Introduction to liquid composites

Michael Bartlett and Robert Style introduce the Soft Matter themed collection on liquid composites.

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.

Elastic ripening and inhibition of liquid-liquid phase separation

Phase separation is a central concept of materials physics [1-3] and has recently emerged as an important route to compartmentalization within living cells [4-6]. Biological phase separation features activity [7], complex compositions [8], and elasticity [9], which reveal important gaps in our understanding of this universal physical phenomenon. Here, we explore the impact of elasticity on phase separation in synthetic polymer networks. We show that compressive stresses in a polymer network can suppress phase separation of the solvent that swells it, stabilizing mixtures well beyond the liquid-liquid phase separation boundary. Network stresses also drive a new form of ripening, driven by transport of solute down stiffness gradients. This elastic ripening can be much faster than conventional surface tension driven Ostwald ripening.

Extreme cavity expansion in soft solids: Damage without fracture

Cavitation is a common damage mechanism in soft solids. Here, we study this using a phase separation technique in stretched, elastic solids to controllably nucleate and grow small cavities by several orders of magnitude. The ability to make stable cavities of different sizes, as well as the huge range of accessible strains, allows us to systematically study the early stages of cavity expansion. Cavities grow in a scale-free manner, accompanied by irreversible bond breakage that is distributed around the growing cavity rather than being localized to a crack tip. Furthermore, cavities appear to grow at constant driving pressure. This has strong analogies with the plasticity that occurs surrounding a growing void in ductile metals. In particular, we find that, although elastomers are normally considered as brittle materials, small-scale cavity expansion is more like a ductile process. Our results have broad implications for understanding and controlling failure in soft solids.

Magnetically Addressable Shape-Memory and Stiffening in a Composite Elastomer

With a specific stimulus, shape-memory materials can assume a temporary shape and subsequently recover their original shape, a functionality that renders them relevant for applications in fields such as biomedicine, aerospace, and wearable electronics. Shape-memory in polymers and composites is usually achieved by exploiting a thermal transition to program a temporary shape and subsequently recover the original shape. This may be problematic for heat-sensitive environments, and when rapid and uniform heating is required. In this work, a soft magnetic shape-memory composite is produced by encasing liquid droplets of magneto-rheological fluid into a poly(dimethylsiloxane) matrix. Under the influence of a magnetic field, this material undergoes an exceptional stiffening transition, with an almost 30-fold increase in shear modulus. Exploiting this transition, fast and fully reversible magnetic shape-memory is demonstrated in three ways, by embossing, by simple shear, and by unconstrained 3D deformation. Using advanced synchrotron X-ray tomography techniques, the internal structure of the material is revealed, which can be correlated with the composite stiffening and shape-memory mechanism. This material concept, based on a simple emulsion process, can be extended to different fluids and elastomers, and can be manufactured with a wide range of methods.

Effects of strain-dependent surface stress on the adhesive contact of a rigid sphere to a compliant substrate

Recent experiments have reported that the surface stress of soft elastic solids can increase rapidly with surface strain. For example, when a small hard sphere in adhesive contact with a soft silicone gel is slowly retracted from its rest position, it was found that the retraction force versus displacement relation cannot be explained either by the Johnson-Kendall-Roberts (JKR) theory or a recent indentation theory based on an isotropic surface stress that is independent of surface strain. In this paper, we address this problem using a finite element method to simulate the retraction process. Our numerical model does not have the restrictions of the aforementioned theories; that is, it can handle large nonlinear elastic deformation as well as a surface-strain-dependent surface stress. Our simulation is in good agreement with experimental force versus displacement data with no fitting parameters. Therefore, our results lend further support to the claim that significant strain-dependent surface stresses can occur in simple soft elastic gels. However, significant challenges remain in the reconciliation of theory and experiments, particularly regarding the geometry of the contact and substrate deformation.

Singular dynamics in the failure of soft adhesive contacts

We characterize the mechanical recovery of compliant silicone gels following adhesive contact failure. We establish broad, stable adhesive contacts between rigid microspheres and soft gels, then stretch the gels to large deformations by pulling quasi-statically on the contact. Eventually, the adhesive contact begins to fail, and ultimately slides to a final contact point on the bottom of the sphere. Immediately after detachment, the gel recoils quickly with a self-similar surface profile that evolves as a power law in time, suggesting that the adhesive detachment point is singular. The singular dynamics we observe are consistent with a relaxation process driven by surface stress and slowed by viscous flow through the porous, elastic network of the gel. Our results emphasize the importance of accounting for both the liquid and solid phases of gels in understanding their mechanics, especially under extreme deformation.

