Finite-wavelength surface-tension-driven instabilities in soft solids, including instability in a cylindrical channel through an elastic solid

Rolling of stimuli-bent cylindrical robots using contact finite element simulations

Curved cylinders, if rigid, cannot roll on a surface like straight cylinders, but soft cylinders bent by specific stimuli can! Studying the autonomous locomotion of these soft robots and their interactions with the environment using finite element analysis is challenging due to the complex multiphysics of stimuli-responsive soft materials and nonlinear contact mechanics. In this pioneering work, we simulate the rolling of stimuli-bent cylinders on a surface using contact finite elements and introduce a simple yet effective pseudo-thermal field method. Our approach successfully reproduces several modes of autonomous locomotion observed experimentally, including phototropic locomotion, phototropic climbing on a slanted surface, steering under partial illumination, and backward rolling under alternating heat-light stimuli. Parametric analysis demonstrates strong agreement between the experiments and our numerical results, validating the effectiveness of our approach. This study reveals the intriguing and highly nonintuitive dynamics of photo- or thermally bent cylindrical soft robots, and serves as a paradigm for modelling and simulating such rolling robots.

Thermally-modulated shape transition at the interface of soft gel filament and hydrophobic substrate

A liquid filament may pinch off into different shapes on interacting with a soft surface, as modulated by the interplay of inertial, capillary, and viscous forces. While similar shape transitions may intuitively be realized for more complex materials such as soft gel filaments as well, their intricate controllability towards deriving precise and stable morphological features remains challenging, as attributed to the complexities stemming from the underlying interfacial interactions over the relevant length and time scales during the sol-gel transition process. Circumventing these deficits in the reported literature, here we report a new means of precisely-controlled fabrication of gel microbeads via exploiting thermally-modulated instabilities of a soft filament atop a hydrophobic substrate. Our experiments reveal that abrupt morphological transitions of the gel material set in at a threshold temperature, resulting in spontaneous capillary thinning and filament breakup. We show that this phenomenon may be precisely modulated by an alteration in the hydration state of the gel material that may be preferentially dictated by its intrinsic glycerol content. Our results demonstrate that the consequent morphological transitions give rise to topologically-selective microbeads as an exclusive signature of the interfacial interactions of the gel material with the deformable hydrophobic interface underneath. Thus, intricate control may be imposed on the spatio-temporal evolution of the deforming gel, facilitating the inception of highly ordered structures of specific shapes and dimensionalities on demand. This is likely to advance the strategies of long shelf-life analytical biomaterial encapsulations via realizing one-step physical immobilization of bio-analytes on the bead surfaces as a new route to controlled materials processing, without demanding any resourced microfabrication facility or delicate consumable materials.

Post-bifurcation behaviour of elasto-capillary necking and bulging in soft tubes

Previous linear bifurcation analyses have evidenced that an axially stretched soft cylindrical tube may develop an infinite-wavelength (localized) instability when one or both of its lateral surfaces are under sufficient surface tension. Phase transition interpretations have also highlighted that the tube admits a final evolved ‘two-phase’ state. How the localized instability initiates and evolves into the final ‘two-phase’ state is still a matter of contention, and this is the focus of the current study. Through a weakly nonlinear analysis conducted for a general material model, the initial sub-critical bifurcation solution is found to be localized bulging or necking depending on whether the axial stretch is greater or less than a certain threshold value. At this threshold value, an exceptionally super-critical kink-wave solution arises in place of localization. A thorough interpretation of the anticipated post-bifurcation behaviour based on our theoretical results is also given, and this is supported by finite-element method simulations.

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.

Ballooning, bulging, and necking: An exact solution for longitudinal phase separation in elastic systems near a critical point

Prominent examples of longitudinal phase separation in elastic systems include elastic necking, the propagation of a bulge in a cylindrical party balloon, and the beading of a gel fiber subject to surface tension. Here we demonstrate that if the parameters of such a system are tuned near a critical point (where the difference between the two phases vanishes), then the behavior of all systems is given by the minimization of a simple and universal elastic energy familiar from Ginzburg-Landau theory in an external field. We minimize this energy analytically, which yields not only the well known interfacial tanh solution, but also the complete set of stable and unstable solutions in both finite and infinite length systems, unveiling the elastic system's full shape evolution and hysteresis. Correspondingly, we also find analytic results for the the delay of onset, changes in criticality, and ultimate suppression of instability with diminishing system length, demonstrating that our simple near-critical theory captures much of the complexity and choreography of far-from-critical systems. Finally, we find critical points for the three prominent examples of phase separation given above, and demonstrate how each system then follows the universal set of solutions.

