Square-well System Research Articles

Nonlinear rheology of dense colloidal systems with short-ranged attraction: A mode-coupling theory analysis

The nonlinear rheology of glass-forming colloidal suspensions with short-ranged attractions is discussed within the integration-through transients framework combined with the mode-coupling theory of the glass transition. Calculations are based on the square-well system, as a model for colloid-polymer mixtures. The high-density regime featuring reentrant melting of the glass upon increasing the attraction strength and the crossover from repulsive glasses formed at weak attraction to attractive glasses formed at strong attraction are discussed. Flow curves are found in qualitative agreement with experimental data, featuring a strong increase in the yield stress and for suitable interaction parameters, the crossover between two yield stresses. The yield strain, defined as the position of the stress overshoot under startup flow, is found to be proportional to the attraction range for strong attraction. At weak and intermediate attraction strength, the combined effects of hard-core caging and attraction-driven bonding result in a richer dependence on the parameters. The first normal-stress difference exhibits a weaker dependence on short-ranged attractions as the shear stress, since the latter is more sensitive than the short-wavelength features of the static structure.

Theoretical investigation on nearsightedness of finite model and molecular systems based on linear response function analysis.

We examined nearsightedness of electronic matter (NEM) of finite systems on the basis of linear response function (LRF). From the computational results of a square-well model system, the behavior of responses obviously depends on the number of electrons (N): as N increases, LRF, δρ(r)/δv(r′), decays rapidly for the distance, |r−r′|. This exemplifies that the principle suggested by Kohn and Prodan holds even for finite systems: the cause of NEM is destructive interference among electron density amplitudes. In addition, we examined double-well model systems, which have low-lying degenerate levels. In this case, there are two types of LRF: the cases of the half-filled and of full-filled in low-lying degenerate levels. The response for the former is delocalized, while that of the later is localized. These behaviors of model systems are discussed in relation to the molecular systems’ counterparts, H2, He22+, and He2 systems. We also see that NEM holds for the dissociated limit of H2, of which the mechanism is similar to that of the insulating state of solids as suggested by Kohn. We also examined LRF of alanine tripeptide system as well as butane and butadiene molecules, showing that NEM of the polypeptide system is caused by sp3 junctions at Cα atoms that prevent propagation of amplitudes of LRF, which is critically different from that of NEM for finite and infinite homogeneous systems.

True Widom line for a square-well system.

In the present paper we propose a van der Waals-like model that allows a purely analytical study of fluid properties including the equation of state, phase behavior, and supercritical fluctuations. We take a square-well system as an example and calculate its liquid-gas transition line and supercritical fluctuations. Employing this model allows us to calculate not only the thermodynamic response functions (isothermal compressibility βT, isobaric heat capacity CP, density fluctuations ζT, and thermal expansion coefficient αT), but also the correlation length in the fluid ξ. It is shown that the bunch of extrema widens rapidly upon departure from the critical point. It seems that the Widom line defined in this way cannot be considered as a real boundary that divides the supercritical region into gaslike and liquidlike regions. As it has been shown recently, a dynamic line on the phase diagram in the supercritical region, namely, the Frenkel line, can be used for this purpose.

Perturbation method to calculate the density of states

Monte Carlo switching moves ("perturbations") are defined between two or more classical Hamiltonians sharing a common ground-state energy. The ratio of the density of states (DOS) of one system to that of another is related to the ensemble averages of the microcanonical acceptance probabilities of switching between these Hamiltonians, analogously to the case of Bennett's acceptance ratio method for the canonical ensemble [C. H. Bennett, J. Comput. Phys. 22, 245 (1976)]. Thus, if the DOS of one of the systems is known, one obtains those of the others and, hence, the partition functions. As a simple test case, the vapor pressure of an anharmonic Einstein crystal is computed, using the harmonic Einstein crystal as the reference system in one dimension; an auxiliary calculation is also performed in three dimensions. As a further example of the algorithm, the energy dependence of the ratio of the DOS of the square-well and hard-sphere tetradecamers is determined, from which the temperature dependence of the constant-volume heat capacity of the square-well system is calculated and compared with canonical Metropolis Monte Carlo estimates. For these cases and reference systems, the perturbation calculations exhibit a higher degree of convergence per Monte Carlo cycle than Wang-Landau (WL) sampling, although for the one-dimensional oscillator the WL sampling is ultimately more efficient for long runs. Last, we calculate the vapor pressure of liquid gold using an empirical Sutton-Chen many-body potential and the ideal gas as the reference state. Although this proves the general applicability of the method, by its inherent perturbation approach the algorithm is suitable for those particular cases where the properties of a related system are well known.