The contact mechanics challenge: tribology meets soft matter

In the fall of 2015, Martin Müser suggested a Contact Mechanics Challenge for the Tribology community. The challenge was an ambitious effort to compare a wide variety of theoretical and computational contact-mechanics approaches, and involved researchers voluntarily tackling the same hypothetical contact problem. The result is an impressive collection of innovative approaches - including a surprise experimental effort - that highlight the continuing importance of surface contact mechanics and the challenges of solving these large-scale problems. Here, we describe how the Contact Mechanics Challenge also reveals exciting opportunities for the Soft Matter community to engage intensely with classical and emerging problems in tribology, surface science, and contact mechanics.

The mechanical equilibrium of soft solids with surface elasticity

Recent experiments have shown that surface stresses in soft materials can have a significant strain-dependence. Here we explore the implications of this surface elasticity to show how, and when, we expect it to arise. We develop the appropriate boundary condition, showing that it simplifies significantly in certain cases. We show that surface elasticity's main role is to stiffen a solid surface's response to in-plane tractions, in particular at length-scales smaller than a characteristic elastocapillary length. We also investigate how surface elasticity affects the Green's-function problem of a line force on a flat, incompressible, linear-elastic substrate. There are significant changes to this solution, especially in that the well-known displacement singularity is regularised. This raises interesting implications for soft phenomena like wetting contact lines, adhesion and friction. Finally, we discuss open questions, future directions, and close ties with existing fields of research.

Surface elastic constants of a soft solid

Solid interfaces have intrinsic elasticity. However, in most experiments, this is obscured by bulk stresses. Through microscopic observations of the contact-line geometry of a partially wetting droplet on an anisotropically stretched substrate, we measure two surface-elastic constants that quantify the linear dependence of the surface stress of a soft polymer gel on its strain. With these two parameters, one can predict surface stresses for general deformations of the material in the linear-elastic limit.

Indentation of a floating elastic sheet: geometry versus applied tension

The localized loading of an elastic sheet floating on a liquid bath occurs at scales from a frog sitting on a lily pad to a volcano supported by the Earth's tectonic plates. The load is supported by a combination of the stresses within the sheet (which may include applied tensions from, for example, surface tension) and the hydrostatic pressure in the liquid. At the same time, the sheet deforms, and may wrinkle, because of the load. We study this problem in terms of the (relatively weak) applied tension and the indentation depth. For small indentation depths, we find that the force-indentation curve is linear with a stiffness that we characterize in terms of the applied tension and bending stiffness of the sheet. At larger indentations, the force-indentation curve becomes nonlinear and the sheet is subject to a wrinkling instability. We study this wrinkling instability close to the buckling threshold and calculate both the number of wrinkles at onset and the indentation depth at onset, comparing our theoretical results with experiments. Finally, we contrast our results with those previously reported for very thin, highly bendable membranes.

Direct measurement of strain-dependent solid surface stress

Surface stress, also known as surface tension, is a fundamental material property of any interface. However, measurements of solid surface stress in traditional engineering materials, such as metals and oxides, have proven to be very challenging. Consequently, our understanding relies heavily on untested theories, especially regarding the strain dependence of this property. Here, we take advantage of the high compliance and large elastic deformability of a soft polymer gel to directly measure solid surface stress as a function of strain. As anticipated by theoretical work for metals, we find that the surface stress depends on the strain via a surface modulus. Remarkably, the surface modulus of our soft gels is many times larger than the zero-strain surface tension. This suggests that surface stresses can play a dominant role in solid mechanics at larger length scales than previously anticipated.Solid surface stress is a fundamental property of solid interfaces. Here authors measure the solid surface stress of a gel, and show its dependence on surface strain through a surface modulus.

