By extending the nonequilibrium potential refinement algorithm and lattice switch method to the semigrand ensemble, the semigrand potentials of the fcc and hcp structures of polydisperse hard-sphere crystals are calculated with the bias sampling scheme. The result shows that the fcc structure is more stable than the hcp structure for polydisperse hard-sphere crystals below the terminal polydispersity.
High temperature approximation (HTA) is used to describe the phase behavior of polydisperse multi-Yukawa hard-sphere chain fluid mixtures with chain length polydispersity. It is demonstrated that in the frames of the HTA the model belongs to the class of ``truncatable free energy models'', i.e. the models with thermodynamical properties (Helmholtz free energy, chemical potential and pressure) defined by the finite number of generalized moments. Using this property we were able to calculate the complete phase diagram (i.e., cloud and shadow curves as well as binodals) and chain length distribut…
The effect of polydispersity on the freezing transition of hard spheres is examined within a moment description. At low polydispersities a single fluid-to-crystal transition is recovered. With increasing polydispersity we find a density above which the crystal melts back into an amorphous phase. The range of densities over which the crystalline phase is stable shrinks with increasing polydispersity until, at a certain level of polydispersity, the crystal disappears completely from the equilibrium phase diagram. The two transitions converge to a single point which we identify as the polydispers…
We report a Monte Carlo simulation study of the properties of highly asymmetric binary hard sphere mixtures. This system is treated within an effective fluid approximation in which the large particles interact through a depletion potential (R. Roth {\em et al}, Phys. Rev. E{\bf 62} 5360 (2000)) designed to capture the effects of a virtual sea of small particles. We generalize this depletion potential to include the effects of explicit size dispersity in the large particles and consider the case in which the particle diameters are distributed according to a Schulz form having degree of polydisp…
Particle size polydispersity can help to inhibit crystallization of the hard-sphere fluid into close-packed structures at high packing fractions and thus is often employed to create model glass-forming systems. Nonetheless, it is known that hard-sphere mixtures with modest polydispersity still have ordered ground states. Here, we demonstrate by computer simulation that hard-sphere mixtures with increased polydispersity fractionate on the basis of particle size, and a bimodal subpopulation favors formation of topologically close-packed C14 and C15 Laves phases in coexistence with a disordered p…
The phase diagram of a polydisperse hard sphere system is examined by numerical minimization of a discretized form of the Ramakrishnan-Yussouff free energy functional. Crystalline and glassy local minima of the free energy are located and the phase diagram in the density-polydispersity plane is mapped out by comparing the free energies of different local minima. The crystalline phase disappears and the glass becomes the equilibrium phase beyond a "terminal" value of the polydispersity. A crystal to glass transition is also observed as the density is increased at high polydispersity. The phase …
We study the theoretical phase behavior of an asymmetric binary mixture of hard spheres, of which the smaller component is monodisperse and the larger component is polydisperse. The interactions are modelled in terms of the second virial coefficient and are assumed to be additive hard sphere interactions. The polydisperse component is subdivided into sub-components and has an average size ten times the size of the monodisperse component. We give the set of equations that defines the phase diagram for mixtures with more than two components in a solvent. We calculate the theoretical liquid-liqui…
We simulate the dynamics of polydisperse hard spheres at high packing fractions, $ϕ$, with an experimentally-realistic particle size distribution (PSD) and other commonly-used PSDs such as gaussian or top hat. We find that the mode of kinetic arrest depends on the PSD's shape and not only on its variance. For the experimentally-realistic PSD, the largest particles undergo an ideal glass transition at $ϕ\sim 0.588$ while the smallest particles remain mobile. Such species-specific localisation was previously observed only in asymmetric binary mixtures. Our findings suggest that the recent observ…
