Square-well Fluid Research Articles

Adsorption of gas-like molecules to self-aligned square-well fluid channels under confinement of chemically patterned substrates

To extend the work of binary fluid mixtures and their associated bridge-like structures, the adsorption of gas-like molecules (interacting via hard-sphere potentials) on self-assembled fluid channels was examined. We examined the morphological evolution of an initial random binary mixture under confinement of chemically patterned substrates with strong, long-range preferential attraction to the pure square-well component. Gas-like molecules were presumed to have a weak attraction to the square-well fluid. The morphology and corresponding density profiles revealed the underlying chemical and physical adsorption of gas-like molecules to off-strip voids and to the interface of the self-assembled fluid channels. The entropic effects drive the non-interaction hard-sphere molecules to assemble or reorganize in the voids left between the self-assembled square-well fluids. Such studies can help in the study of formation of nano-liquid structures and enhanced adsorption of gas-like molecules for storage purposes.

Tunneling conductivity in composites of attractive colloids

In conductor-insulator nanocomposites in which conducting fillers are dispersed in an insulating matrix, the electrical connectedness is established by inter-particle tunneling or hopping processes. These systems are intrinsically non-percolative and a coherent description of the functional dependence of the conductivity σ on the filler properties, and in particular of the conductor-insulator transition, requires going beyond the usual continuum percolation approach by relaxing the constraint of a fixed connectivity distance. In this article, we consider dispersions of conducting spherical particles which are connected to all others by tunneling conductances and which are subjected to an effective attractive square-well potential. We show that the conductor-insulator transition at low contents φ of the conducting fillers does not determine the behavior of σ at larger concentrations, in striking contrast to what is predicted by percolation theory. In particular, we find that at low φ the conductivity is governed almost entirely by the stickiness of the attraction, while at larger φ values σ depends mainly on the depth of the potential well. As a consequence, by varying the range and depth of the potential while keeping the stickiness fixed, composites with similar conductor-insulator transitions may display conductivity variations of several orders of magnitude at intermediate and large φ values. By using a recently developed effective medium theory and the critical path approximation, we explain this behavior in terms of dominant tunneling processes which involve inter-particle distances spanning different regions of the square-well fluid structure as φ is varied. Our predictions could be tested in experiments by changing the potential profile with different depletants in polymer nanocomposites.

Generalized approach to global renormalization-group theory for fluids

The global renormalization-group theory (GRGT) for fluids is derived starting with the square-gradient approximation for the Helmholtz free energy functional such that any mean-field free energy density and direct correlation function can be employed. The new derivation uses Wilson's functions for representing density fluctuations, thereby relaxing the assumption of cosine variation of density fluctuations used in earlier approaches. The generality of the present approach is shown by deriving the relationships to the earlier developments. A qualitative way to infer the free parameters in the present form of GRGT is also suggested. The new theory is applied to square-well fluids of ranges 1.5 and 3.0 (in units of hard-sphere diameter) and Lennard-Jones fluids. It is shown that the present theory produces a flat isotherm in the two-phase region. Thus the theory accounts for fluctuations at all length scales and avoids the use of Maxwell's construction. An analysis of the liquid-vapor phase diagrams and the critical constants obtained for different potentials shows that, with a mean-field free energy density that is accurate away from the critical region and an appropriate coarse graining length for the mean-field theory, GRGT can provide results in good agreement with the simulation and experimental results.

Density functional theory for the selective adsorption of small molecules on a surface modified with polymer brushes

A density functional method based on weighted density approximation is extended to study the selective adsorption of small molecules on a surface modified with end-grafted square-well chains. The excess part of the Helmholtz free energy functional is divided into two components: the hard sphere repulsion and the square-well attraction. The equation of state for hard sphere chain fluids developed by Liu et al. is used to calculate the repulsive part of the excess Helmholtz free energy functional, and the equation of state for square-well chain fluid with variable range developed by Li et al. is employed to calculate the attractive part. With this theoretical model, we examine the physical properties of the grafted polymer and the selective adsorption of small molecules on the modified surface.

