Emergent power-law interactions in near-crystalline membranes.

We derive exact results for the fluctuations in energy produced by microscopic disorder in near-crystalline athermal systems. Our formalism captures the heterogeneity in the elastic energy of polydisperse soft disks in energy-minimized configurations. We use this to predict the distribution of interaction energy between two defects in a disordered background. We show that this interaction energy displays a disorder-averaged power-law behavior 〈δE〉∼Δ^{-4} at large distances Δ between the defects. These interactions upon disorder average also display the sixfold symmetry of the underlying reference crystal. Additionally, we show that the fluctuations in the interaction energy encode the athermal correlations introduced by the disordered background. We verify our predictions with energy-minimized configurations of polydisperse soft disks in two dimensions.

Pattern stabilization in swarms of programmable active matter: A probe for turbulence at large length scales.

We propose an algorithm for creating stable, ordered, swarms of active robotic agents arranged in any given pattern. The strategy involves suppressing a class of fluctuations known as "nonaffine" displacements, viz., those involving nonlinear deformations of a reference pattern, while all (or most) affine deformations are allowed. We show that this can be achieved using precisely calculated, fluctuating, thrust forces associated with a vanishing average power input. A surprising outcome of our study is that once the structure of the swarm is maintained at steady state, the statistics of the underlying flow field is determined solely from the statistics of the forces needed to stabilize the swarm.

Athermal Fluctuations in Disordered Crystals.

We analyze the fluctuations in particle positions and interparticle forces in disordered crystals composed of jammed soft particles in the limit of weak disorder. We demonstrate that such athermal systems are fundamentally different from their thermal counterparts, characterized by constrained fluctuations of forces perpendicular to the lattice directions. We develop a disorder perturbation expansion in polydispersity about the crystalline state, which we use to derive exact results to linear order. We show that constrained fluctuations result as a consequence of local force balance conditions, and are characterized by non-Gaussian distributions, which we derive exactly. We analytically predict several properties of such systems, including the scaling of the average coordination with polydispersity and packing fraction, which we verify with numerical simulations using soft disks with one-sided harmonic interactions.

Nonaffine Displacements Encode Collective Conformational Fluctuations in Proteins.

Identifying subtle conformational fluctuations underlying the dynamics of biomacromolecules is crucial for resolving their free energy landscape. We show that a collective variable, originally proposed for crystalline solids, is able to filter out essential macromolecular motions more efficiently than other approaches. While homogeneous or "affine" deformations of the biopolymer are trivial, biopolymer conformations are complicated by the occurrence of inhomogeneous or "nonaffine" displacements of atoms relative to their positions in the native structure. We show that these displacements encode functionally relevant conformations of macromolecules, and in combination with a formalism based upon time-structured independent component analysis, they quantitatively resolve the free energy landscape of a number of macromolecules of hierarchical complexity. The kinetics of conformational transitions among the basins can now be mapped within the framework of a Markov state model. The nonaffine modes, obtained by projecting out homogeneous fluctuations from the local displacements, are found to be responsible for local structural changes required for transitioning between pairs of macrostates.

Nucleation Theory for Yielding of Nearly Defect-Free Crystals: Understanding Rate Dependent Yield Points.

Experiments and simulations show that when an initially defect-free rigid crystal is subjected to deformation at a constant rate, irreversible plastic flow commences at the so-called yield point. The yield point is a weak function of the deformation rate, which is usually expressed as a power law with an extremely small nonuniversal exponent. We reanalyze a representative set of published data on nanometer sized, mostly defect-free Cu, Ni, and Au crystals in light of a recently proposed theory of yielding based on nucleation of stable stress-free regions inside the metastable rigid solid. The single relation derived here, which is not a power law, explains data covering 15 orders of magnitude in timescales.

Disorder-induced enhancement of local hexatic correlations in two-dimensional fluids.