We study the theoretical phase behavior of an asymmetric binary mixture of hard spheres, of which the smaller component is monodisperse and the larger component is polydisperse. The interactions are modeled in terms of the second virial coefficient and can be additive hard sphere (HS) or non-additive hard sphere (NAHS) interactions. The polydisperse component is subdivided into two sub-components and has an average size ten or three times the size of the monodisperse component. We give the set of equations that defines the phase diagram for mixtures with more than two components in a solvent. …
The phase diagram of a polydisperse mixture of uniaxial rod-like and plate-like hard parallelepipeds is determined for aspect ratios $κ=5$ and 15. All particles have equal volume and polydispersity is introduced in a highly symmetric way. The corresponding binary mixture is known to have a biaxial phase for $κ=15$, but to be unstable against demixing into two uniaxial nematics for $κ=5$. We find that the phase diagram for $κ=15$ is qualitatively similar to that of the binary mixture, regardless the amount of polydispersity, while for $κ=5$ a sufficient amount of polydispersity stabilizes the b…
We introduce a model for a fluid of polydisperse rounded hard rectangles where the length and width of the rectangular core are fixed, while the roundness is taken into account by the convex envelope of a disk displaced along the perimeter of the core. The diameter of the disk has a continuous polydispersity described by a Schultz distribution function. We implemented the scaled particle theory for this model with the aim to studying: (i) the effect of roundness on the phase behavior of the one-component hard-rectangle fluid, and (ii) how polydispersity affects phase transitions between isotro…
We propose an extension of the second-order Barker-Henderson perturbation theory for polydisperse hard-sphere multi-Morse mixture. To verify the accuracy of the theory, we compare its predictions for the limiting case of monodisperse system, with predictions of the very accurate reference hypernetted chain approximation. The theory is used to describe the liquid-gas phase behavior of the mixture with different type and different degree of polydispersity. In addition to the regular liquid-gas critical point, we observe the appearance of the second critical point induced by polydispersity. With …
We discuss phase coexistence of polydisperse colloidal suspensions in the presence of adhesion forces. The combined effect of polydispersity and Baxter's sticky-hard-sphere (SHS) potential, describing hard spheres interacting via strong and very short-ranged attractive forces, give rise, within the Percus-Yevick (PY) approximation, to a system of coupled quadratic equations which, in general, cannot be solved either analytically or numerically. We review and compare two recent alternative proposals, which we have attempted to by-pass this difficulty. In the first one, truncating the density ex…
A theoretical scheme for the calculation of the full phase diagram (including cloud and shadow curves, binodals and distribution functions of the coexisting phases) for colloid-polymer mixtures with polymer chain length polydispersity and hard-sphere colloidal and polymeric monomer sizes polydispersity is proposed. The scheme combines thermodynamic perturbation theory for associating fluids and recently developed method used to determine the phase diagram of polydisperse spherical shape colloidal fluids (L.Bellier-Castella {\it et al.}, {\it J.Chem.Phys.} {\bf 113}, 8337(2000)). By way of illu…
A general Monte Carlo simulation method of calculating the elastic constants of polydisperse hard-sphere colloidal crystal was developed. The elastic constants of a size polydisperse hard sphere fcc crystal is calculated. The pressure and three elastic constants(C11, C12 and C44) increase significantly with the polydispersity. It was also found from extrapolation that there is a mechanical terminal polydispersity above which a fcc crystal will be mechanically unstable.
We develop a cavity-based method which allows to extract thermodynamic properties from position information in hard-sphere/disk systems. So far, there are 'available-volume' and 'free-volume' methods. We add a third one, which we call 'available-volume-after-takeout', and which is shown to be mathematically equivalent to the others. In applications, where data sets are finite, all three methods show limitations, and they do this in different parameter ranges. We illustrate the principal equivalence and the limitations on data from molecular dynamics -- In particular, we test robustness against…
Hard spheres with a polydispersity above approximately 8% are shown to crystallise into two phase-separated solid phases. A polydispersity above 8% is too large to be tolerated by a single solid phase but phase separation produces two fractions with polydispersities sufficiently narrow to allow them to crystallise. It may not be possible to observe this in experiment due to the intervention of a glass transition.