Coarse-grained lattice Monte Carlo simulations with continuous interaction potentials

A coarse-grained lattice Metropolis Monte Carlo (CG-MMC) method is presented for simulating fluid systems described by standard molecular force fields. First, a thermodynamically consistent coarse-grained interaction potential is obtained numerically and automatically from a continuous force field such as Lennard-Jones. The coarse-grained potential then is used to drive CG-MMC simulations of vapor-liquid equilibrium in Lennard-Jones, square-well, and simple point charge water systems. The CG-MMC predicts vapor-liquid phase envelopes, as well as the particle density distributions in both the liquid and vapor phases, in excellent agreement with full-resolution Monte Carlo simulations, at a fraction of the computational cost.

Gibbs–Bogoliubov variational procedure with the square-well reference system

For the first time, the square-well (SW) model is used as a reference system in the thermodynamic variational calculations for liquids. The SW system is taken within the random phase approximation. To apply this approach to simple metals, the local Animalu–Heine model pseudopotential, the Vashishta–Singwi exchange-correlation function, and the Nozieres–Pines exchange-correlation energy are used. Thermodynamics and structure of liquid Na at T = 373 K are studied. We found that there are several areas of the SW parameters where the solution is stable. We chose the best of such areas in which the calculated structure factor is in good agreement with the experimental one. Then the Helmholtz free energy is minimized in this area with respect to the hard-core diameter and the SW depth at the fixed SW width. It was found that the SW variational procedure gives an upper bound of the Helmholtz free energy which is lower than that in the hard-sphere variational procedure, and that the obtained thermodynamic properties are in reasonable agreement with the experiment.

Optimized equation of the state of the square-well fluid of variable range based on a fourth-order free-energy expansion

The free energy of square-well (SW) systems of hard-core diameter sigma with ranges 1 < or = lambda < or = 3 is expanded in a perturbation series. This interval covers most ranges of interest, from short-ranged SW fluids (lambda approximately 1.2) used in modeling colloids to long ranges (lambda approximately 3) where the van der Waals classic approximation holds. The first four terms are evaluated by means of extensive Monte Carlo simulations. The calculations are corrected for the thermodynamic limit and care is taken to evaluate and to control the various sources of error. The results for the first two terms in the series confirm well-known independent results but have an increased estimated accuracy and cover a wider set of well ranges. The results for the third- and fourth-order terms are novel. The free-energy expansion for systems with short and intermediate ranges, 1 < or = lambda < or = 2, is seen to have properties similar to those of systems with longer ranges, 2 < or = lambda < or = 3. An equation of state (EOS) is built to represent the free-energy data. The thermodynamics given by this EOS, confronted against independent computer simulations, is shown to predict accurately the internal energy, pressure, specific heat, and chemical potential of the SW fluids considered and for densities 0 < or = rho sigma(3) < or = 0.9 including subcritical temperatures. This fourth-order theory is estimated to be accurate except for a small region at high density, rho sigma(3) approximately 0.9, and low temperature where terms of still higher order might be needed.

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.

Solvation potentials for flexible chain molecules in solution: On the validity of a pairwise decomposition

The effects of a solvent on the conformation of a flexible n-site solute molecule can be described formally in terms of an n-body solvation potential. Given the practical difficulty in computing such multibody potentials, it is common to carry out a pairwise decomposition in which the n-body potential is approximated by a sum of two-body potentials. Here we investigate the validity of this two-site approximation for short interaction-site chain-in-solvent systems. Using exact expressions for the conformation of an isolated chain, we construct a mapping between the full chain-in-solvent system and its solvation potential representation. We present results for both hard-sphere and square-well systems with n=5 that show that the two-site approximation is sufficient to completely capture the effects of an explicit solvent on chain conformation for a wide range of conditions (which include varying the solvent diameter in the hard-sphere system and varying the chain-solvent coupling in the square-well system). In all cases, a set of two-site potentials (one for each distinct site-site pair) is required. We also show that these two-site solvation potentials can be used to accurately compute a multisite intramolecular correlation function.