Quenched disorder affects translational and orientational correlations in two-dimensional interacting particle systems. Such disorder always suppresses orientational order in crystalline states. Surprisingly, in fluid phases of particles interacting with a power-law repulsive interaction, increasing the strength of a quenched Gaussian random pinning potential appears to enhance hexatic order locally. We propose that nearby pairs of pinned particles lock in the relative orientation of neighbours around them, propagating hexatic orientational order across larger distances than in the unpinned fluid. We test this idea using Monte Carlo simulations of interacting particles in their fluid phase in two dimensions, where two of the particles are constrained to be permanently pinned at a fixed distance from each other. We use Voronoi tesselations of instantaneous particle configurations to demonstrate that these pinned particles create hexatic neighbourhoods that can extend well beyond the range of their separation. This structuring is enhanced when the particle density is increased and is most prominent in the vicinity of the liquid-solid transition. Testing these ideas experimentally using experiments on 2D colloidal systems should be feasible.

Nonequilibrium Biophysical Processes Influence the Large-Scale Architecture of the Cell Nucleus.

Model approaches to nuclear architecture have traditionally ignored the biophysical consequences of ATP-fueled active processes acting on chromatin. However, transcription-coupled activity is a source of stochastic forces that are substantially larger than the Brownian forces present at physiological temperatures. Here, we describe an approach to large-scale nuclear architecture in metazoans that incorporates cell-type-specific active processes. The model predicts the statistics of positional distributions, shapes, and overlaps of each chromosome. Simulations of the model reproduce common organizing principles underlying large-scale nuclear architecture across human cell nuclei in interphase. These include the differential positioning of euchromatin and heterochromatin, the territorial organization of chromosomes (including both gene-density-based and size-based chromosome radial positioning schemes), the nonrandom locations of chromosome territories, and the shape statistics of individual chromosomes. We propose that the biophysical consequences of the distribution of transcriptional activity across chromosomes should be central to any chromosome positioning code.

Exploring the link between crystal defects and nonaffine displacement fluctuations.

We generalize and then use a recently introduced formalism to study thermal fluctuations of atomic displacements in several two- and three-dimensional crystals. We study both close-packed and open crystals with multiatom bases. Atomic displacement fluctuations in a solid, once coarse grained over some neighborhood, may be decomposed into two mutually orthogonal components. In any dimension d there are always d^{2} affine displacements representing local strains and rotations of the ideal reference configuration. In addition, there exist a number of nonaffine localized displacement modes that cannot be represented as strains or rotations. The number of these modes depends on d and the size of the coarse-graining region. All thermodynamic averages and correlation functions concerning the affine and nonaffine displacements may be computed within harmonic theory. We show that for compact crystals, such as the square and triangular crystals in d=2 and the simple body-centered-cubic and face-centered-cubic crystals in d=3, a single set of d-fold degenerate modes always dominates the nonaffine subspace and is separated from the rest by a large gap. These modes may be identified with specific precursor configurations that lead to lattice defects. In open crystals, such as the honeycomb and kagome lattices, there is no prominent gap, although soft nonaffine modes continue to be associated with known floppy modes representing localized defects. Higher-order coupling between affine and nonaffine components of the displacements quantifies the tendency of the lattice to be destroyed by large homogeneous strains. We show that this coupling is larger by almost an order of magnitude for open lattices as compared to compact ones. Deformation mechanisms such as lattice slips and stacking faults in close-packed crystals can also be understood within this framework. The qualitative features of these conclusions are expected to be independent of the details of the atomic interactions.

Orientational correlations in fluids with quenched disorder.

Snapshots of colloidal particles moving on disordered two-dimensional substrates can be used to extract equal-time many-body correlations in their positions. To understand the systematics of these correlations, we perform Monte Carlo simulations of a two-dimensional model fluid placed in a quenched disordered background. We use configurations generated from these simulations to compute translational and orientational two-point correlations at equal time, concentrating on correlations in local orientational order as a function of density and disorder strength. We calculate both the disorder averaged version of conventional two-point correlation functions for orientational order, as well as the disorder averaged version of a novel correlation function of time-averaged disorder-induced inhomogeneities in local orientation analogous to the Edwards-Anderson correlation function in spin systems. We demonstrate that these correlations can exhibit interesting nonmonotonic behavior in proximity to the underlying fluid-solid transition and suggest that this prediction should be experimentally accessible.

On identifying collective displacements in apo-proteins that reveal eventual binding pathways.

