Large Hard Spheres Research Articles

Oscillatory structural forces probed by the nanofilm Macroscopic contact angle

The research on oscillatory structural forces (OSF) has a long history. OSF arise when liquid dispersions of a nano/micellar solution of complex fluids are confined between surfaces. In recent years, significant progress has been made in the measuring and modeling of OSF. Advanced techniques that use tools such as the surface forces apparatus (SFA), colloidal probe atomic force microscope (CP-AFM), and foam film capillary balance (FCB) were developed to gain information about nanofilm interactions in the confinement film geometry as the “layer” and “in-layer” particle entropy structural transition takes place. Henderson’s theory, based on the statistical mechanics approach to model the OSF, proposed analytical results using the Laplace transformation of the radial distribution function (RDF) for two large hard spheres in a fluid to calculate the film free energy; this approach is commonly applied to the model of the nanofilm layering between two solid surfaces. To extend Henderson’s theory to model the “layer” and “in-layer” structural transition of a capillary system, e.g., foam and emulsions, we propose the use of the foam film-meniscus macroscopic contact angle approach to calculate the nanofilm’s OSF.

Effect of particle exchange on the glass transition of binary hard spheres

We investigate the replica theory of the liquid-glass transition for a binary mixture of large and small additive hard spheres. We consider two different ansätze for this problem: the frozen glass ansatz (FGA) in which the exchange of large and small particles in a glass state is prohibited, and the exchange glass ansatz (EGA), in which it is allowed. We calculate the dynamical and thermodynamical glass transition points with the two ansätze. We show that the dynamical transition density of the FGA is lower than that of the EGA, while the thermodynamical transition density of the FGA is higher than that of the EGA. We discuss the algorithmic implications of these results for the density-dependence of the relaxation time of supercooled liquids. We particularly emphasize the difference between the standard Monte Carlo and swap Monte Carlo algorithms. Furthermore, we discuss the importance of particle exchange for estimating the configurational entropy.

Unusual crowding-induced chain looping kinetics in hard-sphere fluids: a contrastive study with polymer solutions.

A theoretical framework is developed to investigate the looping kinetics of a chain in hard-sphere (HS) fluids, based on a generalized Smoluchowski diffusion-reaction equation. A contrastive study with polymer solutions is performed. The crowding-associated effective viscosity and collapse effects are properly taken into account, which obey different scaling relations in HS and polymer fluids. We examine the dependence of the looping time on both concentration and size of crowders, demonstrating unusual and distinct discrepancies in the two crowded media. Firstly, in the solution of large polymers, the looping rate grows monotonically with polymer concentration. On the other hand, in the solution of large HSs, a caging regime can be observed, where the looping time tends to the value in the absence of crowders. Secondly, polymers in moderate size generally impede chain looping due to the enhanced viscosity. However, in HS fluids, the looping time exhibits a rather complicated variation with increasing HS size. We show a possible mechanism where in the case of small crowders with a relatively strong compaction in the probed chain, the looping kinetics can be facilitated. As the crowder size increases, the collapse effect is reduced and looping is dominated by viscosity-induced inhibition. Simultaneously, our theory rationalizes another possibility of the mechanism observed by recent simulation work. We conclude that the looping kinetics in specific systems actually should be governed by the critical competition between the two crowding factors. By giving reasonable measurements of effective viscosity and collapse, our theoretical framework can provide a unified strategy to analyze crowding effects on the looping rate in a systematic manner.

Stick boundary condition at large hard sphere arising from effective attraction in binary hard-sphere mixtures.

We have studied the diffusion of a large hard-sphere solute immersed in binary hard-sphere mixtures. We reveal how the boundary condition at the solute surface is affected by the solvent density around the solute. Solving equations for a binary compressible mixture by perturbation expansions, we obtain the boundary condition depending on the size ratio of binary solvent spheres. When the size ratio is 1:2, the boundary condition lies close to the slip boundary condition. By contrast, when the size ratio becomes large, the boundary condition approaches the stick boundary condition with the addition of larger solvent spheres. We find that the transition to the stick boundary condition is caused by the increase in the solvent density around the solute due to an entropic effect.

