Ellipsoid contact potential: Theory and relation to overlap potentials

Effect of anisotropy on the formation of active particle films

Active colloids belong to a class of nonequilibrium systems where energy uptake, conversion, and dissipation occur at the level of individual colloidal particles, which can lead to particles' self-propelled motion and surprising collective behavior. Examples include coexistence of vapor- and liquid-like steady states for active particles with repulsive interactions only, phenomena known as motility-induced phase transitions. Similarly to motile unicellular organisms, active colloids tend to accumulate at confining surfaces forming dense adsorbed films. In this work, we study the structure and dynamics of aggregates of self-propelled particles near confining solid surfaces, focusing on the effects of the particle anisotropic interactions. We performed Langevin dynamics simulations of two complementary models for active particles: ellipsoidal particles interacting through the Gay-Berne potential and rodlike particles composed of several repulsive Lennard-Jones beads. We observe a nonmonotonic behavior of the structure of clusters formed along the confining surface as a function of the particle aspect ratio, with a film spreading when particles are near-spherical, compact clusters with hedgehog-like particle orientation for more elongated active particles, and a complex dynamical behavior for an intermediate aspect ratio. The stabilization time of cluster formation along the confining surface also displays a nonmonotonic dependence on the aspect ratio, with a local minimum at intermediate values. Additionally, we demonstrate that the hedgehog-like aggregates formed by Gay-Berne ellipsoids exhibit higher structural stability as compared to the ones formed by purely repulsive active rods, which are stable due to the particle activity only.

Measure of overlap between two arbitrary ellipses on a sphere

Various packing problems and simulations of hard and soft interacting particles, such as microscopic models of nematic liquid crystals, reduce to calculations of intersections and pair interactions between ellipsoids. When constrained to a spherical surface, curvature and compactness lead to non-trivial behaviour that finds uses in physics, computer science and geometry. A well-known idealized isotropic example is the Tammes problem of finding optimal non-intersecting packings of equal hard disks. The anisotropic case of elliptic particles remains, on the other hand, comparatively unexplored. We develop an algorithm to detect collisions between ellipses constrained to the two-dimensional surface of a sphere based on a solution of an eigenvalue problem. We investigate and discuss topologically distinct ways two ellipses may touch or intersect on a sphere, and define a contact function that can be used for construction of short- and long-range pair potentials.

A mathematical framework for the analysis and comparison of contact detection methods for ellipses and ellipsoids

The purpose of this research is to provide a framework for the analysis and comparison of contact detection algorithms for pairs of ellipses and ellipsoids. This work focuses primarily on the category of algorithms that are the most computationally efficient and can produce estimates of the separation and the penetration distance between ellipses and ellipsoids. Specifically, only analytic representations of the ellipses and ellipsoids are considered and contact detection for moving pairs of ellipsoids is not treated. The first contribution is a mathematical framework for the study of these algorithms, most notably with existence and uniqueness proofs for classes of contact detection algorithms, formal descriptions of the asymptotics of pairs of ellipses in close contact (or overlap), and a global analysis of constraints on the normals. The framework highlights the key role played by the different definitions of contact found in the literature, independent of the numerical strategies deployed to estimate the separation/penetration distance. Specifically, it is shown that all the studied algorithms can be expressed as minimization problems, with or without non-binding constraints on the normal(s) at the contact point(s), and that the constraints can be used to identify the global minima among the critical points in the minimization problem. Another contribution of this research, based on the mathematical framework introduced, is a better classification of the known algorithms. These algorithms are compared on established test problems, and their strengths and weaknesses are highlighted and explained in terms of their classification. Furthermore, this research provides comparisons in speed and stability between the most efficient algorithms in each category over a large sample size of test problems. Among the other contributions, this research describes inexpensive but effective initial estimates of the contact to be used in iterative algorithms. Finally, the usefulness of the new framework is illustrated with the introduction of a fast algorithm combining some new and old ideas.

