Dynamic and reversible shape response of red blood cells in synthetic liquid crystals.

Mammalian cells are soft, and correct functioning requires that cells undergo dynamic shape changes in vivo. Although a range of diseases are associated with stiffening of red blood cells (RBCs; e.g., sickle cell anemia or malaria), the mechanical properties and thus shape responses of cells to complex viscoelastic environments are poorly understood. We use vapor pressure measurements to identify aqueous liquid crystals (LCs) that are in osmotic equilibrium with RBCs and explore mechanical coupling between RBCs and LCs. When transferred from an isotropic aqueous phase into a LC, RBCs exhibit complex yet reversible shape transformations, from initially biconcave disks to elongated and folded geometries with noncircular cross-sections. Importantly, whereas the shapes of RBCs are similar in isotropic fluids, when strained by LC, a large variance in shape response is measured, thus unmasking cell-to-cell variation in mechanical properties. Numerical modeling of LC and cell mechanics reveals that RBC shape responses occur at constant cell membrane area but with membrane shear moduli that vary between cells from 2 to 16 × 10-6 N/m. Temperature-dependent LC elasticity permits continuous tuning of RBC strains, and chemical cross-linking of RBCs, a model for diseased cells, leads to striking changes in shape responses of the RBCs. Overall, these results provide insight into the coupling of strain between soft mammalian cells and synthetic LCs, and hint at new methods for rapidly characterizing mechanical properties of single mammalian cells in a population and thus cell-to-cell variance.

Geometric localization of thermal fluctuations in red blood cells.

The thermal fluctuations of membranes and nanoscale shells affect their mechanical characteristics. Whereas these fluctuations are well understood for flat membranes, curved shells show anomalous behavior due to the geometric coupling between in-plane elasticity and out-of-plane bending. Using conventional shallow shell theory in combination with equilibrium statistical physics we theoretically demonstrate that thermalized shells containing regions of negative Gaussian curvature naturally develop anomalously large fluctuations. Moreover, the existence of special curves, "singular lines," leads to a breakdown of linear membrane theory. As a result, these geometric curves effectively partition the cell into regions whose fluctuations are only weakly coupled. We validate these predictions using high-resolution microscopy of human red blood cells (RBCs) as a case study. Our observations show geometry-dependent localization of thermal fluctuations consistent with our theoretical modeling, demonstrating the efficacy in combining shell theory with equilibrium statistical physics for describing the thermalized morphology of cellular membranes.

Active matter invasion of a viscous fluid: Unstable sheets and a no-flow theorem.

We investigate the dynamics of a dilute suspension of hydrodynamically interacting motile or immotile stress-generating swimmers or particles as they invade a surrounding viscous fluid. Colonies of aligned pusher particles are shown to elongate in the direction of particle orientation and undergo a cascade of transverse concentration instabilities, governed at small times by an equation that also describes the Saffman-Taylor instability in a Hele-Shaw cell, or the Rayleigh-Taylor instability in a two-dimensional flow through a porous medium. Thin sheets of aligned pusher particles are always unstable, while sheets of aligned puller particles can either be stable (immotile particles), or unstable (motile particles) with a growth rate that is nonmonotonic in the force dipole strength. We also prove a surprising "no-flow theorem": a distribution initially isotropic in orientation loses isotropy immediately but in such a way that results in no fluid flow everywhere and for all time.

Geometrically controlled snapping transitions in shells with curved creases.

