Hard-sphere Chain Fluids Research Articles

Phase Equilibria of Polydisperse Square-Well Chain Fluid Confined in Random Porous Media: TPT of Wertheim and Scaled Particle Theory.

Extension of Wertheim's thermodynamic perturbation theory and its combination with scaled particle theory is proposed and applied to study the liquid-gas phase behavior of polydisperse hard-sphere square-well chain fluid confined in the random porous media. Thermodynamic properties of the reference system, represented by the hard-sphere square-well fluid in the matrix, are calculated using corresponding extension of the second-order Barker-Henderson perturbation theory. We study effects of polydispersity and confinement on the phase behavior of the system. While polydispersity causes increase of the region of phase coexistence due to the critical temperature increase, confinement decreases the values of both critical temperature and critical density making the region of phase coexistence smaller. This effect is enhanced with the increase of the size ratio of the fluid and matrix particles. The increase of the average chain length at fixed values of polydispersity and matrix density shifts the critical point to a higher temperature and a slightly lower density.

Cubic and quartic hard-sphere and Lennard-Jones chain equations of state as foundations for complex fluid modeling

Hard-sphere compressibility factor Z and hard-dimer radial distribution function at contact gHD(σ) are fit to simple packing fraction η expressions and are used in the dimer version of Wertheim’s perturbation theory to obtain five cubic equations of state (EOS’s) for athermal hard-sphere chain fluids. The predicted Zm versus η for various chain lengths m and the reduced second-virial coefficient versus m are compared with simulation data and the original noncubic dimer perturbation theory EOS’s. A cubic analog of the Boublik–Mansoori–Carnahan–Starling (BMCS) EOS successfully represents the unlike pair correlation function g12(σ) and Z versus η for binary hard-sphere mixtures of moderate size difference in the components. The new CTPT-D models are extended to calculate Zm versus η for binary homonuclear hard-sphere chain mixtures. The choice of a suitable EOS dispersion term to simultaneously represent single phase PVT and phase equilibria is briefly explored. A similar TPT approach was used to derive a quartic EOS for Lennard-Jones chain fluids, and Zm versus η isotherms are determined and compared with simulation results for chains up to 100 segments long. The new cubic and quartic EOS’s provide a simple foundation for developing equations of state for complex fluids including multipolar and association effects.

Monte Carlo simulation and equation of state for flexible charged hard-sphere chain fluids: polyampholyte and polyelectrolyte solutions.

The thermodynamic modeling of flexible charged hard-sphere chains representing polyampholyte or polyelectrolyte molecules in solution is considered. The excess Helmholtz energy and osmotic coefficients of solutions containing short polyampholyte and the osmotic coefficients of solutions containing short polyelectrolytes are determined by performing canonical and isobaric-isothermal Monte Carlo simulations. A new equation of state based on the thermodynamic perturbation theory is also proposed for flexible charged hard-sphere chains. For the modeling of such chains, the use of solely the structure information of monomer fluid for calculating the chain contribution is found to be insufficient and more detailed structure information must therefore be considered. Two approaches, i.e., the dimer and dimer-monomer approaches, are explored to obtain the contribution of the chain formation to the Helmholtz energy. By comparing with the simulation results, the equation of state with either the dimer or dimer-monomer approach accurately predicts the excess Helmholtz energy and osmotic coefficients of polyampholyte and polyelectrolyte solutions except at very low density. It also well captures the effect of temperature on the thermodynamic properties of these solutions.

The isotropic-nematic and nematic-nematic phase transition of binary mixtures of tangent hard-sphere chain fluids: an analytical equation of state.