We consider the mass heterogeneity in a gas of polydisperse hard particles as a key to optimizing a dynamical property: the kinetic relaxation rate. Using the framework of the Boltzmann equation, we study the long time approach of a perturbed velocity distribution toward the equilibrium Maxwellian solution. We work out the cases of discrete as well as continuous distributions of masses, as found in dilute fluids of mesoscopic particles such as granular matter and colloids. On the basis of analytical and numerical evidence, we formulate a dynamical equipartition principle that leads to the resu…
The liquid-gas phase diagram for polydisperse dipolar hard-sphere fluid with polydispersity in the hard-sphere size and dipolar moment is calculated using extension of the recently proposed thermodynamic perturbation theory for central force (TPT-CF) associating potential. To establish the connection with the phase behavior of ferrocolloidal dispersions it is assumed that the dipole moment is proportional to the cube of the hard-sphere diameter. We present and discuss the full phase diagram, which includes cloud and shadow curves, binodals and distribution functions of the coexisting daughter …
We compute the equilibrium concentration of stacking faults and point defects in polydisperse hard-sphere crystals. We find that, while the concentration of stacking faults remains similar to that of monodisperse hard sphere crystals, the concentration of vacancies decreases by about a factor two. Most strikingly, the concentration of interstitials in the maximally polydisperse crystal may be some six orders of magnitude larger than in a monodisperse crystal. We show that this dramatic increase in interstitial concentration is due to the increased probability of finding small particles and tha…
The melting of a polydisperse hard disk system is investigated by Monte Carlo simulations in the semigrand canonical ensemble. This is done in the context of possible continuous melting by a dislocation unbinding mechanism, as an extension of the 2D hard disk melting problem. We find that while there is pronounced fractionation in polydispersity, the apparent density-polydispersity gap does not increase in width, contrary to 3D polydisperse hard spheres. The point where the Young's modulus is low enough for the dislocation unbinding to occur moves with the apparent melting point, but stays wit…
We study theoretically the equilibrium phase behaviour of a mixture of polydisperse hard-sphere colloids and monodisperse polymers, modelled using the Asakura-Oosawa model within the free volume approximation of Lekkerkerker et al. We compute full phase diagrams in the plane of colloid and polymer volume fractions, using the moment free energy method. The intricate features of phase separation in pure polydisperse colloids combine with the appearance of polymer-induced gas-liquid coexistence to give a rich variety of phase diagram topologies as the polymer-colloid size ratio and the colloid po…
We study the effects of size polydispersity on the gas-liquid phase behaviour of mixtures of sticky hard spheres. To achieve this, the system of coupled quadratic equations for the contact values of the partial cavity functions of the Percus-Yevick solution is solved within a perturbation expansion in the polydispersity, i.e. the normalized width of the size distribution. This allows us to make predictions for various thermodynamic quantities which can be tested against numerical simulations and experiments. In particular, we determine the leading-order effects of size polydispersity on the cl…
Phase behavior of the Yukawa hard-sphere polydisperse mixture with high degree of polydispersity is studied using high temperature approximation (HTA) and mean spherical approximation (MSA). We have extended and applied the scheme developed to calculate the phase diagrams of polydisperse mixtures described by the truncatable free energy models, i.e., the models with Helmholtz free energy defined by the finite number of the moments of the species distribution function. At high degree of polydispersity, several new features in the topology of the two-phase diagram have been observed: the cloud a…
Motivated by old experiments on colloidal suspensions, we report molecular dynamics simulations of assemblies of hard spheres, addressing crystallization and glass formation. The simulations cover wide ranges of polydispersity s (standard deviation of the particle size distribution divided by its mean) and particle concentration. No crystallization is observed for s > 0.07. For 0.02 < s < 0.07, we find that increasing the polydispersity at a given concentration slows down crystal nucleation. The main effect here is that polydispersity reduces the supersaturation since it tends to stabilise the…
We apply a recently proposed density functional for mixtures of parallel hard cylinders, based on Rosenfeld's fundamental measure theory, to study the effect of length-polydispersity on the relative stability between the smectic and columnar liquid crystal phases.To this purpose we derive from this functional an expression for the direct correlation function and use it to perform a bifurcation analysis. We compare the results with those obtained with a second and a third virial approximation of this function. All three approximations lead to the same conclusion: there is a terminal polydispers…
The solid-solid coexistence of a polydisperse hard sphere system is studied by using the Monte Carlo simulation. The results show that for large enough polydispersity the solid-solid coexistence state is more stable than the single-phase solid. The two coexisting solids have different composition distributions but the same crystal structure. Moreover, there is evidence that the solid-solid transition terminates in a critical point as in the case of the fluid-fluid transition.