Peristaltic Elastic Instability in an Inflated Cylindrical Channel

A long cylindrical cavity through a soft solid forms a soft microfluidic channel, or models a vascular capillary. We observe experimentally that, when such a channel bears a pressurized fluid, it first dilates homogeneously, but then becomes unstable to a peristaltic elastic instability. We combine theory and numerics to fully characterize the instability in a channel with initial radius a through an incompressible bulk neo-Hookean solid with shear modulus μ. We show instability occurs supercritically with wavelength 12.278…a when the cavity pressure exceeds 2.052…μ. In finite solids, the wavelength for peristalsis lengthens, with peristalsis ultimately being replaced by a long-wavelength bulging instability in thin-walled cylinders. Peristalsis persists in Gent strain-stiffening materials, provided the material can sustain extension by more than a factor of 6. Although naively a pressure driven failure mode of soft channels, the instability also offers a route to fabricate periodically undulating channels, producing, e.g., waveguides with photonic or phononic stop bands.

Plateau-Rayleigh instability in solids is a simple phase separation

A long elastic cylinder, with radius a and shear-modulus μ, becomes unstable given sufficient surface tension γ. We show this instability can be simply understood by considering the energy, E(λ), of such a cylinder subject to a homogenous longitudinal stretch λ. Although E(λ) has a unique minimum, if surface tension is sufficient [Γ≡γ/(aμ)>sqrt[32]] it loses convexity in a finite region. We use a Maxwell construction to show that, if stretched into this region, the cylinder will phase-separate into two segments with different stretches λ_{1} and λ_{2}. Our model thus explains why the instability has infinite wavelength and allows us to calculate the instability's subcritical hysteresis loop (as a function of imposed stretch), showing that instability proceeds with constant amplitude and at constant (positive) tension as the cylinder is stretched between λ_{1} and λ_{2}. We use full nonlinear finite-element calculations to verify these predictions and to characterize the interface between the two phases. Near Γ=sqrt[32] the length of such an interface diverges, introducing a new length scale and allowing us to construct a one-dimensional effective theory. This treatment yields an analytic expression for the interface itself, revealing that its characteristic length grows as l_{wall}∼a/sqrt[Γ-sqrt[32]].

Electro-elastocapillary Rayleigh-plateau instability in dielectric elastomer films

We demonstrate, using both finite element simulations and a linear stability analysis, the emergence of an electro-elastocapillary Rayleigh-plateau instability in dielectric elastomer (DE) films under 2D, plane strain conditions. When subject to an electric field, the DEs exhibit a buckling instability for small elastocapillary numbers. For larger elastocapillary numbers, the DEs instead exhibit the Rayleigh-plateau instability. The stability analysis demonstrates the critical effect of the electric field in causing the Rayleigh-plateau instability, which cannot be induced solely by surface tension in DE films. Overall, this work demonstrates the effects of geometry, boundary conditions, and multi-physical coupling on a new example of Rayleigh-plateau instability in soft solids.

Sphere-To-Tube Transition toward Nanotube Formation: A Universal Route by Inverse Plateau-Rayleigh Instability

Nanotube formation in low-temperature solution has attracted intense interest since the 1990s. How to disclose the in-depth physicochemical nature of nanotubes and pursue new available chemical strategies is still highly desirable but remains a challenge. Here, we report that sphere-to-tube transition triggered by inverse Plateau-Rayleigh instability can be a chemical route for scalable production of nanotubes. As a proof of concept, formation of a phosphorus nitride (PN) nanotube and various hierarchical nanotube architectures by coalescence of the PN hollow spheres is achieved under systematic solvothermal reaction. The combination of theoretical analysis and dynamic simulation elucidates that the inverse Plateau-Rayleigh instability driven by the competition between curvature elasticity and surface energy is responsible for the PN nanotube formation observed in experiments. We anticipate that the sphere-to-tube transition provides a paradigm for nanotube synthesis for practical applications.