Asymptotic orderings and approximations of the Master kinetic equation for large hard spheres systems

In this paper the problem is posed of determining the physically-meaningful asymptotic orderings holding for the statistical description of a large N-body system of hard spheres, i.e., formed by N≡1ε≫1 particles, which are allowed to undergo instantaneous and purely elastic unary, binary or multiple collisions. Starting point is the axiomatic treatment recently developed [Tessarotto et al., 2013–2016] and the related discovery of an exact kinetic equation realized by Master equation which advances in time the 1-body probability density function (PDF) for such a system. As shown in the paper the task involves introducing appropriate asymptotic orderings in terms of ε for all the physically-relevant parameters. The goal is that of identifying the relevant physically-meaningful asymptotic approximations applicable for the Master kinetic equation, together with their possible relationships with the Boltzmann and Enskog kinetic equations, and holding in appropriate asymptotic regimes. These correspond either to dilute or dense systems and are formed either by small-size or finite-size identical hard spheres, the distinction between the various cases depending on suitable asymptotic orderings in terms of ε.

Effects of interactions between depletants in phase diagrams of binary hard-sphere systems

Fluid-solid phase diagrams for binary hard-sphere systems were calculated to study the effects of interactions between depletants. Two effective potentials between large hard spheres were examined. One was the Asakura-Oosawa (AO) potential, and the other was an effective potential obtained by using an integral equation theory (IE potential). The IE potential has oscillations caused by the interactions between depletants, whereas in the AO potential, the inter-depletant correlation is ignored. The phase diagrams were obtained by using the thermodynamic perturbation theory with the density functional theory. The phase diagrams for the IE potential systems differed from those for the AO potential systems because of the inter-depletant correlation. The differences in the present calculation were caused by the first minimum of the effective potential.

Generic phase diagram of binary superlattices

Emergence of a large variety of self-assembled superlattices is a dramatic recent trend in the fields of nanoparticle and colloidal sciences. Motivated by this development, we propose a model that combines simplicity with a remarkably rich phase behavior applicable to a wide range of such self-assembled systems. Those systems include nanoparticle and colloidal assemblies driven by DNA-mediated interactions, electrostatics, and possibly, controlled drying. In our model, a binary system of large and small hard spheres (L and S, respectively) interacts via selective short-range ("sticky") attraction. In its simplest version, this binary sticky sphere model features attraction only between S and L particles. We show that, in the limit when this attraction is sufficiently strong compared with [Formula: see text], the problem becomes purely geometrical: the thermodynamically preferred state should maximize the number of [Formula: see text] contacts. A general procedure for constructing the phase diagram as a function of system composition f and particle size ratio r is outlined. In this way, the global phase behavior can be calculated very efficiently for a given set of plausible candidate phases. Furthermore, the geometric nature of the problem enables us to generate those candidate phases through a well-defined and intuitive construction. We calculate the phase diagrams for both 2D and 3D systems and compare the results with existing experiments. Most of the 3D superlattices observed to date are featured in our phase diagram, whereas several more are predicted for future discovery.

Tuning the bridging attraction between large hard particles by the softness of small microgels.

In this study, the attraction between large hard polystyrene (PS) spheres is studied by using three types of small microgels as bridging agents. One is a purely soft poly(N-isopropylacrylamide) (PNIPAM) microgel, the other two have a non-deformable PS hard core surrounded by a soft PNIPAM shell but are different in the core-shell ratio. The affinity for bridging the large PS spheres is provided and thus affected by the PNIPAM constituent in the microgels. The bridging effects caused by the microgels can be indirectly incorporated into their influence on the effective attraction interaction between the large hard spheres, since the size of the microgels is very small in comparison to the size of the PS hard spheres. At a given volume fraction of large PS spheres, they behave essentially as hard spheres in the absence of small microgels. By gradually adding the microgels, the large spheres are connected to each other through the bridging of small particles until the attraction strength reaches a maximum value, after which adding more small particles slowly decreases the effective attraction strength and eventually the large particles disperse individually when saturated adsorption is achieved. The aggregation and gelation behaviors triggered by these three types of small microgels are compared and discussed. A way to tune the strength and range of the short-range attractive potential via changing the softness of bridging microgels (which can be achieved either by using core-shell microgels or by changing the temperature) is proposed.

