Square-well Interaction Research Articles

Dynamical Arrest, Percolation, Gelation, and Glass Formation in Model Nanoparticle Dispersions with Thermoreversible Adhesive Interactions

We report an experimental study of the dynamical arrest transition for a model system consisting of octadecyl coated silica suspended in n-tetradecane from dilute to concentrated conditions spanning the state diagram. The dispersion's interparticle potential is tuned by temperature affecting the brush conformation leading to a thermoreversible model system. The critical temperature for dynamical arrest, T*, is determined as a function of dispersion volume fraction by small-amplitude dynamic oscillatory shear rheology. We corroborate this transition temperature by measuring a power-law decay of the autocorrelation function and a loss of ergodicity via fiber-optic quasi-elastic light scattering. The structure at T* is measured using small-angle neutron scattering. The scattering intensity is fit to extract the interparticle pair-potential using the Ornstein-Zernike equation with the Percus-Yevick closure approximation, assuming a square-well interaction potential with a short-range interaction (1% of particle diameter). (1) The strength of attraction is characterized using the Baxter temperature (2) and mapped onto the adhesive hard sphere state diagram. The experiments show a continuous dynamical arrest transition line that follows the predicted dynamical percolation line until ϕ ≈ 0.41 where it subtends the predictions toward the mode coupling theory attractive-driven glass line. An alternative analysis of the phase transition through the reduced second virial coefficient B(2)* shows a change in the functional dependence of B(2)* on particle concentration around ϕ ≈ 0.36. We propose this signifies the location of a gel-to-glass transition. The results presented herein differ from those observed for depletion flocculated dispersion of micrometer-sized particles in polymer solutions, where dynamical arrest is a consequence of multicomponent phase separation, suggesting dynamical arrest is sensitive to the physical mechanism of attraction.

Predicting the self-assembly of a model colloidal crystal

We investigate the self-assembly (crystallisation) of particles with hard cores and isotropic, square-well interactions, using a Monte Carlo scheme to simulate overdamped Langevin dynamics. We measure correlation and response functions during the early stages of assembly, and we analyse the results using fluctuation-dissipation theorems, aiming to predict which systems will self-assemble successfully and which will get stuck in disordered states. The early-time correlation and response measurements are made before significant crystallisation has taken place, indicating that dynamical measurements are valuable in measuring a system's propensity for kinetic trapping.

Advances in generalised van der Waals approaches for the isotropic–nematic fluid phase equilibria of thermotropic liquid crystals–an algebraic equation of state for attractive anisotropic particles with the Onsager trial function

In this contribution we provide a review and reformulation of perturbation theories (generalised van der Waals approaches) for the description of the fluid phase behaviour and orientational ordering transitions of thermotropic nematic liquid crystals. Free-energy functionals of the basic Onsager form are used as the platform for the development of a general formulation that reduces to the specific forms of the various theories that have found common use. A novel closed analytical description of the thermodynamic properties and degree of nematic order is then obtained by employing the Onsager trial function to represent the orientational distribution function in terms of a single parameter. The latter essentially constitutes an algebraic equation of state for the nematic phase appropriate for use in engineering applications. The description of ordering transitions with scaled-Onsager theories and suitable trial functions has already been illustrated by its application to systems of hard spherocylinders (HSCs), indicating that the approach provides an excellent representation of the orientational order of the hard-core (athermal) system [Mol. Phys. 106, 649 (2008)]. Here, the hard-body model is extended to account for attractive interactions (treated at the van der Waals level) of a general isotropic/anisotropic form (e.g., Lennard-Jonesium (LJ), square-well (SW), Maier–Saupe (MS), etc.). The adequacy of our generalised van der Waals–Onsager theory is exemplified by an analysis of the vapour–liquid, liquid–nematic, and vapour–nematic phase equilibria for hard spherocylinders with attractive square-well interactions (HSC-SW). The effect of the potential range and molecular aspect ratio on the vapour–liquid–nematic equilibria and orientational ordering transitions is examined to investigate the van der Waals limit (corresponding states) of the ordering phase behaviour. In the case of systems with an aspect ratio of ∼5 the corresponding-states limit is reached when the range is about 16 times the molecular diameter. For progressively longer molecules with an attractive range that follows their long dimension, the fluid–nematic equilibrium is enhanced to the point that the vapour–liquid boundary becomes metastable relative to fluid–nematic equilibria. In the case of molecules of very large aspect ratio (∼50), an additional region of nematic–nematic coexistence is exhibited by the system.