Ordering of Oblate Hard Particles between Hybrid Penetrable Walls

We report a Monte Carlo (MC) simulation study of a model discotic liquid crystal (DLC) confined between hybrid walls with controllable penetrability. The model consists of oblate hard Gaussian overlap (HGO) particles. Particle-substrate interactions are modeled as follows: each substrate sees a particle as a disc of zero thickness and diameter D less than or equal to that of the actual particle, σ0, embedded inside the particle and located halfway along, and perpendicular to, its minor axis. This allows us to control the anchoring properties of the substrates, from planar (edge-on) for D ≈ 0 to homeotropic (face-on) for D ≈ σ0, which can be done independently at either substrate. Depending on the values of Ds ≡ D/σ0 at the top (Dst) and bottom (Dsb) substrates, we find domains in (Dsb, Dst) space in which particle alignment is uniform planar (UP), is uniform homeotropic (UH), or varies linearly from planar at one substrate to homeotropic at the other (Lin). These domains are separated by regions of bistability (P-Lin and H-Lin), which appear to be wider than for prolate HGOs, and there may be also a small tristable (P-H-Lin) region. Results are compared with the predictions of density functional theory, implemented at the level of Onsager's second-virial approximation with Parsons-Lee rescaling. As in the case of symmetric confinement studied previously, the agreement between theory and simulation is substantially less good than for prolate HGOs: in particular, for the investigated substrate separation L = 6σ0, the Lin configuration is never predicted. These discrepancies are likely a consequence of the fact that Onsager's theory is less accurate for discs than for rods.

Computing the Closest Approach Distance of Two Ellipsoids

This paper presents two practical methods for computing the closest approach distance of two ellipsoids in their inter-center direction. The closest approach distance is crucial for collision handling in the dynamic simulation of rigid and deformable bodies approximated with ellipsoids. To find the closest approach distance, we formulate a set of equations for two ellipsoids contacting each other externally in terms of the inter-center distance, contact point, and normal vector. The equations are solved robustly and efficiently using a hybrid of the fixed-point iteration method and bisection method with root bracketing, and a hybrid of Newton’s method and the bisection method. In addition to a stopping criterion expressed with the progress of the solution, we introduce a novel criterion expressed in terms of the error in distance. This criterion can be effectively employed in real-time applications such as computer games by allowing an unnoticeable error. Experimental results demonstrate the robustness and efficiency of the proposed methods in various experiments.

General patchy ellipsoidal particle model for the aggregation behaviors of shape- and/or surface-anisotropic building blocks

We present a general patchy ellipsoidal particle model suitable for conducting dynamics simulations of the aggregation behaviors of various shape- and/or surface-anisotropic colloids, especially patchy ellipsoids with continuously variable shape and tunable patchiness. To achieve higher computational efficiency in dynamics simulations, we employ a multi-GPU acceleration technique based on a domain decomposition algorithm. The validation and performance evaluation of this GPU-assisted model are performed by simulating several typical benchmark systems of non-patchy and patchy ellipsoids. Given the generality and efficiency of our GPU-assisted patchy ellipsoidal particle model, it will provide a highly feasible dynamics simulation framework to investigate the aggregation behaviors of anisotropic soft matter systems comprised of shape- and/or surface-anisotropic building blocks.

Interparticle Interactions in Colloidal Systems: Toward a Comprehensive Mesoscale Model

Intermolecular interactions control the collective behavior of colloidal clusters. The overwhelming majority of literature focuses on cohesive attributes of intermolecular forces as they govern the jamming process. However, the overlooked sliding component plays a critical role in the slow relaxation dynamics of colloidal aggregates and the emergence of discontinuous shear thickening in dense suspensions. Here, we use crystalline calcium-silicate-hydrate (C-S-H) as a model system to explore synergistic cohesive and sliding interactions. We use the free energy perturbation approach to reconstruct potential of mean force profile between two finite-sized nanolayers in an aqueous environment. We show that sliding free energy barriers decay exponentially as the separation distance increases. The characteristic length scale of the decay is related to the interface corrugation. We introduce a simple yet effective mechanism to capture the sliding behavior of colloids with surface roughness. Moreover, we develop a global free energy landscape model by juxtaposing cohesive and sliding interactions. This model enables us to measure the height of energy barriers, which is essential to elucidate deformation mechanism and dynamics of colloidal aggregates. For cohesive colloids, our approach predicts a sublinear relation between applied normal and shear stress at the onset of sliding that is in contrast to Amontons' laws of friction. We demonstrate that the sublinear trend is due to the adhesion and nature of soft contact at the nanoscale. The proposed framework provides a new route to draw a more realistic picture of intermolecular interactions in nanoparticulate systems such as geomaterials, cementitious systems, complex colloidal assemblies, and dense suspensions.