Depletion gels from dense soft colloids: Rheology and thermoreversible melting

Upon addition of small nonadsorbing linear polymers, colloidal glasses composed of large hard spheres melt and eventually revitrify into the so-called attractive glass regime. We show that, when replacing the hard spheres by star polymers representing model soft particles, a reentrant gel is formed. This is the result of compression and depletion of the stars due to the action of the osmotic pressure from the linear homopolymers. The viscoelastic properties of the soft dense gel were studied with emphasis on the shear-induced yielding process, which involved localized breaking of elements with a size of the order of the correlation length. Based on these results, a phenomenological attempt was made at describing the universal rheological features of colloid/nonadsorbing polymer mixtures of varying softness. The star gel was found to undergo thermoreversible melting, despite the fact that conventional hard-sphere depletion gels are invariant to heating. This phenomenon is attributed to the hybrid internal microstructure of the stars, akin to a dry-to-wet brush transition, and is characterized by slow kinetics, on the time scale of the osmotic gel formation process. These results may be useful in finding generic features in colloidal gelation, as well as in the molecular design of new soft composite materials.

Perturbation Theory of Large-Particle Diffusion in a Binary Solvent Mixture

We study the diffusion of a large spherical particle immersed in a binary compressive liquid mixture using a perturbation theory. We focus on the breakdown of the Stokes–Einstein (SE) relation caused by the microscopic solvation structure of binary solvent particles around a solute particle. In order to consider the solvation structure, we solve multicomponent generalized Langevin equations by singular perturbation expansion. Then, we assume that solvent particles are much smaller than the solute particle. Solving the equations, we express the diffusion coefficient analytically using the radial distribution functions of a binary mixture. The expression shows the breakdown of the SE relation if the density distribution of a binary solvent is inhomogeneous around a solute particle. Actually, we show that the SE relation breaks down when a large hard sphere diffuses in a binary hard-sphere mixture. We observe the large deviation from the SE relation, which is a result specific to the binary solvent.

Thermal melting in depletion gels of hairy nanoparticles

Under certain conditions, the addition of small non-adsorbing polymers to a suspension of large hard spheres yields a depletion gel. It is expected that, provided that solvent quality issues do not lead to phase separation, thermodynamics imposes an invariance of the hard-sphere depletion gel upon heating. We show here that this is not the case for depletion gels of soft hairy nanoparticles. Using multiarm star polymers as a model that encompasses the key features of a huge range of grafted colloids, we show that their microstructural details make the difference. In particular, their hybrid nature, consisting of a carbosilane core and a polymeric shell, gives rise to a mismatch in thermal expansion upon heating, which affects the nanoparticle's conformation in a way analogous to a dry-to-wet brush transition, and hence leads to thermal melting of the gel. These results open the route for the molecular design of new materials, providing new opportunities to colloid chemistry and processing.

Thermodynamically consistent closure approximation for hard spheres systems

We present a new closure relation that is an extension of the Percus-Yevick approximation. In the proposed closure, we introduce an additional term and a mixing coefficient that can be determined by imposing a condition of thermodynamic self-consistency. Moreover, the mixing coefficient is calculated analytically within a linear approximation. In the case of a monodisperse system of hard spheres, we compare the results of our model to well-established thermodynamic expressions and also to the structural properties of fairly known closure approximations. In the second case, and using an equivalent scheme, the new closure relation is extended to the depletion potential between two large hard spheres immersed in a liquid of small hard spheres. In both cases, the results of our model are in good agreement with numerical simulations performed at intermediate concentrations.

