Small Hard Spheres Research Articles

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...

Pressure effects on structures formed by entropically driven self-assembly: Illustration for denaturation of proteins

We propose a general framework of pressure effects on the structures formed by the self-assembly of solute molecules immersed in solvent. The integral equation theory combined with the morphometric approach is employed for a hard-body model system. Our picture is that protein folding and ordered association of proteins are driven by the solvent entropy: At low pressures, the structures almost minimizing the excluded volume (EV) generated for solvent particles are stabilized. Such structures appear to be even more stabilized at high pressures. However, it is experimentally known that the native structure of a protein is unfolded, and ordered aggregates such as amyloid fibrils and actin filaments are dissociated by applying high pressures. This initially puzzling result can also be elucidated in terms of the solvent entropy. A clue to the basic mechanism is in the phenomenon that, when a large hard-sphere solute is immersed in small hard spheres forming the solvent, the small hard spheres are enriched near the solute and this enrichment becomes greater as the pressure increases. We argue that "attraction" is entropically provided between the solute surface and solvent particles, and the attraction becomes higher with rising pressure. Due to this effect, at high pressures, the structures possessing the largest possible solvent-accessible surface area together with sufficiently small EV become more stable in terms of the solvent entropy. To illustrate this concept, we perform an analysis of pressure denaturation of three different proteins. It is shown that only the structures that have the characteristics described above exhibit interesting behavior. They first become more destabilized relative to the native structure as the pressure increases, but beyond a threshold pressure the relative instability begins to decrease and they eventually become more stable than the native structure.

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.

Molecular dynamics results for the radial distribution functions of highly asymmetric hard sphere mixtures

Molecular dynamics (MD) results for the radial distribution functions of mixtures of large and small hard spheres are reported for size ratios whose (large/small) values are 1, 2.5, 5, 7.5, and 10 in the region where the concentration of the large spheres is very small. The MD contact values of these functions are compared with formulae due to Boublik, Mansoori, Carnahan, Starling, Leland, Grundke, and Henderson, Viduna and Smith, Henderson, Trokhymchuk, Woodcock, and Chan, as well as new formulae that are considered here. The new formulae give good agreement for the large–small contact values and reasonably good agreement for the large–large contact values. The Viduna–Smith formula is satisfactory for the small–small contact value and quite reasonable for the small–large contact value. Undoubtedly, further improvements are possible. These results give insight into what may be called the colloidal limit, where the size ratio is exceedingly large while the concentration of the large spheres is exceedingly small, and into the passage to this limit.

A spherical model with directional interactions. I. Static properties

We introduce a simple spherical model whose structural properties are similar to the ones generated by models with directional interactions, by employing a binary mixture of large and small hard spheres, with a square-well attraction acting only between particles of different sizes. The small particles provide the bonds between the large ones. With a proper choice of the interaction parameters, as well as of the relative concentration of the two species, it is possible to control the effective valence. Here we focus on a specific choice of the parameters which favors tetrahedral ordering and study the equilibrium static properties of the system in a large window of densities and temperatures. Upon lowering the temperature we observe a progressive increase in local order, accompanied by the formation of a four-coordinated network of bonds. Three different density regions are observed: At low density the system phase separates into a gas and a liquid phase; at intermediate densities a network of fully bonded particles develops; at high densities--due to the competition between excluded volume and attractive interactions--the system forms a defective network. The very same behavior has been previously observed in numerical studies of nonspherical models for molecular liquids, such as water, and in models of patchy colloidal particles. Different from these models, theoretical treatments devised for spherical potentials, e.g., integral equations and ideal mode coupling theory for the glass transition, can be applied in the present case, opening the way for a deeper understanding of the thermodynamic and dynamic behavior of low valence molecules and particles.

Hypersonic acoustic excitations in binary colloidal crystals: Big versus small hard sphere control

The phononic band structure of two binary colloidal crystals, at hypersonic frequencies, is studied by means of Brillouin light scattering and analyzed in conjunction with corresponding dispersion diagrams of the single colloidal crystals of the constituent particles. Besides the acoustic band of the average medium, the authors' results show the existence of narrow bands originating from resonant multipole modes of the individual particles as well as Bragg-type modes due to the (short-range) periodicity. Strong interaction, leading to the occurrence of hybridization gaps, is observed between the acoustic band and the band of quadrupole modes of the particles that occupy the largest fractional volume of the mixed crystal; the effective radius is either that of the large (in the symmetric NaCl-type crystalline phase) or the small (in the asymmetric NaZn(13)-type crystalline phase) particles. The possibility to reveal a universal behavior of the phononic band structure for different single and binary colloidal crystalline suspensions, by representing in the dispersion diagrams reduced quantities using an appropriate length scale, is discussed.