Most of the liquid-state theories, including glass-transition theories, are constructed on the basis of two-body density correlations. However, we have recently shown that many-body correlations, in particular bond orientational correlations, play a key role in both the glass transition and the crystallization transition. Here we show, with numerical simulations of supercooled polydisperse hard spheres systems, that the lengthscale associated with any two-point spatial correlation function does not increase toward the glass transition. A growing lengthscale is instead revealed by considering m…
The effect of polydispersity on the phase behaviour of hard spheres is examined using a moment projection method. It is found that the Boublik-Mansoori-Carnahan-Starling-Leland equation of state shows a spinodal instability for a bimodal distribution if the large spheres are sufficiently polydisperse, and if there is sufficient disparity in mean size between the small and large spheres. The spinodal instability direction points to the appearance of a very dense phase of large spheres.
We describe a simulation method for the accurate study of the equilibrium freezing properties of polydisperse fluids under the experimentally relevant condition of fixed polydispersity. The approach is based on the phase switch Monte Carlo method of Wilding and Bruce [Phys. Rev. Lett. {\bf 85}, 5138 (2000)]. This we have generalized to deal with particle size polydispersity by incorporating updates which alter the diameter $σ$ of a particle, under the control of a distribution of chemical potential differences $\tildeμ(σ)$. Within the resulting isobaric semi-grand canonical ensemble, we detail…
I study the effect of length polydispersity in the surface phase diagram of hard rods interacting with a hard wall. The properly extended interface Gibbs-Duhem equation for a polydisperse system allows us to predict the behaviour of the surface tension as a function of the bulk density at the the wall-isotropic interface. Two groups of qualitative different bulk and surface phase diagrams are calculated from two families of parametrized length distribution functions $p(l)$. This parameterization controls the law of decay at large $l$. I also study the segregation due to polydispersity at the i…
We use polydispersity in size as a control parameter to explore certain aspects of melting and freezing transitions in a system of Lennard-Jones spheres. Both analytical theory and computer simulations are employed to establish a potentially interesting relationship between observed terminal polydispersity in Lennard-Jones polydisperse spheres and prediction of the same in the integral equation based theoretical analysis of liquid-solid transition. As we increase polydispersity, solid becomes inherently unstable because of the strain built up due to the size disparity. This aspect is studied h…
We analyse the effect of a continuous spread of particle lengths on the phase behavior of rodlike particles, using the Onsager theory of hard rods. Our aim is to establish whether ``unusual'' effects such as isotropic-nematic-nematic (I-N-N) phase separation can occur even for length distributions with a single peak. We focus on the onset of I-N coexistence. For a log-normal distribution we find that a finite upper cutoff on rod lengths is required to make this problem well-posed. The cloud curve, which tracks the density at the onset of I-N coexistence as a function of the width of the length…
The structural properties of polydisperse hard spheres in the presence of a hard wall are investigated via Monte Carlo simulation and density functional theory (DFT). Attention is focussed on the local density distribution $ρ(σ,z)$, measuring the number density of particles of diameter $σ$ at a distance $z$ from the wall. The form of $ρ(σ,z)$ is obtained for bulk volume fractions $η_b=0.2$ and $η_b=0.4$ for two choices of the bulk parent distribution: a top-hat form, which we study for degrees of polydispersity $δ=11.5%$ and $δ=40.4%$, and a truncated Schulz form having $δ=40.7%$. Excellent ov…
We used the Density Functional formalism, in particular the Scaled Particle Theory, applied to a length-polydisperse hard-rectangular fluid to study its phase behavior as a function of the mean particle aspect ratio ($κ_0$) and polydispersity ($Δ_0$). The numerical solutions of the coexistence equations were calculated by transforming the original problem with infinite degrees of freedoms to a finite set of equations for the amplitudes of the Fourier expansion of the moments of the density profiles. We divided the study into two parts: The first one is devoted to the calculation of the phase d…
The structure of polydisperse hard sphere fluids, in the presence of a wall, is studied by the Rosenfeld density functional theory. Within this approach, the local excess free energy depends on only four combinations of the full set of density fields. The case of continuous polydispersity thereby becomes tractable. We predict, generically, an oscillatory size segregation close to the wall, and connect this, by a perturbation theory for narrow distributions, with the reversible work for changing the size of one particle in a monodisperse reference fluid.
