Granular solids transmit stress as two-phase composites.

A basic problem in the science of realistic granular matter is the plethora of heuristic models of the stress field in the absence of a first-principles theory. Such a theory is formulated here, based on the idea that static granular assemblies can be regarded as two-phase composites. A thought experiment is described, demonstrating that the state of such materials can be varied continuously from marginal stability, via a two-phase granular assembly, then porous structure, and finally be made perfectly elastic. For completeness, I review briefly the condition for marginal stability in infinitely large assemblies. The general solution for the stress equations in d=2 is reviewed in detail and shown to be consistent with the two-phase idea. A method for identifying the phases of finite regions in larger systems is constructed, providing a stability parameter that quantifies the "proximity" to the marginally stable state. The difficulty involved in deriving stress fields in such composites is a unique constraint on the boundary between phases, and, to highlight it, a simple case of a stack of plates of alternating phase is solved explicitly. An effective medium approximation, which satisfies this constraint, is then developed and analyzed in detail. This approach forms a basis for the extension of the stress theory to general granular solids that are not marginally stable or at the yield threshold.

'Phase transitions' in bacteria - From structural transitions in free living bacteria to phenotypic transitions in bacteria within biofilms.

Phase transitions are common in inanimate systems and have been studied extensively in natural sciences. Less explored are the rich transitions that take place at the micro- and nano-scales in biological systems. In conventional phase transitions, large-scale properties of the media change discontinuously in response to continuous changes in external conditions. Such changes play a significant role in the dynamic behaviours of organisms. In this review, we focus on some transitions in both free-living and biofilms of bacteria. Particular attention is paid to the transitions in the flagellar motors and filaments of free-living bacteria, in cellular gene expression during the biofilm growth, in the biofilm morphology transitions during biofilm expansion, and in the cell motion pattern transitions during the biofilm formation. We analyse the dynamic characteristics and biophysical mechanisms of these phase transition phenomena and point out the parallels between these transitions and conventional phase transitions. We also discuss the applications of some theoretical and numerical methods, established for conventional phase transitions in inanimate systems, in bacterial biofilms.

Cell surface fluctuations regulate early embryonic lineage sorting.

In development, lineage segregation is coordinated in time and space. An important example is the mammalian inner cell mass, in which the primitive endoderm (PrE, founder of the yolk sac) physically segregates from the epiblast (EPI, founder of the fetus). While the molecular requirements have been well studied, the physical mechanisms determining spatial segregation between EPI and PrE remain elusive. Here, we investigate the mechanical basis of EPI and PrE sorting. We find that rather than the differences in static cell surface mechanical parameters as in classical sorting models, it is the differences in surface fluctuations that robustly ensure physical lineage sorting. These differential surface fluctuations systematically correlate with differential cellular fluidity, which we propose together constitute a non-equilibrium sorting mechanism for EPI and PrE lineages. By combining experiments and modeling, we identify cell surface dynamics as a key factor orchestrating the correct spatial segregation of the founder embryonic lineages.

Disorder Criterion and Explicit Solution for the Disc Random Packing Problem.

Predicting the densest random disc packing fraction is an unsolved paradigm problem relevant to a number of disciplines and technologies. One difficulty is that it is ill defined without setting a criterion for the disorder. Another is that the density depends on the packing protocol and the multitude of possible protocol parameters has so far hindered a general solution. A new approach is proposed here. After formulating a well-posed form of the general protocol-independent problem for planar packings of discs, a systematic criterion is proposed to avoid crystalline hexagonal order as well as further topological order. The highest possible random packing fraction is then derived exactly: ϕ_{RCP}=0.852 525…. The solution is based on the cell order distribution that is shown to (i) yield directly the packing fraction; (ii) parametrize all possible packing protocols; (iii) make it possible to define and limit all topological disorder. The method is further useful for predicting the highest packing fraction in specific protocols, which is illustrated for a family of simply sheared packings that generate maximum-entropy cell order distributions.

Erratum: Disorder Criterion and Explicit Solution for the Disc Random Packing Problem [Phys. Rev. Lett. 127, 118002 (2021)].

This corrects the article DOI: 10.1103/PhysRevLett.127.118002.

Friction-Controlled Entropy-Stability Competition in Granular Systems.

