Hard Nonspherical Particles Research Articles

Enhanced two-dimensional nematic order in slit-like pores

The effect of out-of-plane positional freedom is examined on the stability of two-dimensional (2D) nematic order of hard non-spherical particles using the second virial density-functional theory. The particles are allowed to move and rotate freely in the plane of confining walls and can move between the two parallel walls. The wall-to-wall distance (H) is varied between the strictly 2D and the two-layer forming cases, i.e. σ < H < 2σ, where σ is the particle’s shortest length. As expected, we observe that more and more particles are required for the formation of 2D nematics with increasing H when the rod-like particles are hard ellipsoids. Surprisingly, we found that the opposite tendency is observed in the case of hard cylinders, i.e. fewer and fewer particles are needed to stabilize the nematic order with increasing H. This paradox can be understood by projecting the three-dimensional system into a 2D mixture of particles having position-dependent aspect ratios and molecular areas. However, the complex phase behaviour found for plate-like cylindrical particles with increasing H cannot be explained in terms of the same simple geometrical arguments.

Predicting maximally random jammed packing density of non-spherical hard particles via analytical continuation of fluid equation of state.

Dense packings of hard particles are useful models for condensed matters including crystalline and glassy state of solids, simple liquids, granular materials and composites. It is very challenging to devise predictive theories of random packings, due to the intrinsic non-equilibrium and non-local nature of the system. Here, we develop a formalism for accurately predicting the density (i.e., fraction of space covered by the particles) ηMRJ of the maximally random jammed (MRJ) packing state of a wide spectrum of congruent non-spherical hard particles in three-dimensional Euclidean space [Doublestruck R]3, via analytical continuation of the corresponding fluid equation of state (EOS). This formalism is based on the assumption that the fluid branch of the EOS can be analytically extended into the meta-stable region, which leads to a diverging pressure at the jamming point (i.e., the MRJ state). This allows us to estimate ηMRJ as the pole in the EOS, which can be expressed in terms of the virial coefficients encoding intrinsic local n-body packing information of the particles, and depending alone on particle shape. The accuracy of our formalism is verified using the hard sphere system and is subsequently applied to a wide spectrum of non-spherical shapes. The predictions are compared to numerical results whenever possible, and excellent agreements are found.

On the Non-uniqueness of Physical Scattering for Hard Non-spherical Particles

We prove the existence of uncountably-many physical scattering maps for non-spherical hard particles which, when used to construct global-in-time weak solutions of Newton’s equations of motion, conserve the total linear momentum, angular momentum and kinetic energy of the particle system for all time. We prove this result by first exhibiting the non-uniqueness of a classical solution to a constrained Monge–Ampere equation posed on Euclidean space, and then applying the deep existence theory of Ballard for hard particle dynamics. In the final section of the article, we briefly discuss the relevance of our observations to the statistical mechanics of hard particle systems.

Fundamental measure theory for non-spherical hard particles: predicting liquid crystal properties from the particle shape

Density functional theory (DFT) for hard bodies provides a theoretical description of the effect of particle shape on inhomogeneous fluids. We present improvements of the DFT framework fundamental measure theory (FMT) for hard bodies and validate these improvements for hard spherocylinders. To keep the paper self-contained, we first discuss the recent advances in FMT for hard bodies that lead to the introduction of fundamental mixed measure theory (FMMT) in our previous paper (2015 Europhys. Lett. 109 26003). Subsequently, we provide an efficient semi-empirical alternative to FMMT and show that the phase diagram for spherocylinders is described with similar accuracy in both versions of the theory. Finally, we present a semi-empirical modification of FMMT whose predictions for the phase diagram for spherocylinders are in excellent quantitative agreement with computer simulation results.

Neural-network approach to modeling liquid crystals in complex confinement.