Thermodynamic Description of Liquid-State Limits

A state of random close packing (RCP) of spheres is found to have a thermodynamic status and a fundamental role in the description of liquid-state equilibria. The RCP limiting amorphous ground state, with reproducible density and well-characterized structure, is obtained by well-defined irreversible and reversible processes. The limiting packing fraction y(RCP) = 0.6366 ± 0.0005 (Buffon's constant within the uncertainty), and a residual entropy per sphere ΔS((RCP-FCC)) is approximately equal to k(B) (Boltzmann's constant). Since the Mayer virial expansion does not represent dense fluid equations-of-state for densities exceeding the available-volume percolation transition (ρ(pa)), we infer that a RCP state belongs to the same thermodynamic phase as prepercolation equilibrium dense hard-sphere fluid and likewise for hard-core fluids with attractive forces. Monte Carlo (MC) calculation of the liquid-state coexistence properties of square-well (SW) attractive spheres, together with existing MC results for liquid-vapor coexistence in the SW fluid, support this conclusion. Further findings for liquid-vapor coexistence limits are reported. The extremely weak second-order available-volume percolation transition of the hard-sphere fluid is strengthened by square-well perturbation as temperature is reduced. At the critical temperature, this transition becomes first order, whereupon a liquid at the percolation density coexists in thermodynamic equilibrium with its vapor at a lower density. The critical coexisting vapor density relates to the extended-volume bonded cluster percolation transition ρ(pe)(λ) defined for given well width (λ). Taking experimental liquid argon data as an example, it can be seen that the thermodynamic description of the coexistence limits, found here for square-well fluids, applies to real liquids.

Equation of state of square-well chain fluid with variable range for CO<sub>2</sub> + physical absorption solvent

采用变阱宽方阱链流体(SWCF-VR)状态方程关联了CO 2 在几种常规物理吸收溶剂中的溶解度数据, 得到了二元交互作用参数, 建立了二元交互作用参数与温度的关联式. 结果表明, 采用一个二元交互作用参数, SWCF-VR 方程均能很好地描述CO 2 在常规流体中的溶解度, 尤其能满意再现甲醇低温洗工艺中CO 2 -甲醇的相行为. 利用建立的二元交互作用参数与温度的关联式, 可将SWCF-VR 状态方程拓展应用于预测二元系统气液两相的密度以及CO 2 -物理吸收溶剂多元系统的气液平衡.

Capturing Thermodynamic Behavior of Ionic Liquid Systems: Correlations with the SWCF-VR Equation

An equation of state for square-well chain fluids with variable well-width range (SWCF-VR EoS) [Li et al. Fluid Phase Equilib.2009, 276, 57] was applied to ionic liquid (IL) systems. ILs were treat...

Modeling pVT Properties and Vapor-Liquid Equilibrium of Ionic Liquids Using Cubic-plus-association Equation of State

Combining Peng-Robinson (PR) equation of state (EoS) with an association model derived from shield-sticky method (SSM) by Liu et al., a new cubic-plus-association (CPA) EoS is proposed to describe the thermodynamic properties of pure ionic liquids (ILs) and their mixtures. The new molecular parameters for 25 ILs are obtained by fitting the experimental density data over a wide temperature and pressure range, and the overall average deviation is 0.22%. The model parameter b for homologous ILs shows a good linear relationship with their molecular mass, so the number of model parameters is reduced effectively. Using one temperature-independent binary adjustable parameter k ij , satisfactory correlations of vapor-liquid equilibria (VLE) for binary mixtures of ILs + non-associating solvents and + associating solvents are obtained with the overall average deviation of vapor pressure 2.91% and 7.01%, respectively. In addition, VLE results for ILs + non-associating mixtures from CPA, lattice-fluid (LF) and square-well chain fluids with variable range (SWCF-VR) EoSs are compared.

Modeling of Surface Tension and Viscosity for Non-electrolyte Systems by Means of the Equation of State for Square-well Chain Fluids with Variable Interaction Range

The equation of state (EOS) for square-well chain fluid with variable range (SWCF-VR) developed in our previous work based on statistical mechanical theory for chemical association is employed for the correlations of surface tension and viscosity of common fluids and ionic liquids (ILs). A model of surface tension for multi-component mixtures is presented by combining the SWCF-VR EOS and the scaled particle theory and used to produce the surface tension of binary and ternary mixtures. The predicted surface tensions are in excellent agreement with the experimental data with an overall average absolute relative deviation (AAD) of 0.36%. A method for the calculation of dynamic viscosity of common fluids and ILs at high pressure is presented by combining Eyring's rate theory of viscosity and the SWCF-VR EOS. The calculated viscosities are in good agreement with the experimental data with the overall AAD of 1.44% for 14 fluids in 84 cases. The salient feature is that the molecular parameters used in these models are self-consistent and can be applied to calculate different thermodynamic properties such as pVT, vapor-liquid equilibrium, caloric properties, surface tension, and viscosity.

Augmented van der Waals equations of state: SAFT-VR versus Yukawa based van der Waals equation

Performance of the SAFT-VR equation of state developed for the hard sphere based simple fluids, namely the square-well, Sutherland and Yukawa fluids, is examined by comparing its results with simulation data and an augmented van der Waals (vdW) equation based on a Yukawa (Y) reference. Its shown that both for the equilibrium vapor–liquid data and data along selected isotherms in the liquid and supercritical fluid phases the vdW(Y) equation provides better results, particularly when going to lower temperatures.

