is proposed.

Thermodynamic perturbation theory for associating fluids with small bond angles: Effects of steric hindrance, ring formation, and double bonding

We develop a comprehensive approach to model associating fluids with small bond angles using Wertheim's perturbation theory. We show theoretically and through Monte Carlo simulations that as bond angle is varied various modes of association become dominant. The theory is shown to be in excellent agreement with Monte Carlo simulation for the prediction of the internal energy, pressure, and fractions in rings and chains, double bonded over the full range of bond angles.

Three new branched chain equations of state based on Wertheim's perturbation theory

In this work, we present three new branched chain equations of state (EOS) based on Wertheim's perturbation theory. The first represents a slightly approximate general branched chain solution of Wertheim's second order perturbation theory (TPT2) for athermal hard chains, and the second represents the extension of first order perturbation theory with a dimer reference fluid (TPT1-D) to branched athermal hard chain molecules. Each athermal branched chain EOS was shown to give improved results over their linear counterparts when compared to simulation data for branched chain molecules with the branched TPT1-D EOS being the most accurate. Further, it is shown that the branched TPT1-D EOS can be extended to a Lennard-Jones dimer reference system to obtain an equation of state for branched Lennard-Jones chains. The theory is shown to accurately predict the change in phase diagram and vapor pressure which results from branching as compared to experimental data for n-octane and corresponding branched isomers.

Quantitative description of the self-assembly of patchy particles into chains and rings

We numerically study a simple fluid composed of particles having a hard-core repulsion complemented by two patchy attractive sites on the particle poles. An appropriate choice of the patch angular width allows for the formation of ring structures which, at low temperatures and low densities, compete with the growth of linear aggregates. The simplicity of the model makes it possible to compare simulation results and theoretical predictions based on the Wertheim perturbation theory, specialized to the case in which ring formation is allowed. Such a comparison offers a unique framework for establishing the quality of the analytic predictions. We find that the Wertheim theory describes remarkably well the simulation results.

Phase Diagrams for Systems Containing Hyperbranched Polymers

Hyperbranched polymers show an outstanding potential for applications ranging from chemistry over nanotechnology to pharmacy. In order to take advantage of this potential, the underlying phase behaviour must be known. From the thermodynamic point of view, the modelling of these phase diagrams is quite challenging, because the thermodynamic properties depend on the architecture of the hyperbranched polymer as well as on the number and kind of present functional end groups. The influence of architecture can be taken into account via the lattice cluster theory (LCT) as an extension of the well-known Flory–Huggins theory. Whereas the Flory–Huggins theory is limited to linear polymer chains, the LCT can be applied to an arbitrary chain architecture. The number and the kind of functional groups can be handled via the Wertheim perturbation theory, applicable for directed forces between the functional groups and the surrounding solvent molecules. The combination of the LCT and the Wertheim theory can be established for the modelling or even prediction of the liquid-liquid equilibria (LLE) of polymer solutions in a single solvent or in a solvent mixture or polymer blends, where the polymer can have an arbitrary structure. The applied theory predicts large demixing regions for mixtures of linear polymers and hyperbranched polymers, as well as for mixtures made from two hyperbranched polymers. The introduction of empty lattice sites permits the theoretical investigation of pressure effects on phase behaviour. The calculated phase diagrams were compared with own experimental data or to experimental data taken from literature.

Supramolecular assembly and surfactant behavior of triblock rod-coil amphiphiles at liquid interfaces using classical density functional theory

In the framework of Wertheim's perturbation theory we present a classical density functional theory for rod-coil molecules with one rigid block and one or two flexible blocks. The theory is validated against molecular simulation for the case of an amphiphilic molecule in a selective pore. The theory is then applied to study the interfacial properties of the oil–water interface in the presence of triblock rod-coil amphiphiles (TBRCA's) consisting of a center rigid hydrophobic rod portion with flexible hydrophilic tails attached to each end. It is shown that TBRCA's preferentially adsorb to the interface in either a parallel or perpendicular orientation. The fraction of TBRCA's in each orientation can be manipulated by varying the length of the flexible hydrophilic tails, temperature, length of the hydrophobic rod or the total number of TBRCA's at the interface. Also it is shown that the structural properties of TBRCA's can be adjusted to yield a broad range of surfactant behavior.

Equilibrium self-assembly of colloids with distinct interaction sites: Thermodynamics, percolation, and cluster distribution functions

We calculate the equilibrium thermodynamic properties, percolation threshold, and cluster distribution functions for a model of associating colloids, which consists of hard spherical particles having on their surfaces three short-ranged attractive sites (sticky spots) of two different types, A and B. The thermodynamic properties are calculated using Wertheim's perturbation theory of associating fluids. This also allows us to find the onset of self-assembly, which can be quantified by the maxima of the specific heat at constant volume. The percolation threshold is derived, under the no-loop assumption, for the correlated bond model: In all cases it is two percolated phases that become identical at a critical point, when one exists. Finally, the cluster size distributions are calculated by mapping the model onto an effective model, characterized by a--state-dependent--functionality f and unique bonding probability p. The mapping is based on the asymptotic limit of the cluster distributions functions of the generic model and the effective parameters are defined through the requirement that the equilibrium cluster distributions of the true and effective models have the same number-averaged and weight-averaged sizes at all densities and temperatures. We also study the model numerically in the case where BB interactions are missing. In this limit, AB bonds either provide branching between A-chains (Y-junctions) if epsilon(AB)/epsilon(AA) is small, or drive the formation of a hyperbranched polymer if epsilon(AB)/epsilon(AA) is large. We find that the theoretical predictions describe quite accurately the numerical data, especially in the region where Y-junctions are present. There is fairly good agreement between theoretical and numerical results both for the thermodynamic (number of bonds and phase coexistence) and the connectivity properties of the model (cluster size distributions and percolation locus).

