Extreme cavity expansion in soft solids: Damage without fracture

Effect of Polymer Gel Elasticity on Complex Coacervate Phase Behavior

Gels are key materials in biological systems such as tissues and may control biocondensate formation and structure. To further understand the effects of elastic environments on biomacromolecular assembly, we have investigated the phase behavior and radii of complex coacervate droplets in polyacrylamide (PAM) networks as a function of gel modulus. Poly-l-lysine (PLL) and sodium hyaluronate (HA) complex coacervate phases were prepared in PAM gels with moduli varying from 0.035 to 15.0 kPa. The size of the complex coacervate droplets is reported from bright-field microscopy and confocal fluorescence microscopy. Overall, the complex coacervate droplet volume decreases inversely with the modulus. Fluorescence microscopy is also used to determine the phase behavior and concentration of fluorescently tagged HA in the complex coacervate phases as a function of ionic strength (100-270 mM). We find that the critical ionic strength and complex coacervate stability are nonmonotonic as a function of the network modulus and that the local gel concentration can be used to control phase behavior and complex coacervate droplet size scale. By understanding how elastic environments influence simple electrostatic assembly, we can further understand how biomacromolecules exist in complex, crowded, and elastic cellular environments.

Liquid-liquid phase separation within fibrillar networks

Complex fibrillar networks mediate liquid-liquid phase separation of biomolecular condensates within the cell. Mechanical interactions between these condensates and the surrounding networks are increasingly implicated in the physiology of the condensates and yet, the physical principles underlying phase separation within intracellular media remain poorly understood. Here, we elucidate the dynamics and mechanics of liquid-liquid phase separation within fibrillar networks by condensing oil droplets within biopolymer gels. We find that condensates constrained within the network pore space grow in abrupt temporal bursts. The subsequent restructuring of condensates and concomitant network deformation is contingent on the fracture of network fibrils, which is determined by a competition between condensate capillarity and network strength. As a synthetic analog to intracellular phase separation, these results further our understanding of the mechanical interactions between biomolecular condensates and fibrillar networks in the cell.

Phase separation and ripening in a viscoelastic gel

The process of phase separation in elastic solids and viscous fluids is of fundamental importance to the stability and function of soft materials. We explore the dynamics of phase separation and domain growth in a viscoelastic material such as a polymer gel. Using analytical theory and Monte Carlo simulations, we report a domain growth regime in which the domain size increases algebraically with a ripening exponent [Formula: see text] that depends on the viscoelastic properties of the material. For a prototypical Maxwell material, we obtain [Formula: see text], which is markedly different from the well-known Ostwald ripening process with [Formula: see text]. We generalize our theory to systems with arbitrary power-law relaxation behavior and discuss our findings in the context of the long-term stability of materials as well as recent experimental results on phase separation in cross-linked networks and cytoskeleton.

Active foam: the adaptive mechanics of 2D air-liquid foam under cyclic inflation

Foam is a canonical example of disordered soft matter where local force balance leads to the competition of many metastable configurations. We present an experimental and theoretical framework for "active foam" where an individual voxel inflates and deflates periodically. Local periodic activity leads to irreversible and reversible T1 transitions throughout the foam, eventually reaching a reversible limit cycle. Individual vertices displace outwards and subsequently return back to their approximate original radial position; this radial displacement follows an inverse law. Surprisingly, each return trajectory does not retrace its outbound path but encloses a finite area, with a clockwise (CW) or counterclockwise (CCW) direction, which we define as a local swirl. These swirls form coherent patterns spanning the scale of the material. Using a dynamical model, we demonstrate that swirl arises from disorder in the local micro-structure. We demonstrate that disorder and strain-rate control a crossover between cooperation and competition between swirls in adjacent vertices. Over 5-10 cycles, the region around the active voxel structurally adapts from a higher-energy metastable state to a lower-energy state, locally ordering and stiffening the structure. The coherent domains of CW/CCW swirl become smaller as the system stabilizes, indicative of a process similar to the Hall-Petch effect. Finally, we introduce a statistical model that evolves edge lengths with a set of rules to explore how this class of materials adapts as a function of initial structure. Adding activity to foam couples structural disorder and adaptive dynamics to encourage the development of a new class of abiotic, cellularized active matter.