Mechanical stability of particle-stabilized droplets under micropipette aspiration

We investigate the mechanical behavior of particle-stabilized droplets using micropipette aspiration. We observe that droplets stabilized with amphiphilic dumbbell-shaped particles exhibit a two-stage response to increasing suction pressure. Droplets first drip, then wrinkle and buckle like an elastic shell. While particles have a dramatic impact on the mechanism of failure, the mechanical strength of the droplets is only modestly increased. On the other hand, droplets coated with the molecular surfactant sodium dodecyl sulfate are even weaker than bare droplets. In all cases, the magnitude of the critical pressure for the onset of instabilities is set by the fluid surface tension.

It's Harder to Splash on Soft Solids

Droplets splash when they impact dry, flat substrates above a critical velocity that depends on parameters such as droplet size, viscosity, and air pressure. By imaging ethanol drops impacting silicone gels of different stiffnesses, we show that substrate stiffness also affects the splashing threshold. Splashing is reduced or even eliminated: droplets on the softest substrates need over 70% more kinetic energy to splash than they do on rigid substrates. We show that this is due to energy losses caused by deformations of soft substrates during the first few microseconds of impact. We find that solids with Young's moduli ≲100  kPa reduce splashing, in agreement with simple scaling arguments. Thus, materials like soft gels and elastomers can be used as simple coatings for effective splash prevention. Soft substrates also serve as a useful system for testing splash-formation theories and sheet-ejection mechanisms, as they allow the characteristics of ejection sheets to be controlled independently of the bulk impact dynamics of droplets.

Surface tension and the Mori-Tanaka theory of non-dilute soft composite solids

Eshelby's theory is the foundation of composite mechanics, allowing calculation of the effective elastic moduli of composites from a knowledge of their microstructure. However, it ignores interfacial stress and only applies to very dilute composites-i.e. where any inclusions are widely spaced apart. Here, within the framework of the Mori-Tanaka multiphase approximation scheme, we extend Eshelby's theory to treat a composite with interfacial stress in the non-dilute limit. In particular, we calculate the elastic moduli of composites comprised of a compliant, elastic solid hosting a non-dilute distribution of identical liquid droplets. The composite stiffness depends strongly on the ratio of the droplet size, R, to an elastocapillary lengthscale, L. Interfacial tension substantially impacts the effective elastic moduli of the composite when [Formula: see text]. When R<3L/2 (R=3L/2) liquid inclusions stiffen (cloak the far-field signature of) the solid.

Solid capillarity: when and how does surface tension deform soft solids?

Soft solids differ from stiff solids in an important way: their surface stresses can drive large deformations. Based on a topical workshop held in the Lorentz Center in Leiden, this Opinion highlights some recent advances in the growing field of solid capillarity and poses key questions for its advancement.

Interfacial tension and a three-phase generalized self-consistent theory of non-dilute soft composite solids

In the dilute limit Eshelby's inclusion theory captures the behavior of a wide range of systems and properties. However, because Eshelby's approach neglects interfacial stress, it breaks down in soft materials as the inclusion size approaches the elastocapillarity length L≡γ/E. Here, we use a three-phase generalized self-consistent method to calculate the elastic moduli of composites comprised of an isotropic, linear-elastic compliant solid hosting a spatially random monodisperse distribution of spherical liquid droplets. As opposed to similar approaches, we explicitly capture the liquid-solid interfacial stress when it is treated as an isotropic, strain-independent surface tension. Within this framework, the composite stiffness depends solely on the ratio of the elastocapillarity length L to the inclusion radius R. Independent of inclusion volume fraction, we find that the composite is stiffened by the inclusions whenever R < 3L/2. Over the same range of parameters, we compare our results with alternative approaches (dilute and Mori-Tanaka theories that include surface tension). Our framework can be easily extended to calculate the composite properties of more general soft materials where surface tension plays a role.

Adsorption of soft particles at fluid interfaces

Soft particles can be better emulsifiers than hard particles because they stretch at fluid interfaces. This deformation can increase adsorption energies by orders of magnitude relative to rigid particles. The deformation of a particle at an interface is governed by a competition of bulk elasticity and surface tension. When particles are partially wet by the two liquids, deformation is localized within a material-dependent distance L from the contact line. At the contact line, the particle morphology is given by a balance of surface tensions. When the particle radius R≪L, the particle adopts a lenticular shape identical to that of an adsorbed fluid droplet. Particle deformations can be elastic or plastic, depending on the relative values of the Young modulus, E, and yield stress, σp. When surface tensions favour complete spreading of the particles at the interface, plastic deformation can lead to unusual fried-egg morphologies. When deformable particles have surface properties that are very similar to one liquid phase, adsorption can be extremely sensitive to small changes of their affinity for the other liquid phase. These findings have implications for the adsorption of microgel particles at fluid interfaces and the performance of stimuli-responsive Pickering emulsions.