Curvature and mechanics are intimately connected for thin materials, and this coupling between geometry and physical properties is readily seen in folded structures from intestinal villi and pollen grains to wrinkled membranes and programmable metamaterials. While the well-known rules and mechanisms behind folding a flat surface have been used to create deployable structures and shape transformable materials, folding of curved shells is still not fundamentally understood. Shells naturally deform by simultaneously bending and stretching, and while this coupling gives them great stability for engineering applications, it makes folding a surface of arbitrary curvature a nontrivial task. Here we discuss the geometry of folding a creased shell, and demonstrate theoretically the conditions under which it may fold smoothly. When these conditions are violated we show, using experiments and simulations, that shells undergo rapid snapping motion to fold from one stable configuration to another. Although material asymmetry is a proven mechanism for creating this bifurcation of stability, for the case of a creased shell, the inherent geometry itself serves as a barrier to folding. We discuss here how two fundamental geometric concepts, creases and curvature, combine to allow rapid transitions from one stable state to another. Independent of material system and length scale, the design rule that we introduce here explains how to generate snapping transitions in arbitrary surfaces, thus facilitating the creation of programmable multistable materials with fast actuation capabilities.

Wrinkling of milk skin is mediated by evaporation.

Wrinkling of thin films and membranes can occur due to various mechanisms such as growth and/or mismatch between the mechanical properties of the film and substrate. However, the physical origins of dynamic wrinkling in soft membranes are still not fully understood. Here we use milk skin as a tractable experimental system to investigate the physics of wrinkle formation in a thin, poroelastic film. Upon heating milk, a micron-thick hydrogel of denatured proteins and fat globules forms at the air-water interface. Over time, we observe an increase in the total length of wrinkles. By confocal imaging and profilometry, we determine that the composition and thickness of the milk skin appears to be homogeneous over the length scale of the wrinkles, excluding differences in milk skin composition as a major contributor to wrinkling. To explain the physical origins of wrinkle growth, we describe theory that considers the milk skin as a thin, poroelastic film where pressure is generated by the evaporative-driven flow of solvent across the film; this imparts in-plane stresses in the milk skin, which cause wrinkling. Viscous effects can explain the time-dependent growth of wrinkles. Our theoretical predictions of the effects of relative humidity on the total length of wrinkles over time are consistent with our experimental results. Our findings provide insight into the physics of the common phenomenon of milk skin wrinkling, and identify hydration gradients as another physical mechanism that can drive morphological instabilities in soft matter.

Origami structures with a critical transition to bistability arising from hidden degrees of freedom.

Origami is used beyond purely aesthetic pursuits to design responsive and customizable mechanical metamaterials. However, a generalized physical understanding of origami remains elusive, owing to the challenge of determining whether local kinematic constraints are globally compatible and to an incomplete understanding of how the folded sheet's material properties contribute to the overall mechanical response. Here, we show that the traditional square twist, whose crease pattern has zero degrees of freedom (DOF) and therefore should not be foldable, can nevertheless be folded by accessing bending deformations that are not explicit in the crease pattern. These hidden bending DOF are separated from the crease DOF by an energy gap that gives rise to a geometrically driven critical bifurcation between mono- and bistability. Noting its potential utility for fabricating mechanical switches, we use a temperature-responsive polymer-gel version of the square twist to demonstrate hysteretic folding dynamics at the sub-millimetre scale.

Topological Mechanics of Origami and Kirigami.

Origami and kirigami have emerged as potential tools for the design of mechanical metamaterials whose properties such as curvature, Poisson ratio, and existence of metastable states can be tuned using purely geometric criteria. A major obstacle to exploiting this property is the scarcity of tools to identify and program the flexibility of fold patterns. We exploit a recent connection between spring networks and quantum topological states to design origami with localized folding motions at boundaries and study them both experimentally and theoretically. These folding motions exist due to an underlying topological invariant rather than a local imbalance between constraints and degrees of freedom. We give a simple example of a quasi-1D folding pattern that realizes such topological states. We also demonstrate how to generalize these topological design principles to two dimensions. A striking consequence is that a domain wall between two topologically distinct, mechanically rigid structures is deformable even when constraints locally match the degrees of freedom.

Photothermally reprogrammable buckling of nanocomposite gel sheets.

