Stable Frank-Kasper phases of self-assembled, soft matter spheres

Single molecular species can self-assemble into Frank Kasper (FK) phases, finite approximants of dodecagonal quasicrystals, defying intuitive notions that thermodynamic ground states are maximally symmetric. FK phases are speculated to emerge as the minimal-distortional packings of space-filling spherical domains, but a precise quantitation of this distortion and how it affects assembly thermodynamics remains ambiguous. We use two complementary approaches to demonstrate that the principles driving FK lattice formation in diblock copolymers emerge directly from the strong-stretching theory of spherical domains, in which minimal inter-block area competes with minimal stretching of space-filling chains. The relative stability of FK lattices is studied first using a diblock foam model with unconstrained particle volumes and shapes, which correctly predicts not only the equilibrium {\sigma} lattice, but also the unequal volumes of the equilibrium domains. We then provide a molecular interpretation for these results via self-consistent field theory, illuminating how molecular stiffness regulates the coupling between intra-domain chain configurations and the asymmetry of local packing. These findings shed new light on the role of volume exchange on the formation of distinct FK phases in copolymers, and suggest a paradigm for formation of FK phases in soft matter systems in which unequal domain volumes are selected by the thermodynamic competition between distinct measures of shape asymmetry.Comment: 40 pages, 22 figure

Contact graphs of ball packings

A contact graph of a packing of closed balls is a graph with balls as vertices and pairs of tangent balls as edges. We prove that the average degree of the contact graph of a packing of balls (with possibly different radii) in $\mathbb{R}^3$ is not greater than $13.955$. We also find new upper bounds for the average degree of contact graphs in $\mathbb{R}^4$ and $\mathbb{R}^5$

Computational Approaches to Lattice Packing and Covering Problems

We describe algorithms which address two classical problems in lattice geometry: the lattice covering and the simultaneous lattice packing-covering problem. Theoretically our algorithms solve the two problems in any fixed dimension d in the sense that they approximate optimal covering lattices and optimal packing-covering lattices within any desired accuracy. Both algorithms involve semidefinite programming and are based on Voronoi's reduction theory for positive definite quadratic forms, which describes all possible Delone triangulations of Z^d. In practice, our implementations reproduce known results in dimensions d <= 5 and in particular solve the two problems in these dimensions. For d = 6 our computations produce new best known covering as well as packing-covering lattices, which are closely related to the lattice (E6)*. For d = 7, 8 our approach leads to new best known covering lattices. Although we use numerical methods, we made some effort to transform numerical evidences into rigorous proofs. We provide rigorous error bounds and prove that some of the new lattices are locally optimal.Comment: (v3) 40 pages, 5 figures, 6 tables, some corrections, accepted in Discrete and Computational Geometry, see also http://fma2.math.uni-magdeburg.de/~latgeo