Simulation of Chemical Potentials and Phase Equilibria in Two- and Three-Dimensional Square-Well Fluids: Finite Size Effects

We study the simulation cell size dependence of chemical potential isotherms in subcritical square-well fluids by means of series of canonical ensemble Monte Carlo simulations with increasing numbers of particles, for both three-dimensional bulk systems and two-dimensional planar layers, using Widom-like particle insertion methods. By estimating the corresponding vapor/liquid coexistence densities using a Maxwell-like equal area rule for the subcritical chemical potential isotherms, we are able to study the influence of system size not only on chemical potentials but also on the coexistence properties. The chemical potential versus density isotherms show van der Waals-like loops in the subcritical vapor/liquid coexistence range that exhibit distinct finite size effects for both two- and three-dimensional fluids. Generally, in agreement with recent findings for related studies of Lennard-Jones fluids, the loops shrink with increasing number of particles. In contrast to the subcritical isotherms themselves, the equilibrium vapor/liquid densities show only a weak system size dependence and agree quantitatively with the best-known literature values for three-dimensional fluids. This allows our approach to be used to accurately predict the phase coexistence properties. Our resulting phase equilibrium results for two-dimensional square-well fluids are new. Knowledge concerning finite size effects of square-well systems is important not only for the simulation of thermodynamic properties of simple fluids, but also for the simulation of models of more complex fluids (such as aqueous or polymer fluids) involving square-well interactions.

Long tail correlations between hydrophobic solutes in a model solvent

Recently, a one-dimensional model with nearest-neighbour square-well interactions has been studied, in which parameters were chosen to mimic some of the thermodynamic properties of solutions of non-polar solutes in water. Here we investigate the properties of the pair distribution functions, comparing correlations between solvent molecules with those between solute molecules in the limit of infinite dilution. Under some appropriate thermodynamic conditions, both contain the same underlying–exponentially monotonic–decay, with a decay length that is characteristically long-tailed. But while this length is barely of interest for the pure solvent by virtue of the minuteness of the correlations at such long distances, and is almost undetectable for all practical purposes, it appears dramatically in the case of solute–solute correlations, becoming visible when one can access large length separations. The implications of this unexpected feature are discussed, also in relation to the mean force between solute molecules and the osmotic second virial coefficient for the solute. The case of square-well interactions is discussed at some length, and references are made to analogous results for the case of a triangular-well interaction potential.

Effects of Macromolecular Crowding on Biochemical Reaction Equilibria: A Molecular Thermodynamic Perspective

A molecular thermodynamic model is developed to investigate the effects of macromolecular crowding on biochemical reactions. Three types of reactions, representing protein folding/conformational isomerization, coagulation/coalescence, and polymerization/association, are considered. The reactants, products, and crowders are modeled as coarse-grained spherical particles or as polymer chains, interacting through hard-sphere interactions with or without nonbonded square-well interactions, and the effects of crowder size and chain length as well as product size are examined. The results predicted by this model are consistent with experimentally observed crowding effects based on preferential binding or preferential exclusion of the crowders. Although simple hard-core excluded-volume arguments do in general predict the qualitative aspects of the crowding effects, the results show that other intermolecular interactions can substantially alter the extent of enhancement or reduction of the equilibrium and can even change the direction of the shift. An advantage of the approach presented here is that competing reactions can be incorporated within the model.

Theoretical prediction of multiple fluid-fluid transitions in monocomponent fluids

The authors use the analytical equation of state obtained by the discrete perturbation theory [A. L. Benavides and A. Gil-Villegas, Mol. Phys. 97, 1225 (1999)] to study the phase diagram of fluids with discrete spherical potentials formed by a repulsive square-shoulder plus an attractive square-well interaction (SS+SW). This interaction is characterized by the usual energy and size parameters plus three dimensionless parameters: two of them measuring the widths of the SS and the SW and the third the relative height of the SS. The matter of interest is that, for certain values of the interaction parameters, the SS+SW systems exhibit more than one first-order fluid-fluid transition. The evidence that several real substances (such as water, phosphorus, carbon, and silica, among others) exhibit an extra liquid-liquid transition has drawn interest into the study of interactions responsible for this behavior. The simple SS+SW fluid is one of the systems that, in spite of being spherically symmetric, shows multiple fluid-fluid transitions. In this work the authors investigate systematically the effect on the phase diagram of varying the interaction parameters. The use of an analytical free-energy equation gives a clear thermodynamic picture of the emergence of different types of critical points, throwing new light on the phase behavior of these fluids and thus clarifying previous results obtained by other techniques. The interplay of attractive and repulsive forces with several scale lengths produces very rich phase diagrams, including cases with three critical points. The region of the interaction-parameter space where multiple critical points appear is mapped for various families of interactions.