Patterning deformation within the plane of thin elastic sheets represents a powerful tool for the definition of complex and stimuli-responsive 3D buckled shapes. Previous experimental methods, however, have focused on sheets that access a limited number of shapes pre-programmed into the sheet, restricting the degree of dynamic control. Here, we demonstrate on-demand reconfigurable buckling of poly(N-isopropylacrylamide-co-acrylic acid) (PNIPAM) hydrogel network films containing gold nanoparticles (AuNPs) by patterned photothermal deswelling. Predictable, easily controllable, and reversible transformations from a single flat gel sheet to numerous different three-dimensional forms are shown. Importantly, the response time is limited by poroelastic mass transport, rather than photochemical switching kinetics, enabling reconfiguration of shape on timescales of several seconds, with further increases in speed possible by reducing film thickness.

Reflection and refraction of flexural waves at geometric boundaries.

We present a theory of flexural wave propagation on elastic shells having nontrivial geometry and develop an analogy to geometric optics. The transport of momentum within the shell itself is anisotropic due to the curvature, and as such complex classical effects such as birefringence are generically found. We determine the equations of reflection and refraction of such waves at boundaries between different local geometries, showing that waves are totally internally reflected, especially at boundaries between regions of positive and negative Gaussian curvature. We verify these effects by using finite element simulations and discuss the ramifications of these effects for the statistical mechanics of thin curved materials.

Measurement of monolayer viscosity using noncontact microrheology.

Microrheological studies of phospholipid monolayers, bilayers, and other Langmuir monolayer systems are traditionally performed by observing the thermal fluctuations of tracers attached to the membrane or interface. Measurements of this type obtain surface moduli that are orders of magnitude different from those obtained using macroscopic or active techniques. These large discrepancies can result from uncertainties in the tracer's coupling to the monolayer or the local disruption of the monolayer by the tracer. To avoid such problems, we perform a microrheological experiment with the tracer particle placed at a known depth beneath the monolayer; this avoids the issues mentioned at the cost of generating a weaker, purely hydrodynamic coupling between the tracer and the monolayer. We calculate the appropriate response functions for this submerged particle microrheology and demonstrate the technique on three model monolayer systems.

Adhesion transition of flexible sheets.

Intermolecular forces are known to precipitate adhesion events between solid bodies. Inspired by a macroscale experiment showing the hysteretic adhesion of a piece of flexible tape over a plastic substrate, we develop here a model of far-field dry adhesion between two flexible sheets interacting via a power-law potential. We show that phase transitions from unadhered to adhered states occur as dictated by a dimensionless bending parameter representing the ratio of interaction strength to bending stiffness. The order of the adhesion transitions, as well as their hysteretic nature, is shown to depend on the form of the interaction potential between the flexible sheets. When three or more sheets interact, additional geometrical considerations determine the hierarchical or sequential nature of the adhesion transitions.

Lattice mechanics of origami tessellations.

Origami-based design holds promise for developing materials whose mechanical properties are tuned by crease patterns introduced to thin sheets. Although there have been heuristic developments in constructing patterns with desirable qualities, the bridge between origami and physics has yet to be fully developed. To truly consider origami structures as a class of materials, methods akin to solid mechanics need to be developed to understand their long-wavelength behavior. We introduce here a lattice theory for examining the mechanics of origami tessellations in terms of the topology of their crease pattern and the relationship between the folds at each vertex. This formulation provides a general method for associating mechanical properties with periodic folded structures and allows for a concrete connection between more conventional materials and the mechanical metamaterials constructed using origami-based design.

Programming reversibly self-folding origami with micropatterned photo-crosslinkable polymer trilayers.

Self-folding microscale origami patterns are demonstrated in polymer films with control over mountain/valley assignments and fold angles using trilayers of photo-crosslinkable copolymers with a temperature-sensitive hydrogel as the middle layer. The characteristic size scale of the folds W = 30 µm and figure of merit A/ W (2) ≈ 5000, demonstrated here represent substantial advances in the fabrication of self-folding origami.