Using cyclic shear to drive a two-dimensional granular system, we determine the structural characteristics for different interparticle friction coefficients. These characteristics are the result of a competition between mechanical stability and entropy, with the latter's effect increasing with friction. We show that a parameter-free maximum-entropy argument alone predicts an exponential cell order distribution, with excellent agreement with the experimental observation. We show that friction only tunes the mean cell order and, consequently, the exponential decay rate and the packing fraction. We further show that cells, which can be very large in such systems, are short-lived, implying that our systems are liquidlike rather than glassy.

Force-based three-dimensional model predicts mechanical drivers of cell sorting.

Many biological processes, including tissue morphogenesis, are driven by cell sorting. However, the primary mechanical drivers of sorting in multicellular aggregates (MCAs) remain controversial, in part because there is no appropriate computational model to probe mechanical interactions between cells. To address this important issue, we developed a three-dimensional, local force-based simulation based on the subcellular element method. In our method, cells are modelled as collections of locally interacting force-bearing elements. We use the method to investigate the effects of tension and cell-cell adhesion on MCA sorting. We predict a minimum level of adhesion to produce inside-out sorting of two cell types, which is in excellent agreement with observations in several developmental systems. We also predict the level of tension asymmetry needed for robust sorting. The generality and flexibility of the method make it applicable to tissue self-organization in a myriad of other biological processes, such as tumorigenesis and embryogenesis.

Support of modified Archimedes' law theory in granular media.

We study the resistance force of cylindrical objects penetrating quasi-statically into granular media experimentally and numerically. Simulations are validated against experiments. In contrast to previous studies, we find in both experiments and simulations that the force-depth relation consists of three regimes, rather than just two: transient and steady-state. The three regimes are driven by different dynamics: an initial matter compression, a developing stagnant zone, and an increase in steady-state force with a fully developed stagnant zone. By simulations, we explored the effects of a wide range of parameters on the penetration dynamics. We find that the initial packing fraction, the inter-granular sliding friction coefficient, and the grain shape (aspect ratio) have a significant effect on the gradient Kφ of the force-depth relation in the steady-state regime, while the rolling friction coefficient noticeably affects only the initial compression regime. Conversely, Kφ is not sensitive to the following grain properties: size, size distribution, shear modulus, density, and coefficient of restitution. From the stress fields observed in the simulations, we determine the internal friction angles φ, using the Mohr-Coulomb yield criterion, and use these results to test the recently-proposed modified Archimedes' law theory. We find excellent agreement, with the results of all the simulations falling very close to the predicted curve of φ vs. Kφ. We also examine the extreme case of frictionless spheres and find that, although no stagnant zone develops during penetration into such media, the value of their internal friction angle, φ = 9° ± 1°, also falls squarely on the theoretical curve. Finally, we use the modified Archimedes' law theory and an expression for the time-dependent growth of the stagnant zone to propose an explicit constitutive relation that fits excellently the force-depth curve throughout the entire penetration process.

Equally probable positive and negative Poisson's ratios in disordered planar systems.

Auxetic materials, characterised by a negative Poisson's ratio, have properties that are different from most conventional materials. These are a result of the constraints on the kinematics of the material's basic structural components, and have important technological implications. Models of these materials have been studied extensively, but theoretical descriptions have remained largely limited to materials with an ordered microstructure. Here we investigate whether negative Poisson's ratios can arise spontaneously in disordered systems. To this end, we develop a quantitative description of the structure in systems of connected basic elements, which enables us to analyse the local and global responses to small external tensile forces. We find that the Poisson's ratios in these disordered systems are equally likely to be positive or negative on both the element and system scales. Separating the strain into translational, rotational and expansive components, we find that the translational strains of neighbouring basic structural elements are positively correlated, while their rotations are negatively correlated. There is no correlation in this type of system between the local auxeticity and local structural characteristics. Our results suggest that auxeticity is more common in disordered structures than the ubiquity of positive Poisson's ratios in macroscopic materials would suggest.

Archimedes' law explains penetration of solids into granular media.

Understanding the response of granular matter to intrusion of solid objects is key to modelling many aspects of behaviour of granular matter, including plastic flow. Here we report a general model for such a quasistatic process. Using a range of experiments, we first show that the relation between the penetration depth and the force resisting it, transiently nonlinear and then linear, is scalable to a universal form. We show that the gradient of the steady-state part, K ϕ , depends only on the medium's internal friction angle, ϕ, and that it is nonlinear in µ = tan ϕ, in contrast to an existing conjecture. We further show that the intrusion of any convex solid shape satisfies a modified Archimedes' law and use this to: relate the zero-depth intercept of the linear part to K ϕ and the intruder's cross-section; explain the curve's nonlinear part in terms of the stagnant zone's development.