Purely entropic self-assembly of the bicontinuous Ia3d gyroid phase in equilibrium hard-pear systems

We investigate a model of hard pear-shaped particles which forms the bicontinuous Ia[Formula: see text]d structure by entropic self-assembly, extending the previous observations of Barmes et al. (2003 Phys. Rev. E68, 021708. (doi:10.1103/PhysRevE.68.021708)) and Ellison et al. (2006 Phys. Rev. Lett.97, 237801. (doi:10.1103/PhysRevLett.97.237801)). We specifically provide the complete phase diagram of this system, with global density and particle shape as the two variable parameters, incorporating the gyroid phase as well as disordered isotropic, smectic and nematic phases. The phase diagram is obtained by two methods, one being a compression-decompression study and the other being a continuous change of the particle shape parameter at constant density. Additionally, we probe the mechanism by which interdigitating sheets of pears in these systems create surfaces with negative Gauss curvature, which is needed to form the gyroid minimal surface. This is achieved by the use of Voronoi tessellation, whereby both the shape and volume of Voronoi cells can be assessed in regard to the local Gauss curvature of the gyroid minimal surface. Through this, we show that the mechanisms prevalent in this entropy-driven system differ from those found in systems which form gyroid structures in nature (lipid bilayers) and from synthesized materials (di-block copolymers) and where the formation of the gyroid is enthalpically driven. We further argue that the gyroid phase formed in these systems is a realization of a modulated splay-bend phase in which the conventional nematic has been predicted to be destabilized at the mesoscale due to molecular-scale coupling of polar and orientational degrees of freedom.

Coarse-graining the structure of polycyclic aromatic hydrocarbons clusters

Clusters of polycyclic aromatic hydrocarbons are essential components of soot and may concentrate a significant fraction of carbon matter in the interstellar medium.

Phase behavior of shape-changing spheroids

We introduce a simple model for a biaxial nematic liquid crystal. This consists of hard spheroids that can switch shape between prolate (rodlike) and oblate (platelike) subject to an energy penalty Δε. The spheroids are approximated as hard Gaussian overlap particles and are treated at the level of Onsager's second-virial description. We use both bifurcation analysis and a numerical minimization of the free energy to show that, for additive particle shapes, (i) there is no stable biaxial phase even for Δε=0 (although there is a metastable biaxial phase in the same density range as the stable uniaxial phase) and (ii) the isotropic-to-nematic transition is into either one of two degenerate uniaxial phases, rod rich or plate rich. We confirm that even a small amount of shape nonadditivity may stabilize the biaxial nematic phase.

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.

Constraints and vibrations in static packings of ellipsoidal particles

We numerically investigate the mechanical properties of static packings of frictionless ellipsoidal particles in two and three dimensions over a range of aspect ratio and compression Δφ. While amorphous packings of spherical particles at jamming onset (Δφ=0) are isostatic and possess the minimum contact number z_{iso} required for them to be collectively jammed, amorphous packings of ellipsoidal particles generally possess fewer contacts than expected for collective jamming (z<z_{iso}) from naive counting arguments, which assume that all contacts give rise to linearly independent constraints on interparticle separations. To understand this behavior, we decompose the dynamical matrix M=H-S for static packings of ellipsoidal particles into two important components: the stiffness H and stress S matrices. We find that the stiffness matrix possesses 2N(z_{iso}-z) eigenmodes e[over ̂]_{0} with zero eigenvalues even at finite compression, where N is the number of particles. In addition, these modes e[over ̂]_{0} are nearly eigenvectors of the dynamical matrix with eigenvalues that scale as Δφ, and thus finite compression stabilizes packings of ellipsoidal particles. At jamming onset, the harmonic response of static packings of ellipsoidal particles vanishes, and the total potential energy scales as δ^{4} for perturbations by amplitude δ along these "quartic" modes, e[over ̂]_{0}. These findings illustrate the significant differences between static packings of spherical and ellipsoidal particles.

Deposition of general ellipsoidal particles

We present a systematic overview of granular deposits composed of ellipsoidal particles with different particle shapes and size polydispersities. We study the density and anisotropy of such deposits as functions of small to moderate size polydispersity and two shape parameters that fully describe the shape of a general ellipsoid. Our results show that, while shape influences significantly the macroscopic properties of the deposits, polydispersity in the studied range plays apparently a secondary role. The density attains a maximum for a particular family of nonsymmetrical ellipsoids, larger than the density observed for prolate or oblate ellipsoids. As for anisotropy measures, the contact forces are increasingly preferred along the vertical direction as the shape of the particles deviates from a sphere. The deposits are constructed by means of a molecular dynamics method, where the contact forces are efficiently and accurately computed. The main results are discussed in the light of applications for porous media models and sedimentation processes.