Siloxane Molecules: Nonlinear Elastic Behavior and Fracture Characteristics

Fracture phenomena in soft materials span multiple length and time scales. This poses a major challenge in computational modeling and predictive materials design. To pass quantitatively from molecular to continuum scales, a precise representation of the material response at the molecular level is vital. Here, we derive the nonlinear elastic response and fracture characteristics of individual siloxane molecules using molecular dynamics (MD) studies. For short chains, we find deviations from classical scalings for both the effective stiffness and mean chain rupture times. A simple model of a nonuniform chain of Kuhn segments captures the observed effect and agrees well with MD data. We find that the dominating fracture mechanism depends on the applied force scale in a nonmonotonic fashion. This analysis suggests that common polydimethylsiloxane (PDMS) networks fail at cross-linking points. Our results can be readily lumped into coarse-grained models. Although focusing on PDMS as a model system, our study presents a general procedure to pass beyond the window of accessible rupture times in MD studies employing mean first passage time theory, which can be exploited for arbitrary molecular systems.

The crucial role of elasticity in regulating liquid-liquid phase separation in cells

Liquid-liquid phase separation has emerged as a fundamental mechanism underlying intracellular organization, with evidence for it being reported in numerous different systems. However, there is a growing concern regarding the lack of quantitative rigor in the techniques employed to study phase separation, and their ability to account for the complex nature of the cellular milieu, which affects key experimentally observable measures, such as the shape, size and transport dynamics of liquid droplets. Here, we bridge this gap by combining recent experimental data with theoretical predictions that capture the subtleties of nonlinear elasticity and fluid transport. We show that within a biologically accessible range of material parameters, phase separation is highly sensitive to elastic properties and can thus be used as a mechanical switch to rapidly transition between different states in cellular systems. Furthermore, we show that this active mechanically mediated mechanism can drive transport across cells at biologically relevant timescales and could play a crucial role in promoting spatial localization of condensates; whether cells exploit such mechanisms for transport of their constituents remains an open question.

Leveraging ultra-low interfacial tension and liquid-liquid phase separation in embedded 3D bioprinting

Many recently developed 3D bioprinting strategies operate by extruding aqueous biopolymer solutions directly into a variety of different support materials constituted from swollen, solvated, aqueous, polymer assemblies. In developing these 3D printing methods and materials, great care is often taken to tune the rheological behaviors of both inks and 3D support media. By contrast, much less attention has been given to the physics of the interfaces created when structuring one polymer phase into another in embedded 3D printing applications. For example, it is currently unclear whether a dynamic interfacial tension between miscible phases stabilizes embedded 3D bioprinted structures as they are shaped while in a liquid state. Interest in the physics of interfaces between complex fluids has grown dramatically since the discovery of liquid-liquid phase separation (LLPS) in living cells. We believe that many new insights coming from this burst of investigation into LLPS within biological contexts can be leveraged to develop new materials and methods for improved 3D bioprinting that leverage LLPS in mixtures of biopolymers, biocompatible synthetic polymers, and proteins. Thus, in this review article, we highlight work at the interface between recent LLPS research and embedded 3D bioprinting methods and materials, and we introduce a 3D bioprinting method that leverages LLPS to stabilize printed biopolymer inks embedded in a bioprinting support material.

Cavitation in a soft porous material

We study the collapse and expansion of a cavitation bubble in a deformable porous medium. We develop a continuum-scale model that couples compressible fluid flow in the pore network with the elastic response of a solid skeleton. Under the assumption of spherical symmetry, our model can be reduced to an ordinary differential equation that extends the Rayleigh-Plesset equation to bubbles in soft porous media. The extended Rayleigh-Plesset equation reveals that finite-size effects lead to the breakdown of the universal scaling relation between bubble radius and time that holds in the infinite-size limit. Our data indicate that the deformability of the porous medium slows down the collapse and expansion processes, a result with important consequences for wide-ranging phenomena, from drug delivery to spore dispersion.