A modified soft-core fluid model for the direct correlation function of the square-shoulder and square-well fluids

We propose a simple analytical expression of the direct correlation function for the square-shoulder and square-well fluids. Our approximation is based on an ansatz for the direct correlation function of a modified soft-core fluid, whose parameters are adjusted by fitting the data obtained from Monte Carlo computer simulations. Moreover, it is complemented with a Wertheim-like parametrization to reproduce correctly the direct correlation inside the hard-core. We demonstrate that this approach is in quantitative agreement with the numerical solution of the Ornstein–Zernike equation within the Percus–Yevick approximation. We also show that our results are accurate in a large regime of densities for different interaction ranges and potential strengths. Therefore, this opens up the possibility of introducing the square-shoulder or the square-well potentials as new reference systems in advanced theoretical approximations.

The penetrable square-well model: extensive versus non-extensive phases

The phase diagram of the penetrable square-well fluid is investigated through Monte Carlo simulations of various nature. This model was proposed as the simplest possibility of combining bounded repulsions at short scale and short-range attractions. We prove that the model is thermodynamically stable for sufficiently low values of the penetrability parameter, and in this case the system behaves similarly to the square-well model. For larger penetration, there exists an intermediate region where the system is metastable, with well-defined fluid–fluid and fluid–solid transitions, at finite size, but eventually it becomes unstable in the thermodynamic limit. We characterize the unstable non-extensive phase appearing at high penetrability, where the system collapses into an isolated blob of a few clusters of many overlapping particles each.

Modeling of the Thermodynamic Properties of Aqueous Ionic Liquid Solutions with an Equation of State for Square-Well Chain Fluid with Variable Range

A new thermodynamic model based on an equation of state for square-well chain fluid with variable range (SWCF-VR) is proposed to describe the thermodynamic properties of aqueous solutions of ionic liquids. In this approach the electrostatic interactions are characterized via the mean spherical approximation and the ionic associations between the cations and anions of the ionic liquid are considered through the shield-sticky approach. The liquid densities, osmotic coefficients, and vapor pressures of aqueous solutions of nine ionic liquids (ILs) containing [Cxmim][Br] (x = 2, ..., 6), [C4mim][BF4] and [Cxmim][MSO4] (x = 1, 3, 4) have been modeled with the new model. Two ionic parameters for each anion and three for each cation of the ionic liquids were adjusted to experimental liquid densities and osmotic coefficients, and the corresponding average deviations are only 0.87% and 2.49%, respectively. Using the same ionic parameters, the vapor pressures of ionic liquid solutions are accurately predicted. The ...

A perturbative density functional theory for square-well fluids

We report a perturbative density functional theory for quantitative description of the structural and thermodynamic properties of square-well fluids in the bulk or at inhomogeneous conditions. The free-energy functional combines a modified fundamental measure theory to account for the short-range repulsion and a quadratic density expansion for the long-range attraction. The long-correlation effects are taken into account by using analytical expressions of the direct correlation functions of bulk fluids recently obtained from the first-order mean-spherical approximation. The density functional theory has been calibrated by extensive comparison with simulation data from this work and from the literature. The theory yields good agreement with simulation results for the radial distribution function of bulk systems and for the density profiles of square-well fluids near the surfaces of spherical cavities or in slit pores over a broad range of the parameter space and thermodynamic conditions.

Application of a renormalization-group treatment to the statistical associating fluid theory for potentials of variable range (SAFT-VR)

An accurate prediction of phase behavior at conditions far and close to criticality cannot be accomplished by mean-field based theories that do not incorporate long-range density fluctuations. A treatment based on renormalization-group (RG) theory as developed by White and co-workers has proven to be very successful in improving the predictions of the critical region with different equations of state. The basis of the method is an iterative procedure to account for contributions to the free energy of density fluctuations of increasing wavelengths. The RG method has been combined with a number of versions of the statistical associating fluid theory (SAFT), by implementing White's earliest ideas with the improvements of Prausnitz and co-workers. Typically, this treatment involves two adjustable parameters: a cutoff wavelength L for density fluctuations and an average gradient of the wavelet function Φ. In this work, the SAFT-VR (variable range) equation of state is extended with a similar crossover treatment which, however, follows closely the most recent improvements introduced by White. The interpretation of White's latter developments allows us to establish a straightforward method which enables Φ to be evaluated; only the cutoff wavelength L then needs to be adjusted. The approach used here begins with an initial free energy incorporating only contributions from short-wavelength fluctuations, which are treated locally. The contribution from long-wavelength fluctuations is incorporated through an iterative procedure based on attractive interactions which incorporate the structure of the fluid following the ideas of perturbation theories and using a mapping that allows integration of the radial distribution function. Good agreement close and far from the critical region is obtained using a unique fitted parameter L that can be easily related to the range of the potential. In this way the thermodynamic properties of a square-well (SW) fluid are given by the same number of independent intermolecular model parameters as in the classical equation. Far from the critical region the approach provides the correct limiting behavior reducing to the classical equation (SAFT-VR). In the critical region the β critical exponent is calculated and is found to take values close to the universal value. In SAFT-VR the free energy of an associating chain fluid is obtained following the thermodynamic perturbation theory of Wertheim from the knowledge of the free energy and radial distribution function of a reference monomer fluid. By determining L for SW fluids of varying well width a unique equation of state is obtained for chain and associating systems without further adjustment of critical parameters. We use computer simulation data of the phase behavior of chain and associating SW fluids to test the accuracy of the new equation.

