Exceptionally dense and resilient critically jammed polydisperse disk packings.

Understanding the way disordered particle packings transition between jammed (rigid) and unjammed (fluid) states is of both great practical importance and strong fundamental interest. The values of critical packing fraction (and other state variables) at the jamming transition are protocol dependent. Here, we demonstrate that this variability can be systematically traced to structural measures of packing, as well as to energy measures inside the jammed regime. A novel generalized simultaneous particle swap algorithm constructs overjammed states of desired energy, which upon decompression lead to predictable critical packing fractions. Thus, for a given set of particle sizes, states with extraordinarily high critical packing fractions can be found efficiently, which sustain substantial shear strain and preserve their special structure over the entire jammed domain. The close relation revealed here between the energy landscape of overjammed soft-particle packings and the behavior near the jamming transition points towards new ways of understanding and constructing disordered materials with exceptional properties.

DNA self-organization controls valence in programmable colloid design.

Just like atoms combine into molecules, colloids can self-organize into predetermined structures according to a set of design principles. Controlling valence-the number of interparticle bonds-is a prerequisite for the assembly of complex architectures. The assembly can be directed via solid "patchy" particles with prescribed geometries to make, for example, a colloidal diamond. We demonstrate here that the nanoscale ordering of individual molecular linkers can combine to program the structure of microscale assemblies. Specifically, we experimentally show that covering initially isotropic microdroplets with N mobile DNA linkers results in spontaneous and reversible self-organization of the DNA into Z(N) binding patches, selecting a predictable valence. We understand this valence thermodynamically, deriving a free energy functional for droplet-droplet adhesion that accurately predicts the equilibrium size of and molecular organization within patches, as well as the observed valence transitions with N Thus, microscopic self-organization can be programmed by choosing the molecular properties and concentration of binders. These results are widely applicable to the assembly of any particle with mobile linkers, such as functionalized liposomes or protein interactions in cell-cell adhesion.

An unrecognized inertial force induced by flow curvature in microfluidics.

Modern inertial microfluidics routinely employs oscillatory flows around localized solid features or microbubbles for controlled, specific manipulation of particles, droplets, and cells. It is shown that theories of inertial effects that have been state of the art for decades miss major contributions and strongly underestimate forces on small suspended objects in a range of practically relevant conditions. An analytical approach is presented that derives a complete set of inertial forces and quantifies them in closed form as easy-to-use equations of motion, spanning the entire range from viscous to inviscid flows. The theory predicts additional attractive contributions toward oscillating boundaries, even for density-matched particles, a previously unexplained experimental observation. The accuracy of the theory is demonstrated against full-scale, three-dimensional direct numerical simulations throughout its range.

Structural Measures as Guides to Ultrastable States in Overjammed Packings.

Jammed, disordered packings of given sets of particles possess a multitude of equilibrium states with different mechanical properties. Identifying and constructing desired states, e.g., of superior stability, is a complex task. Here, we show that in two-dimensional particle packings the energy of all metastable states (inherent structures) is reliably classified by simple scalar measures of local steric packing. These structural measures are insensitive to the particle interaction potential and so robust that they can be used to guide a modified swap algorithm that anneals polydisperse packings toward low-energy metastable states exceptionally fast. The low-energy states are extraordinarily stable against applied shear, so that the approach also efficiently identifies ultrastable packings.

Predicting the characteristics of defect transitions on curved surfaces.

The energetically optimal position of lattice defects on intrinsically curved surfaces is a complex function of shape parameters. For open surfaces, a simple condition predicts the critical size for which a central disclination yields lower energy than a boundary disclination. In practice, this transition is modified by activation energies or more favorable intermediate defect positions. Here it is shown that these transition characteristics (continuous or discontinuous, first or second order) can also be inferred from analytical, general criteria evaluated from the surface shape. A universal scale of activation energy is found, and the criteria are generalized to predict transition order as surface shape symmetry is broken. The results give practical insight into structural transitions to disorder in many cellular materials of technological and biological importance.

Simple, General Criterion for Onset of Disclination Disorder on Curved Surfaces.