Finding the structure of a confined liquid crystal is a difficult task since both the density and order parameter profiles are nonuniform. Starting from a microscopic model and density-functional theory, one has to either (i) solve a nonlinear, integral Euler-Lagrange equation, or (ii) perform a direct multidimensional free energy minimization. The traditional implementations of both approaches are computationally expensive and plagued with convergence problems. Here, as an alternative, we introduce an unsupervised variant of the multilayer perceptron (MLP) artificial neural network for minimizing the free energy of a fluid of hard nonspherical particles confined between planar substrates of variable penetrability. We then test our algorithm by comparing its results for the structure (density-orientation profiles) and equilibrium free energy with those obtained by standard iterative solution of the Euler-Lagrange equations and with Monte Carlo simulation results. Very good agreement is found and the MLP method proves competitively fast, flexible, and refinable. Furthermore, it can be readily generalized to the richer experimental patterned-substrate geometries that are now experimentally realizable but very problematic to conventional theoretical treatments.

Equilibrium Phase Behavior and Maximally Random Jammed State of Truncated Tetrahedra

Numerous recent investigations have been devoted to the determination of the equilibrium phase behavior and packing characteristics of hard nonspherical particles, including ellipsoids, superballs, and polyhedra, to name but just a few shapes. Systems of hard nonspherical particles exhibit a variety of stable phases with different degrees of translational and orientational order, including isotropic liquid, solid crystal, rotator and a variety of liquid crystal phases. In this paper, we employ a Monte Carlo implementation of the adaptive-shrinking-cell (ASC) numerical scheme and free-energy calculations to ascertain with high precision the equilibrium phase behavior of systems of congruent Archimedean truncated tetrahedra over the entire range of possible densities up to the maximal nearly space-filling density. In particular, we find that the system undergoes two first-order phase transitions as the density increases: first a liquid-solid transition and then a solid-solid transition. The isotropic liquid phase coexists with the Conway-Torquato (CT) crystal phase at intermediate densities, verifying the result of a previous qualitative study [ J. Chem. Phys. 2011 , 135 , 151101 ]. The freezing- and melting-point packing fractions for this transition are respectively ϕF = 0.496 ± 0.006 and ϕM = 0.591 ± 0.005. At higher densities, we find that the CT phase undergoes another first-order phase transition to one associated with the densest-known crystal, with coexistence densities in the range ϕ ∈ [0.780 ± 0.002, 0.802 ± 0.003]. We find no evidence for stable rotator (or plastic) or nematic phases. We also generate the maximally random jammed (MRJ) packings of truncated tetrahedra, which may be regarded to be the glassy end state of a rapid compression of the liquid. Specifically, we systematically study the structural characteristics of the MRJ packings, including the centroidal pair correlation function, structure factor and orientational pair correlation function. We find that such MRJ packings are hyperuniform with an average packing fraction of 0.770, which is considerably larger than the corresponding value for identical spheres (≈ 0.64). We conclude with some simple observations concerning what types of phase transitions might be expected in general hard-particle systems based on the particle shape and which would be good glass formers.

Surface tension of isotropic-nematic interfaces: Fundamental measure theory for hard spherocylinders

A fluid constituted of hard spherocylinders is studied using a density functional theory for non-spherical hard particles, which can be written as a function of weighted densities. This is based on an extended deconvolution of the Mayer f-function for arbitrarily shaped convex hard bodies in tensorial weight functions, which depend each only on the shape and orientation of a single particle. In the course of an examination of the isotropic-nematic interface at coexistence the functional is applied to anisotropic and inhomogeneous problems for the first time. We find good qualitative agreement with other theoretical predictions and also with Monte Carlo simulations.

Organizing principles for dense packings of nonspherical hard particles: Not all shapes are created equal

We have recently devised organizing principles to obtain maximally dense packings of the Platonic and Archimedean solids and certain smoothly shaped convex nonspherical particles [Torquato and Jiao, Phys. Rev. E 81, 041310 (2010)]. Here we generalize them in order to guide one to ascertain the densest packings of other convex nonspherical particles as well as concave shapes. Our generalized organizing principles are explicitly stated as four distinct propositions. All of our organizing principles are applied to and tested against the most comprehensive set of both convex and concave particle shapes examined to date, including Catalan solids, prisms, antiprisms, cylinders, dimers of spheres, and various concave polyhedra. We demonstrate that all of the densest known packings associated with this wide spectrum of nonspherical particles are consistent with our propositions. Among other applications, our general organizing principles enable us to construct analytically the densest known packings of certain convex nonspherical particles, including spherocylinders, "lens-shaped" particles, square pyramids, and rhombic pyramids. Moreover, we show how to apply these principles to infer the high-density equilibrium crystalline phases of hard convex and concave particles. We also discuss the unique packing attributes of maximally random jammed packings of nonspherical particles.