Affine and topogical structural entropies in granular statistical mechanics: Explicit calculations and equation of state.

We identify two orthogonal sources of structural entropy in rattler-free granular systems: affine, involving structural changes that only deform the contact network, and topological, corresponding to different topologies of the contact network. We show that a recently developed connectivity-based granular statistical mechanics separates the two naturally by identifying the structural degrees of freedom with spanning trees on the graph of the contact network. We extend the connectivity-based formalism to include constraints on, and correlations between, degrees of freedom as interactions between branches of the spanning tree. We then use the statistical mechanics formalism to calculate the partition function generally and the different entropies in the high-angoricity limit. We also calculate the degeneracy of the affine entropy and a number of expectation values. From the latter, we derive an equipartition principle and an equation of state relating the macroscopic volume and boundary stress to the analog of the temperature, the contactivity.

Fundamental structural characteristics of planar granular assemblies: Self-organization and scaling away friction and initial state.

The microstructural organization of a granular system is the most important determinant of its macroscopic behavior. Here we identify the fundamental factors that determine the statistics of such microstructures, using numerical experiments to gain a general understanding. The experiments consist of preparing and compacting isotropically two-dimensional granular assemblies of polydisperse frictional disks and analyzing the emergent statistical properties of quadrons-the basic structural elements of granular solids. The focus on quadrons is because the statistics of their volumes have been found to display intriguing universal-like features [T. Matsushima and R. Blumenfeld, Phys. Rev. Lett. 112, 098003 (2014)PRLTAO0031-900710.1103/PhysRevLett.112.098003]. The dependence of the structures and of the packing fraction on the intergranular friction and the initial state is analyzed, and a number of significant results are found. (i) An analytical formula is derived for the mean quadron volume in terms of three macroscopic quantities: the mean coordination number, the packing fraction, and the rattlers fraction. (ii) We derive a unique, initial-state-independent relation between the mean coordination number and the rattler-free packing fraction. The relation is supported numerically for a range of different systems. (iii) We collapse the quadron volume distributions from all systems onto one curve, and we verify that they all have an exponential tail. (iv) The nature of the quadron volume distribution is investigated by decomposition into conditional distributions of volumes given the cell order, and we find that each of these also collapses onto a single curve. (v) We find that the mean quadron volume decreases with increasing intergranular friction coefficients, an effect that is prominent in high-order cells. We argue that this phenomenon is due to an increased probability of stable irregularly shaped cells, and we test this using a herewith developed free cell analytical model. We conclude that, in principle, the microstructural characteristics are governed mainly by the packing procedure, while the effects of intergranular friction and initial states are details that can be scaled away. However, mechanical stability constraints suppress slightly the occurrence of small quadron volumes in cells of order ≥6, and the magnitude of this effect does depend on friction. We quantify in detail this dependence and the deviation it causes from an exact collapse for these cells. (vi) We argue that our results support strongly the view that ensemble granular statistical mechanics does not satisfy the uniform measure assumption of conventional statistical mechanics. Results (i)-(iv) have been reported in the aforementioned reference, and they are reviewed and elaborated on here.

Theory-based design of sintered granular composites triples three-phase boundary in fuel cells.

Solid-oxide fuel cells produce electric current from energy released by a spontaneous electrochemical reaction. The efficiency of these devices depends crucially on the microstructure of their electrodes and in particular on the three-phase boundary (TPB) length, along which the energy-producing reaction occurs. We present a systematic maximization of the TPB length as a function of four readily controllable microstructural parameters, for any given mean hydraulic radius, which is a conventional measure of the permeability to gas flow. We identify the maximizing parameters and show that the TPB length can be increased by a factor of over 300% compared to current common practices. We support this result by calculating the TPB of several numerically simulated structures. We also compare four models for a single intergranular contact in the sintered electrode and show that the model commonly used in the literature is oversimplified and unphysical. We then propose two alternatives.

Failure of the Volume Function in Granular Statistical Mechanics and an Alternative Formulation.