Determining the positions of lattice defects on bounded elastic surfaces with Gaussian curvature is a nontrivial task of mechanical energy optimization. We introduce a simple way to predict the onset of disclination disorder from the shape of the surface. The criterion fixes the value of a weighted integral Gaussian curvature to a universal constant and proves accurate across a great variety of shapes. It provides improved understanding of the limitations to crystalline order in many natural and engineering contexts, such as the assembly of viral capsids.

A simple landscape of metastable state energies for two-dimensional cellular matter.

The mechanical behavior of cellular matter in two dimensions can be inferred from geometric information near its energetic ground state. Here it is shown that the much larger set of all metastable state energies is universally described by a systematic expansion in moments of the joint probability distribution of size (area) and topology (number of neighbors). The approach captures bounds to the entire range of metastable state energies and quantitatively identifies any such state. The resulting energy landscape is invariant across different classes of energy functionals, across simulation techniques, and across system polydispersities. The theory also finds a threshold in tissue adhesion beyond which no metastable states are possible. Mechanical properties of cellular matter in biological and technological applications can thus be identified by visual information only.

Universal Features of Metastable State Energies in Cellular Matter.

Mechanical equilibrium states of cellular matter are overwhelmingly metastable and separated from each other by topology changes. Using theory and simulations, it is shown that for a wide class of energy functionals in 2D, including those describing tissue cell layers, local energy differences between neighboring metastable states as well as global energy differences between initial states and ground states are governed by simple, universal relations. Knowledge of instantaneous length of an edge undergoing a T1 transition is sufficient to predict local energy changes, while the initial edge length distribution yields a successful prediction for the global energy difference. An analytical understanding of the model parameters is provided.

Hexagonal Patterning of the Insect Compound Eye: Facet Area Variation, Defects, and Disorder.

The regular hexagonal array morphology of facets (ommatidia) in the Drosophila compound eye is accomplished by regulation of cell differentiation and planar cell polarity during development. Mutations in certain genes disrupt regulation, causing a breakdown of this perfect symmetry, so that the ommatidial pattern shows onset of disorder in the form of packing defects. We analyze a variety of such mutants and compare them to normal (wild-type), finding that mutants show increased local variation in ommatidial area, which is sufficient to induce a significant number of defects. A model formalism based on Voronoi construction is developed to predict the observed correlation between ommatidium size variation and the number of defects, and to study the onset of disorder in this system with statistical tools. The model uncovers a previously unknown large-scale systematic size variation of the ommatidia across the eye of both wild-type and mutant animals. Such systematic variation of area, as well as its statistical fluctuations, are found to have distinct effects on eye disorder that can both be quantitatively modeled. Furthermore, the topological order is also influenced by the internal structure of the ommatidia, with cells of greater relative mechanical stiffness providing constraints to ommatidial deformation and thus to defect generation. Without free parameters, the simulation predicts the size-topology correlation for both wild-type and mutant eyes. This work develops formalisms of size-topology correlation that are very general and can be potentially applied to other cellular structures near the onset of disorder.

Cell shapes and patterns as quantitative indicators of tissue stress in the plant epidermis.

In a confluent, single-cell tissue layer, we show that cell shapes and statistics correlate directly with the tissue's mechanical properties, described by an energy functional with generic interfacial terms only. Upon increasing the cohesive component of the model, we observe a clear transition from a tense state with isotropic cells to a relaxed state with anisotropic cells. Signatures of the transition are present in the interfacial mechanics, the domain geometry, and the domain statistics, thus linking all three fields of study. This transition persists for all cell size distributions, but its exact position is crucially dependent on fluctuations in the parameter values of the functional (quenched disorder). The magnitude of fluctuations can be matched to the observed shape distribution of cells, so that visual observation of cell shapes and statistics provides information about the mechanical state of the tissue. Comparing with experimental data from the Cucumis epidermis, we find that the system is located right at the transition, allowing the tissue to relieve most of the local stress while maintaining integrity.

Heterogeneous vesicles: an analytical approach to equilibrium shapes.

