Elastocapillarity can control the formation and the morphology of beads-on-string structures in solid fibers

Comparison of hydroxyapatite and honeycomb micro-structure in bone tissue engineering using electrospun beads-on-string fibers

Thick honeycomb-like electrospun scaffold with nanoparticles of hydroxyapatite (nHA) recently demonstrated its potential to promote proliferation and differentiation of a murine embryonic cell line (C3H10T1/2) to osteoblasts. In order to distinguish the respective effects of the structure and the composition on cell differentiation, beads-on-string fibers were used to manufacture thick honeycomb-like scaffolds without nHA. Mechanical and biological impacts of those beads-on string fibers were evaluated. Uniaxial tensile test showed that beads-on-string fibers decreased the Young Modulus and maximal stress but kept them appropriate for tissue engineering. C3H10T1/2 were seeded and cultured for 6 days on the scaffolds without any growth factors. Viability assays revealed the biocompatibility of the beads-on-string scaffolds, with adequate cells-materials interactions observed by confocal microscopy. Alkaline phosphatase staining was performed at day 6 in order to compare the early differentiation of cells to bone fate. The measure of stained area and intensity confirmed the beneficial effect of both honeycomb structure and nHA, independently. Finally, we showed that honeycomb-like electrospun scaffolds could be relevant candidates for promoting bone fate to cells in the absence of nHA. It offers an easier and faster manufacture process, in particular in bone-interface tissue engineering, permitting to avoid the dispersion of nHA and their interaction with the other cells.

Plateau-Rayleigh instability of a soft layer coated on a rigid cylinder

We study the Plateau-Rayleigh instability of a soft viscoelastic solid layer coated on a rigid cylinder i.e., a soft fibre with a rigid core. The onset of instability is examined using a linear stability analysis. We find that increasing the rigid cylinder radius or the stiffness of the layer reduces the growth rate of the fastest growing mode. For each rigid cylinder radius, a critical elastocapillary number is found below which all wavelengths of disturbances are stable. The critical value for a soft fibre with a thick rigid cylindrical core can be several orders of magnitudes larger than that for a totally soft fibre (no rigid core). This highlights the strong stabilizing effect of the rigid core on the system. Increasing the relaxation timescale of the viscoelastic material also slows down the growth of disturbances, but has no effect on the critical elastocapillary number. Interestingly, the wavelength of the fastest growing mode is independent of the rigid cylinder radius for the purely elastic case.

Cross-scale mechanobiological regulation of cylindrical compressible liquid inclusion via coating

The double-bag theory in modern anatomy suggests that structures with coatings are commonly found in human body at various length scales, such as osteocyte processes covered by pericellular matrix and bones covered by muscle tissue. To understand the mechanical behaviors and physiological responses of such biological structures, we develop an analytical model to quantify surface effects on the deformation of a coated cylindrical compressible liquid inclusion in an elastic matrix subjected to remote loading. Our analytical solution reveals that coating can either amplify or attenuate the volumetric strain of the inclusion, depending on the relative elastic moduli of inclusion, coating, and matrix. For illustration, we utilize this solution to explore amplification/attenuation of volumetric strain in musculoskeletal systems, nerve cells, and vascular tissues. We demonstrate that coating often plays a crucial role in mechanical regulation of the development and repair of human tissues and cells. Our model provides qualitative analysis of cross-scale mechanical response of coated liquid inclusions, helpful for constructing mechanical microenvironment of cells.

Elastic Rayleigh-Plateau instability: dynamical selection of nonlinear states

A slender thread of elastic hydrogel is susceptible to a surface instability that is reminiscent of the classical Rayleigh-Plateau instability of liquid jets. The final, highly nonlinear states that are observed in experiments arise from a competition between capillarity and large elastic deformations. Combining a slender analysis and fully three-dimensional numerical simulations, we present the phase map of all possible morphologies for an unstable neo-Hookean cylinder subjected to capillary forces. Interestingly, for softer cylinders we find the coexistence of two distinct configurations, namely, cylinders-on-a-string and beads-on-a-string. It is shown that for a given set of parameters, the final pattern is selected via a dynamical evolution. To capture this, we compute the dispersion relation and determine the characteristic wavelength of the dynamically selected profiles. The validity of the "slender" results is confirmed via simulations and these results are consistent with experiments on elastic and viscoelastic threads.

