Carolina Brito

Geometric interpretation of previtrification in hard sphere liquids

We derive a microscopic criterion for the stability of hard sphere configurations and we show empirically that this criterion is marginally satisfied in the glass. This observation supports a geometric interpretation for the initial rapid rise in viscosity with packing fraction or previtrification. It also implies that barely stable soft modes characterize the glass structure, whose spatial extension is estimated. We show that both the short-term dynamics and activation processes occur mostly along those soft modes and we study some implications of these observations. This article synthesizes new and previous results [C. Brito and M. Wyart, Europhys. Lett. 76, 149 (2006); C. Brito and M. Wyart, J. Stat. Mech.: Theory Exp. 2007, L08003] in a unified view.

Heterogeneous dynamics, marginal stability and soft modes in hard sphere glasses

In a recent publication we established an analogy between the free energy of a hard sphere system and the energy of an elastic network. This result enables one to study the free energy landscape of hard spheres, which was previously accessible only via density functional theory. In our formalism normal modes can easily be defined and computed. In this work we use these tools to analyze the activated transitions between meta-basins, both in the ageing regime deep in the glass phase and near the glass transition. We observe numerically that structural relaxation occurs mostly along a very small number of nearly unstable extended modes. This number decays for denser packing and is significantly lowered as the system undergoes the glass transition. This observation supports the assertion that structural relaxation and marginal modes share common properties. In particular, theoretical results show that these modes extend at least on some length scale l*~(c−)−1/2 where c corresponds to the maximum packing fraction, i.e. the jamming transition. This prediction is consistent with very recent numerical observations of dynamical length scales in sheared systems near the jamming threshold, where a similar exponent is found, and with the commonly observed growth of the rearranging regions with compression near the glass transition.

On the rigidity of a hard-sphere glass near random close packing

We study theoretically and numerically the microscopic cause of the rigidity of hard-sphere glasses near their maximum packing. We show that, after coarse-graining over time, the hard-sphere interaction can be described by an effective potential which is exactly logarithmic at the random close packing c. This allows to define normal modes, and to apply recent results valid for elastic networks: rigidity is a non-local property of the packing geometry, and is characterized by some length scale l* which diverges at c (Wyart M., Nagel S. R. and Witten T. A., Europhys. Lett., 72 (2005) 486; Wyart M., Silbert L. E., Nagel S. R. and Witten T. A., Phys. Rev. E, 72 (2005) 051306). We compute the scaling of the bulk and shear moduli near c, and speculate on the possible implications of these results for the glass transition.

Force distribution affects vibrational properties in hard-sphere glasses

Significance How a liquid becomes rigid at the glass transition is a central problem in condensed matter physics. In many scenarios of the glass transition, liquids go through a critical temperature below which minima of free energy appear. However, even in the simplest glass, hard spheres, what confers mechanical stability at large density is highly debated. In this work we show that to quantitatively understand stability at a microscopic level, the presence of weakly interacting pairs of particles must be included. This approach allows us to predict various nontrivial scaling behavior of the elasticity and vibrational properties of colloidal glasses that can be tested experimentally. It also gives a spatial interpretation to recent, exact calculations in infinite dimensions. We theoretically and numerically study the elastic properties of hard-sphere glasses and provide a real-space description of their mechanical stability. In contrast to repulsive particles at zero temperature, we argue that the presence of certain pairs of particles interacting with a small force f soften elastic properties. This softening affects the exponents characterizing elasticity at high pressure, leading to experimentally testable predictions. Denoting ℙ(f)∼fθe, the force distribution of such pairs and ϕc the packing fraction at which pressure diverges, we predict that (i) the density of states has a low-frequency peak at a scale ω*, rising up to it as D(ω)∼ω2+a, and decaying above ω* as D(ω)∼ω−a where a=(1−θe)/(3+θe) and ω is the frequency, (ii) shear modulus and mean-squared displacement are inversely proportional with 〈δR2〉∼1/μ∼(ϕc−ϕ)κ, where κ=2−2/(3+θe), and (iii) continuum elasticity breaks down on a scale ℓc∼1/δz∼(ϕc−ϕ)−b, where b=(1+θe)/(6+2θe) and δz=z−2d, where z is the coordination and d the spatial dimension. We numerically test (i) and provide data supporting that θe≈0.41 in our bidisperse system, independently of system preparation in two and three dimensions, leading to κ≈1.41, a≈0.17, and b≈0.21. Our results for the mean-square displacement are consistent with a recent exact replica computation for d=∞, whereas some observations differ, as rationalized by the present approach.

Architecture and coevolution of allosteric materials

Significance In allosteric proteins, binding a ligand affects activity at a distant site. The physical principles allowing for such an action at a distance are not well understood. Here we introduce a numerical scheme to evolve allosteric materials in which the number of solutions, their spatial architectures, and the correlations among them can be computed. We show that allostery in these materials uses recently discovered elastic edge modes near the active site to transmit information, and that correlations generated during evolution alone can reveal key aspects of this architecture. We introduce a numerical scheme to evolve functional elastic materials that can accomplish a specified mechanical task. In this scheme, the number of solutions, their spatial architectures, and the correlations among them can be computed. As an example, we consider an “allosteric” task, which requires the material to respond specifically to a stimulus at a distant active site. We find that functioning materials evolve a less-constrained trumpet-shaped region connecting the stimulus and active sites, and that the amplitude of the elastic response varies nonmonotonically along the trumpet. As previously shown for some proteins, we find that correlations appearing during evolution alone are sufficient to identify key aspects of this design. Finally, we show that the success of this architecture stems from the emergence of soft edge modes recently found to appear near the surface of marginally connected materials. Overall, our in silico evolution experiment offers a window to study the relationship between structure, function, and correlations emerging during evolution.