Stress accumulation by confined ice in a temperature gradient

When materials freeze, they often undergo damage due to ice growth. Although this damage is commonly ascribed to the volumetric expansion of water upon freezing, it is usually driven by the flow of water toward growing ice crystals that feeds their growth. The freezing of this additional water can cause a large buildup of stress. Here, we demonstrate a technique for characterizing this stress buildup with unprecedented spatial resolution. We create a stable ice-water interface in a controlled temperature gradient and measure the deformation of the confining boundary. Analysis of the deformation field reveals stresses applied to the boundary with [Formula: see text](micrometers) spatial resolution. Globally, stresses increase steadily over time as liquid water is transported to more deeply undercooled regions. Locally, stresses increase until ice growth is stalled by the confining stresses. Importantly, we find a strong localization of stresses, which significantly increases the likelihood of damage caused by the presence of ice, even in apparently benign freezing situations. Ultimately, the limiting stress that the ice exerts is proportional to the local undercooling, in accordance with the Clapeyron equation, which describes the equilibrium between a stressed solid and its melt. Our results are closely connected to the condensation pressure during liquid-liquid phase separation and the crystallization pressure for growing crystals. Thus, they are highly relevant in fields ranging from cryopreservation and frost heave to food science, rock weathering, and art conservation.

Stretchy and disordered: Toward understanding fracture in soft network materials via mesoscopic computer simulations

Soft network materials exist in numerous forms ranging from polymer networks, such as elastomers, to fiber networks, such as collagen. In addition, in colloidal gels, an underlying network structure can be identified, and several metamaterials and textiles can be considered network materials as well. Many of these materials share a highly disordered microstructure and can undergo large deformations before damage becomes visible at the macroscopic level. Despite their widespread presence, we still lack a clear picture of how the network structure controls the fracture processes of these soft materials. In this Perspective, we will focus on progress and open questions concerning fracture at the mesoscopic scale, in which the network architecture is clearly resolved, but neither the material-specific atomistic features nor the macroscopic sample geometries are considered. We will describe concepts regarding the network elastic response that have been established in recent years and turn out to be pre-requisites to understand the fracture response. We will mostly consider simulation studies, where the influence of specific network features on the material mechanics can be cleanly assessed. Rather than focusing on specific systems, we will discuss future challenges that should be addressed to gain new fundamental insights that would be relevant across several examples of soft network materials.

Putting the Squeeze on Phase Separation

Phase separation is a ubiquitous process and finds applications in a variety of biological, organic, and inorganic systems. Nature has evolved the ability to control phase separation to both regulate cellular processes and make composite materials with outstanding mechanical and optical properties. Striking examples of the latter are the vibrant blue and green feathers of many bird species, which are thought to result from an exquisite control of the size and spatial correlations of their phase-separated microstructures. By contrast, it is much harder for material scientists to arrest and control phase separation in synthetic materials with such a high level of precision at these length scales. In this Perspective, we briefly review some established methods to control liquid-liquid phase separation processes and then highlight the emergence of a promising arrest method based on phase separation in an elastic polymer network. Finally, we discuss upcoming challenges and opportunities for fabricating microstructured materials via mechanically controlled phase separation.

Non-specific adhesive forces between filaments and membraneless organelles

Many membraneless organelles are liquid-like domains that form inside the active, viscoelastic environment of living cells through phase separation. To investigate the potential coupling of phase separation with the cytoskeleton, we quantify the structural correlations of membraneless organelles (stress granules) and cytoskeletal filaments (microtubules) in a human-derived epithelial cell line. We find that microtubule networks are substantially denser in the vicinity of stress granules. When microtubules are depolymerized, the sub-units localize near the surface of the stress granules. We interpret these data using a thermodynamic model of partitioning of particles to the surface and bulk of the droplets. In this framework, our data are consistent with a weak (≲k _B T) affinity of the microtubule sub-units for stress granule interfaces. As microtubules polymerize, their interfacial affinity increases, providing sufficient adhesion to deform droplets and/or the network. Our work suggests that proteins and other objects in the cell have a non-specific affinity for droplet interfaces that increases with the contact area and becomes most apparent when they have no preference for the interior of a droplet over the rest of the cytoplasm. We validate this basic physical phenomenon in vitro through the interaction of a simple protein-RNA condensate with microtubules.