A TWO-COMPONENT FLUID MIXTURE OF THE HARD SPHEROCYLINDERS

We study a classical fluid mixture of nonspherical molecules. The components of the mixture are two kinds of the hard spherocylinders with different shape anisotropies L/D. Two different approaches are used to calculate the direct correlation functions (DCF) of this kind of fluids. First, we use a formalism based on the weighted density functional theory (WDFT), introduced by Chamoux and Perera [ J. Chem. Phys.104, 1493 (1996)]. Second, we describe a general approach solving the Percus–Yevick (PY) and the hypernetted chain integral equation numerically for the fluid mixtures of hard nonspherical particles. In the second approach, the pair, total, and DCF of binary molecular fluid mixtures can be calculated simultaneously whereas in the WDFT, the pair and the total correlation functions are calculated indirectly. The obtained correlation functions are compared using these two methods. The pressure of the fluid mixture is also calculated using the Fourier zero components of the DCFs and compared with the Monte Carlo simulation. Finally, the large and small shape anisotropy, are considered and the results are compared with the binary fluid mixture of hard ellipsoids and hard spheres. The results are fairly in agreement.

The influence of shape on the glassy dynamics of hard nonspherical particle fluids. I. Dynamic crossover and elasticity

We extend and apply the center-of-mass version of the microscopic naive mode coupling theory to study the ideal kinetic glass transition of dense fluids and suspensions composed of broad families of one-, two-, and three-dimensional hard nonspherical particles. A kinetic arrest diagram is constructed which indicates a dynamical crossover or onset of activated barrier hopping controlled transport. We find (quasi-) one-dimensional rods and rings form ideal glasses at the lowest volume fractions which decrease strongly with aspect ratio. Two-dimensional disks form ideal glasses at intermediate volume fractions which decrease slowly with the number of particles comprising the planar objects. Compact three-dimensional cluster particles exhibit a subtle nonmonotonic variation of the onset volume fraction that depends on their detailed shape, surface corrugation, and intraparticle interstitial volume. A strong correlation between the ideal kinetic arrest volume fraction and dimensionless compressibility (amplitude of density fluctuations) is predicted. The elastic shear modulus (transient localization length) grows (decreases) exponentially with volume fraction in a manner that becomes stronger as particle dimensionality increases.

The influence of shape on the glassy dynamics of hard nonspherical particle fluids. II. Barriers, relaxation, fragility, kinetic vitrification, and universality

We extend and apply the nonlinear Langevin equation theory of activated barrier hopping dynamics in glassy fluids and colloidal suspensions to study broad families of one-, two-, and three-dimensional hard nonspherical particles. Beyond the ideal kinetic arrest volume fraction, entropic barriers emerge with heights (alpha relaxation times, inverse diffusion constants) that increase nonlinearly (nonexponentially) with volume fraction and in a manner that becomes stronger with particle dimensionality. Partial collapse of the volume fraction dependence of barrier heights and reduced relaxation times of different particle shapes within a fixed dimensionality class are achieved based on a difference volume fraction variable that quantifies the distance from the ideal mode coupling theory dynamic crossover. However, the barrier, alpha relaxation time, and self-diffusion constant results of all shapes are remarkably well collapsed onto a single universal master curve based on a theoretically motivated coupling constant which quantifies the renormalized mean square force on a tagged particle. The latter is determined mainly by the square of the intermolecular site-site contact value of the pair correlation function, thereby providing an explicit microscopic connection between local packing, binary collisions, and slow dynamics. A large variation of the dynamic fragility with particle shape is found with compact cluster particles being the most fragile. A kinetic glass transition map is constructed that is relevant to vitrification of laboratory colloidal suspensions. The possible relevance of the hard particle results for understanding the dynamic fragility of thermal van der Waals liquids is discussed.