Modeling of Droplet Evaporation on Superhydrophobic Surfaces.

When a drop of water is placed on a rough surface, there are two possible extreme regimes of wetting: the one called Cassie-Baxter (CB) with air pockets trapped underneath the droplet and the one called the Wenzel (W) state characterized by the homogeneous wetting of the surface. A way to investigate the transition between these two states is by means of evaporation experiments, in which the droplet starts in a CB state and, as its volume decreases, penetrates the surfaces grooves, reaching a W state. Here we present a theoretical model based on the global interfacial energies for CB and W states that allows us to predict the thermodynamic wetting state of the droplet for a given volume and surface texture. We first analyze the influence of the surface geometric parameters on the droplets final wetting state with constant volume and show that it depends strongly on the surface texture. We then vary the volume of the droplet, keeping the geometric surface parameters fixed to mimic evaporation and show that the drop experiences a transition from the CB to the W state when its volume reduces, as observed in experiments. To investigate the dependency of the wetting state on the initial state of the droplet, we implement a cellular Potts model in three dimensions. Simulations show very good agreement with theory when the initial state is W, but it disagrees when the droplet is initialized in a CB state, in accordance with previous observations which show that the CB state is metastable in many cases. Both simulations and the theoretical model can be modified to study other types of surfaces.

Extracting vibrational modes from fluctuations: a pedagogical discussion

The study of the jamming transition of granular and colloidal systems, has lead to a proliferation of theoretical and numerical results formulated in the language of the eigenspectrum of the dynamical matrix for these disordered systems. Only recently however, have these modes been accessed experimentally in colloidal and granular media, by computing the eigenmodes of the covariance matrix of the particle positions. At the same time, new conceptual and methodological questions regarding the interpretation of these results have appeared. In the present paper, we first give an overview of the theoretical framework which is appropriate to interpret the eigenmodes and eigenvalues of the correlation matrix in terms of the vibrational properties of these systems. We then illustrate several aspects of the statistical and data analysis techniques necessary to extract reliable results from experimental data. Concentrating on the cases of hard sphere simulations, colloidal and granular experiments, we discuss how to test, in turn, for the existence of a metastable state and the statistical independence of the sampling, the effect of experimental resolution, and the harmonic hypothesis underlying the approach; highlighting both the promises and limitations of this approach.

Percolation and cooperation with mobile agents: Geometric and strategy clusters

We study the conditions for persistent cooperation in an off-lattice model of mobile agents playing the Prisoners Dilemma game with pure, unconditional strategies. Each agent has an exclusion radius r(P), which accounts for the population viscosity, and an interaction radius r(int), which defines the instantaneous contact network for the game dynamics. We show that, differently from the r(P)=0 case, the model with finite-sized agents presents a coexistence phase with both cooperators and defectors, besides the two absorbing phases, in which either cooperators or defectors dominate. We provide, in addition, a geometric interpretation of the transitions between phases. In analogy with lattice models, the geometric percolation of the contact network (i.e., irrespective of the strategy) enhances cooperation. More importantly, we show that the percolation of defectors is an essential condition for their survival. Differently from compact clusters of cooperators, isolated groups of defectors will eventually become extinct if not percolating, independently of their size.

Orientational order at finite temperature on curved surfaces

We study the effect of thermal fluctuations in the XY model on surfaces with unequal principal curvatures. Unlike Gaussian curvature that typically frustrates orientational order, the extrinsic curvature of the surface can act as a local field that promotes long-range order at low temperature. We find numerically that the transition from the high temperature isotropic phase to the true long-range ordered phase is characterized by critical exponents consistent with those of the flat space Ising model in two dimensions, up to finite size effects. Our results suggest a versatile strategy to achieve geometric control of liquid crystal order by suitable design of the underlying curvature of a substrate.

Structural signatures of (two) characteristic dynamical temperatures in lithium metasilicate.

We report on the dynamic and structural characterization of lithium metasilicate Li2SiO3, a network forming ionic glass, by means of molecular dynamics simulations. The system is characterized by a network of SiO4 tetrahedra disrupted by Li ions which diffuse through the network. Measures of mean square displacement of Si and O atoms allow us to identify a temperature at which tetrahedra stop moving relative to each other. This temperature Tc ≈ 1500K can be characterized within the framework of mode coupling theory. At a much lower temperature Tg ≈ 1000K, a change in the slope of the volume versus temperature data allows to single out the glass transition. We find signatures of both transitions in structural order parameters, related to the orientation of tetrahedra. Going down in temperature we find that, around the mode coupling transition temperature, a set of order parameters which measure the relative orientation of tetrahedra cease to increase and stay constant below Tc. Another well known measure of orientational order, the bond orientational order parameter, which in the studied system measures local order within single tetrahedrons, is found to continue growing below Tc until Tg, below which it remains constant. Our results allow to relate two characteristic dynamic transitions with corresponding structural transitions, as observed in two different orientational order parameters. Furthermore, the results indicate that the network of thetrahedra continue to relax well below the point where neighboring tetrahedra cannot rearrange relative to each other, and the glass is reached only upon a process of relaxation of atoms which form the thetrahedron, as quantified by the change in the bond orientational order parameters.