Experimental determination and calculation of thermodynamic properties of CO 2 + octane to high temperatures and high pressures

The phase behavior and critical points of carbon dioxide + octane are determined from 313 to 393 K, and their molar volumes and densities are measured both in the subcritical and supercritical (SC) regions using a variable-volume autoclave in this study. The critical curve of carbon dioxide + octane has been predicted with an equation of state (EOS) by Heilig and Franck, in which a repulsion term and a square-well potential attraction term for intermolecular interaction has been used. To use the pairwise combination rule for the square-well molecular interaction potential, three adjustable parameters are required. The thermodynamic properties including mole fractions, densities and molar volumes of this system are also calculated using the Heilig–Franck, the Peng–Robison (PR) and the Peng–Robinson–Stryjek–Vera (PRSV) equation of state. Among the equations of state, the Heilig–Franck equation of state has been found to have the best correlation with binary vapor–liquid equilibrium data of the carbon dioxide + octane system.

Statistical Thermodynamics of Solute Molecule in Model Mixed Fluid

The solvent effect in a mixed solvent on the thermodynamical properties of a solute molecule is studied with a simplified model system. The solvent molecules in the model system are distinguished only by the interaction with the solute molecule, which is assumed as a square-well interaction. The exact relationship between the thermodynamical quantities in the single-component solvent and those in the mixed solvent are derived through the bond-number (BN) distribution of the solute molecule; the bond number is the number of interacting molecules in the potential well of the solute molecule. The BN distributions are evaluated for the hard-sphere fluid using the Monte Carlo method over a wide density region. The mean field approximation does not work well at low mole fraction of the interacting component in the mixed solvent due to the fluctuation of the local composition. The effect of the fluctuation of the local composition was reproduced by the modified independent-site model under the mean field approximation, in which the fluctuation of the composition is assumed to take place independently at hypothetical sites around a solute molecule.

Vapor-liquid equilibrium and critical behavior of the square-well fluid of variable range: A theoretical study

The vapor-liquid phase behavior and the critical behavior of the square-well (SW) fluid are investigated as a function of the interaction range, lambdain [1.25, 3], by means of the self-consistent Ornstein-Zernike approximation (SCOZA) and analytical equations of state based on a perturbation theory [A. L. Benavides and F. del Rio, Mol. Phys. 68, 983 (1989); A. Gil-Villegas, F. del Rio, and A. L. Benavides, Fluid Phase Equilib. 119, 97 (1996)]. For this purpose the SCOZA, which has been restricted up to now to a few model systems, has been generalized to hard-core systems with arbitrary interaction potentials requiring a fully numerical solution of an integro-partial differential equation. Both approaches, in general, describe well the liquid-vapor phase diagram of the square-well fluid when compared with simulation data. SCOZA yields very precise predictions for the coexistence curves in the case of long ranged SW interaction (lambda>1.5), and the perturbation theory is able to predict the binodal curves and the saturated pressures, for all interaction ranges considered if one stays away from the critical region. In all cases, the SCOZA gives very good predictions for the critical temperatures and the critical pressures, while the perturbation theory approach tends to slightly overestimate these quantities. Furthermore, we propose analytical expressions for the critical temperatures and pressures as a function of the square-well range.

Density functional theory and simulation of the columnar phase of a system of parallel hard ellipsoids with attractive interactions

A simple molecular model consisting of parallel hard oblate ellipsoids with superimposed square-well attractive interactions of variable range is considered for the study of the phase behavior of thermotropic discotic molecules. A density functional theory appropriate for nonuniform fluids is formulated in which the hard-core contributions to the free energy are treated within a nonlocal weighted-density approximation (WDA) while the attractive contributions are treated at a mean-field level. It is shown that the columnar phase becomes stable relative to the nematic phase at fluid densities for a range of values of the range of the attractive well. In these cases, the region of stability of the columnar phase is bounded at high temperatures by a nematic-columnar-solid triple point. The calculations show that if the attractions are made too long ranged (lambda/D> or approximately =0.84 for particles of aspect ratio of L/D=0.1, where lambda/D is the range of the attractive interaction in units of the molecular diameter D), columnar ordering becomes unstable and the nematic phase dominates at all fluid densities. It is shown that columnar ordering is also predicted when the density functional theory is supplemented with the smoothed-density approximation (SDA). Computer simulations have also been carried out for a particular choice of model parameters; our simulation data confirm the stabilization of the hexagonal columnar phase between the solid and nematic phases. A comparison with simulation data allows us to conclude that the WDA provides a fairly good description of the columnar phase and very good agreement for the nematic-columnar transition properties. On the other hand, our calculations show that the SDA largely underestimates the transition pressure and predicts a too-strongly first-order nematic-columnar transition