Self-Assembly of a Colloidal Interstitial Solid with Tunable Sublattice Doping

We determine the phase diagram of a binary mixture of small and large hard spheres with a size ratio of 0.3 using free-energy calculations in Monte Carlo simulations. We find a stable binary fluid phase, a pure face-centered-cubic (fcc) crystal phase of the small spheres, and binary crystal structures with LS and LS(6) stoichiometries. Surprisingly, we demonstrate theoretically and experimentally the stability of a novel interstitial solid solution in binary hard-sphere mixtures, which is constructed by filling the octahedral holes of an fcc crystal of large spheres with small spheres. We find that the fraction of octahedral holes filled with a small sphere can be completely tuned from 0 to 1. Additionally, we study the hopping of the small spheres between neighboring octahedral holes, and interestingly, we find that the diffusion increases upon increasing the density of small spheres.

Mesophase formation in a system of top-shaped hard molecules: Density functional theory and Monte Carlo simulation

We present the phase diagram of a system of mesogenic top-shaped molecules based on the Parsons-Lee density functional theory and Monte Carlo simulation. The molecules are modeled as a hard spherocylinder with a hard sphere embedded in its center. The stability of five different phases is studied, namely, isotropic, nematic, smectic A, smectic C, and columnar phases. The positionally ordered phases are investigated only for the case of parallel alignment. It is found that the central spherical unit destabilizes the nematic with respect to the isotropic phase, while increasing the length of the cylinder has the opposite effect. Also, the central hard sphere has a strong destabilizing effect on the smectic A phase, due the inefficient packing of the molecules into layers. For large hard sphere units the smectic A phase is completely replaced by a smectic C structure. The columnar phase is first stabilized with increasing diameter of the central unit, but for very large hard sphere units it becomes less stable again. The density functional results are in good agreement with the simulations.

Entropic potential field formed for a linear-motor protein near a filament: Statistical-mechanical analyses using simple models

We report a new progress in elucidating the mechanism of the unidirectional movement of a linear-motor protein (e.g., myosin) along a filament (e.g., F-actin). The basic concept emphasized here is that a potential field is entropically formed for the protein on the filament immersed in solvent due to the effect of the translational displacement of solvent molecules. The entropic potential field is strongly dependent on geometric features of the protein and the filament, their overall shapes as well as details of the polyatomic structures. The features and the corresponding field are judiciously adjusted by the binding of adenosine triphosphate (ATP) to the protein, hydrolysis of ATP into adenosine diphosphate (ADP)+Pi, and release of Pi and ADP. As the first step, we propose the following physical picture: The potential field formed along the filament for the protein without the binding of ATP or ADP+Pi to it is largely different from that for the protein with the binding, and the directed movement is realized by repeated switches from one of the fields to the other. To illustrate the picture, we analyze the spatial distribution of the entropic potential between a large solute and a large body using the three-dimensional integral equation theory. The solute is modeled as a large hard sphere. Two model filaments are considered as the body: model 1 is a set of one-dimensionally connected large hard spheres and model 2 is a double helical structure formed by two sets of connected large hard spheres. The solute and the filament are immersed in small hard spheres forming the solvent. The major findings are as follows. The solute is strongly confined within a narrow space in contact with the filament. Within the space there are locations with sharply deep local potential minima along the filament, and the distance between two adjacent locations is equal to the diameter of the large spheres constituting the filament. The potential minima form a ringlike domain in model 1 while they form a pointlike one in model 2. We then examine the effects of geometric features of the solute on the amplitudes and asymmetry of the entropic potential field acting on the solute along the filament. A large aspherical solute with a cleft near the solute-filament interface, which mimics the myosin motor domain, is considered in the examination. Thus, the two fields in our physical picture described above are qualitatively reproduced. The factors to be taken into account in further studies are also discussed.

Effective interaction between large colloidal particles immersed in a bidisperse suspension of short-ranged attractive colloids

The effective force between two large hard spheres mimicking lyophobic colloids (solute) immersed in an asymmetric two-component mixture of smaller particles (solvents), interacting via Baxter's sticky hard sphere (SHS) potential, was studied using integral equation theory and Monte Carlo simulation. The theoretical predictions were calculated from the analytic solution of the Percus-Yevick/Ornstein-Zernike integral equation for spatial correlations in a three-component mixture at vanishing solute concentration, while the simulation results were obtained by applying a special simulation technique developed for sampling the hard-sphere collision force. Due to layering of the solvent molecules, the effective force between the particles of the solute oscillates with periods equal to the molecular diameters of both solvent components. The attractive force between the solute particles in the SHS mixture comprising strongly attractive molecules of either component decays slower than that in the mixture with weaker interparticle attraction. Similar features are also observed when inspecting the separate contributions of individual components to the total solute-solute force. At sufficient strength of the interparticle stickiness, these oscillations disappear, the force becoming long ranged and attractive at all separations.