We develop an analytical model to predict equilibrium shapes of two-component heterogeneous vesicles or capsules. Using a free energy functional including the bending energies of the two components and line tension contributions, the model describes shape transitions between spherical and polyhedral (faceted) states, complementing and extending results of previous numerical simulations. In the parameter space of relative area fraction, bending modulus ratio, and line tension, a region of polyhedral shapes occurs for weak line tension and large bending modulus ratio and is very robust towards changes in the modeling assumptions. At large enough line tension, the spherical shape fragments into two components. Within the polyhedral region, we compare the energies of all regular and semiregular polyhedra, as well as those of arbitrary prismatic shapes. We find that the largest bending modulus contrasts together with larger line tension favor polyhedra with small face number as optimal shapes. In this region, we also demonstrate the counter-intuitive result that the most symmetric polyhedra are not energetically optimal, with specific Archimedean solids and specific prismatic shapes beating more isotropic (e.g. Platonic) polyhedra. Furthermore, all polyhedra of lowest energy are found to be three-fold coordinated. The shape transition boundary for polyhedra can be computed analytically. The model can be utilized to predict heterogeneous vesicle shapes and to estimate physical properties of the components constituting observed vesicles.

Lewis' law revisited: the role of anisotropy in size-topology correlations.

Since F T Lewis' pioneering work in the 1920s, a linear correlation between the average in-plane area of domains in a two-dimensional (2D) cellular structure and the number of neighbors of the domains has been empirically proposed, with many supporting and dissenting findings in the ensuing decades. Revisiting Lewis' original experiment, we take a larger set of more detailed data on the cells in the epidermal layer of Cucumis, and analyze the data in the light of recent results on size-topology correlations. We find that the correlation between the number-of-neighbor distribution (topology) and the area distribution is altered over that of many other 2D cellular systems (such as foams or disc packings), and that the systematic deviation can be explained by the anisotropic shape of the Cucumis cells. We develop a novel theory of size-topology correlation taking into account the characteristic aspect ratio of the cells within the framework of a granocentric model, and show that both Lewis' and our experimental data is consistent with the theory. In contrast to the granocentric model for isotropic domains, the new theory results in an approximately linear correlation consistent with Lewis' law. These statistical effects can be understood from the increased number of configurations available to a plane-filling domain system with non-isotropic elements, for the first time providing a firm explanation of why Lewis' law is valid in some systems and fails in others.

Analytical results for size-topology correlations in 2D disk and cellular packings.

Random tilings or packings in the plane are characterized by a size distribution of individual elements (domains) and by the statistics of neighbor relations between the domains. Most systems occurring in nature or technology have a unimodal distribution of both areas and number of neighbors. Empirically, strong correlations between these distributions have been observed and formulated as universal laws. Using only the local, correlation-free granocentric model approach with no free parameters, we construct accurate analytical descriptions for disk crystallization, size-topology correlations, and Lemaître's law.

Cadherin-dependent cell morphology in an epithelium: constructing a quantitative dynamical model.

Cells in the Drosophila retina have well-defined morphologies that are attained during tissue morphogenesis. We present a computer simulation of the epithelial tissue in which the global interfacial energy between cells is minimized. Experimental data for both normal cells and mutant cells either lacking or misexpressing the adhesion protein N-cadherin can be explained by a simple model incorporating salient features of morphogenesis that include the timing of N-cadherin expression in cells and its temporal relationship to the remodeling of cell-cell contacts. The simulations reproduce the geometries of wild-type and mutant cells, distinguish features of cadherin dynamics, and emphasize the importance of adhesion protein biogenesis and its timing with respect to cell remodeling. The simulations also indicate that N-cadherin protein is recycled from inactive interfaces to active interfaces, thereby modulating adhesion strengths between cells.

Physical modeling of cell geometric order in an epithelial tissue.

In multicellular organisms, cells pack together to form tissues of intricate and well defined morphology. How such cell-packing geometries arise is an important open question in biology, because the functionality of many differentiated tissues depends on their reliable formation. We show that combining adhesive forces due to E- and N-cadherin with a quantitative description of cell membrane elasticity in an interfacial energy model explains not only the qualitative neighbor relations, but also the detailed geometry of a tissue. The characteristic cellular geometries in the eyes of both wild-type Drosophila and genetic mutants are accurately reproduced by using a fixed set of few, physically motivated parameters. The model predicts adhesion strengths in the eye epithelium, quantifies their role relative to membrane elasticity, and reveals how simple minimization of interfacial energy can give rise to complex geometric patterns of important biological functionality.