Binding of small molecules to proteins often involves large conformational changes in the latter, which open up pathways to the binding site. Observing and pinpointing these rare events in large scale, all-atom, computations of specific protein-ligand complexes, is expensive and to a great extent serendipitous. Further, relevant collective variables which characterise specific binding or un-binding scenarios are still difficult to identify despite the large body of work on the subject. Here, we show that possible primary and secondary binding pathways can be discovered from short simulations of the apo-protein without waiting for an actual binding event to occur. We use a projection formalism, introduced earlier to study deformation in solids, to analyse local atomic displacements into two mutually orthogonal subspaces-those which are "affine" i.e. expressible as a homogeneous deformation of the native structure, and those which are not. The susceptibility to non-affine displacements among the various residues in the apo- protein is then shown to correlate with typical binding pathways and sites crucial for allosteric modifications. We validate our observation with all-atom computations of three proteins, T4-Lysozyme, Src kinase and Cytochrome P450.

Plastic deformation of a permanently bonded network: Stress relaxation by pleats.

We show that a flat two dimensional network of connected vertices, when stretched, may deform plastically by producing "pleats", system spanning linear structures with width comparable to the lattice spacing, where the network overlaps on itself. To understand the pleating process, we introduce an external field that couples to local non-affine displacements, i.e., those displacements of neighbouring vertices that cannot be represented as a local affine strain. We obtain both zero and finite temperature phase diagrams in the strain-field plane. Pleats occur here as a result of an equilibrium first-order transition from the homogeneous network to a heterogeneous phase where stress is localised within pleats and eliminated elsewhere. We show that in the thermodynamic limit, the un-pleated state is always metastable at vanishing field for infinitesimal strain. Plastic deformation of the initially homogeneous network is akin to the decay of a metastable phase via a dynamical transition. We make predictions concerning local stress distributions and thermal effects associated with pleats which may be observable in suitable experimental systems.

On the existence of thermodynamically stable rigid solids.

Customarily, crystalline solids are defined to be rigid since they resist changes of shape determined by their boundaries. However, rigid solids cannot exist in the thermodynamic limit where boundaries become irrelevant. Particles in the solid may rearrange to adjust to shape changes eliminating stress without destroying crystalline order. Rigidity is therefore valid only in the metastable state that emerges because these particle rearrangements in response to a deformation, or strain, are associated with slow collective processes. Here, we show that a thermodynamic collective variable may be used to quantify particle rearrangements that occur as a solid is deformed at zero strain rate. Advanced Monte Carlo simulation techniques are then used to obtain the equilibrium free energy as a function of this variable. Our results lead to a unique view on rigidity: While at zero strain a rigid crystal coexists with one that responds to infinitesimal strain by rearranging particles and expelling stress, at finite strain the rigid crystal is metastable, associated with a free energy barrier that decreases with increasing strain. The rigid phase becomes thermodynamically stable when an external field, which penalizes particle rearrangements, is switched on. This produces a line of first-order phase transitions in the field-strain plane that intersects the origin. Failure of a solid once strained beyond its elastic limit is associated with kinetic decay processes of the metastable rigid crystal deformed with a finite strain rate. These processes can be understood in quantitative detail using our computed phase diagram as reference.

Translationally invariant colloidal crystal templates.

We show that dynamic, feed-back controlled optical traps, whose positions depend on the instantaneous local configuration of particles in a pre-determined way, can stabilise colloidal particles in finite lattices of any given symmetry. Unlike in a static template, the crystal so formed is invariant under uniform translations and retains all possible zero energy modes. We demonstrate this in silico by stabilising the unstable two-dimensional square lattice in a model soft solid with isotropic interactions.

Contrasting the dynamics of elastic and non-elastic deformations across an experimental colloidal Martensitic transition.

We present a framework to segregate the roles of elastic and non-elastic deformations in the examination of real-space experiments of solid-solid Martensitic transitions. The Martensitic transformation of a body-centred-tetragonal (BCT) to a body-centred-orthorhombic (BCO) crystal structure has been studied in a model system of micron-scale ionic microgel colloids (P. S. Mohanty, P. Bagheri, S. Nöjd, A. Yethiraj and P. Schurtenberger, Phys. Rev. X, 2015, 5, 011030). Non-affine fluctuations, i.e., displacement fluctuations that do not arise from purely elastic (affine) deformations, are detected in particle configurations acquired from the experiment. Tracking these fluctuations serves as a highly sensitive tool in signaling the onset of the Martensitic transition and precisely locating particle rearrangements occurring at length scales of a few particle diameters. Particle rearrangements associated with non-affine displacement modes become increasingly favorable during the transformation process. The nature of the displacement fluctuation modes that govern the transformation are shown to be different from those predominant in an equilibrium crystal. We show that BCO crystallites formed through shear may, remarkably, co-exist with those resulting from local rearrangements within the same sample.