A polydisperse granular gas made of inelastic and rough hard disks is considered. Focus is laid on the kinetic-theory derivation of the partial energy production rates and the total cooling rate as functions of the partial densities and temperatures (both translational and rotational) and of the parameters of the mixture (masses, diameters, moments of inertia, and mutual coefficients of normal and tangential restitution). The results are applied to the homogeneous cooling state of the system and the associated nonequipartition of energy among the different components and degrees of freedom. It…
The equilibrium phase behaviour of hard spheres with size polydispersity is studied theoretically. We solve numerically the exact phase equilibrium equations that result from accurate free energy expressions for the fluid and solid phases, while accounting fully for size fractionation between coexisting phases. Fluids up to the largest polydispersities that we can study (around 14%) can phase separate by splitting off a solid with a much narrower size distribution. This shows that experimentally observed terminal polydispersities above which phase separation no longer occurs must be due to non…
The phase diagram of soft spheres with size dispersion has been studied by means of an optimized Monte Carlo algorithm which allows to equilibrate below the kinetic glass transition for all sizes distribution. The system ubiquitously undergoes a first order freezing transition. While for small size dispersion the frozen phase has a crystalline structure, large density inhomogeneities appear in the highly disperse systems. Studying the interplay between the equilibrium phase diagram and the kinetic glass transition, we argue that the experimentally found terminal polydispersity of colloids is a…
We discuss theoretically, using density-functional theory, the phase stability of nematic and smectic ordering in a suspension of platelets of the same thickness but with a high polydispersity in diameter, and study the influence of polydispersity on this stability. The platelets are assumed to interact like hard objects, but additional soft attractive and repulsive interactions, meant to represent the effect of depletion interactions are also considered. The aspect (diameter to thickness) ratio is taken to be very high, in order to model solutions of mineral platelets recently explored experi…
Hyperuniformity characterizes a state of matter for which density fluctuations diminish towards zero at the largest length scales. However, the task of determining whether or not an experimental system is hyperuniform is experimentally challenging due to finite-resolution, noise and sample-size effects that influence characterization measurements. Here we explore these issues, employing video optical microscopy to study hyperuniformity phenomena in disordered two-dimensional jammed packings of soft spheres. Using a combination of experiment and simulation we characterize the detrimental effect…
The fluid - crystal equilibria of polydisperse mixtures of hard spheres have been studied by computer simulation of the solid phase and using an accurate equation of state for the fluid. A new scheme has been developed to evaluate the composition of crystalline phases in equilibrium with a given polydisperse fluid. Some common assumptions in theoretical approaches and their results are discussed on the light of the simulation results. Finally, no evidence of the existence of a terminal polydispersity in the fluid phase is found for polydisperse hard spheres, the disagreement of this finding wi…
A model polydisperse fluid represents many real fluids such as colloidal suspensions and polymer solutions. In this study, considering a concentrated size-polydisperse hard sphere fluid with size derived from two different distribution functions, namely, uniform and Gaussian and explore the effect of polydispersity and mass scaling on the transport properties in general. A simple analytical solution based on the Boltzmann transport equation is also presented (together with the solution using Chapman-Enskog (CE) method) using which various transport coefficients are obtained. The central idea o…
Formation of cylindrical shells via sphere packing from fluidized beds
Faculdade de Engenharia Mecânica, UNICAMP-Universidade Estadual de Campinas（坎皮纳斯州立大学机械工程学院）；Laboratoire PIMM, CNRS, Arts et Métiers Institute of Technology, Cnam（国立工艺学院PIMM实验室）；Faculty of Physics, University of Duisburg-Essen（杜伊斯堡-埃森大学物理系）
We study the equilibrium behaviour of a mixture of monodisperse hard sphere colloids and polydisperse non-adsorbing polymers at their $θ$-point, using the Asakura-Oosawa model treated within the free-volume approximation. Our focus is the experimentally relevant scenario where the distribution of polymer chain lengths across the system is fixed. Phase diagrams are calculated using the moment free energy method, and we show that the mean polymer size $ξ_{\rm c}$ at which gas-liquid phase separation first occurs decreases with increasing polymer polydispersity $δ$. Correspondingly, at fixed mean…
Both polydispersity and the presence of a gravitational field are inherent to essentially any colloidal experiment. While several theoretical works have focused on the effect of polydispersity on the bulk phase behavior of a colloidal system, little is known about the effect of a gravitational field on a polydisperse colloidal suspension. We extend here sedimentation path theory to study sedimentation-diffusion-equilibrium of a mass-polydisperse colloidal system: the particles possess different buoyant masses but they are otherwise identical. The model helps to understand the interplay between…
The isotropic-nematic phase behavior of length polydisperse hard rods with arbitrary length distributions is calculated. Within a numerical treatment of the polydisperse Onsager model using the Gaussian trial function Ansatz we determine the onset of isotropic-nematic phase separation, coming from a dilute isotropic phase and a dense nematic phase. We focus on parent systems whose lengths can be described by either a Schulz or a `fat-tailed' log-normal distribution with appropriate lower and upper cutoff lengths. In both cases, very strong fractionation effects are observed for parent polydisp…
In the Letter [PRL 102, 085703 (2009)] Brambilla, et al. claimed to observe activated dynamics in colloidal hard spheres above the critical packing fraction of mode coupling theory (MCT). By performing microscopic MCT calculations, we show that polydispersity in their system shifts the critical packing fraction above the value determined by van Megen et al. for less polydisperse samples, and that the data agree with theory except for, possibly, the highest packing fraction.