An analytical equation of state (EoS) is derived to describe the isotropic (I) and nematic (N) phase of linear- and partially flexible tangent hard-sphere chain fluids and their mixtures. The EoS is based on an extension of Onsager's second virial theory that was developed in our previous work [T. van Westen, B. Oyarzún, T. J. H. Vlugt, and J. Gross, J. Chem. Phys. 139, 034505 (2013)]. Higher virial coefficients are calculated using a Vega-Lago rescaling procedure, which is hereby generalized to mixtures. The EoS is used to study (1) the effect of length bidispersity on the I-N and N-N phase behavior of binary linear tangent hard-sphere chain fluid mixtures, (2) the effect of partial molecular flexibility on the binary phase diagram, and (3) the solubility of hard-sphere solutes in I- and N tangent hard-sphere chain fluids. By changing the length bidispersity, two types of phase diagrams were found. The first type is characterized by an I-N region at low pressure and a N-N demixed region at higher pressure that starts from an I-N-N triphase equilibrium. The second type does not show the I-N-N equilibrium. Instead, the N-N region starts from a lower critical point at a pressure above the I-N region. The results for the I-N region are in excellent agreement with the results from molecular simulations. It is shown that the N-N demixing is driven both by orientational and configurational/excluded volume entropy. By making the chains partially flexible, it is shown that the driving force resulting from the configurational entropy is reduced (due to a less anisotropic pair-excluded volume), resulting in a shift of the N-N demixed region to higher pressure. Compared to linear chains, no topological differences in the phase diagram were found. We show that the solubility of hard-sphere solutes decreases across the I-N phase transition. Furthermore, it is shown that by using a liquid crystal mixture as the solvent, the solubility difference can by maximized by tuning the composition. Theoretical results for the Henry's law constant of the hard-sphere solute are in good agreement with the results from molecular simulation.

The isotropic-nematic phase transition of tangent hard-sphere chain fluids—Pure components

An extension of Onsager's second virial theory is developed to describe the isotropic-nematic phase transition of tangent hard-sphere chain fluids. Flexibility is introduced by the rod-coil model. The effect of chain-flexibility on the second virial coefficient is described using an accurate, analytical approximation for the orientation-dependent pair-excluded volume. The use of this approximation allows for an analytical treatment of intramolecular flexibility by using a single pure-component parameter. Two approaches to approximate the effect of the higher virial coefficients are considered, i.e., the Vega-Lago rescaling and Scaled Particle Theory (SPT). The Onsager trial function is employed to describe the orientational distribution function. Theoretical predictions for the equation of state and orientational order parameter are tested against the results from Monte Carlo (MC) simulations. For linear chains of length 9 and longer, theoretical results are in excellent agreement with MC data. For smaller chain lengths, small errors introduced by the approximation of the higher virial coefficients become apparent, leading to a small under- and overestimation of the pressure and density difference at the phase transition, respectively. For rod-coil fluids of reasonable rigidity, a quantitative comparison between theory and MC simulations is obtained. For more flexible chains, however, both the Vega-Lago rescaling and SPT lead to a small underestimation of the location of the phase transition.

An equation of state for the isotropic phase of linear, partially flexible and fully flexible tangent hard-sphere chain fluids

A new equation of state is developed that accurately describes the isotropic phase behaviour of linear, partially flexible and fully flexible tangent hard-sphere chain fluids and their mixtures. The equation of state is based on the equation of state of Liu and Hu [H. Liu and Y. Hu, Fluid Phase Equilibr. 122, 75 (1996)] for fully flexible chain fluids. The effect of molecular flexibility is described by a pure-component parameter that is introduced in the theory at the level of the cavity correlation function of next-to-nearest neighbour segments in a chain molecule. The equation of state contains a total of three adjustable model constants. The extension to partially flexible- and linear chain fluids is based on a refitting of the first model constant to numerical data of the second virial coefficient of partially flexible and linear tangent hard-sphere chain fluids. The numerical data were obtained from an analytical approximation for the pair-excluded volume. The other two parameters were adjusted to molecular simulation data for the pressure of linear tangent hard-sphere chain fluids. For both, pure component systems and mixtures of chains of variable flexibility, the pressure and second virial coefficient obtained from the equation of state, are in excellent agreement with the results from Monte Carlo simulations. A significant improvement to TPT1, TPT2, generalised Flory-dimer theory and scaled particle theory is observed.

