Schedule for: 22w5127 - Equilibrium and non-Equilibrium Pattern Formation in Soft Matter: From Elastic Solids to Complex Fluids

Large floating viscous bubbles whose interior gas is rapidly depressurized exhibit a remarkable dynamics, characterized by a periodic pattern of radial wrinkles that permeate the liquid film in the course of its flattening. This instability was discovered in 1998 by Debregeas et al. [1] and has been attributed to the joint effect of gravity and the expansion of a circular rupture [2]. However, a recent experiment by Oratis et al. [3] demonstrated that the instability appears even in the absence of gravity or rupture, indicating a mechanism dominated solely by viscous and capillary forces. Motivated by these experiments we address Stokes flow in a curved film of a non-inertial incompressible liquid with free surfaces, generated by temporal variation of the Gaussian curvature R [4]. Notwithstanding the close analogy between the Newtonian hydrodynamics of viscous liquids and the Hookean elasticity of solids, often called “Stokes-Rayleigh analogy”, the fact that stress in viscous films is generated by the rate-of-change dtR, rather than by R itself as is the case for elastic sheets, reflects a profound difference between these two branches of non-inertial, yet geometrically nonlinear continuum mechanics. Whereas the rigidity of elastic sheets derives from the existence of a “target” metric, their viscous counterparts are not endowed with a preferred metric. We reveal the experimental observations of Ref. [3] as a dramatic ramification of this distinction and the consequent emergence of a universal, curvature-driven surface dynamics, imparted by viscous resistance to dtR 6= 0. Specifically, rapidly-depressurized viscous bubbles flatten by forming a radially moving front of highly localized dtR that separate a flat core and a spherically-shapes periphery, and become wrinkled due to a hoop-compressive stress field at the wake of the propagating front [5]. This novel surface dynamics has close ties to “Jelium physics”, where topological defects spontaneously emerge to screen elastic stress, similarly to dipoles-mediated screening of electrostatic field in conducting media, thereby extending the classic analogy between Wigner crystals, Abrikosov lattice in type-II superconductors, and 2D elasticity of curved crystals, to non-equilibrium 2D viscous hydrodynamics. A particularly exciting possibility is the emergence of such a universal geometrically nonlinear 2D viscous hydrodynamics in strongly-correlated electronic liquids in 2D crystals. [1] G. Debregeas, P.G. de Gennes, F. Brochard-Wyart, “The life and death of ”bare” viscous bubbles”, Science 279, 1704-1707 (2000). [2] R. da Silviera, S. Chaieb, L. Mahadevan, “Rippling instability of a collapsing bubble”, Science 287, 1468-1471 (2000). [3] A.T. Oratis, J.W.M. Bush, H.A. Stone, J. Bird, “A new wrinkle on liquid sheets: Turning the mechanism of viscous bubble collapse upside down”, Science 369, 685 (2020). [4] P.D. Howell, “Models for thin viscous sheets”, Eur. J. App. Math. 7, 321-343 (1996). [5] B. Davidovitch and A. Klein, “How viscous bubbles collapse: topological and symmetry breaking instabilities in curvature-driven hydrodynamics” (2022).

Here we present the individual dynamics of flexible and Brownian filaments under shear and compression. We use actin filaments as a model system and observe their dynamics in microfluidic flow geometries using fluorescent labeling techniques and microscopic tracking methods. The experimental results are completed with analytical and numerical modeling based on slender body theory. Under shear we characterize successive transitions from tumbling to buckling and finally snake turns as a function of an elasto-viscous number, comparing viscous to elastic forces. Under compression we reveal the formation of three dimensional helicoidal structures and explain their formation from linear and weakly nonlinear stability analysis. In more complex, time dependent or mixed flows, as oscillatory shear flows or transport in porous media, filament morphology is modified and buckling instabilities can be suppressed under certain conditions. Pillar arrays can be used to obtain filament separation according to length and flexibility. Finally, we attempt at linking the microscopic observations to the macroscopic suspension properties with preliminary measurements of the shear viscosity of dilute suspensions of actin filaments in microfluidic rheometers and numerical simulations.

Viscous fingering in Hele-Shaw channels is a canonical example of diffusion-limited interfacial growth phenomena which exhibits a fascinating range of complex dynamics. An important advantage of this system for understanding pattern formation is that the key system information is encapsulated in the interface. When air displaces a viscous fluid in the narrow gap between two parallel plates forming a channel, the initially flat interface is linearly unstable. The destabilisation of this flat interface is followed by the growth and competition of fingers, resulting eventually in the steady propagation of a single finger, i.e., a curved front. When the curved front is in turn perturbed locally with finite amplitude, many more complex modes of propagation including periodic modes can be observed fleetingly, thus suggesting that they are unstable. In this talk, we show that similar pattern forming modes of front propagation can be harnessed by altering the channel geometry. We further explore the conditions required for tip instabilities of propagating curved fronts to promote complex pattern formation in both rigid and compliant systems and the role of these tip instabilities in the generation of disordered front propagation.

