Abstract:
In this article we study the following question: What can be the measure of the minimal solid angle of a simplex in $\mathbb{R}^d$? We show that in dimensions three and four it is not greater than the solid angle of the regular simplex. We also study a similar question for trihedral and dihedral angles of polyhedra compared to those of regular solids.

Abstract:
We study the diffusion of Brownian particles on the surface of a sphere and compute the distribution of solid angles enclosed by the diffusing particles. This function describes the distribution of geometric phases in two state quantum systems (or polarised light) undergoing random evolution. Our results are also relevant to recent experiments which observe the Brownian motion of molecules on curved surfaces like micelles and biological membranes. Our theoretical analysis agrees well with the results of computer experiments.

Abstract:
We look at a lattice's Minkowski reduced basis and the solid angle generated by its vectors, which satisfies strong orthogonality conditions due to the basis's minimality nature. Sharp upper and lower bounds are found for all rank-3 and rank-4 lattices so that a Minkowski reduced basis always exists with solid angle measuring in between. Extreme cases happen when the lattice takes rectangular or face-centered cubic shape. Our proof relies on a formula that expresses the high-dimensional solid angle as the product between the lattice's determinant and a quadratic integral on the unit sphere S^{n-1}. At the end, a 5-dimensional counterexample is supplied where the usual face-centered cubic lattice no longer has the smallest measure for solid angle.

Abstract:
We find sharp absolute constants $C_1$ and $C_2$ with the following property: every well-rounded lattice of rank 3 in a Euclidean space has a minimal basis so that the solid angle spanned by these basis vectors lies in the interval $[C_1,C_2]$. In fact, we show that these absolute bounds hold for a larger class of lattices than just well-rounded, and the upper bound holds for all. We state a technical condition on the lattice that may prevent it from satisfying the absolute lower bound on the solid angle, in which case we derive a lower bound in terms of the ratios of successive minima of the lattice. We use this result to show that among all spherical triangles on the unit sphere in $\mathbb R^N$ with vertices on the minimal vectors of a lattice, the smallest possible area is achieved by a configuration of minimal vectors of the (normalized) face centered cubic lattice in $\mathbb R^3$. Such spherical configurations come up in connection with the kissing number problem.

Abstract:
The contact angle that a liquid drop makes on a soft substrate does not obey the classical Young's relation, since the solid is deformed elastically by the action of the capillary forces. The finite elasticity of the solid also renders the contact angles different from that predicted by Neumann's law, which applies when the drop is floating on another liquid. Here we derive an elasto-capillary model for contact angles on a soft solid, by coupling a mean-field model for the molecular interactions to elasticity. We demonstrate that the limit of vanishing elastic modulus yields Neumann's law or a slight variation thereof, depending on the force transmission in the solid surface layer. The change in contact angle from the rigid limit (Young) to the soft limit (Neumann) appears when the length scale defined by the ratio of surface tension to elastic modulus $\gamma/E$ reaches a few molecular sizes.

Abstract:
A methodology for the determination of the solid-fluid contact angle, to be employed within molecular dynamics (MD) simulations, is developed and systematically applied. The calculation of the contact angle of a fluid drop on a given surface, averaged over an equilibrated MD trajectory, is divided in three main steps: (i) the determination of the fluid molecules that constitute the interface, (ii) the treatment of the interfacial molecules as a point cloud data set to define a geometric surface, using surface meshing techniques to compute the surface normals from the mesh, (iii) the collection and averaging of the interface normals collected from the post-processing of the MD trajectory. The average vector thus found is used to calculate the Cassie contact angle ( i.e., the arccosine of the averaged normal z-component). As an example we explore the effect of the size of a drop of water on the observed solid-fluid contact angle. A single coarse-grained bead representing two water molecules and parameterized using the SAFT-γ Mie equation of state (EoS) is employed, meanwhile the solid surfaces are mimicked using integrated potentials. The contact angle is seen to be a strong function of the system size for small nano-droplets. The thermodynamic limit, corresponding to the infinite size (macroscopic) drop is only truly recovered when using an excess of half a million water coarse-grained beads and/or a drop radius of over 26 nm.

Abstract:
We use a probabilistic interpretation of solid angles to generalize the well-known fact that the inner angles of a triangle sum to 180 degrees. For the 3-dimensional case, we show that the sum of the solid inner vertex angles of a tetrahedron T, divided by 2*pi, gives the probability that an orthogonal projection of T onto a random 2-plane is a triangle. More generally, it is shown that the sum of the (solid) inner vertex angles of an n-simplex S, normalized by the area of the unit (n-1)-hemisphere, gives the probability that an orthogonal projection of S onto a random hyperplane is an (n-1)-simplex. Applications to more general polytopes are treated briefly, as is the related Perles-Shephard proof of the classical Gram-Euler relations.

Abstract:
We study the nanoscale behaviour of the density of a simple fluid in the vicinity of an equilibrium contact line for a wide range of Young contact angles between 40 and 135 degrees. Cuts of the density profile at various positions along the contact line are presented, unravelling the apparent step-wise increase of the film height profile observed in contour plots of the density. The density profile is employed to compute the normal pressure acting on the substrate along the contact line. We observe that for the full range of contact angles, the maximal normal pressure cannot solely be predicted by the curvature of the adsorption film height, but is instead softened -- likely by the width of the liquid-vapour interface. Somewhat surprisingly however, the adsorption film height profile can be predicted to a very good accuracy by the Derjaguin-Frumkin disjoining pressure obtained from planar computations, as was first shown in [Nold et al., Phys. Fluids, 26, 072001, 2014] for contact angles less than 90 degrees, a result which here we show to be valid for the full range of contact angles. This suggests that while two-dimensional effects cannot be neglected for the computation of the normal pressure distribution along the substrate, one-dimensional planar computations of the Derjaguin-Frumkin disjoining pressure are sufficient to accurately predict the adsorption height profile.

Abstract:
We interpret a class of 4k-dimensional instanton solutions found by Ward, Corrigan, Goddard and Kent as four-dimensional instantons at angles. The superposition of each pair of four-dimensional instantons is associated with four angles which depend on some of the ADHM parameters. All these solutions are associated with the group $Sp(k)$ and are examples of Hermitian-Einstein connections on $\bE^{4k}$. We show that the eight-dimensional solutions preserve 3/16 of the ten-dimensional N=1 supersymmetry. We argue that under the correspondence between the BPS states of Yang-Mills theory and those of M-theory that arises in the context of Matrix models, the instantons at angles configuration corresponds to the longitudinal intersecting 5-branes on a string at angles configuration of M-theory.

Abstract:
We consider the following question. Suppose that $d\ge2$ and $n$ are fixed, and that $\theta_1,\theta_2,\dots,\theta_n$ are $n$ specified angles. How many points do we need to place in $\mathbb{R}^d$ to realise all of these angles? A simple degrees of freedom argument shows that $m$ points in $\mathbb{R}^2$ cannot realise more than $2m-4$ general angles. We give a construction to show that this bound is sharp when $m\ge 5$. In $d$ dimensions the degrees of freedom argument gives an upper bound of $dm-\binom{d+1}{2}-1$ general angles. However, the above result does not generalise to this case; surprisingly, the bound of $2m-4$ from two dimensions cannot be improved at all. Indeed, our main result is that there are sets of $2m-3$ of angles that cannot be realised by $m$ points in any dimension.