The phase behavior of linear and partially flexible hard-sphere chain fluids and the solubility of hard spheres in hard-sphere chain fluids

The liquid crystal phase behavior of linear and partially flexible hard-sphere chain fluids and the solubility of hard spheres in hard-sphere chain fluids are studied by constant pressure Monte Carlo simulations. An extensive study on the phase behavior of linear fluids with a length of 7, 8, 9, 10, 11, 12, 13, 14, 15, and 20 beads is carried out. The phase behavior of partially flexible fluids with a total length of 8, 10, 14, and 15 beads and with different lengths for the linear part is also determined. A precise description of the reduced pressure and of the packing fraction change at the isotropic-nematic coexistence was achieved by performing long simulation runs. For linear fluids, a maximum in the isotropic to nematic packing fraction change is observed for a chain length of 15 beads. The infinite dilution solubility of hard spheres in linear and partially flexible hard-sphere chain fluids is calculated by the Widom test-particle insertion method. To identify the effect of chain connectivity and molecular anisotropy on free volume, solubility is expressed relative to that of hard spheres in a hard sphere fluid at same packing fraction as relative Henry's law constants. A linear relationship between relative Henry's law constants and packing fraction is observed for all linear fluids. Furthermore, this linearity is independent of liquid crystal ordering and seems to be independent of chain length for linear chains of 10 beads and longer. The same linear relationship was observed for the solubility of hard spheres in nematic forming partially flexible fluids for packing fractions up to a value slightly higher than the nematic packing fraction at the isotropic-nematic coexistence. At higher packing fractions, the small flexibility of these fluids seems to improve solubility in comparison with the linear fluids.

The effect of inter- and intra-molecular interactions for the equation of state of the hard-sphere chain fluid

We present new equations of state for hard-chain fluids by employing the modified pair correlation function for hard-chain molecules at a contact diameter g(σ), which includes both inter- and intra-molecular interactions. The compressibility factor values determined using the modified pair correlation function are close to those predicted by the SAFT-D1 and SAFT-D2 theories and are in good agreement with Monte Carlo (MC) simulation data. It has also been seen that the intra-molecular interaction plays a dominating role at higher chain length and low density.

Universal correlation of self-diffusion coefficients of model and real fluids based on residual entropy scaling law

In this work, the entropy based scaling laws of Rosenfeld, Dzugutov and Bretonnet, which connect reduced self-diffusion coefficients (D⁎) with residual entropy (named excess entropy in this field), were analysed in order to test their attributed universal character. With this purpose, an extensive database with 1727 molecular dynamic and experimental values was compiled for hard-sphere (HS), Lennard-Jones (LJ), hard-sphere chain (HSC), and real (polar, nonpolar, symmetrical and asymmetrical) fluids. It was shown that these equations fail when tested over the entire range of density and temperature (through residual entropy), even for atomic and simple fluids (e.g., HS and LJ) for which they have been originally proposed. Furthermore, the dependence of the self-diffusivities upon both residual entropy and a molecular chain length parameter (r) was clearly found on the basis of HSC and real data. Accordingly, a new universal correlation for the estimation of D⁎ as function of residual entropy and r was obtained, giving rise to an average absolute relative deviation of 9.13% for all database. It was also devised a very simple and accurate entropy based equation for spherical systems (HS and LJ) which provides only 4.61% of error. The original Rosenfeld, Dzugutov and Bretonnet's expressions attain deviations that are several orders of magnitude higher than our values.

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 properties of the freely jointed tangent homonuclear Yukawa chain fluid

Recently, a simple analytical expression for the first coordination shell (1 ≤ x ≤ 2) of the average site–site intermolecular radial distribution function (ASSIRDF) of the freely jointed tangent hard sphere chain (FJTHSC) fluid has been proposed. This expression was used in the framework of the second-order Barker–Henderson perturbation theory together with the site–site intermolecular hard-core Yukawa potential in order to obtain an equation of state (EOS) for the freely jointed tangent Yukawa chain (FJTYC) fluid. The derived EOS was used for calculating the thermodynamic properties of the FJTYC fluid, with the attraction of variable range, such as compressibility factor, vapor–liquid equilibria and critical constants. The calculated compressibility factors, above the critical temperature, were compared with the available simulation data and the results of the well-known theories for a wide range of densities, temperatures and different values of attractive ranges. The overall agreement was good and the accuracy of the results could compete with the accuracy of those obtained using very famous theories and equations of state in the literature such as the statistical associate fluid theory with the interaction of variable range (SAFT-VR).

