George Jackson

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 ch...

A re‐examination of the phase diagram of hard spherocylinders

The phase transitions exhibited by systems of hard spherocylinders, with a diameter D and cylinder length L, are re‐examined with the isothermal–isobaric Monte Carlo (MC‐NPT) simulation technique. For sufficiently large aspect ratios (L/D) the system is known to form liquid crystalline phases: isotropic (I), nematic (N), smectic‐A (Smu2009A), and solid (K) phases are observed with increasing density. There has been some debate about the first stable liquid crystalline phase to appear as the aspect ratio is increased from the hard‐sphere limit. We show that the smectic‐A phase becomes stable before the nematic phase as the anisotropy is increased. There is a transition directly from the isotropic to the smectic‐A phase for the system with L/D=3.2. For larger aspect ratios, e.g., L/D=4, the smectic‐A phase is preceded by a nematic phase. This means that the hard spherocylinder system exhibits I–Smu2009A–K and I–N–Smu2009A triple points, the latter occurring at a larger aspect ratio. We also confirm the simulation resul...

Phase equilibria and critical behavior of square‐well fluids of variable width by Gibbs ensemble Monte Carlo simulation

The vapor–liquid phase equilibria of square‐well systems with hard‐sphere diameters σ, well‐depths e, and ranges λ=1.25, 1.375, 1.5, 1.75, and 2 are determined by Monte Carlo simulation. The two bulk phases in coexistence are simulated simultaneously using the Gibbs ensemble technique. Vapor–liquid coexistence curves are obtained for a series of reduced temperatures between about Tr=T/Tc=0.8 and 1, where Tc is the critical temperature. The radial pair distribution functions g(r) of the two phases are calculated during the simulation, and the results extrapolated to give the appropriate contact values g(σ), g(λσ−), and g(λσ+). These are used to calculate the vapor‐pressure curves of each system and to test for equality of pressure in the coexisting vapor and liquid phases. The critical points of the square‐well fluids are determined by analyzing the density‐temperature coexistence data using the first term of a Wegner expansion. The dependence of the reduced critical temperature T*c=kTc/e, pressure P*c=Pcσ3/e, number density ρ*c=ρcσ3, and compressibility factor Z=P/(ρkT), on the potential range λ, is established. These results are compared with existing data obtained from perturbation theories. The shapes of the coexistence curves and the approach to criticality are described in terms of an apparent critical exponent β. The curves for the square‐well systems with λ=1.25, 1.375, 1.5, and 1.75 are very nearly cubic in shape corresponding to near‐universal values of β (β≊0.325). This is not the case for the system with a longer potential range; when λ=2, the coexistence curve is closer to quadratic in shape with a near‐classical value of β (β≊0.5). These results seem to confirm the view that the departure of β from a mean‐field or classical value for temperatures well below critical is unrelated to long‐range, near‐critical fluctuations.The vapor–liquid phase equilibria of square‐well systems with hard‐sphere diameters σ, well‐depths e, and ranges λ=1.25, 1.375, 1.5, 1.75, and 2 are determined by Monte Carlo simulation. The two bulk phases in coexistence are simulated simultaneously using the Gibbs ensemble technique. Vapor–liquid coexistence curves are obtained for a series of reduced temperatures between about Tr=T/Tc=0.8 and 1, where Tc is the critical temperature. The radial pair distribution functions g(r) of the two phases are calculated during the simulation, and the results extrapolated to give the appropriate contact values g(σ), g(λσ−), and g(λσ+). These are used to calculate the vapor‐pressure curves of each system and to test for equality of pressure in the coexisting vapor and liquid phases. The critical points of the square‐well fluids are determined by analyzing the density‐temperature coexistence data using the first term of a Wegner expansion. The dependence of the reduced critical temperature T*c=kTc/e, pressure P*c=Pcσ...

Bonded hard‐sphere (BHS) theory for the equation of state of fused hard‐sphere polyatomic molecules and their mixtures

The bonded hard‐sphere (BHS) approach originally developed for diatomic and triatomic molecules is generalized to hard‐sphere polyatomic models which are formed by bonding together their constituent hard spheres. The thermodynamic properties of the polyatomic fluid are obtained from the known properties of a corresponding multicomponent mixture of different sized hard spheres with bonding sites. In the limit of complete bonding, hard‐sphere polyatomic molecules are formed. As well as the general expressions for polyatomic molecules and their mixtures, the equation of state of hard‐sphere chain molecules, which are simple models of homologous series such as the alkanes, perfluoroalkanes, etc., is presented. More specifically, the chain molecules are formed from two different types of hard spheres 1 and 2. Spheres of type 1 make up the backbone of the chain and, in this case, would represent the carbon atoms; spheres of type 2 represent the substituents, i.e., the hydrogen or fluorine atoms. Although the BH...

Liquid crystalline phase behavior in systems of hard-sphere chains

A study of the liquid crystalline phase transitions in a system of hard-sphere chains is presented. The chains comprise m=7 tangentially bonded hard-sphere segments in a linear conformation (LHSC). The isothermal–isobaric Monte Carlo simulation technique is used to obtain the equation of state of the system both by compressing the isotropic (I) liquid and by expanding the solid (K). As well as the usual isotropic and solid phases, nematic and smectic-A liquid crystalline states are seen. A large degree of hysteresis is found in the neighborhood of the I–N transition. The results for the rigid LHSC system were compared with existing data for the corresponding semiflexible hard-sphere chains (FHSC): the flexibility has a large destabilizing effect on the nematic phase and consequently it postpones the I–N transition. The results of the simulations are also compared with rescaled Onsager theories for the I–N transition. It is rather surprising to find that the Parsons approach, which has been so successful for other hard-core models such as spherocylinders and ellipsoids, gives very poor results. The related approach of Vega and Lago gives a good description of the I–N phase transition. The procedure of Vega and Lago, as with all two-body resummations of the Onsager theory, only gives a qualitative description of the nematic order.