When a thin film deposited a solid substrate dries, the resulting tensile stresses tend to induce the successive formation of cracks. Such “channel cracks” notoriously propagate along straight paths and only change their direction to meet older cracks with a right angle. We will discuss a different mode of rupture where fracture collaborates with delamination which dictates the typical width of unexpected self-replicating cracks. A second example of spontaneous pattern formation is related to the propagation of the adhesion front between two plates coated with a layer of viscous liquid. Following classical criteria for viscous fingering, the viscous front pushing away air of low viscosity is expected to be stable. In contrast, the front generally destabilizes into growing fingers of a fixed width. This instability relies on the depletion of the liquid films that feed the front. Do the pattern formation in both configurations share common ingredients?

For the past several years, I have been transgressing the boundary between art and science. I have repurposed my scientific images of pattern formation experiments and pattern-forming natural phenomena by presenting them as art. I have exhibiting images and videos in art galleries and juried art shows. I have brought artists into my research lab for several hands-on workshops. I am the co-organizer of the "ArtSci Salon", an evening meet-up group at the Fields Institute of Mathematical Science in Toronto. I have released a trove of icicle shape data for free use under the Creative Commons. I have collaborated with sound artists and composers to use pattern formation images and videos as input to their creative processes. All these activities can be viewed equally as art-making or as scientific outreach. I claim that the scientific field of pattern formation has developed a distinct aesthetic sensibility, informed by mathematics and physics, but inherently visual and dynamic. We have our own "scientific folk art", whether we realize it or not. I further claim that this aesthetic motivation is essential for scientific work and that mixing science and art can be effective outreach. It is also a lot of fun!

We study swelling-induced buckling of ultra-thin PDMS membranes that form the upper boundary of an otherwise rigid microchannel. The membrane is bonded to the sidewalls of the channel and may initially be deformed by either pressurising or depressurising the channel. Swelling is initiated by rapidly covering the PDMS membrane with silicone oil. Wrinkles that are perpendicular to the long axis of the channel appear along the bonded boundaries within a few seconds. As the membrane swells, the wrinkles grow in amplitude until their period doubles, thus forming a hierarchical structure. We observe two types of coarsened folding patterns once the membrane is fully swollen. For modest levels of initial inflation (i.e. smaller internal pressure), we see a pattern of folds emanating from each side boundary in alternation, while at higher levels of initial inflation (i.e. larger internal pressure), fewer folds are formed but they extend across the entire channel width. In both scenarios, however, this secondary wrinkling only partially relaxes the original wrinkles near the channel walls. We also find that once the membrane is swollen, the system can transition between these patterns if the pressure is suitable adjusted. We discuss the mechanisms underpinning the swelling-induced wrinkling of our clamped membranes in the light of recent literature.

We consider the morphologies adopted by liquid in fibrous assemblies, and in particular the coupled effects of geometry, elasticity and swelling on the liquid distribution. We first consider a fiber placed in a flowing mist; the aerosol droplets are intercepted by the fibers, coalesce and form larger drops sitting on the fibers, until reaching a critical size at which they fall, entraining the drops below them. Due to a growth and coalescence process, for most of the collection process the drops are actually uniformly distributed along the fiber surface. This uniform pattern may be further affected by aerodynamics-mediated interactions between the drops. One way to suppress the growth of drops is to replace the single fiber by a pair of parallel fibers; indeed, the liquid will then form long liquid columns between the fibers rather than drops. When the fibers are flexible, the force exerted by the drop may deform them, and the liquid spreads as a long column between collapsed fibers. We examine situations where there are multiple possible equilibria, and the system dynamically switches between different shapes. In addition, for favorable solvents that are absorbed by the fibers, the induced swelling strongly affects the liquid distribution, for example inducing transient motions or even coalescence of the drops, as well as the spontaneous collapse of the fibers. Finally, I will discuss the cases of more complex fibrous systems such as thin textiles or bundle of fibers.