Crowding-induced anisotropic transport modulates reaction kinetics in nanoscale porous media

We quantify the emergence of persistent anisotropy in the diffusion of spherical tracer particles through a nanoscale porous medium composed of a uniform distribution of purely symmetric crowding particles. We focus on the interior of a biological cell as an example of such a medium and find that diffusion is highly directional for distances comparable to the size of some organelles. We use a geometrical procedure that avoids the standard orientational averaging to quantify the anisotropy of diffusive paths and show that the point source distributions are predominantly of prolate ellipsoidal shape as a result of local volume exclusion. This geometrical symmetry breaking strongly skews the distribution of kinetic rates of diffusion-limited reactions toward small values, leading to the result that, for short to intermediate times, almost 80% of the rates measured in an ensemble of heterogeneous media are smaller than the expected rate in an ideal homogeneous medium of similar excluded volume fraction. This crowding-induced modulation may have implications for our understanding and measurement of diffusion-controlled intracellular reaction kinetics and for experimental nanotechnology applications, such as nanoparticle-based bioimaging and drug delivery, where diffusion plays an important role.

Jamming in systems composed of frictionless ellipse-shaped particles

We study numerically frictionless ellipse packings versus the aspect ratio alpha, and find that the jamming transition is fundamentally different from that for spherical particles. The normal mode spectra possess two gaps and three distinct branches over a range of alpha. The energy from deformations along modes in the lowest-energy branch increases quartically, not quadratically. The quartic modes cause novel power-law scaling of the static shear modulus and their number matches the deviation from isostaticity. These results point to a new critical point at alpha>1 that controls jamming of aspherical particles.

Continuous collision detection for ellipsoids

We present an accurate and efficient algorithm for continuous collision detection between two moving ellipsoids under rational Euclidean or affine motion. We start with a highly optimized implementation of interference testing between two stationary ellipsoids based on an algebraic condition described in terms of the signs of roots of the characteristic equation of two ellipsoids. Then we derive a time-dependent characteristic equation for two moving ellipsoids, which enables us to develop an efficient algorithm for computing the time intervals in which two moving ellipsoids collide. The effectiveness of our approach is demonstrated with several practical examples.

Simulation and theory of hybrid aligned liquid crystal films

We present a study of the effects of nanoconfinement on a system of hard Gaussian overlap particles interacting with planar substrates through the hard-needle-wall potential, extending earlier work by two of us [D. J. Cleaver and P. I. C. Teixeira, Chem. Phys. Lett. 338, 1 (2001)]. Here, we consider the case of hybrid films, where one of the substrates induces strongly homeotropic anchoring, while the other favors either weakly homeotropic or planar anchoring. These systems are investigated using both Monte Carlo simulation and density-functional theory, the latter implemented at the level of Onsager's second-virial approximation with Parsons-Lee rescaling. The orientational structure is found to change either continuously or discontinuously depending on substrate separation, in agreement with earlier predictions by others. The theory is seen to perform well in spite of its simplicity, predicting the positional and orientational structure seen in simulations even for small particle elongations.

Chemistry across scales: from molecules to cells

Many important biological functions are strongly dependent on specific chemical interactions. Modelling how the physicochemical molecular details emerge at much larger scales is an active area of research, currently pursued with a variety of methods. We describe a series of theoretical and computational approaches that aim to derive bottom-up descriptions that capture the specificity that ensues from atomistic detail by extracting relevant features at the different scales. The multiscale models integrate the descriptions at different length and time scales by exploiting the idea of mechanical responses. The methodologies bring together concepts and tools developed in seemingly unrelated areas of mathematics such as algebraic geometry, model reduction, structural graph theory and non-convex optimization. We showcase the applicability of the framework with examples from protein engineering and enzyme catalysis, protein assembly, and with the description of lipid bilayers at different scales. Many challenges remain as it is clear that no single methodology will answer all questions in such multidimensional complex problems.

Helix self-assembly from anisotropic molecules

We explore the potential energy landscape for clusters composed of disklike ellipsoidal particles interacting via an anisotropic potential based on the elliptic contact function. Over a wide range of parameter space we find global potential energy minima consisting of helices composed of one or more strands. Characterizing the potential energy surface in the region of helical global minima reveals a topology associated with "structure-seeking" systems. This result indicates that the helices will self-assemble over a wide range of temperature.

