Extracting the surface tension of soft gels from elastocapillary wave behavior

Crossover of surface waves and capillary-viscous-elastic transition in soft biomaterials detected by resonant acoustic rheometry

Viscoelastic properties of hydrogels are important for their application in science and industry. However, rheological assessment of soft hydrogel biomaterials is challenging due to their complex, rapid, and often time-dependent behaviors. Resonant acoustic rheometry (RAR) is a newly developed technique capable of inducing and measuring resonant surface waves in samples in a non-contact fashion. By applying RAR at high temporal resolution during thrombin-induced fibrin gelation and ultraviolet-initiated polyethylene glycol (PEG) polymerization, we observed distinct changes in both frequency and amplitude of the resonant surface waves as the materials changed over time. RAR detected a series of capillary-elastic, capillary-viscous, and visco-elastic transitions that are uniquely manifested as crossover of different types of surface waves in the temporally evolving materials. These results reveal the dynamic interplay of surface tension, viscosity, and elasticity that is controlled by the kinetics of polymerization and crosslinking during hydrogel formation. RAR overcomes many limitations of conventional rheological approaches by offering a new way to comprehensively and longitudinally characterize soft materials during dynamic processes.

Elastocapillary menisci mediate interaction of neighboring structures at the surface of a compliant solid

Surface stress drives long-range elastocapillary interactions at the surface of compliant solids, where it has been observed to mediate interparticle interactions and to alter transport of liquid drops. We show that such an elastocapillary interaction arises between neighboring structures that are simply protrusions of the compliant solid. For compliant micropillars arranged in a square lattice with spacing p less than an interaction distance p^{*}, the distance of a pillar to its neighbors determines how much it deforms due to surface stress: Pillars that are close together tend to be rounder and flatter than those that are far apart. The interaction is mediated by the formation of an elastocapillary meniscus at the base of each pillar, which sets the interaction distance and causes neighboring structures to deform more than those that are relatively isolated. Neighboring pillars also displace toward each other to form clusters, leading to the emergence of pattern formation and ordered domains.

Sloshing Resonance of an Acoustically Levitated Air-in-Liquid Compound Drop

An acoustically levitated air-in-liquid compound drop is set into an out-of-phase azimuthal sloshing resonance by a modulated frequency with modes n = 4-9. Waveforms of the inner and outer liquid-air interfaces conform to the classical Saffren model. Resonance peaks and their harmonics in the frequency spectrum are found to be a function of drop dimension and resonance modes. Drops with multiple small air bubbles do not resonate in sync because of asymmetry. This work has significant implications in the dynamics of core-shell compound drops.

Faraday Instability in Viscous Fluids Covered with Elastic Polymer Films

Faraday instability has great application value in the fields of controlling polymer processing, micromolding colloidal lattices on structured suspensions, organizing particle layers, and conducting cell culture. To regulate Faraday instability, in this article, we attempt to introduce an elastic polymer film covering the surface of a viscous fluid layer and theoretically study the behaviors of the Faraday instability phenomenon and the effect of the elastic polymer film. Based on hydrodynamic theory, the Floquet theory is utilized to formulate its stability criterion, and the critical acceleration amplitude and critical wave number are calculated numerically. The results show that the critical acceleration amplitude for Faraday instability increases with three increasing bending stiffness of the elastic polymer film, and the critical wave number decreases with increasing bending stiffness. In addition, surface tension and viscosity also have important effects on the critical acceleration amplitude and critical wave number. The strategy of controlling Faraday instability by covering an elastic polymer film proposed in this paper has great application potential in new photonic devices, metamaterials, alternative energy, biology, and other fields.

Oscillations of a soft viscoelastic drop

A soft viscoelastic drop has dynamics governed by the balance between surface tension, viscosity, and elasticity, with the material rheology often being frequency dependent, which are utilized in bioprinting technologies for tissue engineering and drop-deposition processes for splash suppression. We study the free and forced oscillations of a soft viscoelastic drop deriving (1) the dispersion relationship for free oscillations, and (2) the frequency response for forced oscillations, of a soft material with arbitrary rheology. We then restrict our analysis to the classical cases of a Kelvin-Voigt and Maxwell model, which are relevant to soft gels and polymer fluids, respectively. We compute the complex frequencies, which are characterized by an oscillation frequency and decay rate, as they depend upon the dimensionless elastocapillary and Deborah numbers and map the boundary between regions of underdamped and overdamped motions. We conclude by illustrating how our theoretical predictions for the frequency-response diagram could be used in conjunction with drop-oscillation experiments as a "drop vibration rheometer", suggesting future experiments using either ultrasonic levitation or a microgravity environment.