Particle migration and sorting in microbubble streaming flows.

Ultrasonic driving of semicylindrical microbubbles generates strong streaming flows that are robust over a wide range of driving frequencies. We show that in microchannels, these streaming flow patterns can be combined with Poiseuille flows to achieve two distinctive, highly tunable methods for size-sensitive sorting and trapping of particles much smaller than the bubble itself. This method allows higher throughput than typical passive sorting techniques, since it does not require the inclusion of device features on the order of the particle size. We propose a simple mechanism, based on channel and flow geometry, which reliably describes and predicts the sorting behavior observed in experiment. It is also shown that an asymptotic theory that incorporates the device geometry and superimposed channel flow accurately models key flow features such as peak speeds and particle trajectories, provided it is appropriately modified to account for 3D effects caused by the axial confinement of the bubble.

Growth control of sessile microbubbles in PDMS devices.

In a microfluidic environment, the presence of bubbles is often detrimental to the functionality of the device, leading to clogging or cavitation, but microbubbles can also be an indispensable asset in other applications such as microstreaming. In either case, it is crucial to understand and control the growth or shrinkage of these bodies of air, in particular in common soft-lithography devices based on polydimethylsiloxane (PDMS), which is highly permeable to gases. In this work, we study the gas transport into and out of a bubble positioned in a microfluidic device, taking into account the direct gas exchange through PDMS as well as the transport of gas through the liquid in the device. Hydrostatic pressure regulation allows for the quantitative control of growth, shrinkage, or the attainment of a stable equilibrium bubble size. We find that the vapor pressure of the liquid plays an important role for the balance of gas transport, accounting for variability in experimental conditions and suggesting additional means of bubble size control in applications.

Ultrasound-induced microbubble coalescence.

We studied the interaction of ultrasound contrast agent bubbles coated with a layer of lipids, driven by 0.5 MHz ultrasound. High-speed photography on the submicrosecond timescale reveals that some bubbles bounce off each other, while others show very fast coalescence during bubble expansion. This fast coalescence cannot be explained by dissipation-limited film drainage rates. We conclude that the lipid shell ruptures upon expansion, exposing clean free bubble interfaces that support plug flow profiles in the film and inertia-limited drainage whose time scales match those of the observed coalescence.

Drainage of single Plateau borders: direct observation of rigid and mobile interfaces.

Foam drainage varies with surfactant. We present direct measurements of the flow velocity profiles across single Plateau borders, which make up the interconnected channel-like network for liquid flow. For protein foams the interface is rigid, whereas small-surfactant foams show significant interfacial mobility. The results agree with a model that takes into account the shearing of the liquid-gas interface transverse to the flow direction. A significant consequence is that bubble size and liquid volume fraction in a foam affect the relative importance of surface rheology on the drainage behavior.

Dissolution without disappearing: multicomponent gas exchange for CO2 bubbles in a microfluidic channel.

We studied the dissolution dynamics of CO2 gas bubbles in a microfluidic channel, both experimentally and theoretically. In the experiments, spherical CO2 bubbles in a flow of a solution of sodium dodecyl sulfate (SDS) first shrink rapidly before attaining an equilibrium size. In the rapid dissolution regime, the time to obtain a new equilibrium is 30 ms regardless of SDS concentration, and the equilibrium radius achieved varies with the SDS concentration. To explain the lack of complete dissolution, we interpret the results by considering the effects of other gases (O2, N2) that are already dissolved in the aqueous phase, and we develop a multicomponent dissolution model that includes the effect of surface tension and the liquid pressure drop along the channel. Solutions of the model for a stationary gas bubble show good agreement with the experimental results, which lead to our conclusion that the equilibrium regime is obtained by gas exchange between the bubbles and liquid phase. Also, our observations from experiments and model calculations suggest that SDS molecules on the gas-liquid interface form a diffusion barrier, which controls the dissolution behaviour and the eventual equilibrium radius of the bubble.