Molecular Dynamics Studies of Two Particles in a Rectangular Box: Hard-Sphere and Square-Well Interactions

Molecular dynamics simulations have been carried out for the simple few-body systems of two particles confined within a 2-dimensional rectangular box. The two different sets of interaction potentials between the colliding particles were employed in these studies: hard-sphere and square-well interactions. Wall pressures, the wall/particle and the particle/particle collision frequencies, and the position autocorrelation functions were computed to examine the thermodynamic, structural and time-dependent properties of such systems. Detailed dynamic effects are discussed to describe configurational particle trajectories including fast/slow relaxation processes observed in the position autocorrelation functions.

Smectic phase in a system of hard ellipsoids with isotropic attractive interactions

The smectic phase is studied for a thermotropic fluid model consisting of aligned hard ellipsoids with superimposed square-well attractive interactions of variable range. The system is analyzed using a density functional theory in which the hard-core contributions to the free-energy functional are treated within a nonlocal weighted density approximation and the attractive contributions are considered at a mean-field level. In the absence of attractions the model reduces, under appropriate scaling, to a fluid of hard spheres and therefore does not exhibit smectic ordering. It is shown that above a certain value of the square-well range, smectic ordering is stable relative to the nematic state at densities well inside the fluid region. The nematic-smectic-A transition is found to be continuous at high temperatures and first order at low temperatures, these two regimes being separated by a tricritical point at an intermediate temperature. These predictions have been confirmed by computer simulation of the model fluid. The results highlight that smectic ordering can be stabilized by coupling anisotropic short-range repulsions with the isotropic contribution of the soft attractive interactions. By increasing the pressure, the range of stability of the smectic phase is seen to decrease. At sufficiently high pressure, the smectic phase is suppressed, and the solid phase dominates. Our calculations show that smectic ordering is no longer stable if the range of the attractions is made too long ranged.

Configurational bias Monte Carlo simulation of phase segregation in block copolymer networks.

Cross-linked block copolymers are used as adhesives in fiber-reinforced composite material manufactures for automotive applications. Good adhesion between the polymer matrix and fibers in the interphase region is required for the structural integrity of these materials. Experimental evidence indicates that superior adhesion is obtained when phase segregation occurs between the two matrix phase block copolymers. It is therefore desirable to predict the conditions under which phase segregation is expected to occur. Configurational bias Monte Carlo simulations of two-component, trifunctional block copolymer networks were carried out to investigate phase segregation in these materials. The effects of four principal parameters on phase segregation were examined: the weight fractions of the two components, the cross-link length, the connectivity of the network, and the ratio of the square-well interactions. The molecular simulation results confirmed trends observed in laboratory measurements.

Calculation of Gas−Liquid Critical Curves for Carbon Dioxide−1-Alkanol Binary Systems

The critical curves of the four binary systems from CO2−methanol to CO2−1-butanol at temperatures from 300 to 600 K and pressures from 4 to 18 MPa have been calculated. An equation of state that uses a perturbation type containing a repulsion term and a square-well potential attraction term for intermolecular interaction has been used. To use the pairwise combination rule for the square-well molecular interaction potential, three adjustable parameters are required. The experimental critical point data are compared with the values predicted from the equation of state. Good agreement is obtained between the experimental critical pressure−composition data and the calculated values.