Equilibrium and dynamic pleating of a crystalline bonded network.

We describe a phase transition that gives rise to structurally non-trivial states in a two-dimensional ordered network of particles connected by harmonic bonds. Monte Carlo simulations reveal that the network supports, apart from the homogeneous phase, a number of heterogeneous "pleated" phases, which can be stabilised by an external field. This field is conjugate to a global collective variable quantifying "non-affineness," i.e., the deviation of local particle displacements from local affine deformation. In the pleated phase, stress is localised in ordered rows of pleats and eliminated from the rest of the lattice. The kinetics of the phase transition is unobservably slow in molecular dynamics simulation near coexistence, due to very large free energy barriers. When the external field is increased further to lower these barriers, the network exhibits rich dynamic behaviour: it transforms into a metastable phase with the stress now localised in a disordered arrangement of pleats. The pattern of pleats shows ageing dynamics and slow relaxation to equilibrium. Our predictions may be checked by experiments on tethered colloidal solids in dynamic laser traps.

Colloids exposed to random potential energy landscapes: From particle number density to particle-potential and particle-particle interactions.

Colloidal particles were exposed to a random potential energy landscape that has been created optically via a speckle pattern. The mean particle density as well as the potential roughness, i.e., the disorder strength, were varied. The local probability density of the particles as well as its main characteristics were determined. For the first time, the disorder-averaged pair density correlation function g((1))(r) and an analogue of the Edwards-Anderson order parameter g((2))(r), which quantifies the correlation of the mean local density among disorder realisations, were measured experimentally and shown to be consistent with replica liquid state theory results.

Excess vibrational density of states and the brittle to ductile transition in crystalline and amorphous solids.

The conditions which determine whether a material behaves in a brittle or ductile fashion on mechanical loading are still elusive and comprise a topic of active research among materials physicists and engineers. In this study, we present the results of in silico mechanical deformation experiments from two very different model solids in two and three dimensions. The first consists of particles interacting with isotropic potentials and the other has strongly direction dependent interactions. We show that in both cases, the excess vibrational density of states is one of the fundamental quantities which characterizes the ductility of the material. Our results can be checked using careful experiments on colloidal solids.

Structural phase transitions in trigonal Selenium induce the formation of a disordered phase.

Arguments based on the Mermin-Wagner theorem suggest that the quasi-1D trigonal phase of Se should be unstable against long wavelength perturbations. Consisting of parallel Se-Se chains, this essentially fragile solid undergoes a partial transition to a monoclinic structure (consisting of 8-membered rings) at low temperatures (≈50 K), and to a distorted trigonal phase at moderate pressures (≈3GPa). Experimental investigations on sub-millimeter-sized single crystals provide clear evidence that these transitions occur via a novel and counter-intuitive route. This involves the reversible formation of an intermediate, disordered structure that appears as a minority phase with increasing pressure as well as with decreasing temperature. The formation of the disordered state is indicated by: (a) a 'Boson-peak' that appears at low temperatures in the specific heat and resonance Raman data, and (b) a decrease in the intensity of Raman lines over a relatively narrow pressure range. We complement the experimental results with a phenomenological model that illustrates how a first order structural transition may lead to disorder. Interestingly, nanocrystals of trigonal Se do not undergo any structural transition in the parameter space studied; neither do they exhibit signs of disorder, further underlining the role of disorder in this type of structural transition.

Pre-yield non-affine fluctuations and a hidden critical point in strained crystals.

A crystalline solid exhibits thermally induced localised non-affine droplets in the absence of external stress. Here we show that upon an imposed shear, the size of these droplets grow until they percolate at a critical strain, well below the value at which the solid begins to yield. This critical point does not manifest in most thermodynamic or mechanical properties, but is hidden and reveals itself in the onset of inhomogeneities in elastic moduli, marked changes in the appearance and local properties of non-affine droplets and a sudden enhancement in defect pair concentration. Slow relaxation of stress and an-elasticity appear as observable dynamical consequences of this hidden criticality. Our results may be directly verified in colloidal crystals with video microscopy techniques but are expected to have more general validity.