Solid-Liquid Host-Guest Composites: The Marriage of Porous Solids and Functional Liquids

Composite materials can provide remarkable improvements over the individual constituents. Especially, with a liquid component introduced into a solid porous host, solid-liquid host-guest composites have recently come to the forefront with exceptional functions that promise them for a wealth of applications. Combining the unprecedented dynamic, transparent, omniphobic, self-healing, diffusive and adaptive nature of functional liquid with inherent solid host's property, solid-liquid host-guest composites can realize the ease of fabrication, long-term stability, and a broad spectrum of enhanced properties, which cannot be fully met by conventional solid-solid composites or liquid-liquid composites. This review presents the state-of-the-art progress in solid-liquid host-guest composites. Initially, the concept, classification, design strategy, as well as fabrication methods as a path forward to develop the composites are unraveled, and further it is elaborated on how the functionality of porous solid and functional liquid can be harnessed to create composites with a broad range of unique properties, especially, the optical, thermal, electric, mechanical, sorption, and separation properties. With these fascinating properties, a myriad of emerging applications such as optical devices, thermal management, electromagnetic-interference shielding, soft electronics, gas capture and release, and multiphase separations are touched upon, inspiring more frontier researches in materials science, interfacial chemistry, membrane science, engineering, and multidisciplinary. Finally, this review provides the perspective on the future directions of solid-liquid host-guest composites and assesses the challenges and opportunities ahead.

Cavitation controls droplet sizes in elastic media

Biological cells use droplets to separate components and spatially control their interior. Experiments demonstrate that the complex, crowded cellular environment affects the droplet arrangement and their sizes. To understand this behavior, we here construct a theoretical description of droplets growing in an elastic matrix, which is motivated by experiments in synthetic systems where monodisperse emulsions form during a temperature decrease. We show that large droplets only form when they break the surrounding matrix in a cavitation event. The energy barrier associated with cavitation stabilizes small droplets on the order of the mesh size and diminishes the stochastic effects of nucleation. Consequently, the cavitated droplets have similar sizes and highly correlated positions. In particular, we predict the density of cavitated droplets, which increases with faster cooling, as in the experiments. Our model also suggests how adjusting the cooling protocol and the density of nucleation sites affects the droplet size distribution. In summary, our theory explains how elastic matrices affect droplets in the synthetic system, and it provides a framework for understanding the biological case.

Morphogenesis and cell ordering in confined bacterial biofilms

Biofilms are aggregates of bacterial cells surrounded by an extracellular matrix. Much progress has been made in studying biofilm growth on solid substrates; however, little is known about the biophysical mechanisms underlying biofilm development in three-dimensional confined environments in which the biofilm-dwelling cells must push against and even damage the surrounding environment to proliferate. Here, combining single-cell imaging, mutagenesis, and rheological measurement, we reveal the key morphogenesis steps of Vibrio cholerae biofilms embedded in hydrogels as they grow by four orders of magnitude from their initial size. We show that the morphodynamics and cell ordering in embedded biofilms are fundamentally different from those of biofilms on flat surfaces. Treating embedded biofilms as inclusions growing in an elastic medium, we quantitatively show that the stiffness contrast between the biofilm and its environment determines biofilm morphology and internal architecture, selecting between spherical biofilms with no cell ordering and oblate ellipsoidal biofilms with high cell ordering. When embedded in stiff gels, cells self-organize into a bipolar structure that resembles the molecular ordering in nematic liquid crystal droplets. In vitro biomechanical analysis shows that cell ordering arises from stress transmission across the biofilm-environment interface, mediated by specific matrix components. Our imaging technique and theoretical approach are generalizable to other biofilm-forming species and potentially to biofilms embedded in mucus or host tissues as during infection. Our results open an avenue to understand how confined cell communities grow by means of a compromise between their inherent developmental program and the mechanical constraints imposed by the environment.

