Tangent Sphere Research Articles

Entropy-Driven Crystallization of Hard Colloidal Mixtures of Polymers and Monomers.

The most trivial example of self-assembly is the entropy-driven crystallization of hard spheres. Past works have established the similarities and differences in the phase behavior of monomers and chains made of hard spheres. Inspired by the difference in the melting points of the pure components, we study, through Monte Carlo simulations, the phase behavior of athermal mixtures composed of fully flexible polymers and individual monomers of uniform size. We analyze how the relative number fraction and the packing density affect crystallization and the established ordered morphologies. As a first result, a more precise determination of the melting point for freely jointed chains of tangent hard spheres is extracted. A synergetic effect is observed in the crystallization leading to synchronous crystallization of the two species. Structural analysis of the resulting ordered morphologies shows perfect mixing and thus no phase separation. Due to the constraints imposed by chain connectivity, the local environment of the individual spheres, as quantified by the Voronoi polyhedron, is systematically more spherical and more symmetric compared to that of spheres belonging to chains. In turn, the local environment of the ordered phase is more symmetric and more spherical compared to that of the initial random packing, demonstrating the entropic origins of the phase transition. In general, increasing the polymer content reduces the degree of crystallinity and increases the melting point to higher volume fractions. According to the present findings, relative concentration is another determining factor in controlling the phase behavior of hard colloidal mixtures based on polymers.

Notes on the geometry of cotangent bundle and unit cotangent sphere bundle

Let $(\mathtt{N},\mathfrak{g})$ be a Riemannian manifold, by using the musical isomorphisms ♪ and $\natural$ induced by $\mathfrak{g}$, we built a bridge between the geometry of the tangent bundle $\mathtt{TN}$ (resp. the unit tangent sphere bundle $\mathtt{T}_{1}\mathtt{N}$) equipped with the Sasaki metric $\mathfrak{g}_{S}$ (resp. the induced Sasaki metric $\bar{\mathfrak{g}}_{S}$) and that of the cotangent bundle $\mathtt{T}^{\ast}\mathtt{N}$ (resp. the unit cotangent sphere bundle $\mathtt{T}_{1}^{\ast}\mathtt{N}$) endowed with the Sasaki metric $\mathfrak{g}_{\widetilde{S}}$ (resp. the induced Sasaki metric $\tilde{\mathfrak{g}}_{\widetilde{S}}$). Moreover, we prove that $\mathtt{T}_{1}^{\ast}\mathtt{N}$ carries a contact metric structure and study some of its proprieties.

An excess free energy and chemical potential for hard homonuclear diatomics from integral equation approach

It has been shown that an improved prediction of the thermodynamic quantity for the hard homonuclear diatomics can be performed with an interpolation scheme that relates the thermodynamic property of the hard sphere to that of tangent hard spheres at the same density. Using the analytic expressions based on the solution of an integral equation an excess Helmholtz free energy per particle and chemical potential for hard homonuclear diatomic fluid have been computed. Calculations are carried out for hard homonuclear diatomics with reduced internuclear separations of 0.1 to 1 at reduced densities of 0.2 to 0.9. Our findings for the excess Helmholtz free energy from the Percus–Yevickand Martynov–Sarkisov approximations presents good agreement with available accurate data, having maximum deviations of 15% from it over the separation and density ranges of the calculations. The excess chemical potential from the Martynov–Sarkisov approximation shows better agreement with accurate available data than those from the Percus–Yevick approximation. For the excess chemical potential, a maximum deviation of 9.5% over the range of the calculations has been shown up for the Percus–Yevick approximation.

Minimal Legendrian surfaces in the tangent sphere bundle of 𝕊³

In this paper we study minimal Legendrian surfaces Σ \Sigma immersed in tangent sphere bundle T 1 S 3 T_1{\mathbb {S}}^3 . We classify (1) totally geodesic Legendrian surfaces, (2) closed minimal Legendrian surfaces of genus smaller than or equal to one and complete minimal Legendrian surfaces with non-negative Gauss curvature.

Splay-induced order in systems of hard tapers.