We first show that the currently accepted statistical mechanics for granular matter is flawed. The reason is that it is based on the volume function, which depends only on a minute fraction of all the structural degrees of freedom and is unaffected by most of the configurational microstates. Consequently, the commonly used partition function underestimates the entropy severely. We then propose a new formulation, replacing the volume function with a connectivity function that depends on all the structural degrees of freedom and accounts correctly for the entire entropy. We discuss the advantages of the new formalism and derive explicit results for two- and three-dimensional systems. We test the formalism by calculating the entropy of an experimental two-dimensional system, as a function of system size, and showing that it is an extensive variable.

Vertical dynamics of a horizontally oscillating active object in a two-dimensional granular medium.

We use a discrete-element method simulation and analytical considerations to study the dynamics of a self-energized object, modeled as a disk, oscillating horizontally within a two-dimensional bed of denser and smaller particles. We find that, for given material parameters, the immersed object (IO) may rise, sink, or not change depth, depending on the oscillation amplitude and frequency, as well as on the initial depth. With time, the IO settles at a specific depth that depends on the oscillation parameters. We construct a phase diagram of this behavior in the oscillation frequency and velocity amplitude variable space. We explain the observed rich behavior by two competing effects: climbing on particles, which fill voids opening under the disk, and sinking due to bed fluidization. We present a cavity model that allows us to derive analytically general results, which agree very well with the observations and explain quantitatively the phase diagram. Our specific analytical results are the following. (i) Derivation of a critical frequency, f_{c}, above which the IO cannot float up against gravity. We show that this frequency depends only on the gravitational acceleration and the IO size. (ii) Derivation of a minimal amplitude, A_{min}, below which the IO cannot rise even if the frequency is below f_{c}. We show that this amplitude also depends only on the gravitational acceleration and the IO size. (iii) Derivation of a critical value, g_{c}, of the IO's acceleration amplitude, below which the IO cannot sink. We show that the value of g_{c} depends on the characteristics of both the IO and the granular bed, as well as on the initial IO's depth.

Universal structural characteristics of planar granular packs.

The dependence of structural self-organization of granular materials on preparation and grain parameters is key to predictive modeling. We study 60 different mechanically equilibrated polydisperse disc packs, generated numerically by two protocols. We show that, for same-variance disc size distributions (DSDs), (1) the mean coordination number of rattler-free packs versus the packing fraction is a function independent of initial conditions, friction, and the DSD, and (2) all quadron volume and cell order distributions collapse to universal forms, also independent of the above. This apparent universality suggests that, contrary to common wisdom, equilibrated granular structures may be determined mainly by the packing protocol and higher moments of the DSD.

Interdependence of the volume and stress ensembles and equipartition in statistical mechanics of granular systems.

We discuss the statistical mechanics of granular matter and derive several significant results. First, we show that, contrary to common belief, the volume and stress ensembles are interdependent, necessitating the use of both. We use the combined ensemble to calculate explicitly expectation values of structural and stress-related quantities for two-dimensional systems. We thence demonstrate that structural properties may depend on the angoricity tensor and that stress-based quantities may depend on the compactivity. This calls into question previous statistical mechanical analyses of static granular systems and related derivations of expectation values. Second, we establish the existence of an intriguing equipartition principle-the total volume is shared equally amongst both structural and stress-related degrees of freedom. Third, we derive an expression for the compactivity that makes it possible to quantify it from macroscopic measurements.

On granular stress statistics: compactivity, angoricity, and some open issues.

We discuss the microstates of compressed granular matter in terms of two independent ensembles: one of volumes and another of boundary force moments. The former has been described in the literature and gives rise to the concept of compactivity: a scalar quantity that is the analogue of temperature in thermal systems. The latter ensemble gives rise to another analogue of the temperature: an angoricity tensor. We discuss averages under either of the ensembles and their relevance to experimental measurements. We also chart the transition from the microcanoncial to a canonical description for granular materials and show that one consequence of the traditional treatment is that the well-known exponential distribution of forces in granular systems subject to external forces is an immediate consequence of the canonical distribution, just as in the microcanonical description E = H leads to exp (-H/kT). We also put this conclusion in the context of observations of nonexponential forms of decay. We then present a Boltzmann-equation and Fokker-Planck approaches to the problem of diffusion in dense granular systems. Our approach allows us to derive, under simplifying assumptions, an explicit relation between the diffusion constant and the value of the hitherto elusive compactivity. We follow with a discussion of several unresolved issues. One of these issues is that the lack of ergodicity prevents convenient translation between time and ensemble averages, and the problem is illustrated in the context of diffusion. Another issue is that it is unclear how to make use in the statistical formalism the emerging ability to exactly predict stress fields for given structures of granular systems.