BHS theory and computer simulations of linear heteronuclear triatomic hard-sphere molecules

As an extension to a recent study of model diatomic molecules, an equation of state is determined for linear heteronuclear triatomics formed from three tangent hard spheres with diameters σ1, σ2 and σ3. Spheres 1 and 3 are positioned at each end with sphere 2 in the centre so that the 1–2 and 2–3 bond lengths are l 12 = (σ1 + σ2)/2 and l 23 = (σ2 + σ3)/2. The bonded hard-sphere (BHS) approach provides a route to the thermodynamic properties of the triatomic fluid. An equation of state is obtained from the corresponding expression for an equimolar ternary mixture of different-sized hard spheres with bonding sites. In the limit of complete bonding, the heteronuclear triatomic molecules are formed. The cases investigated are the homonuclear system with σ1 = σ2 = σ3, the symmetrical heteronuclear systems with σ1 = σ3 = σ2/4, σ1 = σ3 = 3σ2/5 and σ1 = σ3 = 4σ2, and the asymmetrical heteronuclear systems with σ1 = σ2 = 2σ3 and σ1 = 2σ2 = 4σ3. Isothermal-isobaric Monte Carlo (MC-NPT) simulations are performed for...

Describing the Properties of Chains of Segments Interacting Via Soft-Core Potentials of Variable Range with the SAFT-VR Approach

We present a general development for the equation of state (EOS) of chain molecules composed of tangent spherical segments interacting with a soft repulsive potential and an attractive well. The method is based on a recent version of the statistical associating fluid theory for chain molecules with interaction potentials of variable range (SAFT-VR). In this communication we focus our attention on the properties of Lennard–Jones chains (LJC), using SAFT-VR and a sample recipe for the evaluation of the chain free energy that requires only a knowledge of the contact value of the cavity function of a Sutherland-6 system. We study the liquid–vapor coexistence properties for different values of the chain length. The results obtained are of similar accuracy to other EOS for LJC, but our approach is simpler and more general. We show that standard perturbation theories developed for simple liquids can also be used for chain molecules.

Theory of closed-loop liquid-liquid immiscibility in mixtures of molecules with directional attractive forces

The phase equilibria of a binary mixture of equal-sized hard spheres (diameters σ1 = σ2) with mean-field attractive forces between like species (a 11 = a 22) and non between unlike species (a 12 = 0) are determined using an ‘augmented’ van der Waals equation of state. This system is found to have a symmetrical phase diagram exhibiting a large extent of liquid-liquid immiscibility and a tricritical point. Directional attractive forces between the unlike species and then introduced in the form of off-centre, square-well bonding sites, and a perturbation term representing the contribution due to bonding is added to the equation of state. The phase behaviour of systems with varying degrees of association is investigated. As the strength of the site-site bonding interaction is increased relative to that of the mean-field attractions, closed-loop liquid-liquid immiscibility is found with the corresponding lower and upper critical solution points. The region of liquid-liquid coexistence decreases with increasing...

Theory and computer simulations of heteronuclear diatomic hard-sphere molecules (hard dumbbells)

The isothermal-isobaric Monte Carlo (MC-NPT) method is used to determine the PVT behaviour of model heteronuclear diatomic molecules. The systems of particular interest are hard-dumbbell models consisting of two tangent hard spheres with diameters σ1 and σ2 so that the bond distance between the centres of the spheres is l = (σ1 + σ2)/2. Computer simulations are performed for molecules with diameter ratios of R = σ2/σ1 = 1/4, 1/2, 3/4, and 1 over a range of packing fractions in the fluid state (η = π(σ3 1 + σ3 2)N m/(6V) = 0·0 to 0·45). The ‘exact’ results obtained from the simulations are compared with an equation of state derived using a thermodynamic perturbation theory for fluids with highly directional bonding forces. The diatomic hard-sphere molecules can be constructed by bonding together the two components of an equivalent binary mixture of hard spheres with bonding sites. In the limit of infinite bonding, an equation of state for the heteronuclear hard-dumbbell fluid is obtained in terms of g hs 1...

The theoretical prediction of the cricital points of alkanes, perfluoroalkanes, and their mixtures using bonded hard-sphere (BHS) theory

The adequacy of the recently developed bonded hard-sphere (BHS) theory in describing the critical behavior of the homologous series of the alkanes and perfluoroalkanes is examined in this work. A simple united atom model, formed from chains of tangent hard spheres, reproduces the major experimental trends and provides good quantitative agreement for systems with two or more carbon atoms. This simple model cannot, however, reproduce the anomalous behavior of the critical pressure of the alkane series: the values of the critical pressure and temperature for methane are smaller than expected. A more sophisticated distributed-site model, which takes explicit account of the backbone and substituent atoms, reproduces this anomalous behavior. The BHS theory has also been used to predict the upper critical solution temperatures of alkane + perfluoroalkane mixtures. For most systems, the segment-segment parameters are fitted to the butane + perfluorobutane system, although in the case of mixtures containing methane, methane + perfluoromethane parameters must be used. Excellent qualitative agreement with experimental data is seen. This indicates the strength of the BHS approach as a type of group contribution method.