Surface tension and the mechanics of liquid inclusions in compliant solids

Eshelby's theory of inclusions has wide-reaching implications across the mechanics of materials and structures including the theories of composites, fracture, and plasticity. However, it does not include the effects of surface stress, which has recently been shown to control many processes in soft materials such as gels, elastomers and biological tissue. To extend Eshelby's theory of inclusions to soft materials, we consider liquid inclusions within an isotropic, compressible, linear-elastic solid. We solve for the displacement and stress fields around individual stretched inclusions, accounting for the bulk elasticity of the solid and the surface tension (i.e. isotropic strain-independent surface stress) of the solid-liquid interface. Surface tension significantly alters the inclusion's shape and stiffness as well as its near- and far-field stress fields. These phenomena depend strongly on the ratio of the inclusion radius, R, to an elastocapillary length, L. Surface tension is significant whenever inclusions are smaller than 100L. While Eshelby theory predicts that liquid inclusions generically reduce the stiffness of an elastic solid, our results show that liquid inclusions can actually stiffen a solid when R<3L/2. Intriguingly, surface tension cloaks the far-field signature of liquid inclusions when R=3L/2. These results are have far-reaching applications from measuring local stresses in biological tissue, to determining the failure strength of soft composites.

Traction force microscopy in physics and biology

Adherent cells, crawling slugs, peeling paint, sessile liquid drops, bearings and many other living and non-living systems apply forces to solid substrates. Traction force microscopy (TFM) provides spatially-resolved measurements of interfacial forces through the quantification and analysis of the deformation of an elastic substrate. Although originally developed for adherent cells, TFM has no inherent size or force scale, and can be applied to a much broader range of mechanical systems across physics and biology. In this paper, we showcase the wide range of applicability of TFM, describe the theory, and provide experimental details and code so that experimentalists can rapidly adopt this powerful technique.

Universal deformation of soft substrates near a contact line and the direct measurement of solid surface stresses

Droplets deform soft substrates near their contact lines. Using confocal microscopy, we measure the deformation of silicone gel substrates due to glycerol and fluorinated-oil droplets for a range of droplet radii and substrate thicknesses. For all droplets, the substrate deformation takes a universal shape close to the contact line that depends on liquid composition, but is independent of droplet size and substrate thickness. This shape is determined by a balance of interfacial tensions at the contact line and provides a novel method for direct determination of the surface stresses of soft substrates. Moreover, we measure the change in contact angle with droplet radius and show that Young's law fails for small droplets when their radii approach an elastocapillary length scale. For larger droplets the macroscopic contact angle is constant, consistent with Young's law.

Ice-lens formation and geometrical supercooling in soils and other colloidal materials

We present a physically intuitive model of ice-lens formation and growth during the freezing of soils and other dense, particulate suspensions. Motivated by experimental evidence, we consider the growth of an ice-filled crack in a freezing soil. At low temperatures, ice in the crack exerts large pressures on the crack walls that will eventually cause the crack to split open. We show that the crack will then propagate across the soil to form a new lens. The process is controlled by two factors: the cohesion of the soil and the geometrical supercooling of the water in the soil, a new concept introduced to measure the energy available to form a new ice lens. When the supercooling exceeds a critical amount (proportional to the cohesive strength of the soil) a new ice lens forms. This condition for ice-lens formation and growth does not appeal to any ad hoc, empirical assumptions, and explains how periodic ice lenses can form with or without the presence of a frozen fringe. The proposed mechanism is in good agreement with experiments, in particular explaining ice-lens pattern formation and surges in heave rate associated with the growth of new lenses. Importantly for systems with no frozen fringe, ice-lens formation and frost heave can be predicted given only the unfrozen properties of the soil. We use our theory to estimate ice-lens growth temperatures obtaining quantitative agreement with the limited experimental data that are currently available. Finally we suggest experiments that might be performed in order to verify this theory in more detail. The theory is generalizable to complex natural-soil scenarios and should therefore be useful in the prediction of macroscopic frost-heave rates.