Underconstrained jammed packings of nonspherical hard particles: Ellipses and ellipsoids

Continuing on recent computational and experimental work on jammed packings of hard ellipsoids [Donev, Science 303, 990 (2004)] we consider jamming in packings of smooth strictly convex nonspherical hard particles. We explain why an isocounting conjecture, which states that for large disordered jammed packings the average contact number per particle is twice the number of degrees of freedom per particle (Z[over]=2d{f}) , does not apply to nonspherical particles. We develop first- and second-order conditions for jamming and demonstrate that packings of nonspherical particles can be jammed even though they are underconstrained (hypoconstrained, Z[over]<2d{f}). We apply an algorithm using these conditions to computer-generated hypoconstrained ellipsoid and ellipse packings and demonstrate that our algorithm does produce jammed packings, even close to the sphere point. We also consider packings that are nearly jammed and draw connections to packings of deformable (but stiff) particles. Finally, we consider the jamming conditions for nearly spherical particles and explain quantitatively the behavior we observe in the vicinity of the sphere point.

Calculating the free energy of nearly jammed hard-particle packings using molecular dynamics

We present a new event-driven molecular dynamics (MD) algorithm for measuring the free energy of nearly jammed packings of spherical and non-spherical hard particles. This Bounding Cell Molecular Dynamics (BCMD) algorithm exactly calculates the free-energy of a single-occupancy cell (SOC) model in which each particle is restricted to a neighborhood of its initial position using a hard-wall bounding cell. Our MD algorithm generalizes previous ones in the literature by enabling us to study non-spherical particles as well as to measure the free-energy change during continuous irreversible transformations. Moreover, we make connections to the well-studied problem of computing the volume of convex bodies in high dimensions using random walks. We test and verify the numerical accuracy of the method by comparing against rigorous asymptotic results for the free energy of jammed and isostatic disordered packings of both hard spheres and ellipsoids, for which the free energy can be calculated directly as the volume of a high-dimensional simplex. We also compare our results to previously published Monte Carlo results for hard-sphere crystals near melting and jamming and find excellent agreement. We have successfully used the BCMD algorithm to determine the configurational and free-volume contributions to the free energy of glassy states of binary hard disks [A. Donev, F.H. Stillinger, S. Torquato, Do binary hard disks exhibit an ideal glass transition? Phys. Rev. Lett. 96 (22) (2006) 225502]. The algorithm can also be used to determine phases with locally- or globally-minimal free energy, to calculate the free-energy cost of point and extended crystal defects, or to calculate the elastic moduli of glassy or crystalline solids, among other potential applications.

Detailed examination of the calculation of the pressure in simulations of systems with discontinuous interactions from the mechanical and thermodynamic perspectives

We consider some fundamental aspects of the calculation of the pressure from simulations of systems characterized by discontinuous interactions. The usual approach to the computation of the pressure in such systems is based on the mechanical (or virial) expression, which in turn can be expressed in terms of the value of the radial distribution function at contact or the number of overlaps that would occur under a virtual compression of the system. A less widespread, and formally equivalent, approach invokes the thermodynamic definition of the pressure and involves the estimate of the change in free energy associated with a small volume perturbation. There are, however, important differences in the way the virial and thermodynamic evaluations of the pressure are implemented, and, as a consequence, in the corresponding errors associated with each technique. The bulk pressure is determined with the virial and the thermodynamic routes for the hard-sphere system in the fluid and solid states. Good agreement is found between the two approaches, though the virial route is generally found to provide a greater accuracy. This is also borne out from simulations of hard non-spherical particles characterized by the hard Gaussian overlap potential in isotropic and anisotropic (nematic) fluid states. We show that the tensorial components of the pressure can be determined by a simple extension of the approaches developed for the bulk pressure. The diagonal components evaluated in this way for the hard-sphere solid are found to be equivalent and equal to the bulk pressure within the accuracy of the calculation. The most stringent test of the virial and thermodynamic approaches is the computation of the normal and tangential components (and hence the interfacial tension) of a hard-sphere fluid confined between two structureless hard walls.

Entropic forces in dilute colloidal systems

Depletion forces can be accounted for by a contraction of the description in the framework of the integral equations theory of simple liquids. This approach includes, in a natural way, the effects of the concentration on the depletion forces, as well as energetic contributions. In this paper we systematically study this approach in a large variety of dilute colloidal systems composed of spherical and nonspherical hard particles, in two and in three dimensions, in the bulk and in front of a hard wall with a relief pattern. We show by this way the form in which concentration and geometry determine the entropic interaction between colloidal particles. The accuracy of our results is corroborated by comparison with computer simulations.