A new simple analytical expression for the average site–site radial distribution function of hard sphere chain fluid

A new simple analytical expression for the first coordination shell (1 ≤ x ≤ 2) of the average site–site intermolecular radial distribution function (ASSIRDF) of freely jointed tangent hard sphere chain fluid (FJTHSC) is proposed. The proposed expression is a polynomial expansion of packing fraction. The dependence of each polynomial coefficient on the radial coordinate and segment number is described by a polynomial expansion of nonlinear base function and a simple relation proposed by Liu and Hu, respectively. The coefficients involved in the expression of ASSIRDF are optimized by fitting the numerical values of Percus–Yevick (PY) integral equation theory of Chiew. The proposed expression of ASSIRDF is applied to an analytical perturbative equation of state (EOS) in order to modify it and improve its accuracy in predicting the compressibility factor and vapor–liquid equilibria of freely jointed tangent square-well chain (FJTSWC) fluids of variable well-widths (1 ≤ λ ≤ 2). The predicted results are in good agreement with the simulation data and also compared with that of other well-known equations of state.

Phase coexistence in the hard-sphere Yukawa chain fluid with chain length polydispersity: Dimer thermodynamic perturbation theory

An extension of the dimer version of Wertheim's thermodynamic perturbation theory is proposed and used to treat polydisperse mixture of the hard-sphere Yukawa chain fluid with chain length polydispersity. The structure and thermodynamic properties of the reference system, represented by multicomponent mixture of the Yukawa hard-sphere dimers, are described using polymer mean spherical approximation. Explicit analytical expressions for the Helmholtz free energy, chemical potential, and pressure in terms of the two chain length distribution function moments are derived. The theory is used to calculate the full liquid-gas phase diagram, including critical binodal, cloud and shadow curves, and distribution functions of the coexisting phases. Effects of fractionation in terms of the distribution function and its first and second moments are studied. Predictions of the theory for these effects are in qualitative agreement with the corresponding experimental predictions, obtained recently for the polydisperse mixture of polymers in a single solvent. In particular, both theory and experiment predict that longer chain polymers equilibrate to the liquid phase while shorter chain polymers are predominantly encountered in the gas phase.

Accuracy of an analytic equation of state for hard-sphere multi-Yukawa chain fluids based on the first-order mean spherical approximation

The accuracy of analytical expressions of the compressibility factor and internal energy of chain fluids composed of m freely-jointed tangent hard-sphere monomers interacting through a hard-core n-Yukawa (HCnY) potential is further assessed. These expressions were obtained by combining the first-order mean spherical approximation (FMSA) for the monomer thermodynamic properties, the simplified exponential approximation (SEXP) for the monomer radial distribution function (RDF) at contact with the hard core, and the statistical associating fluid theory (SAFT) [Fluid Phase Equilibr. 218 (2004) 47]. The resulting expressions, which are simple to use for any m and n, are in overall good agreement with recent extensive Monte Carlo (MC) simulations performed on new HC2Y chain fluids with m = 2, 4, 8, 16. The absolute average deviations (AAD%) for the reduced pressure and internal energy are respectively within 6.0% and 3.8% of the simulation data.

Phase coexistence in the hard-sphere Yukawa chain fluid with chain length polydispersity: High temperature approximation

High temperature approximation (HTA) is used to describe the phase behavior of polydisperse Yukawa hard-sphere chain fluid mixture with chain length polydispersity. It is demonstrated that in the frames of the HTA the model belongs to the class of ‘truncatable free energy models’, i.e. the models with thermodynamical properties defined by the finite number of generalized moments. Using this property we were able to calculate the complete phase diagram (i.e. cloud and shadow curves as well as binodals) and chain length distribution functions of the coexisting phases.