How short-range attractions impact the structural order, self-diffusivity, and viscosity of a fluid

We present molecular simulation data for viscosity, self-diffusivity, and the local structural ordering of (i) a hard-sphere fluid and (ii) a square-well fluid with short-range attractions. The latter fluid exhibits a region of dynamic anomalies in its phase diagram, where its mobility increases upon isochoric cooling, which is found to be a subset of a larger region of structural anomalies, in which its pair correlations strengthen upon isochoric heating. This "cascade of anomalies" qualitatively resembles that found in recent simulations of liquid water. The results for the hard-sphere and square-well systems also show that the breakdown of the Stokes-Einstein relation upon supercooling occurs for conditions where viscosity and self-diffusivity develop different couplings to the degree of pairwise structural ordering of the liquid. We discuss how these couplings reflect dynamic heterogeneities. Finally, we note that the simulation data suggest how repulsive and attractive glasses may generally be characterized by two distinct levels of short-range structural order.

Kinetic arrest and glass-glass transition in short-ranged attractive colloids

A thermally reversible repulsive hard-sphere to sticky-sphere transition was studied in a model colloidal system over a wide volume fraction range. The static microstructure was obtained from high resolution small angle x-ray scattering, the colloid dynamics was probed by dynamic x-ray and light scattering, and supplementary mechanical properties were derived from bulk rheology. At low concentration, the system shows features of gas-liquid type phase separation. The bulk phase separation is presumably interrupted by a gelation transition at the intermediate volume fraction range. At high volume fractions, fluid-attractive glass and repulsive glass-attractive glass transitions are observed. It is shown that the volume fraction of the particles can be reliably deduced from the absolute scattered intensity. The static structure factor is modeled in terms of an attractive square-well potential, using the leading order series expansion of Percus-Yevick approximation. The ensemble-averaged intermediate scattering function shows different levels of frozen components in the attractive and repulsive glassy states. The observed static and dynamic behavior are consistent with the predictions of a mode-coupling theory and numerical simulations for a square-well attractive system.

Liquid–vapour equilibrium of multipolar square-well fluids.Gibbs ensemble simulations and equation of state

Simulation results for a system comprising a square well plus either a point dipole or a point quadrupole are presented. The properties obtained are the vapour–liquid equilibrium densities and the critical properties. Critical densities are not very sensitive to the values of dipole or quadrupole, while critical temperatures increase significantly when the multipole strength rises. A comparison with a perturbation theory for multipolar square-well systems is presented. Overall agreement between simulated and theoretical values is good when comparison is restricted to quadrupoles or dipoles corresponding to the most relevant real polar substances but is only moderate for the largest multipolar strengths considered.

Logarithmic relaxation in a colloidal system.

The slow dynamics for a colloidal suspension of particles interacting with a hard-core repulsion complemented by a short-ranged attraction is discussed within the frame of mode-coupling theory for ideal glass transitions for parameter points near a higher-order glass-transition singularity. The solutions of the equations of motion for the density correlation functions are solved for the square-well system in quantitative detail by asymptotic expansion using the distance of the three control parameters-packing fraction, attraction strength and attraction range-from their critical values as small parameters. For given wave vectors, distinguished surfaces in parameter space are identified where the next-to-leading-order contributions for the expansion vanish so that the decay functions exhibit a logarithmic decay over large time intervals. For both coherent and tagged particle dynamics the leading-order logarithmic decay is accessible in the liquid regime for wave vectors of several times the principal peak in the structure factor. The logarithmic decay in the correlation function is manifested in the mean-squared displacement as a subdiffusive power law with an exponent varying sensitively with the control parameters. Shifting parameters through the distinguished surfaces, the correlation functions and the logarithm of the mean-squared displacement considered as functions of the logarithm of the time exhibit a crossover from concave to convex behavior, and a similar scenario is obtained when varying the wave vector.

Theory and computer simulation of the zero- and first-order perturbative contributions to the pair correlation function of square-well fluids

We have performed extensive Monte Carlo (MC) simulations in the NVT ensemble of the pair correlation function (p.c.f.) g( r) for square-well (SW) fluids for several densities, temperatures and potential widths. In addition, using the procedure developed by Smith et al. [J. Chem. Phys. 55 (1971) 4027], we have determined by MC–NVT simulations the zero- and first-order terms in the expansion of the p.c.f. in power series of the inverse of the reduced temperature at the same densities. Furthermore, we have compared the values of g( r) obtained from the expansion truncated beyond the first-order term, with those obtained directly by MC simulations performed on the SW system. The aim was mainly to test whether the perturbation expansion converges quickly and whether this procedure, without any theoretical approximation, is accurate enough. We have found that this approximation is very accurate for any value of the potential width and of the reduced temperature at high densities, except perhaps for small potential widths at very low temperatures. For lower densities the convergence slows down more markedly the lower are the values of the potential width and/or the reduced density. In addition, we have compared the theoretical predictions for the first-order contribution obtained from the integral equation theory of Tang and Lu (TL) [J. Chem. Phys. 100 (1994) 6665] with the simulation data. We have found that the theory is fairly accurate for distances larger than the width of the potential, whereas for shorter distances the accuracy decreases as the width of the potential decreases.