Rigidity and Fluidity in Living and Nonliving Matter

Many of the standard equilibrium statistical mechanics techniques do not readily apply to non-equilibrium phase transitions such as the fluid-to-disordered solid transition found in repulsive particulate systems. Examples of repulsive particulate systems are sand grains and colloids. The first part of this thesis contributes to methods beyond equilibrium statistical mechanics to ultimately understand the nature of the fluid-to-disordered solid transition, or jamming, from a microscopic basis. In Chapter 2 we revisit the concept of minimal rigidity as applied to frictionless, repulsive soft sphere packings in two dimensions with the introduction of the jamming graph. Minimal rigidity is a purely combinatorial property encoded via Laman\u27s theorem in two dimensions. It constrains the global, average coordination number of the graph, for instance. Minimal rigidity, however, does not address the geometry of local mechanical stability. The jamming graph contains both properties of global mechanical stability at the onset of jamming and local mechanical stability. We demonstrate how jamming graphs can be constructed using local rules via the Henneberg construction such that these graphs are of the constraint percolation type, where percolation is the study of connected structures in disordered networks. We then probe how jamming graphs destabilize, or become fluid-like, by deleting an edge/contact in the graph and computing the resulting rigid cluster distribution. We also uncover a new potentially diverging lengthscale associated with the random deletion of contacts. In Chapter 3 we study several constraint percolation models, such as k-core percolation and counter-balance percolation, on hyperbolic lattices to better understand the role of loops in such models. The constraints in these percolation models incorporate aspects of local mechanical rigidity found in jammed systems. The expectation is that since these models are indeed easier to analyze than the more complicated problem of jamming, we will gain insight into which constraints affect the nature of the jamming transition and which do not. We find that k = 3-core percolation on the hyperbolic lattice remains a continuous phase transition despite the fact that the loop structure of hyperbolic lattices is different from Euclidean lattices. We also contribute towards numerical techniques for analyzing percolation on hyperbolic lattices. In Chapters 4 and 5 we turn to living matter, which is also nonequilibrium in a very local way in that each constituent has its own internal energy supply. In Chapter 4 we study the fluidity of a cell moving through a confluent tissue, i.e. a group of cells with no gaps between them, via T1 transitions. A T1 transition allows for an edge swap so that a cell can come into contact with new neighbors. Cell migration is then generated by a sequence of such swaps. In a simple four cell system we compute the energy barriers associated with this transition. We then find that the energy barriers in a larger system are rather similar to the four cell case. The many cell case, however, more easily allows for the collection of statistics of these energy barriers given the disordered packings of cell observed in experiments. We find that the energy barriers are exponentially distributed. Such a finding implies that glassy dynamics is possible in a confluent tissue. Finally, in chapter 5 we turn to single cell migration in the extracellular matrix, another native environment of a cell. Experiments suggest that the migration of some cells in the three-dimensional ext ra cellular matrix bears strong resemblance to one-dimensional cell migration. Motivated by this observation, we construct and study a minimal one-dimensional model cell made of two beads and an active spring moving along a rigid track. The active spring models the stress fibers with their myosin-driven contractility and alpha-actinin-driven extendability, while the friction coefficients of the two beads describe the catch/slip bond behavior of the integrins in focal adhesions. Net motion arises from an interplay between active contractility (and passive extendability) of the stress fibers and an asymmetry between the front and back of the cell due to catch bond behavior of integrins at the front of the cell and slip bond behavior of integrins at the back. We obtain reasonable cell speeds with independently estimated parameters. Our model highlights the role of alpha-actinin in three-dimensional cell motility and does not require Arp2/3 actin filament nucleation for net motion

Stress Propagation through Frictionless Granular Material

We examine the network of forces to be expected in a static assembly of hard, frictionless spherical beads of random sizes, such as a colloidal glass. Such an assembly is minimally connected: the ratio of constraint equations to contact forces approaches unity for a large assembly. However, the bead positions in a finite subregion of the assembly are underdetermined. Thus to maintain equilibrium, half of the exterior contact forces are determined by the other half. We argue that the transmission of force may be regarded as unidirectional, in contrast to the transmission of force in an elastic material. Specializing to sequentially deposited beads, we show that forces on a given buried bead can be uniquely specified in terms of forces involving more recently added beads. We derive equations for the transmission of stress averaged over scales much larger than a single bead. This derivation requires the Ansatz that statistical fluctuations of the forces are independent of fluctuations of the contact geometry. Under this Ansatz, the $d(d+1)/2$-component stress field can be expressed in terms of a d-component vector field. The procedure may be generalized to non-sequential packings. In two dimensions, the stress propagates according to a wave equation, as postulated in recent work elsewhere. We demonstrate similar wave-like propagation in higher dimensions, assuming that the packing geometry has uniaxial symmetry. In macroscopic granular materials we argue that our approach may be useful even though grains have friction and are not packed sequentially.=17Comment: 15 pages, 4 figures, revised vertion for Phys. Rev.

Geometric origin of mechanical properties of granular materials

Some remarkable generic properties, related to isostaticity and potential energy minimization, of equilibrium configurations of assemblies of rigid, frictionless grains are studied. Isostaticity -the uniqueness of the forces, once the list of contacts is known- is established in a quite general context, and the important distinction between isostatic problems under given external loads and isostatic (rigid) structures is presented. Complete rigidity is only guaranteed, on stability grounds, in the case of spherical cohesionless grains. Otherwise, the network of contacts might deform elastically in response to load increments, even though grains are rigid. This sets an uuper bound on the contact coordination number. The approximation of small displacements (ASD) allows to draw analogies with other model systems studied in statistical mechanics, such as minimum paths on a lattice. It also entails the uniqueness of the equilibrium state (the list of contacts itself is geometrically determined) for cohesionless grains, and thus the absence of plastic dissipation. Plasticity and hysteresis are due to the lack of such uniqueness and may stem, apart from intergranular friction, from small, but finite, rearrangements, in which the system jumps between two distinct potential energy minima, or from bounded tensile contact forces. The response to load increments is discussed. On the basis of past numerical studies, we argue that, if the ASD is valid, the macroscopic displacement field is the solution to an elliptic boundary value problem (akin to the Stokes problem).Comment: RevTex, 40 pages, 26 figures. Close to published paper. Misprints and minor errors correcte