In recent years we have developed a method to produce microscopic monodisperse oil droplets in an aqueous environment. With an attractive interaction between the droplets, these monodisperse droplets form perfect crystalline aggregates, while a blend of small and large droplets allows us to prepare a disordered glass. By carefully tuning the adhesion forces between the droplets, the aggregates provide model systems for studying various physical phenomena that are not accessible by investigating molecular systems. I will provide a brief overview of experiments we have carried out to address two fundamental questions. First, how does a system transition from crystal to glass, when blending large and small droplets? And second, how does a system transition from granular, when there are a few particles, to many particles, where continuum models are valid. These experiments enable us to study broad questions which relate to real-world problems like predicting the failure and fracture of materials and the size and time distribution of avalanches.

When stressed sufficiently, amorphous materials yield and deform plastically via reorganization of microscopic constituents. Despite much effort, understanding the interdependence of yielding, plasticity, and microscopic structure in non-equilibrium states (i.e. under stress) remains a major challenge. In this talk, I explore this interdependence by cyclic shearing a dense colloidal suspension using a custom-built interfacial stress rheometer. This setup allows for simultaneous measurement of the bulk rheology (G’, G’’) and characterization of the suspension microstructure and dynamics. Sample microstructure is characterized using the concept of excess entropy, an averaged quantity that quantifies particle caging. We find that structural relaxation induced by plastic flow depends on and scale with the strain-rate and microscopic order measured at earlier and later times, respectively. Thus, measurement of sample static structure (excess entropy) provides insight about both strain-rate and constituent rearrangement dynamics in the sample at earlier times. Moreover, our results reveal a direct relation between excess entropy and energy dissipation, that is insensitive to the nature of interactions among particles. We use this relation to build a physically informed model that connects rheology to microstructure. Our findings suggest a framework for tailoring the rheological response of disordered materials by tuning microstructural properties.

We discuss how and why striped pattern mosaics arising from gradient microscopic pattern forming systems exhibit universal behavior. Their order parameters evolve according to gradient flows with energies determined by coordinate invariant combinations of the first two two forms, the metric ("strain") and curvature ("bending"), of a well defined phase surface. The Jacobian of the map from physical to order parameter space has interesting properties that both help define the invariant indices associated with the canonical point (in 2D) and loop (in 3D) defects and allow for an effective linearization of the order parameter equations. It is intriguing that systems that start out with only the symmetries of translation and rotation can, when stressed, naturally produce objects with fractional invariants of 1, 1/2 and 1/3. I will discuss many open challenges.

The viscous fingering instability occurs when a low viscosity fluid displaces a higher viscosity one within a confined geometry. The prototypical experimental setup to study such a system is the Hele-Shaw cell: two large, flat plates separated by a small gap. Typically, the first step in analyzing such a system is to average out this smallest dimension and consider the flow to be two-dimensional. However, for miscible pairs of fluids, the inner fluid does not fill the entirety of the gap and instead forms a tongue that only occupies a fraction of the gap, providing a three-dimensional structure. I will show how the thickness of this tongue is related to increased stability when the viscosity ratio of the fluids approaches unity but is still in a regime where immiscible fluids would be unstable. This leads us to consider what it is about the gap structure for miscible fluids that impacts the system’s stability. To interrogate this I will discuss different ways that one can perturb this structure. I will show how (i) allowing diffusion to disrupt the gap structure leads to the emergence of a new morphology of fingering patterns that is accompanied by a regime of stability and (ii) how actively perturbing the gap structure with shear can change characteristics of the onset of pattern formation.

Micron-scale surface modulations such as wrinkles or folds underpin a number of modern engineering applications, such as photonic structures in photovoltaics and flexible metasurfaces. Controlled and precise fabrication of these modulations is a challenge for human manufacturing techniques. In stark contrast, biological systems robustly utilize morphological changes in their developmental program to create multi-germ bodies, hairs and scales on spatial scales which would be costly to replicate with human manufacturing. In this talk I will present recent measurements of in-vivo butterfly scale development exhibiting wrinkling. The observations are rationalized with a numerical finite element simulation and a parsimonious continuum mechanics model.

Complex fluids exhibit a variety of exotic flow behaviors under high stresses, such as shear thickening and shear jamming. Rheology is a powerful tool to characterize these flow behaviors over the bulk of the fluid. However, this technique is limited in its ability to probe fluid behavior in a spatially resolved way. Here, I will show how we can utilize ultrahigh-speed imaging and the free-surface geometry in drop impact as a new tool for studying the flow of dense colloidal suspensions. In addition to observing Newtonian-like spreading and bulk shear jamming, we observe the transition between these regimes in the form of localized patches of jammed suspension in the spreading drop. This system offers a unique lens with which to study shear-thickening fluids, allowing us to obtain flow information in a spatially-localized manner, so that we can observe coexisting solid and liquid phases. Furthermore, we capture shear jamming as it occurs via a solidification front traveling from the impact point, and show that the speed of this front is set by how far the impact conditions are beyond the shear thickening transition.