Self-consistent Ornstein–Zernike approximation compared with Monte Carlo results for two-dimensional lattice gases

The self-consistent Ornstein–Zernike approach (SCOZA) is solved numerically for a lattice gas or Ising model on the simple square lattice in two dimensions. Interactions of varying range are considered, and the results are compared with corresponding simulation ones. We focus especially upon the location of the critical temperature Tc which is identified with the maximum of the specific heat. The maximum remains finite for the finite-sized simulation sample and also for SCOZA, which treats infinite lattices in two dimensions as though they were finite samples. We also investigate the influence of the precise form of the interaction, first using an interaction that extends the nearest-neighbor case in a simple way and then considering the square-well interactions used in the simulations. We find that the shift in Tc away from its mean-field value is governed primarily by the range of interaction. Other specific features of the interaction leave a smaller influence but are relevant to a quantitative comparison with simulations. The SCOZA yields accurate results, and the influence of the precise form of the attractive interaction plays a significant role in SCOZA’s success.

Gibbs ensemble simulation of the vapour-liquid equilibrium of square well spherocylinders

Gibbs ensemble Monte Carlo simulations have been performed for systems of square-well spherocylinders of different length-to-breadth ratio. The results are used to test a recent perturbation theory proposed for this kind of system. In addition, the results are compared to similar simulations performed for a Kihara fluid of elongated molecules. An unexpected good agreement is found for the coexistence thermodynamic and structural properties of both model fluids, hence suggesting that the hard spherocylinder plus square-well interaction should be considered as a reference potential for a perturbative treatment of more complex fluid models.

Folding of a model three-helix bundle protein: a thermodynamic and kinetic analysis

The kinetics and thermodynamics of an off-lattice model for a three-helix bundle protein are investigated as a function of a bias gap parameter that determines the energy difference between native and non-native contacts. A simple dihedral potential is used to introduce the tendency to form right-handed helices. For each value of the bias parameter, 100 trajectories of up to one microsecond are performed. Such statistically valid sampling of the kinetics is made possible by the use of the discrete molecular dynamics method with square-well interactions. This permits much faster simulations for off-lattice models than do continuous potentials. It is found that major folding pathways can be defined, although ensembles with considerable structural variation are involved. The large gap models generally fold faster than those with a smaller gap. For the large gap models, the kinetic intermediates are non-obligatory, while both obligatory and non-obligatory intermediates are present for small gap models. Certain large gap intermediates have a two-helix microdomain with one helix extended outward (as in domain-swapped dimers); the small gap intermediates have more diverse structures. The importance of studying the kinetic, as well as the thermodynamics, of folding for an understanding of the mechanism is discussed and the relation between kinetic and equilibrium intermediates is examined. It is found that the behavior of this model system has aspects that encompass both the “new” view and the “old” view of protein folding.

Vapor–liquid equilibria of vibrating square well chains

Vibrating square well (SW) 2-mer, 4-mer, and 8-mer with average reduced bond lengths of 0.97±0.03, 0.60±0.03, and 0.40±0.03 were studied by discontinuous molecular dynamics (DMD) simulation in the NVE ensemble. Average bond angles for the reduced bond length of 0.4 were constrained to 127±16° while the longer bond lengths were freely jointed. Vapor–liquid equilibria of the vibrating SW fluids were determined based on DMD simulation by isochoric integration and compared to that of rigid SW chains from Gibbs ensemble Monte Carlo (MC) simulation. The binodals of vibrating chains show a shift to higher temperatures relative to rigid chains, reflecting their less repulsive (more attractive) nature. Vapor pressures of the vibrating chains were computed through isochoric integration with Clausius–Clapeyron consistency to 5% or better. Vapor pressure behavior for each chain model was characterized in terms of critical temperature, critical pressure, and acentric factor. The trend in acentric factor vs. chain length showed that shorter bond lengths gave improved agreement with the experimental trend for n-alkanes. Nevertheless, the trends in acentric factor did not support any molecular model for alkanes which represented methylene segments as individual SW interaction sites. If SW chains are to be applied as models of alkanes, each interaction site must be assigned more than one methylene segment.

Pure fluids of homonuclear and heteronuclear square-well diatomics I. Computer simulation study

Pure fluids of both homonuclear and heteronuclear square-well diatomics have been studied using Monte Carlo simulations in NVT and Gibbs ensembles. For the homonuclear models, three values of the elongation have been considered to complement the existing data for the range of the square-well interaction λ = 1.5. The same range has been considered also for the heteronuclear models whose heterogeneity arise from (i) the difference in the hard-sphere diameters, and (ii) the difference in the strength of the site—site interactions. The computed structural properties include the complete set of the site—site correlation functions and their low density limits. The thermodynamic properties include, in addition to the equation of state and internal energy on three isotherms, and the 2nd and 3rd virial coefficients, also the coexistence properties and an approximate location of the critical point.