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.

A one-dimensional model for elasto-capillary necking

We derive a nonlinear one-dimensional (1d) strain gradient model predicting the necking of soft elastic cylinders driven by surface tension, starting from three-dimensional (3d) finite-strain elasticity. It is asymptotically correct: the microscopic displacement is identified by an energy method. The 1d model can predict the bifurcations occurring in the solutions of the 3d elasticity problem when the surface tension is increased, leading to a localization phenomenon akin to phase separation. Comparisons with finite-element simulations reveal that the 1d model resolves the interface separating two phases accurately, including well into the localized regime, and that it has a vastly larger domain of validity than 1d models proposed so far.

Circumferential wrinkling of polymer nanofibers

Surface wrinkles are commonly observed in soft polymer nanofibers produced in electrospinning. This paper studies the conditions of circumferential wrinkling in polymer nanofibers under axial stretching. A nonlinear continuum mechanics model is formulated to take into account the combined effects of surface energy and nonlinear elasticity of the nanofibers on wrinkling initiation, in which the soft nanofibers are treated as incompressible, isotropically hyperelastic neo-Hookean solid. The critical condition to trigger circumferential wrinkling is determined and its dependencies upon the surface energy, mechanical properties, and geometries of the nanofibers are examined. In the limiting case of spontaneous circumferential wrinkling, the theoretical minimum radius of soft nanofibers producible in electrospinning is determined, which is related closely to the intrinsic length l_{0}=γ/E of the polymer (γ: the surface energy; E: a measure of the elastic modulus), and compared with that of spontaneous longitudinal wrinkling in polymer nanofibers. The present study provides a rational understanding of surface wrinkling in polymer nanofibers and a technical approach for actively tuning the surface morphologies of polymer nanofibers for applications, e.g., high-grade filtration, oil-water separation, tissue scaffolding, etc.

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.

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

We deploy linear stability analysis to find the threshold wavelength (λ) and surface tension (γ) of Rayleigh-Plateau type "peristaltic" instabilities in incompressible neo-Hookean solids in a range of cylindrical geometries with radius R_{0}. First we consider a solid cylinder, and recover the well-known, infinite-wavelength instability for γ≥6μR_{0}, where μ is the solid's shear modulus. Second, we consider a volume-conserving (e.g., fluid filled and sealed) cylindrical cavity through an infinite solid, and demonstrate infinite-wavelength instability for γ≥2μR_{0}. Third, we consider a solid cylinder embedded in a different infinite solid, and find a finite-wavelength instability with λ∝R_{0}, at surface tension γ∝μR_{0}, where the constants depend on the two solids' modulus ratio. Finally, we consider an empty cylindrical channel (or filled with expellable fluid) through an infinite solid, and find an instability with finite wavelength, λ≈2R_{0}, for γ≥2.543...μR_{0}. Using finite-strain numerics, we show such a channel jumps at instability to a highly peristaltic state, likely precipitating it's blockage or failure. We argue that finite wavelengths are generic for elastocapillary instabilities, with the simple cylinder's infinite wavelength being the exception rather than the rule.

Adhesion-induced instabilities and pattern formation in thin films of elastomers and gels

A hydrostatically stressed soft elastic film circumvents the imposed constraint by undergoing a morphological instability, the wavelength of which is dictated by the minimization of the surface and the elastic strain energies of the film. While for a single film, the wavelength is entirely dependent on its thickness, a co-operative energy minimization dictates that the wavelength depends on both the elastic moduli and thicknesses of two contacting films. The wavelength can also depend on the material properties of a film if its surface tension has a pronounced effect in comparison to its elasticity. When such a confined film is subjected to a continually increasing normal displacement, the morphological patterns evolve into cracks, which, in turn, govern the adhesive fracture behavior of the interface. While, in general, the thickness provides the relevant length scale underlying the well-known Griffith-Kendall criterion of debonding of a rigid disc from a confined film, it is modified non-trivially by the elasto-capillary number for an ultra-soft film. Depending upon the degree of confinement and the spatial distribution of external stress, various analogs of the canonical instability patterns in liquid systems can also be reproduced with thin confined elastic films.