The main objective of this work is to clarify the role that taper-shaped elongated molecules, i.e., molecules with one end wider than the other, can play in stabilizing orientational order. The focus is exclusively on entropy-driven self-organization induced by purely excluded volume interactions. Drawing an analogy to RM734 (4-[(4-nitrophenoxy)carbonyl]phenyl-2,4-dimethoxybenzoate), which is known to stabilize ferroelectric nematic (N_{F}) and nematic splay (N_{S}) phases, and assuming that molecular biaxiality is of secondary importance, we consider monodisperse systems composed of hard molecules. Each molecule is modeled using six collinear tangent spheres with linearly decreasing diameters. Through hard-particle, constant-pressure Monte Carlo simulations, we study the emergent phases as functions of the ratio between the smallest and largest diameters of the spheres (denoted as d) and the packing fraction (η). To analyze global and local molecular orderings, we examine molecular configurations in terms of nematic, smectic, and hexatic order parameters. Additionally, we investigate the radial pair distribution function, polarization correlation function, and the histogram of angles between molecular axes. The last characteristic is utilized to quantify local splay. The findings reveal that splay-induced deformations drive unusual long-range orientational order at relatively high packing fractions (η>0.5), corresponding to crystalline phases. When η<0.5, only short-range order is affected, and in addition to the isotropic liquid, only the standard nematic and smectic-A liquid crystalline phases are stabilized. However, for η>0.5, apart from the ordinary nonpolar hexagonal crystal, three additional frustrated crystalline polar blue phases with long-range splay modulation are observed: antiferroelectric splay crystal (Cr_{S}P_{A}), antiferroelectric double-splay crystal (Cr_{DS}P_{A}), and ferroelectric double-splay crystal (Cr_{DS}P_{F}). Finally, we employ Onsager-Parsons-Lee local density functional theory to investigate whether any sterically induced (anti)ferroelectric nematic or smectic-A type of ordering is possible for our system, at least in a metastable regime.

Invariant contact metric structures on tangent sphere bundles of compact symmetric spaces

A new characterization is provided for the class of compact rank-one symmetric spaces. Such spaces are the only symmetric spaces of compact type for which the standard vector field xi ^{S} on their sphere bundles is Killing with respect to some invariant Riemannian metric. The set of all these metrics is determined, as well as the set of all those invariant contact metric structures with characteristic vector field xi ^{S}. Moreover, on tangent sphere bundles of compact symmetric spaces with rank greater than or equal to two, a family of invariant contact metric structures, which contains the standard structure, is obtained.

A Schur type lemma for the mean Berwald curvature in Finsler geometry

In this short paper, we study a symmetric covariant tensor in Finsler geometry, which is called the mean Berwald curvature. We first investigate the geometry of the fibres as the submanifolds of the tangent sphere bundle on a Finsler manifold. Then we prove that if the mean Berwald curvature is isotropic along fibres, then the Berwald scalar curvature is constant along fibres.

Densest packing of flexible polymers in 2D films.

How dense objects, particles, atoms, and molecules can be packed is intimately related to the properties of the corresponding hosts and macrosystems. We present results from extensive Monte Carlo simulations on maximally compressed packings of linear, freely jointed chains of tangent hard spheres of uniform size in films whose thickness is equal to the monomer diameter. We demonstrate that fully flexible chains of hard spheres can be packed as efficiently as monomeric analogs, within a statistical tolerance of less than 1%. The resulting ordered polymer morphology corresponds to an almost perfect hexagonal triangular (TRI) crystal of the p6m wallpaper group, whose sites are occupied by the chain monomers. The Flory scaling exponent, which corresponds to the maximally dense polymer packing in 2D, has a value of ν = 0.62, which lies between the limits of 0.50 (compact and collapsed state) and 0.75 (self-avoiding random walk).

Magnetic unit vector fields

We show that a unit vector field on an oriented Riemannian manifold is a critical point of the Landau Hall functional if and only if it is a critical point of the Dirichlet energy functional. Therefore, we provide a characterization for a unit vector field to be a magnetic map into its unit tangent sphere bundle. Then, we classify all magnetic left invariant unit vector fields on $3$-dimensional Lie groups.

Local and Global Order in Dense Packings of Semi-Flexible Polymers of Hard Spheres.

The local and global order in dense packings of linear, semi-flexible polymers of tangent hard spheres are studied by employing extensive Monte Carlo simulations at increasing volume fractions. The chain stiffness is controlled by a tunable harmonic potential for the bending angle, whose intensity dictates the rigidity of the polymer backbone as a function of the bending constant and equilibrium angle. The studied angles range between acute and obtuse ones, reaching the limit of rod-like polymers. We analyze how the packing density and chain stiffness affect the chains' ability to self-organize at the local and global levels. The former corresponds to crystallinity, as quantified by the Characteristic Crystallographic Element (CCE) norm descriptor, while the latter is computed through the scalar orientational order parameter. In all cases, we identify the critical volume fraction for the phase transition and gauge the established crystal morphologies, developing a complete phase diagram as a function of packing density and equilibrium bending angle. A plethora of structures are obtained, ranging between random hexagonal closed packed morphologies of mixed character and almost perfect face centered cubic (FCC) and hexagonal close-packed (HCP) crystals at the level of monomers, and nematic mesophases, with prolate and oblate mesogens at the level of chains. For rod-like chains, a delay is observed between the establishment of the long-range nematic order and crystallization as a function of the packing density, while for right-angle chains, both transitions are synchronized. A comparison is also provided against the analogous packings of monomeric and fully flexible chains of hard spheres.

