arXiv subjects

Mark J. Bowick

Publications and source records attributed to Mark J. Bowick.

At least 37 records · Page 2Linked to original sources

arXiv2020

Multi-defect Dynamics in Active Nematics

Recent experiments and numerical studies have drawn attention to the dynamics of active nematics. Two-dimensional active nematics flow spontaneously and exhibit spatiotemporal chaotic flows with proliferation of topological defects in the nematic texture. It has been proposed that the dynamics of active nematics can be understood in terms of the dynamics of interacting defects, propelled by active stress. Previous work has derived effective equations of motion for individual defects as quasi-particles moving in the mean field generated by other defects, but an effective theory governing multi-defect dynamics has remained out of reach. In this paper, we examine the dynamics of 2D active nematics in the limit of strong order and overdamped compressible flow. The activity-induced defect dynamics is formulated as a perturbation of the manifold of quasi-static nematic textures explicitly parameterized by defect positions. This makes it possible to derive a set of coupled ordinary differential equations governing defect (and therefore texture) dynamics. Interestingly, because of the non-orthogonality of textures associated with individual defects, their motion is coupled through a position dependent ``collective mobility" matrix. In addition to the familiar active self-propulsion of the $+1/2$ defect, we obtain new collective effects of activity that can be interpreted in terms of non-central and non-reciprocal interactions between defects.

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arXiv2020

Shape and size changes of adherent elastic epithelia

Epithelial tissues play a fundamental role in various morphogenetic events during development and early embryogenesis. Although epithelial monolayers are often modeled as two-dimensional (2D) elastic surfaces, they distinguish themselves from conventional thin elastic plates in three important ways - the presence of an apical-basal polarity, spatial variability of cellular thickness, and their nonequilibrium active nature. Here, we develop a minimal continuum model of a planar epithelial tissue as an active elastic material that incorporates all these features. We start from a full three-dimensional (3D)description of the tissue and derive an effective 2D model that captures, through the curvature of the apical surface, both the apical-basal asymmetry and the spatial geometry of the tissue. Crucially, variations of active stresses across the apical-basal axis lead to active torques that can drive curvature transitions. By identifying four distinct sources of activity, we find that bulk active stresses arising from actomyosin contractility and growth compete with boundary active tensions due to localized actomyosin cables and lamellipodial activity to generate the various states spanning the morphospace of a planar epithelium. Our treatment hence unifies 3D shape deformations through the coupled mechanics of apical curvature change and in-plane expansion/contraction of substrate-adhered tissues. Finally, we discuss the implications of our results for some biologically relevant processes such as tissue folding at the onset of lumen formation.

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arXiv2019

Non-uniform Curvature and Anisotropic Deformation Control Wrinkling Patterns on Tori

We investigate wrinkling patterns in a tri-layer torus consisting of an expanding thin outer layer, an intermediate soft layer and an inner core with a tunable shear modulus, inspired by pattern formation in developmental biologies, such as follicle pattern formation during the development of chicken embryos. We show from large-scale finite element simulations that hexagonal wrinkling patterns form for stiff cores whereas stripe wrinkling patterns develop for soft cores. Hexagons and stripes co-exist to form hybrid patterns for cores with intermediate stiffness. The governing mechanism for the pattern transition is that the stiffness of the inner core controls the degree to which the major radius of the torus expands this has a greater effect on deformation in the long direction as compared to the short direction of the torus. This anisotropic deformation alters stress states in the outer layer which change from biaxial (preferred hexagons) to uniaxial (preferred stripes) compression as the core stiffness is reduced. As the outer layer continues to expand, stripe and hexagon patterns will evolve into Zigzag and segmented labyrinth, respectively. Stripe wrinkles are observed to initiate at the inner surface of the torus while hexagon wrinkles start from the outer surface as a result of curvature-dependent stresses in the torus. We further discuss the effects of elasticities and geometries of the torus on the wrinkling patterns.

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arXiv2018

Topology and ground state degeneracy of tetrahedral smectic vesicles

Chemical design of block copolymers makes it possible to create polymer vesicles with tunable microscopic structure. Here we focus on a model of a vesicle made of smectic liquid-crystalline block copolymers at zero temperature. The vesicle assumes a faceted tetrahedral shape and the smectic layers arrange in a stack of parallel straight lines with topological defects localized at the vertices. We counted the number of allowed states at $T=0$. For any fixed shape, we found a two-dimensional countable degeneracy in the smectic pattern depending on the tilt angle between the smectic layers and the edge of the tetrahedral shell. For most values of the tilt angle, the smectic layers contain spiral topological defects. The system can spontaneously break chiral symmetry when the layers organize into spiral patterns, composed of a bound pair of $+1/2$ disclinations. Finally, we suggest possible applications of tetrahedral smectic vesicles in the context of functionalizing defects and the possible consequences of the spiral structures for the rigidity of the vesicle.