Phase Transitions in a Mixture of Hard-Sphere Dipoles and Neutral Hard Spheres

Using the reference hypernetted chain (RHNC) integral equation theory and a rigorous stability analysis method, we investigate the phase behavior of a mixture of hard-sphere dipoles and neutral hard spheres based on the correlations of the homogeneous isotropic phase. Lowering the temperature down to the points where the RHNC equations fail to have a solution, several fluctuations strongly increase. At low densities our results indicate the onset of chain formation, which is similar with the pure DHS system. At high densities, the results indicate the appearance of isotropic-to-ferroelectric transitions (small neutral hard spheres concentrations) and demixing transitions (large neutral hard spheres concentrations).

Prediction of binary hard-sphere crystal structures

We present a method based on a combination of a genetic algorithm and Monte Carlo simulations to predict close-packed crystal structures in hard-core systems. We employ this method to predict the binary crystal structures in a mixture of large and small hard spheres with various stoichiometries and diameter ratios between 0.4 and 0.84. In addition to known binary hard-sphere crystal structures similar to NaCl and AlB2, we predict additional crystal structures with the symmetry of CrB, gammaCuTi, alphaIrV, HgBr2, AuTe2, Ag2Se, and various structures for which an atomic analog was not found. In order to determine the crystal structures at infinite pressures, we calculate the maximum packing density as a function of size ratio for the crystal structures predicted by our GA using a simulated annealing approach.

Three-Dimensional Density Profiles of Small and Medium Spheres near a Pair of Large Spheres: Relevance to Entropic Interaction Induced between Large Spheres

In the system we consider, a pair of large hard spheres, whose surface separation is fixed at a prescribed value, is immersed in a binary mixture comprising small and medium hard spheres. The density profiles of the medium and small spheres near the large spheres are calculated using the three-dimensional integral equation theory. When the surface separation equals the medium-sphere diameter, a medium sphere is held between the large spheres in contact with both of their surfaces. This is represented by an extremely sharp, high peak in the reduced density profile of medium spheres. It is found that the profile can be reproduced by the Kirkwood superposition approximation using the reduced density profile of medium spheres around a single large sphere, which is calculated using the radial-symmetric integral equation theory. The small spheres are excluded by the medium sphere held as described above, and the thermal pressure due to the small spheres acting on each large-sphere surface becomes highly asymmet...

Transient ordering in a quasi-two-dimensional binary liquid near freezing

We report the results of a theoretical study of locally ordered fluctuations in a quasi-two-dimensional (quasi-2D) binary hard sphere mixture as monitored by the aperture cross-correlation function. Systems with thickness less than two large hard sphere diameters were studied, over a range of large hard sphere density and relative concentration of large and small hard spheres in the liquid state, for two ratios of sphere diameters, 0.2 and 0.3. In several major respects the occurrence and character of the structured fluctuations in a quasi-2D binary hard sphere mixture have similarities with those in a quasi-2D one-component hard sphere fluid, as reported by Sheu and Rice [J. Chem. Phys. 128, 244517 (2008)]. Specifically, our studies establish that structured fluctuations with both hexagonal and square symmetries can be found in the liquid phase well below the freezing density and that the occurrence of particular structured fluctuations is correlated with the symmetry of the solid to which the liquid freezes. However, there is also a major difference, specifically, the presence of the smaller spheres in the binary mixture can suppress the occurrence of all structured fluctuations. We attribute the affect of a small volume fraction of small spheres in a dense binary mixture of small and large spheres on the occurrence of structured fluctuations to the unfavorable entropy change associated with demixing of the small and large spheres.