Ground-state shapes and structures of colloidal domains

In charged colloidal suspensions, the competition between square-well attraction and long-range Yukawa repulsion leads to various stable domains and Wigner supercrystals. Using a continuum model and symmetry arguments, a phase diagram of spheres, cylinders, and lamellae is obtained as a function of two control parameters, the volume fraction and the ratio between the surface tension and repulsion. Above a critical value of the ratio, the microphase cannot be supported by the Yukawa repulsion and macroscopic phase separation occurs. This finding quantitatively explains the lack of pattern formation in simple liquids because of the small hard sphere diameter in comparison with the size of macromolecules. The phase diagram also predicts microphase separation at zero value of the ratio, suggesting the possibility of self-assembly in repulsive systems.

Depletion interaction in a quasi-two-dimensional binary colloid mixture: Monte Carlo simulations

We report the results of extensive calculations of the depletion interaction between the large hard spheres in a quasi-two-dimensional (q2D) binary mixture of large and small hard spheres. Two definitions of the depletion interaction have been examined, and the dependencies of both on the large and small sphere densities, and the confining wall separation have been explored. The results of the simulations show that the depletion interaction is enhanced relative to its magnitude in a three-dimensional binary mixture with the same density, composition and sphere diameter ratio and that it has a complex dependence on the large sphere-large sphere separation. There are qualitative differences between the properties of q2D and mathematical 2D systems that are relevant to the interpretation of experimental data.

Isostructural solid–solid transitions in binary asymmetrical hard sphere system: Based on solvent-mediated potential

Size dependence is imparted onto a modified bridge functional, adopted for a recently proposed semi-analytical hard sphere reference system theory for calculation of solvent-mediated potential (SMP). The SMP for two large hard sphere particles immersed in a small hard sphere solvent bath predicted by the present improved version is in satisfactory agreement with the prediction from a theoretically based fitting formula. Isostructural solid–solid transitions in the binary asymmetrical hard sphere system are investigated based on a single-component macrofluid approximation combined with the improved version. It is found that the isostructural solid–solid transition appears when size asymmetry increases. The limiting asymmetry size ratio is near 1 / 8 . As the size asymmetry increases, critical density for both large and small hard sphere components for the fcc isostructural solid–solid transition increases and decreases, respectively.

Simulation and approximate formulae for the radial distribution functions of highly asymmetric hard sphere mixtures

The Henderson and Chan (HC) formulae for the contact values of the radial distribution functions (RDFs) of a highly asymmetric hard sphere mixture are reconsidered in light of a recent formula of Roth, Evans and Dietrich for the RDF of a pair of exceedingly large spheres at zero concentration in a solvent of small hard spheres. Two modifications of the HC formulae using this result give a large sphere–large sphere contact value that is considerably smaller than that of the original formulation. These new HC results are compared with the molecular dynamics simulations of Lue and Woodcock, for a diameter ratio of 1:10, supplemented by a few new results that are reported here. The new HC formulae are in much better agreement with the MD results than is the popular Boublik–Mansoori–Carnahan–Starling–Leland formula. Also, some simulation results for the RDFs as functions of separation are reported.

Interaction between a macrosphere and a flat wall mediated by a hard-sphere colloidal suspension.

The interaction between the macrosphere and the flat wall immersed in a binary fluid comprising small and large (size ratio around 1:10) hard spheres has been investigated. We find that the presence of the highly size-asymmetric particles qualitatively modifies the induced excluded-volume interaction between the macrosphere and the flat wall compared to that observed in a single-component suspending fluid comprised of only large or only small species. The role in the interaction between a macrosphere and a flat wall played by species of the fine component that usually is approximated by a continuum medium (primitive description of a bidisperse fluid) is emphasized. Particularly, we show that taking into account the smaller component of a bidisperse suspending fluid modifies significantly the macrosphere-wall depletion attraction that is predicted when only large particles are taken into account. The depletion attraction force created by the presence of small fluid particles enhances the adsorption of large particles and their structuring on the wall.

Effective Depletion Potential of Colloidal Spheres

A new semianalytical method, which is a combination of thedensity functional theory with Rosenfeld density functional andthe Ornstein–Zernike equation, is proposed for the calculationof the effective depletion potentials between a pair of bigspheres immersed in a small hard sphere fluid. The calculatedresults are almost identical to the integral equation method withthe Percus–Yevick approximation, and are also in agreement well with theMonte Carlo simulation results.