Plateau-Rayleigh instability in a soft viscoelastic material

A soft cylindrical interface endowed with surface tension can be unstable to wavy undulations. This is known as the Plateau-Rayleigh instability (PRI) and for solids the instability is governed by the competition between elasticity and capillarity. A dynamic stability analysis is performed for the cases of a soft (i) cylinder and (ii) cylindrical cavity assuming the material is viscoelastic with power-law rheology. The governing equations are made time-independent through the Laplace transform from which a solution is constructed using displacement potentials. The dispersion relationships are then derived, which depend upon the dimensionless elastocapillary number, solid Deborah number, and compressibility number, and the static stability limit, critical disturbance, and maximum growth rate are computed. This dynamic analysis recovers previous literature results in the appropriate limits. Elasticity stabilizes and compressibility destabilizes the PRI. For an incompressible material, viscoelasticity does not affect stability but does decrease the growth rate and shift the critical wavenumber to lower values. The critical wavenumber shows a more complex dependence upon compressibility for the cylinder but exhibits a predictable trend for the cylindrical cavity.

Dirac cones and chiral selection of elastic waves in a soft strip

We study the propagation of in-plane elastic waves in a soft thin strip, a specific geometrical and mechanical hybrid framework which we expect to exhibit a Dirac-like cone. We separate the low frequencies guided modes (typically 100 Hz for a 1-cm-wide strip) and obtain experimentally the full dispersion diagram. Dirac cones are evidenced together with other remarkable wave phenomena such as negative wave velocity or pseudo-zero group velocity (ZGV). Our measurements are convincingly supported by a model (and numerical simulation) for both Neumann and Dirichlet boundary conditions. Finally, we perform one-way chiral selection by carefully setting the source position and polarization. Therefore, we show that soft materials support atypical wave-based phenomena, which is all of the more interesting as they make most of the biological tissues.

Experimental observation of Faraday waves in soft gels

We report the experimental observation of Faraday waves on soft gels. These were obtained using agarose in a mechanically vibrated cylindrical container. Low driving frequencies induce subharmonic standing waves with spatial structure that conforms to the geometry of the container. We report the experimental observation of the first 15 resonant Faraday wave modes that can be defined by the mode number (n,ℓ) pair. We also characterize the shape of the instability tongue and show the complex dependence upon material properties can be understood as an elastocapillary effect.

Faraday waves in soft elastic solids

Recent experiments have observed the emergence of standing waves at the free surface of elastic bodies attached to a rigid oscillating substrate and subjected to critical values of forcing frequency and amplitude. This phenomenon, known as Faraday instability, is now well understood for viscous fluids but surprisingly eluded any theoretical explanation for soft solids. Here, we characterize Faraday waves in soft incompressible slabs using the Floquet theory to study the onset of harmonic and subharmonic resonance eigenmodes. We consider a ground state corresponding to a finite homogeneous deformation of the elastic slab. We transform the incremental boundary value problem into an algebraic eigenvalue problem characterized by the three dimensionless parameters, that characterize the interplay of gravity, capillary and elastic waves. Remarkably, we found that Faraday instability in soft solids is characterized by a harmonic resonance in the physical range of the material parameters. This seminal result is in contrast to the subharmonic resonance that is known to characterize viscous fluids, and opens the path for using Faraday waves for a precise and robust experimental method that is able to distinguish solid-like from fluid-like responses of soft matter at different scales.

Spinning elastic beads: a route for simultaneous measurements of the shear modulus and the interfacial energy of soft materials

Large deformations of soft elastic beads spinning at high angular velocity in a denser background fluid are investigated theoretically, numerically, and experimentally using millimeter-size polyacrylamide hydrogel particles introduced in a spinning drop tensiometer. We determine the equilibrium shapes of the beads from the competition between the centrifugal force and the restoring elastic and surface forces. Considering the beads as neo-Hookean up to large deformations, we show that their elastic modulus and interfacial energy constant can be simultaneously deduced from their equilibrium shape. Also, our results provide further support to the scenario in which interfacial energy and interfacial tension coincide for amorphous polymer gels.

How capillarity affects the propagation of elastic waves in soft gels

Elastic waves propagating at the interface of soft solids can be altered by the presence of external forces such as capillarity and gravity. We measure the dispersion relation of waves at the free surface of agarose gels with great accuracy, revealing the existence of multiple modes as well as an apparent dispersion. We disentangle the role of capillarity and elasticity by considering the three-dimensional nature of mechanical waves, achieving quantitative agreement between theoretical predictions and experiments. Notably, our results show that capillarity plays an important role for wave numbers smaller than expected from balancing elastic and capillary forces. We further confirm the efficiency of our approach by including the effect of gravity in our predictions and quantitatively comparing it to experiments.