Self-assembly of patchy particles into polymer chains: A parameter-free comparison between Wertheim theory and Monte Carlo simulation

The authors numerically study a simple fluid composed of particles having a hard-core repulsion, complemented by two short-ranged attractive (sticky) spots at the particle poles, which provides a simple model for equilibrium polymerization of linear chains. The simplicity of the model allows for a close comparison, with no fitting parameters, between simulations and theoretical predictions based on the Wertheim perturbation theory. This comparison offers a unique framework for the analytic prediction of the properties of self-assembling particle systems in terms of molecular parameters and liquid state correlation functions. The Wertheim theory has not been previously subjected to stringent tests against simulation data for ordering across the polymerization transition. The authors numerically determine many of the thermodynamic properties governing this basic form of self-assembly (energy per particle, order parameter or average fraction of particles in the associated state, average chain length, chain length distribution, average end-to-end distance of the chains, and the static structure factor) and find that predictions of the Wertheim theory accord remarkably well with the simulation results.

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)

Critical temperature of infinitely long chains from Wertheim's perturbation theory

Wertheim's theory is used to determine the critical properties of chains formed by m tangent spheres interacting through the pair potential u(r). It is shown that within Wertheim's theory the critical temperature and compressibility factor reach a finite non-zero value for infinitely long chains, whereas the critical density and pressure vanish as m -1.5. Analysing the zero density limit of Wertheim's equation or state for chains it is found that the critical temperature of the infinitely long chain can be obtained by solving a simple equation which involves the second virial coefficient of the reference monomer fluid and the second virial coefficient between a monomer and a dimer. According to Wertheim's theory, the critical temperature of an infinitely long chain (i.e. the Θ temperature) corresponds to the temperature where the second virial coefficient of the monomer is equal to 2/3 of the second virial coefficient between a monomer and dimer. This is a simple and useful result. By computing the second virial coefficient of the monomer and that between a monomer and a dimer, we have determined the Θ temperature that follows from Wertheim's theory for several kinds of chains. In particular, we have evaluated Θ for chains made up of monomer units interacting through the Lennard-Jones potential, the square well potential and the Yukawa potential. For the square well potential, the Θ temperature that follows from Wertheim's theory is given by a simple analytical expression. It is found that the ratio of Θ to the Boyle and critical temperatures of the monomer decreases with the range of the potential.

Statistical associating fluid theory for chain molecules with attractive potentials of variable range

A version of the statistical associating fluid theory (SAFT) is developed for chain molecules of hard-core segments with attractive potentials of variable range (SAFT-VR). The different contributions to the Helmholtz free energy are evaluated according to the Wertheim perturbation theory. The monomer properties are obtained from a high-temperature expansion up to second order, using a compact expression for the first-order perturbation term (mean-attractive energy) a1. Making use of the mean-value theorem, a1 is given as the van der Waals attractive constant and the Carnahan and Starling contact value for the hard-sphere radial distribution function in terms of an effective packing fraction. The second-order perturbation term a2 is evaluated with the local compressibility approximation. The monomer cavity function, required for the calculation of the free energy due to the formation of the chains and the contribution due to association, is given as a function of a1. We analyse the equation of state for chain molecules with three different types of monomer hard-core potentials with variable attractive range: square-well (SW), Yukawa (Y), and Sutherland (S). The theory for the hard-core potentials can easily be generalised to soft-core systems: we develop a simple equation of state for Mie m−n potentials, of which the Lennard-Jones (LJ) 6-12 potential is a particular case. The equations of state, expressed in terms of reduced variables, are explicit functions of the reduced temperature, the packing fraction, the number of monomers segments forming the chain, and the parameter λ which characterises the range of the the attractive potential. The relevance of the last parameter in the application of the theory to n-alkanes and n-perfluoroalkanes is explicitly shown with the SW expressions. An accurate description of the vapour pressure and the saturated liquid densities is obtained, with a simple dependence of the parameters of the monomer potential on the number of carbons. The extension of our SAFT-VR expressions to mixtures is also presented in terms of a simple expression for the mean-attractive energy for mixtures, based on a straightforward generalisation of the theory for pure components.

Equation of state for the square-well chain fluid based on the dimer version of Wertheim's perturbation theory

An equation of state for the freely jointed square-well chain fluid developed using Wertheim's thermodynamic perturbation theory and incorporating structural information on the square-well dimer fluid (TPT-D) is tested against the results of the Monte Carlo (MC) simulations. The performance of Wertheim's first order thermodynamic perturbation theory (TPT1) and the TPT-D theory are evaluated using MC simulation results for the square-well monomers and square-well dimers to avoid introducing errors due to approximations in evaluating the thermodynamic properties of the reference fluids. The dimer version of Wertheim's thermodynamic perturbation theory (TPT-D) increases the accuracy of the predicted value of compressibility factor of chain fluids in all ranges of density and temperature studied. To obtain the thermodynamic properties needed, NVT MC simulations were performed for the bulk phase of the square-well monomer, and for dimer, 4-mer, 8-mer and 16-mer square-well chain fluids. These MC simulation res...