Entropy-driven demixing in spherocylinder binary mixtures.

The stability of binary fluid mixtures, with respect to a demixing transition, is examined within the framework of the geometrical approximation of the direct correlation for hard nonspherical particles. In this theory, the direct correlation function is essentially written in terms of the geometrical properties of the individual molecules, and those of the overlap region between two different molecules, taken at fixed separation and orientations. Within the present theory, the demixing spinodal line in the (rho(1),rho(2)) concentration plane is obtained analytically, and shown to be a quadratic function of the total packing fraction and the compositions. The theory is applied herein to binary mixtures of hard spherocylinders in the isotropic phase. Isotropic fluid-fluid demixing can be predicted for a large variety of sizes and aspect ratios, and the necessary condition for entropic demixing is a sufficiently large thickness difference between the two particles that belong to each of the fluids in the mixture. As the theory reduces exactly to the Percus-Yevick approximation for a hard sphere mixture, accordingly it will not predict fluid-fluid demixing for this particular case. Demixing is also forbidden in two other cases; for a mixture of spherocylinders and small spheres, and for mixtures of equally thin spherocylinders. The influence and competition of an ordering instability on the demixing is also examined. The ordering of a fluid will always be displaced toward higher packing fractions by the addition of a nonordering fluid, and in some cases the entropic demixing can dominate the entire fluid range. Although the present theory merges exactly with the correct Onsager limit, it is shown that, for intermediate cases, the results can be significantly different from predictions of Onsager type approaches. These discrepancies are analyzed in particular for the needle plus spherocylinder mixture. Finally, in view of the nature of the theory, it is conjectured that the predicted demixing densities values are rather upper bounds to what should be expected.

External field induced tricritical phenomenon in the isotropicnematic phase transition of hard non-spherical particle systems

The effect of static external field is studied on the isotropic–nematic phase transition of a system of hard non-spherical particles (rods or platelets) with negative anisotropic polarizability (susceptabilities). On the basis of Onsager theory, the phase coexistence curve is calculated numerically without any approximation. It is found that a weakly ordered nematic phase (uniaxial planar) is in coexistence with a highly ordered biaxial nematic phase which ends at a tricritical point. In the limit of infinite field strength, the orientations of the particles are confined in a plane perpendicular to the field and continuous isotropic–nematic phase transition takes place.

RESEARCH NOTE Square-well fluid based decoupling approximation for system of hard non-spherical particles with spherical square-wells

RESEARCH NOTE Square-well fluid based decoupling approximation for system of hard non-spherical particles with spherical square-wells

Simplified equation of state for non-spherical hard particles: an optimized shape factor approach

Based on an analysis using the Carnahan–Müller shape factor approach, mathematically simple and physically sound equations of state for fluids of non-spherical hard particles are proposed. The analysis has been performed using two versions of the statistical associating fluid theory for chains of hard-spheres, the hard convex body equation of state of Boublík and Nezbeda (Collect. Czech. Chem. Commun., 1986, 51, 2301) and the Flory-dimer model of Honnell and Hall (J. Chem. Phys., 1989, 90, 1841). The combination of the shape factor concept with the recently developed generalized biquadratic equation of state BQ75 gives access to simple biquadratic equations of state for different molecular shapes. With the combination of these two methods it is possible to simplify macroscopic models for molecular fluids. Furthermore the generalized method proposed here can be used to obtain simple equations of state from data generated by molecular simulations.

Orientational order in dipolar fluids consisting of nonspherical hard particles

We investigate fluids of dipolar hard particles by a certain variant of density-functional theory. The proper treatment of the long range of the dipolar interactions yields a contribution to the free energy which favors ferromagnetic order. This corrects previous theoretical analyses. We determine phase diagrams for dipolar ellipsoids and spherocylinders as a function of the aspect ratio of the particles and their dipole moment. In the nonpolar limit the results for the phase boundary between the isotropic and nematic phase agree well with simulation data. Adding a longitudinal dipole moment favors the nematic phase. For oblate or slightly elongated particles we find a ferromagnetic liquid phase, which has also been detected in computer simulations of fluids consisting of spherical dipolar particles. The detailed structure of the phase diagram and its evolution upon changing the aspect ratio are discussed in detail.