Fluid flows containing plant pathogens are major factors in crop production loss. In particular, the releasing mechanism of pathogenic or allergic particles deposited on flexible plant surfaces has not been understood well. In this talk, we will study how vibrating leaf motion can generate a coherent and long-lasting pattern with periodically shedding vortices that enhances particle mixing and spatial transport. First, we extract Lagrangian coherent structures (LCS) that emerge from the unsteady, oscillatory flow induced by a fluttering plant leaf. Such coherent structures exhibit expansive and advective motions, set by a balance between a leaf’s elasticity and external inertia. The study further completes the picture of fungal spore escape by applying LCS diagnostics metrics to inspect advection in coherent airflow structures.

Lane formation is a paradigmatic example of spontaneous organization occurring in two component counterflows, which has been observed in diverse contexts including pedestrian traffic and driven colloids. A typical experimental or simulation set-up comprises two groups moving in opposite directions who, as a result of collisions or collision avoidance manoeuvres, achieve segregation into lanes parallel to the direction of motion. In my talk I will discuss a new kinetic theory which gives insight into the physical origin of laning and make predictions about the rate at which the lanes emerge from a homogeneous crowd. To complement the theoretical discussion, I will also present an experiment with human crowds confirming some new dynamical phenomena predicted by the theory.

The simple act of twisting a flat sheet has been used widely by humanity in making catgut bow strings and surgical sutures going back thousands of years and has been proposed more recently as a strategy to make yarns from novel ultra-thin sheets. Twisting is further reflected and utilized in the act of wringing water out of a towel, wrapping candy, upcycling plastic sheets into ropes, and turbans found in the Indian subcontinent made from a long fabric which are redeployed for everything from retrieving water from a well to sheltering from the elements. Despite this ubiquity and importance, a formal description or even recognition of the underlying twist-fold transformations has not been reported. By harnessing the power of state-of-the-art micro-focus x-ray scanning to noninvasively image the fine internal structures, we show that a sheet is found to accordion fold onto itself templated by the transverse instability as the twist is increased to half a turn, and with subsequent instabilities resulting in a dramatic decrease in crosssection and hierarchically folded yarn-like structures. Going beyond charting the patterning route taken by the sheet depending on its extensibility, we will discuss an analytical tensional twist-folding model which incorporates significant stretching and self-contact. We will highlight the usefulness of simple compression induced folding and origami kinematics in understanding shape transformation even when sheets undergo significant stretching. Time permitting, we will discuss extensions of our formulation to filament tangling. Work in collaboration with Drs. Julien Chopin and Animesh Biswas

Mechanism-based mechanical metamaterials are designer materials that experience microscale buckling in response to mechanical deformation. Their analysis challenges our understanding of the relationship between microscopic and macroscopic behavior in geometrically nonlinear mechanical systems. I will focus on a particular example, the “Kagome metamaterial.” Paradoxically, it has a huge variety of energy-free deformations (mechanisms), but its macroscopic elastic energy vanishes only for compressive conformal maps. The analysis of its behavior provides insight on – and reveals open questions about – many fundamental questions, including the relationship between mechanisms and Guest-Hutchinson modes, and degree to which the effective energy governs such a system’s macroscopic mechanical response. This is joint work with Xuenan Li (theory) and Katia Bertoldi and Bolei Deng (simulation).

Kirigami is the art of cutting paper to make it articulated and deployable, allowing for it to be shaped into complex two and three-dimensional geometries. The mechanical response of a kirigami sheet when it is pulled at its ends is enabled and limited by the presence of cuts that serve to guide the possible non-planar deformations. Inspired by the geometry of this artform, we ask two questions: (i) What is the shortest path between points at which forces are applied? (ii) What is the nature of the ultimate shape of the sheet when it is strongly stretched? Mathematically, these questions are related to the nature and form of geodesics in the Euclidean plane with linear obstructions (cuts), and the nature and form of isometric immersions of the sheet with cuts when it can be folded on itself, and are related to a class of questions associated with isometric immersions in differential geometry and geodesics in discrete and computational geometry. It is not hard to observe that a geodesic connecting given two points in the kirigamized plane is piecewise polygonal. We provide a constructive proof that the family of all such polygonal geodesics can be simultaneously rectified into a straight line by flat-folding the sheet so that its configuration is a (non-unique) piecewise affine planar isometric immersion. This is a joint work with Q. Han (Notre Dame University) and L. Mahadevan (Harvard University).