3D fluorescent mapping of invisible molecular damage after cavitation in hydrogen exposed elastomers

Elastomers saturated with gas at high pressure suffer from cavity nucleation, inflation, and deflation upon rapid or explosive decompression. Although this process often results in undetectable changes in appearance, it causes internal damage, hampers functionality (e.g., permeability), and shortens lifetime. Here, we tag a model poly(ethyl acrylate) elastomer with π-extended anthracene-maleimide adducts that fluoresce upon network chain scission, and map in 3D the internal damage present after a cycle of gas saturation and rapid decompression. Interestingly, we observe that each cavity observable during decompression results in a damaged region, the shape of which reveals a fracture locus of randomly oriented penny-shape cracks (i.e., with a flower-like morphology) that contain crack arrest lines. Thus, cavity growth likely proceeds discontinuously (i.e., non-steadily) through the stable and unstable fracture of numerous 2D crack planes. This non-destructive methodology to visualize in 3D molecular damage in polymer networks is novel and serves to understand how fracture occurs under complex 3D loads, predict mechanical aging of pristine looking elastomers, and holds potential to optimize cavitation-resistance in soft materials.

Droplets Sit and Slide Anisotropically on Soft, Stretched Substrates

Anisotropically wetting substrates enable useful control of droplet behavior across a range of applications. Usually, these involve chemically or physically patterning the substrate surface, or applying gradients in properties like temperature or electrical field. Here, we show that a flat, stretched, uniform soft substrate also exhibits asymmetric wetting, both in terms of how droplets slide and in their static shape. Droplet dynamics are strongly affected by stretch: glycerol droplets on silicone substrates with a 23% stretch slide 67% faster in the direction parallel to the applied stretch than in the perpendicular direction. Contrary to classical wetting theory, static droplets in equilibrium appear elongated, oriented parallel to the stretch direction. Both effects arise from droplet-induced deformations of the substrate near the contact line.

Dynamic asymmetry and why chromatin defies simple physical definitions

Recent experiments have demonstrated a nucleus where chromatin is molded into stable, interwoven loops. Yet, many of the proteins, which shape chromatin structure, bind only transiently. In those brief encounters, these dynamic proteins temporarily crosslink chromatin loops. While, on the average, individual crosslinks do not persist, in the aggregate, they are sufficient to create and maintain stable chromatin domains. Owing to the asymmetry in size and speed of molecules involved, this type of organization imparts unique biophysical properties-the slow (chromatin) component can exhibit gel-like behaviors, whereas the fast (protein) component allows domains to respond with liquid-like characteristics.

Forming Anisotropic Crystal Composites: Assessing the Mechanical Translation of Gel Network Anisotropy to Calcite Crystal Form

The promise of crystal composites with direction-specific properties is an attractive prospect for diverse applications; however, synthetic strategies for realizing such composites remain elusive. Here, we demonstrate that anisotropic agarose gel networks can mechanically "mold" calcite crystal growth, yielding anisotropically structured, single-crystal composites. Drying and rehydration of agarose gel films result in the affine deformation of their fibrous networks to yield fiber alignment parallel to the drying plane. Precipitation of calcium carbonate within these anisotropic networks results in the formation of calcite crystal composite disks oriented parallel to the fibers. The morphology of the disks, revealed by nanocomputed tomography imaging, evolves with time and can be described by linear-elastic fracture mechanics theory, which depends on the ratio between the length of the crystal and the elastoadhesive length of the gel. Precipitation of calcite in uniaxially deformed agarose gel cylinders results in the formation of rice-grain-shaped crystals, suggesting the broad applicability of the approach. These results demonstrate how the anisotropy of compliant networks can translate into the desired crystal composite morphologies. This work highlights the important role organic matrices can play in mechanically "molding" biominerals and provides an exciting platform for fabricating crystal composites with direction-specific and emergent functional properties.