A semi-empirical hard-sphere chain equation of state: Pure and mixture

The thermodynamic perturbation dimer theory (TPT-D) is modified by scaling the site–site correlation function at contact for hard-dimers using the contact value of site-site correlation function for hard-spheres. With the exception of a constant, the resulting modified TPT-D is identical with the thermodynamic perturbation theory (TPT). The modified TPT-D, however, predicts the Monte Carlo (MC) compressibility factors for hard-sphere chains more accurately than TPT. The average absolute percent deviations (%AAD) was found to be 2.75 for the modified TPT-D and 6.24 for TPT. The modified TPT-D was then extended to mixtures of hard-sphere chain fluids. The modified TPT-D accurately predicts compressibility factors for binary mixtures of hard-sphere chains or mixtures containing hard-sphere chain and hard-sphere fluids. When the theories were used to predict compressibility factors for 89 mixtures containing hard-spheres and hard-sphere chains with different lengths and number of segments the %AAD was found to be 1.49 for the modified TPT-D and 2.78 for TPT.

Polydisperse hard-sphere yukawa chain fluid: High temperature approximation

The high temperature approximation (HTA) for a polydisperse hard-sphere Yukawa chain fluid is proposed. A closed form analytical expression for Laplace transform of the reference system sitesite radial distribution function, which is used as an input in the theoretical scheme developed, is obtained from the solution of the averaged version of the multidensity Ornstein‐Zernike equation. It is demonstrated that in the frames of the HTA polydisperse hard-sphere Yukawa chain fluid model belongs to a class of “truncatable free energy models”, i.e. the models with thermodynamical properties (Helmholtz free energy, chemical potential and pressure) defined by a finite number of generalized moments. The accuracy of the theory is accessed by comparison of its liquid-gas phase diagram predictions obtained in the monodisperse limiting case, with the corresponding predictions of the statistical associating fluid theory for the potentials of variable range (SAFT‐ VR) and thermodynamic perturbation theory dimer (TPTD).

SIMPLIFIED HARD-SPHERE AND HARD-SPHERE CHAIN EQUATIONS OF STATE FOR ENGINEERING APPLICATIONS

A simple hard-sphere equation of state is proposed. This hard-sphere equation is a ratio of second-order polynomials that meets the ideal gas and close-packed density limits. It predicts the compressibility of hard-sphere fluids at low and medium densities to within a degree of quality similar to the widely used Carnahan-Starling equation. In addition, the proposed equation performs better at high densities, particularly near the close-packed density. An expression is also derived to relate the site-site correlation function at contact for hard dimers with the site-site correlation function at contact for hard spheres. With this relationship, the thermodynamic perturbation-dimer theory (TPT-D) of hard-sphere chains is simplified. The new theory performs comparably with the TPT-D when the compressibility factors of hard-sphere chain fluids containing up to 201-mer are predicted, however, it has the advantages of both simplicity and accuracy. From a practical perspective, this theory can be used to construct equations of state for polymer solutions or fluid systems containing short- and long-chain molecules.

Equation of state and liquid-vapor equilibria of one- and two-Yukawa hard-sphere chain fluids: theory and simulation.

The accuracy of several theories for the thermodynamic properties of the Yukawa hard-sphere chain fluid are studied. In particular, we consider the polymer mean spherical approximation (PMSA), the dimer version of thermodynamic perturbation theory (TPTD), and the statistical associating fluid theory for potentials of variable attractive range (SAFT-VR). Since the original version of SAFT-VR for Yukawa fluids is restricted to the case of one-Yukawa tail, we have extended SAFT-VR to treat chain fluids with two-Yukawa tails. The predictions of these theories are compared with Monte Carlo (MC) simulation data for the pressure and phase behavior of the chain fluid of different length with one- and two-Yukawa tails. We find that overall the PMSA and TPTD give more accurate predictions than SAFT-VR, and that the PMSA is slightly more accurate than TPTD.

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.