An efficient way to introduce elastic energy that can bias an origami structure toward desired shapes is to allow curved tiles between the creases. (Think: a Frank Gehry building with creases, that folds up spontaneously from a flat sheet.) Isometric bending of the tiles then supplies the energy. The $h^3$ scaling of the energy of thin sheets ($h=$ thickness) spans a broad energy range, that is also consistent with a single origami design. And with a given design, different tiles can have different values of $h$. Even a single tile can have differing values of $h$. In this lecture we present a theory and systematic design methods for quite general curved-tile origami structures that can be folded from a flat sheet. Unlike the standard approach to origami design (which is Eulerian), we find it useful to develop Lagrangian methods. A group orbit method using discrete isometry groups enables the design of complex structures from simple calculations. Kirchhoff’s nonlinear plate theory is ideal for accurate calculation of the energy. Joint work with Huan Liu.

It is well-known that crack front complexity - essentially arising from planar symmetry breaking - leads to an enhancement of strain energy required to drive the crack; however, less is known about how this process happens. This is primarily on account of a paucity of experimental data. Here, using direct, 3D confocal microscopy, we probe the local fracture toughness at the tip of both complex and planar cracks in brittle hydrogels. The hydrogel samples are fluorescently labeled for contrast, ensuring that we resolve the material interface down to the sub-micron diffraction limit of our microscope. Using the 3D data of the loaded crack surfaces, we directly probe the crack tip opening displacement (CTOD), and by proxy, evaluate the effective stress intensity factor directly at the crack tip. We find that the CTOD depends on the conformation of the crack tip, and that the CTOD increases significantly with greater crack tip complexity; furthermore, the CTOD can vary spatially along the crack front, reflecting the broken planar symmetry of the complex crack fronts. These data are compared directly with symmetric crack fronts for reference values in identical material, ensuring that our results are not artifacts of material preparation.

Shape-morphing finds widespread utility, from the deployment of small stents and large solar sails to actuation and propulsion in soft robotics. Origami and kirigami - patterns of cuts and folds in a sheet - are versatile platforms for shape-morphing, inspiring the design of many morphing structures and devices. However, it remains a challenge to design patterns that morph into a specified surface on demand, and to predict their response to a broad range of loads and stimuli. This talk explores general design and modeling principles for origami and kirigami structures. In the first part of the talk, we develop an efficient algorithm that explicitly characterizes the designs and deformations of a large class of easily deployable origami. We then employ this algorithm in an inverse design framework to approximate a targeted surface. In the second part of the talk, we describe a coarse-graining procedure to determine all the slighty stressed (soft) modes of deformation of a large class of periodic and planar kirigami. The procedure gives a system of nonlinear partial differential equations (PDE) expressing geometric compatibility of angle functions related to the motion of individual slits. Leveraging known solutions of the PDE and FEM modeling, we present illuminating agreement between simulations and experiments across kirigami designs. Our results reveal a dichotomy of designs that deform with persistent versus decaying slit actuation, which we explain using the Poisson's ratio of the unit cell.

Liquid crystal elastomers (LCEs) are active solids that can have significant shape changes upon external stimuli such as heat or illumination. The shape change is determined by its nematic director that encodes the direction of molecular alignment. In this talk, I will mainly present a shape programming problem and a pattern formation problem in LCEs. The former concerns how to design nematic director patterns in 2D to obtain shape-programable curved folds after actuation. Unlike traditional curved-fold origami, the LCE curved folds carry non-zero programmable Gaussian curvature, which determines the mechanical response of the folds. More specifically, we find that the total curvature of the folds scales as t^(-1/7), where t is the thickness of the LCE sheet. In the latter problem, we use a relaxed energy model for non-ideal LCEs to explain an unusual pattern formation in LCE sheets confined on solid substrates. The model reveals that double twinning laminates play key roles in the coexistence of a small-scale parallel wrinkling and a large-scale 45-degree wrinkling in LCEs, which is not observed in the squeezing of ordinary soft solids. Looking forward, LCEs can stimulate both promising engineering applications and interesting scientific problems.

Buckling in thin structures is generally considered as a first step towards failure. Instead, we view mechanical and interfacial instabilities in structures as opportunities for scalable, reversible, and robust mechanisms that must first be understood predictively, and then harvested for their function. This new design paradigm – building with instabilities – calls for an improved understanding of instabilities and pattern formation in complex media. Three examples will be presented: (1) fluid-instability based approaches for digitally fabricating geometrically complex uniformly sized structures, (2) flexible fabric-based gripper that contracts radially upon inflation (3) deployable structures inspired by insect wing expansion. The main common feature underlying these various problems is the prominence of geometry, and its interplay with mechanics, in dictating complex mechanical behavior that is relevant and applicable over a wide range of length scales.