Growth of gas-filled penny-shaped cracks in decompressed hydrogels

We report that the decompression of soft brittle materials can lead to the growth of internal gas-filled cracks. These cracks are oblate spheroids ('penny shape'), whose major radius grows linearly in time, irreversibly fracturing the surrounding material. Our optical measurements in hydrogels characterise and quantify the three-dimensional crack geometry and growth rate. These results are in good agreement with our analytical model coupling fracture mechanics and gas diffusion, and predicting the dependence on the mechanical properties, gas diffusivity and super-saturation conditions (gas pressure, solubility, temperature). Our results suggest a new potential mechanism for decompression sickness in scuba diving and for indirect optical measurements of the fracture properties of hydrogels.

Modeling Elastically Mediated Liquid-Liquid Phase Separation

We propose a continuum theory of the liquid-liquid phase separation in an elastic network, where phase-separated microscopic droplets rich in one fluid component can form as an interplay of fluids mixing, droplet nucleation, network deformation, thermodynamic fluctuation, etc. We find that the size of the phase-separated droplets decreases with the shear modulus of the elastic network in the form of ∝[modulus]^{-1/3} and the number density of the droplet increases almost linearly with the shear modulus ∝[modulus], which are verified by the experimental observations. Phase diagrams in the space of (fluid constitution, mixture interaction, network modulus) are provided, which can help to understand similar phase separations in biological cells and also to guide fabrications of synthetic cells with desired phase properties.

Designing the Morphology of Separated Phases in Multicomponent Liquid Mixtures

Phase separation of multicomponent liquid mixtures plays an integral part in many processes ranging from industry to cellular biology. In many cases the morphology of coexisting phases is crucially linked to the function of the separated mixture, yet it is unclear what determines the morphology when multiple phases are present. We developed a graph theory approach to predict the topology of coexisting phases from a given set of surface energies, enumerate all topologically distinct morphologies, and reverse engineer conditions for surface energies that produce the target morphology.

Modeling of surface mechanical behaviors of soft elastic solids: theory and examples

Surfaces of soft solids can have significant surface stress, extensional modulus and bending stiffness. Previous theoretical studies have usually examined cases in which both the surface stress and bending stiffness are constant, assuming small deformation. In this work we consider a general formulation in which the surface can support large deformation and carry both surface stresses and surface bending moments. We demonstrate that the large deformation theory can be reduced to the classical linear theory (Shuttleworth equation). We obtain exact solutions for problems of an inflated cylindrical shell and bending of a plate with a finite thickness. Our analysis illustrates the different manners in which surface stiffening and surface bending stabilize these structures. We discuss how the complex surface constitutive behaviors affect the stress field of the bulk. Our calculation provides insights into effects of strain-dependent surface stress and surface bending in the large deformation regime, and can be used as a model to implement surface finite elements to study large deformation of complex structures.

Theory of droplet ripening in stiffness gradients

Liquid droplets embedded in soft solids are a new composite material whose properties are not very well explored. In particular, it is unclear how the elastic properties of the matrix affect the dynamics of the droplets. Here, we study theoretically how stiffness gradients influence droplet growth and arrangement. We show that stiffness gradients imply concentration gradients in the dilute phase, which transport droplet material from stiff to soft regions. Consequently, droplets dissolve in the stiff region, creating a dissolution front. Using a mean-field theory, we predict that the front emerges where the curvature of the elasticity profile is large and that it propagates diffusively. This elastic ripening can occur at much higher rates than classical Ostwald ripening, thus driving the dynamics. Our work shows how gradients in elastic properties control the arrangement of droplets, which has potential applications in soft matter physics and biological cells.

Elastic stresses reverse Ostwald ripening

When liquid droplets nucleate and grow in a polymer network, compressive stresses can significantly increase their internal pressure, reaching values that far exceed the Laplace pressure. When droplets have grown in a polymer network with a stiffness gradient, droplets in relatively stiff regions of the network tend to dissolve, favoring growth of droplets in softer regions. Here, we show that this elastic ripening can be strong enough to reverse the direction of Ostwald ripening: large droplets can shrink to feed the growth of smaller ones. To numerically model these experiments, we generalize the theory of elastic ripening to account for gradients in solubility alongside gradients in mechanical stiffness.