Fluid–fluid coexistence in colloidal systems with short-ranged strongly directional attraction

We present a systematic numerical study of the phase behavior of square-well fluids with a “patchy” short-ranged attraction. In particular, we study the effect of the size and number of attractive patches on the fluid–fluid coexistence. The model that we use is a generalization of the hard sphere square well model. The systems that we study have a stronger tendency to form gels than the isotropic square-well system. For this reason, we had to combine Gibbs ensemble simulations of the fluid–fluid coexistence with a parallel tempering scheme. For moderate directionality, changes of the critical density and the width of coexistence curves are small. For strong directionality, however, we find clear deviations from the extended law of corresponding states: in contrast to isotropic attractions, the critical point is not characterized by a universal value of the reduced second virial coefficient. Furthermore, as the directionality increases, multiparticle bonding affects the critical temperature. We discuss implications for the phase behavior, and possibly crystallization, of globular proteins.

Higher-order glass-transition singularities in systems with short-ranged attractivepotentials

Within the mode-coupling theory for the evolution of structural relaxation, theA4-glass-transitionsingularities are identified for systems of particles interacting with a hard-sphererepulsion complemented by different short-ranged potentials: Baxter’ssingular potential regularized by a large-wavevector cut-off, a model for theAsakura–Oosawa depletion attraction, a triangular potential, a Yukawaattraction, and a square-well potential. The regular potentials yield criticalpacking fractions, critical Debye–Waller factors, and critical amplitudes very closeto each other. The elastic moduli and the particle localization lengths forcorresponding states of the Yukawa system and the square-well system may differby up to 20 and 10%, respectively.

Self-consistent nonperturbative theory: Application to a two-dimensional square-well system

A self-consistent, nonperturbative theory, developed to describe the structure and thermodynamics of a classical system of particles and presented in a previous paper [Phys. Rev. E 65, 016131 (2002)], is generalized to a two-dimensional system and applied to the square-well potential. The theory predicts a phase diagram which turns out to be in very good agreement with that obtained by computer simulations performed by us. This is a consequence of the very accurate results of the theory as concerns the angle-averaged two-body distribution function and the Helmholtz free energy, which we also present and compare with computer simulations. By contrast, a first-order perturbation theory only provides qualitative agreement, showing that higher-order terms play an important role and that these terms are well accounted for by the nonperturbative theory.

Vapour—liquid equilibrium of the square-well fluid of variable range via a hybrid simulation approach

The equilibrium between vapour and liquid in a square-well system has been determined by a hybrid simulation approach combining chemical potentials calculated via the Gibbs ensemble Monte Carlo technique with pressures calculated by the standard NVT Monte Carlo method. The phase equilibrium was determined from the thermodynamic conditions of equality of pressure and chemical potential between the two phases. The results of this hybrid approach were tested by independent NPT and μPT calculations and are shown to be of much higher accuracy than those of conventional GEMC simulations. The coexistence curves, vapour pressures and critical points were determined for SW systems of interaction ranges λ = 1.25, 1.5, 1.75 and 2. The new results show a systematic dependence on the range λ, in agreement with results from perturbation theory where previous work had shown more erratic behaviour.

Ideal glass in attractive systems with differentpotentials

We discuss the ideal glass transition for two types of potentialmodel of attractive colloidal systems, i.e. the square-well system and the Yukawa hard-sphere fluid. We use the framework ofthe ideal mode-coupling theory and we mostly focus our attentionon the nature of the singularities predicted by the theory. We alsostudy the phenomena that arise by varying the range of the attraction,since this parameter has been identified as one of the key parameters incolloidal systems.

Fast and accurate thermodynamics of square-well systems from umbrella-sampling simulations of hard-sphere systems

A new method for calculating thermodynamics of any square-well systems by umbrella sampling of the corresponding hard-sphere systems is proposed. The removal of energetic barriers leads to an efficient configurational sampling. As an illustrative example, the method is applied to a square-well chain of freely-jointed 64 beads. Results indicate that it produces fast and more accurate thermodynamic results than either the direct molecular dynamics simulation or the umbrella sampling of the square-well chain. The new method, which provides a new route for accurate heat-capacity and free energy calculations, can be easily extended to a system interacting with the Lennard-Jones or any other continuous potentials.