“An enhanced and optimized Monte Carlo method to calculate view factors in packed beds”

Comprehensive radiation studies on packed beds require the estimation of view factors between the surfaces, which often computationally expensive. This paper proposes a novel method based on Monte Carlo Ray Tracing for high-precision and low time computation processing of view factors in random assemblies of cylindrical packed beds. The Monte Carlo Ray Tracing method is improved by coupling it with Kowsary’s tangent sphere method under the parallel computation processing through Compute Unified Device Architecture, showing a reduction of the computational time in approximately 153 times less than a traditional Monte Carlo Ray Tracing method on the hardware used and obtaining a maximum relative error of 0.39% for configurations evaluated in the literature. Furthermore, a calculation methodology for view factors between particle–particle, particle–wall, and particle-lid under an original layer concept, is presented and applied on a set of randomly assembled monosized packed beds generated with the LIGGGHTS software, covering an average porosity range from 0.38 to 0.50 for the arrays. Finally, detailed analysis and discussion of the results are performed, allowing to find characteristic view factors for the particles according to their positions and correlations for view factors particle–wall and particle-lid which is defined as layer view factor method, considering error intervals for each interaction respectively, defining a view factor calculation methodology for packed beds within the studied porosity range.

Resistivity contribution tensor for nonconductive sphere doublets

• The resistivity contribution tensor for nonconductive sphere doublets is analytically evaluated. • Making reference to tangent sphere coordinates, the temperature fields are expressed by means of convergent integrals. • Based on a stream function, the axial component of the resistivity contribution tensor is found in closed form. • The transverse component of the resistivity contribution tensor is numerically assessed based on an Euler shooting method. • Comparison with respect equivalent spheroids is provided too. The distribution of the temperature and heat flux fields around a couple of unequal nonconductive tangent spherical inhomogeneities (or pores) embedded in an infinite medium under a steady-state and remotely applied heat flux is addressed in the present work. Owing to the 3D geometrical layout of the inhomogeneity, use is made of the tangent sphere coordinate system. A corrective temperature field expressed in terms of convergent integrals is superposed to the fundamental one to fulfill the BCs at the surfaces of the spheres. When the heat flux is aligned to the symmetry axis (axisymmetric problem), the solution can be found straightforwardly by introducing a stream function, which allows for transforming the Neumann BCs into a Dirichlet boundary value problem. Conversely, for the transversal heat flux (non-axisymmetric problem), the problem is formulated in terms of temperature, thus leading to a system of two ODEs which is handled numerically through a Euler shooting method, after preliminary asymptotic expansions. Once the temperature fields are known, the components of the resistivity contribution tensor are assessed varying the aspect ratio of the two spheres. It is found that the extrema of the thermal resistivity are achieved for spheres of equal size. The study allows assessing the effective thermal conductivity of a wide range of smart composites involving insulating inhomogeneities resembling sphere doublets.

Sasakian Structures on Tangent Sphere Bundles of Compact Rank-One Symmetric Spaces

A positive answer is given to the existence of Sasakian structures on the tangent sphere bundle of some Riemannian manifold whose sectional curvature is not constant. Among other results, it is proved that the tangent sphere bundle T_{r}(G/K), for any r> 0, of a compact rank-one symmetric space G/K, not necessarily of constant sectional curvature, admits a unique G-invariant K-contact structure whose characteristic vector field is the standard field of T(G/K). Such a structure is in fact Sasakian and it can be expressed as an induced structure from an almost Hermitian structure on the punctured tangent bundle T(G/K){setminus } {text{ zero } text{ section }}.

In silico study of liquid crystalline phases formed by bent-shaped molecules with excluded volume type interactions

What impact does the mesogens’ shape have on the formation of the liquid crystalline phase? Using Monte Carlo and molecular dynamics simulations we numerically studied a liquid composed of achiral, bent-shaped molecules built of tangent spheres. The system is known to spontaneously break mirror symmetry, as it forms a macroscopically chiral, twist-bend nematic phase [Phys. Rev. Lett. 115, 147801 (2015)]. We examined a full phase diagram by altering the molecules’ curvature along with packing fractions and observed several phases characterized by the orientational and/or translational ordering of molecules. Apart from conventional nematic, smectic A, and the aforementioned twist-bend nematic phase, we identified splay-bend smectic phase. For large densities and strongly curved molecules, another smectic phase emerged, where the polarization vector rotates within a single smectic layer.

