The effect of boundary adaptivity on hexagonal ordering and bistability in circularly confined quasi hard discs

The behaviour of materials under spatial confinement is sensitively dependent on the nature of the confining boundaries. In two dimensions, confinement within a hard circular boundary inhibits the hexagonal ordering observed in bulk systems at high density. Using colloidal experiments and Monte Carlo simulations, we investigate two model systems of quasi hard discs under circularly symmetric confinement. The first system employs an adaptive circular boundary, defined experimentally using holographic optical tweezers. We show that deformation of this boundary allows, and indeed is required for, hexagonal ordering in the confined system. The second system employs a circularly symmetric optical potential to confine particles without a physical boundary. We show that, in the absence of a curved wall, near perfect hexagonal ordering is possible. We propose that the degree to which hexagonal ordering is suppressed by a curved boundary is determined by the `strictness' of that wall.Comment: 10 pages, 8 figure

Correlation between crystalline order and vitrification in colloidal monolayers

We investigate experimentally the relationship between local structure and dynamical arrest in a quasi-2d colloidal model system which approximates hard discs. We introduce polydispersity to the system to suppress crystallisation. Upon compression, the increase in structural relaxation time is accompanied by the emergence of local hexagonal symmetry. Examining the dynamical heterogeneity of the system, we identify three types of motion : "zero-dimensional" corresponding to beta-relaxation, "one-dimensional" or stringlike motion and "two-dimensional" motion. The dynamic heterogeneity is correlated with the local order, that is to say locally hexagonal regions are more likely to be dynamically slow. However we find that lengthscales corresponding to dynamic heterogeneity and local structure do not appear to scale together approaching the glass transition.Comment: 13 papes, to appear in J. Phys.: Condens. Matte

Evidence for the formation of comet 67P/Churyumov-Gerasimenko through gravitational collapse of a bound clump of pebbles

The processes that led to the formation of the planetary bodies in the Solar System are still not fully understood. Using the results obtained with the comprehensive suite of instruments on-board ESA’s Rosetta mission, we present evidence that comet 67P/Churyumov-Gerasimenko likely formed through the gentle gravitational collapse of a bound clump of mm-sized dust aggregates (“pebbles”), intermixed with microscopic ice particles. This formation scenario leads to a cometary make-up that is simultaneously compatible with the global porosity, homogeneity, tensile strength, thermal inertia, vertical temperature profiles, sizes and porosities of emitted dust, and the steep increase in water-vapour production rate with decreasing heliocentric distance, measured by the instruments on-board the Rosetta spacecraft and the Philae lander. Our findings suggest that the pebbles observed to be abundant in protoplanetary discs around young stars provide the building material for comets and other minor bodies

Analysing local algorithms in location-aware quasi-unit-disk graphs

A local algorithm with local horizon r is a distributed algorithm that runs in r synchronous communication rounds; here r is a constant that does not depend on the size of the network. As a consequence, the output of a node in a local algorithm only depends on the input within r hops from the node. We give tight bounds on the local horizon for a class of local algorithms for combinatorial problems on unit-disk graphs (UDGs). Most of our bounds are due to a refined analysis of existing approaches, while others are obtained by suggesting new algorithms. The algorithms we consider are based on network decompositions guided by a rectangular tiling of the plane. The algorithms are applied to matching, independent set, graph colouring, vertex cover, and dominating set. We also study local algorithms on quasi-UDGs, which are a popular generalisation of UDGs, aimed at more realistic modelling of communication between the network nodes. Analysing the local algorithms on quasi-UDGs allows one to assume that the nodes know their coordinates only approximately, up to an additive error. Despite the localisation error, the quality of the solution to problems on quasi-UDGs remains the same as for the case of UDGs with perfect location awareness. We analyse the increase in the local horizon that comes along with moving from UDGs to quasi-UDGs.Peer reviewe

Annulus Graphs in R^<i>d</i>

A d-dimensional annulus graph with radii R1 and R2 (here R2≥R1≥0) is a graph embeddable in Rd so that two vertices u and v form an edge if and only if their images in the embedding are at distance in the interval [R1, R2]. In this paper we show that the family Ad(R1, R2) of d-dimensional annulus graphs with radii R1 and R2 is uniquely characterised by R2/R1 when this ratio is sufficiently large. Moreover, as a step towards a better understanding of the structure of Ad(R1, R2), we show that supG∈Ad(R1, R2)χ(G)/ω(G) is given by exp(O(d)) for all R1, R2 satisfying R2≥R1&gt;0 and also exp(Ω(d)) if moreover R2/R1≥1.2

Annulus Graphs in R^<i>d</i>

A d-dimensional annulus graph with radii R1 and R2 (here R2≥R1≥0) is a graph embeddable in Rd so that two vertices u and v form an edge if and only if their images in the embedding are at distance in the interval [R1, R2]. In this paper we show that the family Ad(R1, R2) of d-dimensional annulus graphs with radii R1 and R2 is uniquely characterised by R2/R1 when this ratio is sufficiently large. Moreover, as a step towards a better understanding of the structure of Ad(R1, R2), we show that supG∈Ad(R1, R2)χ(G)/ω(G) is given by exp(O(d)) for all R1, R2 satisfying R2≥R1&gt;0 and also exp(Ω(d)) if moreover R2/R1≥1.2