On pair additivity of the depletion force

The depletion interaction between two and three big hard spheres in a small hard sphere fluid with volumefraction π/10 is calculated both from Monte Carlo simulations and the Asakura–Oosawa approximation.The Monte Carlo results are in good agreement with previous DFT results. The three-bodyinteraction at this volume fraction of small hard spheres is found to be very small, whichindicates that the pair additivity of the depletion force is a good approximation.

Stacking faults in close-packed clusters

Ground state geometries of small hard sphere clusters were studied using two different type of contact interaction, a pair-potential and a many-atom interaction. Monte Carlo method in an FCC lattice with all possible (111) stacking faults was used to obtain the minimum energy geometries for clusters up to 59 atoms. Due to the surface energy, FCC packing is generally favoured as opposite to the HCP structure. However, in most cluster sizes the ground state obtained with the many-atom interaction has one or more stacking faults. The most symmetric geometry is usually not the ground state. Clusters with 59 and 100 atoms were studied due the possibility of a high symmetry cluster with stacking faults in all four directions. The size dependence of the total energy has similarities with that of the average moment of inertia.

A Simple Calculation of Structural and Depletion Forces for Fluids/Suspensions Confined in a Film

By use of the analytic result for the Laplace transform of the radial distribution function for two large hard spheres dispersed in a fluid of the smaller hard spheres, simple equations for the film interaction energy and film disjoining pressure, due to the structural and depletion forces, are derived. The proposed equations satisfy some known exact results and explicitly express the energy and pressure as a function of the film thickness and volume fraction of the small hard spheres. The predicted results for the film energy and film disjoining pressure are compared with existing computer simulation data. The proposed simple analytical expressions can be applied to understand the stability of liquid films containing colloidal particles and the stability of suspensions, foams, and emulsions.

Effects of size polydispersity on depletion interactions

The depletion potentials between two infinite planes, and between two large spheres, due to polydisperse mixtures of non-interacting polymer coils and of interacting small hard spheres, are calculated over a wide range of polydispersities and of polymer or colloid concentrations. The case of non-interacting polymers can be treated analytically within a generalization of the Asakura-Oosawa model. The more difficult case of a polydisperse bath of hard spheres is treated within Rosenfield's ‘fundamental measure’ formulation of density functional theory. Polydispersity is found to have little effect on the depletion attraction, but to strongly reduce the repulsive barrier due to correlation and to damp the subsequent oscillations in the depletion potential.

Depletion Forces near Curved Surfaces

Based on density functional theory the influence of curvature on the depletion potential of a single big hard sphere immersed in a fluid of small hard spheres with packing fraction ${\ensuremath{\eta}}_{s}$ either inside or outside of a hard spherical cavity of radius ${R}_{c}$ is calculated. The relevant features of this potential are analyzed as a function of ${\ensuremath{\eta}}_{s}$ and ${R}_{c}$. There is a very slow convergence towards the flat wall limit ${R}_{c}\ensuremath{\rightarrow}\ensuremath{\infty}$. Our results allow us to discuss the strength of depletion forces acting near membranes both in normal and lateral directions and to make contact with recent experimental results.

The stability limit of the fluid phase of polydisperse sticky spheres

It has been shown by Stell (1991, J. statist. Phys., 63, 1203) that at low temperature mono-disperse sticky spheres collapse to form coexisting close-packed solid and infinitely dilute gases. We show that polydisperse sticky spheres also collapse and calculate the collapse temperature. The polydisperse spheres separate into fractions with narrower polydispersities which can then solidify. This is perhaps the first example of a single-peaked polydisperse mixture phase solidifying and separating. It implies that a mixture of polydisperse large hard spheres with much smaller hard spheres does not show fluid—fluid coexistence.

The stability limit of the fluid phase of polydisperse sticky spheres

It has been shown by Stell (1991, J. statist. Phys., 63, 1203) that at low temperature mono-disperse sticky spheres collapse to form coexisting close-packed solid and infinitely dilute gases. We show that polydisperse sticky spheres also collapse and calculate the collapse temperature. The polydisperse spheres separate into fractions with narrower polydispersities which can then solidify. This is perhaps the first example of a single-peaked polydisperse mixture phase solidifying and separating. It implies that a mixture of polydisperse large hard spheres with much smaller hard spheres does not show fluid—fluid coexistence.