Crystallization of Flexible Chains of Tangent Hard Spheres under Full Confinement.

We present results from extensive Monte Carlo simulations on the crystallization of athermal polymers under full confinement. Polymers are represented as freely jointed chains of tangent hard spheres of uniform size. Confinement is applied through the presence of flat, parallel, and impenetrable walls in all dimensions. We analyze crystallization as the summation of two contributions: one that occurs in the bulk volume of the system (bulk crystallization), and one on the wall surfaces (surface crystallization). Depending on volume fraction initially amorphous (disordered) hard-sphere chain packings transit to the stable crystal phase. The established ordered morphologies consist primarily of hexagonal close-packed (HCP) crystals in the bulk volume and of triangular (TRI) crystals on the surface. As in the case of athermal packings in the bulk (without confinement), a structural competition is observed between the 5-fold local symmetry and the formation of close-packed crystallites. Effectively, the full confinement inside a cube favors the growth of the HCP crystal, as the FCC one is quite incompatible with the imposed spatial constraints. Consequently, we observe the formation of noncompact ordered motifs which grow from the surface to the inner volume of the simulation cell. We further compare the 2D and 3D crystals formed by monomeric hard spheres under the same simulation conditions. Significant differences are observed at low densities that tend to diminish as concentration increases.

Enriques surfaces and an Apollonian packing in eight dimensions

AbstractWe call a packing of hyperspheres in n dimensions an Apollonian sphere packing if the spheres intersect tangentially or not at all; they fill the n-dimensional Euclidean space; and every sphere in the packing is a member of a cluster of $n+2$ mutually tangent spheres (and a few more properties described herein). In this paper, we describe an Apollonian packing in eight dimensions that naturally arises from the study of generic nodal Enriques surfaces. The $E_7$ , $E_8$ and Reye lattices play roles. We use the packing to generate an Apollonian packing in nine dimensions, and a cross section in seven dimensions that is weakly Apollonian. Maxwell described all three packings but seemed unaware that they are Apollonian. The packings in seven and eight dimensions are different than those found in an earlier paper. In passing, we give a sufficient condition for a Coxeter graph to generate mutually tangent spheres and use this to identify an Apollonian sphere packing in three dimensions that is not the Soddy sphere packing.

On unit sphere tangent bundles over complex Grassmannians

On unit sphere tangent bundles over complex Grassmannians

Horizontal Delaunay surfaces with constant mean curvature in $\mathbb{S}^2 \times \mathbb{R}$ and $\mathbb{H}^2 \times \mathbb{R}$

We obtain a $1$-parameter family of horizontal Delaunay surfaces with positive constant mean curvature in $\mathbb{S}^2\times\mathbb{R}$ and $\mathbb{H}^2\times\mathbb{R}$, being the mean curvature larger than $\frac{1}{2}$ in the latter case. These surfaces are not equivariant but singly periodic, lie at bounded distance from a horizontal geodesic, and complete the family of horizontal unduloids previously given by the authors. We study in detail the geometry of the whole family and show that horizontal unduloids are properly embedded in $\mathbb H^2\times\mathbb{R}$. We also find (among unduloids) families of embedded constant mean curvature tori in $\mathbb S^2\times\mathbb{R}$ which are continuous deformations from a stack of tangent spheres to a horizontal invariant cylinder. In particular, we find the first non-equivariant examples of embedded tori in $\mathbb{S}^2\times\mathbb{R}$, which have constant mean curvature $H>\frac12$. Finally, we prove that there are no properly immersed surface with constant mean curvature $H\leq\frac{1}{2}$ at bounded distance from a horizontal geodesic in $\mathbb{H}^2\times\mathbb{R}$.

On the Classifying of the Tangent Sphere Bundle with Almost Contact B-Metric Structure

On the Classifying of the Tangent Sphere Bundle with Almost Contact B-Metric Structure

On the completeness of total spaces of horizontally conformal submersions

Abstract In this paper, we address the completeness problem of certain classes of Riemannian metrics on vector bundles. We first establish a general result on the completeness of the total space of a vector bundle when the projection is a horizontally conformal submersion with a bound condition on the dilation function, and in particular when it is a Riemannian submersion. This allows us to give completeness results for spherically symmetric metrics on vector bundle manifolds and eventually for the class of Cheeger-Gromoll and generalized Cheeger-Gromoll metrics on vector bundle manifolds. Moreover, we study the completeness of a subclass of g-natural metrics on tangent bundles and we extend the results to the case of unit tangent sphere bundles. Our proofs are mainly based on techniques of metric topology and on the Hopf-Rinow theorem.