Sequential quenching of randomly deposited ellipsoids: Anisotropy and spatial patterns

We studied the sequential quenching of prolate ellipsoids on a homogeneous surface by using our proposed pair potential whose repulsive part was that of hard ellipses and attractive part was the r-6 pairwise attraction. Both the strength and range parameters for the attraction were functions of the orientations of the pair of ellipses and related by epsilon proportional, variantsigma-lambda. The parameter lambda determines the relative strength of the side-by-side and end-to-end attractions and thus plays an important role in determining the alignment of the particles. We adopted the value lambda=2.19 by using point-energy additivity to compute the minimum energies for both of these configurations for a pair of ellipsoids of revolution with aspect ratio of 2:1. We investigated the effect of temperature and the parameter lambda on the alignments of ellipses. Both radial distribution function, g(r), and orientational correlation function, G(r), show the expected longer ranges of orientational correlation at lower temperatures and show higher degree of orientational order for lambda=3.5 than lambda=2.19 and 0.10. This can also be seen in the examples of configurations showing that for lambda=3.5, ellipses are more aligned than lambda=2.19 and 0.1.

Elasticity of a system with noncentral potentials

We derive expressions for determination of the stress and the elastic constants in systems composed of particles interacting via noncentral two-body potentials as thermal averages of products of first and second partial derivatives of the interparticle potentials and components of the interparticle separation vectors. These results are adapted to hard potentials, where the stress and the elastic constants are expressed as thermal averages of the components of normals to contact surfaces between the particles and components of vectors separating their centers. The averages require knowledge of the simultaneous contact probabilities of two pairs of particles. We apply the expressions to particles for which a contact function can be defined, and demonstrate the feasibility of the method by computing the stress and the elastic constants of a two-dimensional system of hard ellipses using Monte Carlo simulations.

Coarse-grained interaction potentials for anisotropic molecules

We have proposed an efficient parametrization method for a recent variant of the Gay Berne potential for dissimilar and biaxial particles [Phys. Rev. E 67, 041710 (2003)] and demonstrated it for a set of small organic molecules. Compared with the previously proposed coarse-grained models, the new potential exhibits a superior performance in close contact and large distant interactions. The repercussions of thermal vibrations and elasticity have been studied through a statistical method. The study justifies that the potential of mean force is representable with the same functional form, extending the application of this coarse-grained description to a broader range of molecules. Moreover, the advantage of employing coarse-grained models over truncated atomistic summations with large distance cutoffs has been briefly studied.

The directional contact distance of two ellipsoids: Coarse-grained potentials for anisotropic interactions

We obtain the distance of closest approach of the surfaces of two arbitrary ellipsoids valid at any orientation and separation measured along their intercenter vector. This directional distance is derived from the elliptic contact function. The geometric meaning behind this approach is clarified. An elliptic pair potential for modeling arbitrary mixtures of elliptic particles, whether hard or soft, is proposed based on this distance. Comparisons with Gay-Berne potentials are discussed. Analytic expressions for the forces and torques acting on the elliptic particles are given.

Computer simulations of hard pear-shaped particles

We report results obtained from Monte Carlo simulations investigating mesophase formation in two model systems of hard pear-shaped particles. The first model considered is a hard variant of the truncated Stone-expansion model previously shown to form nematic and smectic mesophases when embedded within a 12-6 Gay-Berne-like potential [R. Berardi, M. Ricci, and C. Zannoni, ChemPhysChem 7, 443 (2001)]. When stripped of its attractive interactions, however, this system is found to lose its liquid crystalline phases. For particles of length to breadth ratio k=3, glassy behavior is seen at high pressures, whereas for k=5 several bi- layerlike domains are seen, with high intradomain order but little interdomain orientational correlation. For the second model, which uses a parametric shape parameter based on the generalized Gay-Berne formalism, results are presented for particles with elongation k=3, 4, and 5. Here, the systems with k=3 and 4 fail to display orientationally ordered phases, but the system with k=5 shows isotropic, nematic and, unusual for a hard-particle model, interdigitated smectic A2 phases.

Interaction potentials for soft and hard ellipsoids

Using results from colloid science we derive interaction potentials for computer simulations of mixtures of soft or hard ellipsoids of arbitrary shape and size. Our results are in many respects reminicent of potentials of the Gay-Berne type but have a well-defined microscopic interpretation and no adjustable parameters. Since our potentials require the calculation of similar variables, the modification of existing simulation codes for Gay-Berne potentials is straightforward. The computational performance should remain unaffected.