Applied origami. Using origami design principles to fold reprogrammable mechanical metamaterials.

Although broadly admired for its aesthetic qualities, the art of origami is now being recognized also as a framework for mechanical metamaterial design. Working with the Miura-ori tessellation, we find that each unit cell of this crease pattern is mechanically bistable, and by switching between states, the compressive modulus of the overall structure can be rationally and reversibly tuned. By virtue of their interactions, these mechanically stable lattice defects also lead to emergent crystallographic structures such as vacancies, dislocations, and grain boundaries. Each of these structures comes from an arrangement of reversible folds, highlighting a connection between mechanical metamaterials and programmable matter. Given origami's scale-free geometric character, this framework for metamaterial design can be directly transferred to milli-, micro-, and nanometer-size systems.

High-energy deformation of filaments with internal structure and localized torque-induced melting of DNA.

We develop a continuum elastic approach to examining the bending mechanics of semiflexible filaments with a local internal degree of freedom that couples to the bending modulus. We apply this model to study the nonlinear mechanics of a double-stranded DNA oligomer (shorter than its thermal persistence length) whose free ends are linked by a single-stranded DNA chain. This construct, studied by H. Qu and G. Zocchi [Europhys. Lett. 94, 18003 (2011)], displays nonlinear strain softening associated with the local melting of the double-stranded DNA under applied torque and serves as a model system with which to study the nonlinear elasticity of DNA under large energy deformations. We show that one can account quantitatively for the observed bending mechanics using an augmented wormlike chain model, the helix-coil wormlike chain. We also predict that the highly bent and partially molten dsDNA should exhibit particularly large end-to-end fluctuations associated with the fluctuation of the length of the molten region, and propose appropriate experimental tests. We suggest that the augmented wormlike chain model discussed here is a useful analytic approach to the nonlinear mechanics of DNA or other biopolymer systems.

Fluid transport by active elastic membranes.

A flexible membrane deforming its shape in time can self-propel in a viscous fluid. Alternatively, if the membrane is anchored, its deformation will lead to fluid transport. Past work in this area focused on situations where the deformation kinematics of the membrane were prescribed. Here we consider models where the deformation of the membrane is not prescribed, but instead the membrane is internally forced. Both the time-varying membrane shape and the resulting fluid motion result then from a balance between prescribed internal active stresses, internal passive resistance, and external viscous stresses. We introduce two specific models for such active internal forcing: one where a distribution of active bending moments is prescribed, and one where active inclusions exert normal stresses on the membrane by pumping fluid through it. In each case, we asymptotically calculate the membrane shape and the fluid transport velocities for small forcing amplitudes, and recover our results using scaling analysis.

Propulsion by passive filaments and active flagella near boundaries.

Confinement and wall effects are known to affect the kinematics and propulsive characteristics of swimming microorganisms. When a solid body is dragged through a viscous fluid at constant velocity, the presence of a wall increases fluid drag, and thus the net force required to maintain speed has to increase. In contrast, recent optical trapping experiments have revealed that the propulsive force generated by human spermatozoa is decreased by the presence of boundaries. Here, we use a series of simple models to analytically elucidate the propulsive effects of a solid boundary on passively actuated filaments and model flagella. For passive flexible filaments actuated periodically at one end, the presence of the wall is shown to increase the propulsive forces generated by the filaments in the case of displacement-driven actuation, while it decreases the force in the case of force-driven actuation. In the case of active filaments as models for eukaryotic flagella, we demonstrate that the manner in which a solid wall affects propulsion cannot be known a priori, but is instead a nontrivial function of the flagellum frequency, wavelength, its material characteristics, the manner in which the molecular motors self-organize to produce oscillations (prescribed activity model or self-organized axonemal beating model), and the boundary conditions applied experimentally to the tethered flagellum. In particular, we show that in some cases, the increase in fluid friction induced by the wall can lead to a change in the waveform expressed by the flagella, which results in a decrease in their propulsive force.