A method for determining surface tension, viscosity, and elasticity of gels via ultrasonic levitation of gel drops

A method for obtaining the elasticity, surface tension, and viscosity of ultrasonically levitated gel drops is presented. The drops examined were made of agarose, a hydrogel. In contrast to previous studies where fluid properties are obtained using ultrasonic levitation of a liquid drop, herein the material studied was a gel which has a significant elasticity. The work presented herein is significant in that gels are of growing importance in biomedical applications and exhibit behaviors partially determined by their elasticities and surface tensions. Obtaining surface tension for these substances is important but challenging since measuring this quantity using the standard Wilhelmy plate or DuNuoy ring methods is not possible due to breakage of the gel. The experiments were conducted on agarose gels having elasticities ranging from 12.2 to 200.3 Pa. A method is described for obtaining elasticity, surface tension, and viscosity, and the method is experimentally demonstrated for surface tension and viscosity. For the range of elasticities explored, the measured surface tension ranged from 0.1 to 0.3 N/m, and the viscosity ranged from 0.0084 to 0.0204 Pa s. The measurements of surface tension are, to the authors' knowledge, the first obtained of a gel using ultrasonic levitation.

The elastic Rayleigh drop

Bioprinting technologies rely on the formation of soft gel drops for printing tissue scaffolds and the dynamics of these drops can affect the process. A model is developed to describe the oscillations of a spherical gel drop with finite shear modulus, whose interface is held by surface tension. The governing elastodynamic equations are derived and a solution is constructed using displacement potentials decomposed into a spherical harmonic basis. The resulting nonlinear characteristic equation depends upon two dimensionless numbers, elastocapillary and compressibility, and admits two types of solutions, (i) spheroidal (or shape change) modes and (ii) torsional (rotational) modes. The torsional modes are unaffected by capillarity, whereas the frequency of shape oscillations depend upon both the elastocapillary and compressibility numbers. Two asymptotic dispersion relationships are derived and the limiting cases of the inviscid Rayleigh drop and elastic globe are recovered. For a fixed polar wavenumber, there exists an infinity of radial modes that each transition from an elasticity wave to a capillary wave upon increasing the elastocapillary number. At the transition, there is a qualitative change in the deformation field and a set of recirculation vortices develop at the free surface. Two special modes that concern volume oscillations and translational motion are characterized. A new instability is documented that reflects the balance between surface tension and compressibility effects due to the elasticity of the drop.

Mechanisms for bacterial gliding motility on soft substrates

The motility mechanism of certain prokaryotes has long been a mystery, since their motion, known as gliding, involves no external appendages. The physical principles behind gliding still remain poorly understood. Using myxobacteria as an example of such organisms, we identify here the physical principles behind gliding motility and develop a theoretical model that predicts a 2-regime behavior of the gliding speed as a function of the substrate stiffness. Our theory describes the elasto-capillary-hydrodynamic interactions between the membrane of the bacteria, the slime it secretes, and the soft substrate underneath. Defining gliding as the horizontal translation under zero net force, we find the 2-regime behavior is due to 2 distinct mechanisms of motility thrust. On mildly soft substrates, the thrust arises from bacterial shape deformations creating a flow of slime that exerts a pressure along the bacterial length. This pressure in conjunction with the bacterial shape provides the necessary thrust for propulsion. On very soft substrates, however, we show that capillary effects must be considered that lead to the formation of a ridge at the slime-substrate-air interface, thereby creating a thrust in the form of a localized pressure gradient at the bacterial leading edge. To test our theory, we perform experiments with isolated cells on agar substrates of varying stiffness and find the measured gliding speeds in good agreement with the predictions from our elasto-capillary-hydrodynamic model. The mechanisms reported here serve as an important step toward an accurate theory of friction and substrate-mediated interactions between bacteria proliferating in soft media.

Elastocapillary Transition in Gel Drop Oscillations

We report experimental observations of surface oscillations in an ultrasoft agarose gel drop. Ultrasonic levitation is used to excite shape oscillations in the gel drop and we report the natural frequency of the drop as it depends upon a nondimensional elastocapillary number, which we define as the ratio of the elastocapillary length to drop size. Our experiments span a wide range of experimental parameters and we recover the appropriate scaling laws in the elastic and capillary wave limits. The crossover between these two limits is observed and agrees well with a proposed frequency relationship.

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.