Thermoresponsive poly(N-isopropylacrylamide) (PNIPAM) particles of a nearly constant swelling ratio and with polydispersity indices (PDIs) varying over a wide range (7.4% - 48.9%) are synthesized to study the effects of polydispersity on the dynamics of suspensions of soft PNIPAM colloidal particles. The PNIPAM particles are characterized using dynamic light scattering (DLS) and scanning electron microscopy (SEM). The zero shear viscosity ($η_{0}$) data of these colloidal suspensions, estimated from rheometric experiments as a function of the effective volume fraction $ϕ_{eff}$ of the suspensi…
We recently found that crystallization of monodisperse hard spheres from the bulk fluid faces a much higher free energy barrier in four than in three dimensions at equivalent supersaturation, due to the increased geometrical frustration between the simplex-based fluid order and the crystal [J.A. van Meel, D. Frenkel, and P. Charbonneau, Phys. Rev. E 79, 030201(R) (2009)]. Here, we analyze the microscopic contributions to the fluid-crystal interfacial free energy to understand how the barrier to crystallization changes with dimension. We find the barrier to grow with dimension and we identify t…
We study the phase behaviour of the Zwanzig model of suspensions of hard rods, allowing for polydispersity in the lengths of the rods. In spite of the simplified nature of the model (rods are restricted to lie along one of three orthogonal axes), the results agree qualitatively with experimental observations: the coexistence region broadens significantly as the polydispersity increases, and strong fractionation occurs, with long rods found preferentially in the nematic phase. These conclusions are obtained from an analysis of the exact phase equilibrium equations. In the second part of the pap…
We show that an analogy between crowding in fluid and jammed phases of hard spheres captures the density dependence of the kissing number for a family of numerically generated jammed states. We extend this analogy to jams of mixtures of hard spheres in $d=3$ dimensions, and thus obtain an estimate of the random close packing (RCP) volume fraction, $ϕ_{\textrm{RCP}}$, as a function of size polydispersity. We first consider mixtures of particle sizes with discrete distributions. For binary systems, we show agreement between our predictions and simulations, using both our own and results reported…
We extend previous analysis of data for the melt-nucleus interfacial free energy, $γ$, gained from optical experiments on suspensions of charged colloidal spheres, which crystallize with body centred cubic (bcc) crystal structures. Compiling data from five pure species with different polydispersities and one binary mixture, we find the equilibrium melt-crystal interfacial energy to be considerably larger than the hard sphere reference value. Both this quantity and the entropy of freezing decrease with increasing polydispersity. Moreover, we give a first experimental determination of the Turnbu…
We use a Fundamental-Measure density functional for hard board-like polydisperse particles, in the restricted-orientation approximation, to explain the phase behaviour of platelet colloidal suspensions studied in recent experiments. In particular, we focus our attention on the behavior of the total packing fraction of the mixture, $η$, in the region of two-phase isotropic-nematic coexistence as a function of mean aspect ratio, polydispersity and fraction of total volume $γ$ occupied by the nematic phase. In our model, platelets are polydisperse in the square section, of side length $σ$, but ha…