Drying in a capped capillary

We use a model Density Functional, based on the White-Bear version of Fundamental Measure theory, to test recent predictions, due to Evans and co-workers, that capillary condensation in a capped capillary-slit is a continuous interfacial critical phenomenon related to the complete wetting transition. Using a model with a square-well intermolecular fluid–fluid attraction we first determine accurately the location of the first-order capillary evaporation transition in an infinite (open) hard-wall capillary slit. Extending the density functional model to allow for a two-dimensional order-parameter profile, we then study the adsorption as the chemical potential is reduced to capillary evaporation but now in a capillary-slit that is capped at one end. The equilibrium density profiles obtained show that, sufficiently close to the phase boundary, a meniscus separating liquid-like and vapour-like phases forms near the capped end, and that as capillary evaporation is approached, continuously unbinds from the capped end. Our numerical results indicate that the divergence of the adsorption due to the unbinding of the meniscus is logarithmic and is the same as for the complete wetting transition in systems with short-ranged forces.

Perturbation theory of liquids for short-ranged hard-core potentials: Structure and thermodynamics of short-ranged square-well fluids

A new method is devised to derive a thermodynamic perturbation theory that is suitable for short-ranged potentials. We have calculated structure and thermodynamic of short-ranged square-well potential with good agreement with computer simulation data even at low temperatures. We have also compared the results with other perturbation theories. It is found that the proposed theory gives better results when used to compute the structure and thermodynamic of short-ranged hard-core potentials.

Percolation line and response functions in simple supercritical fluids

The question of a physical relevance (meaning) of the percolation line in simple supercritical fluids has been addressed using both extensive Monte Carlo simulations and accurate analytic equations of state. Thermodynamic and structural properties of two qualitatively different fluids, the Lennard-Jones (continuous model) and square-well fluid (stepwise model), have been studied in the vicinity of their percolation lines over a range of pressures in order to find potential changes which may take place when an infinite cluster is detected in the system. Two different criteria for the occurrence of an infinite cluster, physical and configurational criteria, have also been used to assess their effect on the observed properties. It is found that the lines of extremes of various response functions run close to the percolation lines. Accounting for uncertainties in the definition of bonds and determination of the percolation line one may conjecture that extremes in some response functions occur when crossing the percolation line.

Selective adsorption of small molecules on grafted polymers

A density functional approach to the description of the selective adsorption of a binary mixture of small molecules on a surface grafted with the square-well chains is proposed. The excess Helmholtz free energy functional of the system is divided into two contributions from the hard-sphere repulsion and the square-well attraction respectively. In the bulk phase, the equation of state of Liu and Hu (Liu H L, Hu Y 1996 Feuid Phase Equilibra 122 75) is used to calculate the free energy of the repulsive part and the equation of state of Li and He (Li J L and He H H 2009 Feuid Phase Equilibria 276 57) is used to calculate the free energy of the attractive part. A simple weighted density approximation is adopted on both parts to construct the functional. Using this theoretical approach, the structures of the grafted polymers and the selective adsorptions of binary square-well fluids on the grafted layer at different temperatures are investigated. The theory predicts that the brush thickness increases linearly with the grafting density but non-linearly with temperature and the brush thickness tends to sateration at high temperature. Given the chain length and grafting density, the theoretical calculation also reveals that the polymer brush has strong selective adsorption cap ability at low temperature and the cap ability will weaken greatly when the polymer brush is heated above the temperature of sateration.

Vapour–liquid phase equilibria of simple fluids confined in patterned slit pores

We present the influence of surface heterogeneity on the vapour–liquid phase behaviour of square-well fluids in slit pores using grand-canonical transition-matrix Monte Carlo simulations along with the histogram-reweighting method. Properties such as phase coexistence envelopes, critical properties and local density profiles of the confined SW fluid are reported for chemically and physically patterned slit surfaces. It is observed that in the chemically patterned pores, fluid–fluid and surface attraction parameters along with the width of attractive and inert stripes play fundamentally different roles in the phase coexistence and critical properties. On the other hand, pillar gap and height significantly affect the vapour–liquid equilibria in the physically patterned slit pores. We also present the effect of chemically and physically patterned slit surfaces on the spreading pressure.