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arXiv2018

Defect unbinding in active nematics

We formulate the statistical dynamics of topological defects in the active nematic phase, formed in two dimensions by a collection of self-driven particles on a substrate. An important consequence of the non-equilibrium drive is the spontaneous motility of strength +1/2 disclinations. Starting from the hydrodynamic equations of active nematics, we derive an interacting particle description of defects that includes active torques. We show that activity, within perturbation theory, lowers the defect-unbinding transition temperature, determining a critical line in the temperature-activity plane that separates the quasi-long-range ordered (nematic) and disordered (isotropic) phases. Below a critical activity, defects remain bound as rotational noise decorrelates the directed dynamics of +1/2 defects, stabilizing the quasi-long-range ordered nematic state. This activity threshold vanishes at low temperature, leading to a re-entrant transition. At large enough activity, active forces always exceed thermal ones and the perturbative result fails, suggesting that in this regime activity will always disorder the system. Crucially, rotational diffusion being a two-dimensional phenomenon, defect unbinding cannot be described by a simplified one-dimensional model.

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arXiv2018

Nonlinear mechanics of thin frames

The dramatic effect kirigami, such as hole cutting, has on the elastic properties of thin sheets invites a study of the mechanics of thin elastic frames under an external load. Such frames can be thought of as modular elements needed to build any kirigami pattern. Here we develop the technique of elastic charges to address a variety of elastic problems involving thin sheets with perforations, focusing on frames with sharp corners. We find that holes generate elastic defects (partial disclinations) which act as sources of geometric incompatibility. Numerical and analytic studies are made of three different aspects of loaded frames - the deformed configuration itself, the effective mechanical properties in the form of force-extension curves and the buckling transition triggered by defects. This allows us to understand generic kirigami mechanics in terms of a set of force-dependent elastic charges with long-range interactions.

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arXiv2018

Kirigami mechanics as stress relief by elastic charges

We develop a geometric approach to understand the mechanics of perforated thin elastic sheets, using the method of strain-dependent image elastic charges. This technique recognizes the buckling response of a hole under external load as a geometrically tuned mechanism of stress relief. We use a diagonally pulled square paper frame as a model system to quantitatively test and validate our approach. Specifically, we compare nonlinear force-extension curves and global displacement fields in theory and experiment. We find a strong softening of the force response accompanied by curvature localization at the inner corners of the buckled frame. Counterintuitively, though in complete agreement with our theory, for a range of intermediate hole sizes, wider frames are found to buckle more easily than narrower ones. Upon extending these ideas to many holes, we demonstrate that interacting elastic image charges can provide a useful kirigami design principle to selectively relax stresses in elastic materials.

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arXiv2018

Geometric frustration and solid-solid transitions in model 2D tissue

We study the mechanical behavior of two-dimensional cellular tissues by formulating the continuum limit of discrete vertex models based on an energy that penalizes departures from a target area $A_0$ and a target perimeter $P_0$ for the component cells of the tissue. As the dimensionless target shape index $s_0 = \frac{P_0}{\sqrt{A_0}}$ is varied, we find a transition from a soft elastic regime for compatible target perimeter and area to a stiffer nonlinear elastic regime frustrated by geometric incompatibility. We show that the ground state in the soft regime has a family of degenerate solutions associated with zero modes for the target area and perimeter. The onset of geometric incompatibility at a critical $s_0^c$ lifts this degeneracy. The resultant energy gap leads to a nonlinear elastic response distinct from that obtained in classical elasticity models. We draw an analogy between cellular tissues and anelastic deformations in solids.

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arXiv2017

Topological Sound and Flocking on Curved Surfaces

Active systems on curved geometries are ubiquitous in the living world. In the presence of curvature orientationally ordered polar flocks are forced to be inhomogeneous, often requiring the presence of topological defects even in the steady state due to the constraints imposed by the topology of the underlying surface. In the presence of spontaneous flow the system additionally supports long-wavelength propagating sound modes which get gapped by the curvature of the underlying substrate. We analytically compute the steady state profile of an active polar flock on a two-sphere and a catenoid, and show that curvature and active flow together result in symmetry protected topological modes that get localized to special geodesics on the surface (the equator or the neck respectively). These modes are the analogue of edge states in electronic quantum Hall systems and provide unidirectional channels for information transport in the flock, robust against disorder and backscattering.

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arXiv2017

Stiffening Thermal Membranes by Cutting

Two-dimensional crystalline membranes have recently been realized experimentally in such systems as graphene and molybdenum disulfide, sparking a resurgence in interest in their statistical properties. Thermal fluctuations can significantly affect the effective mechanical properties of properly thermalized membranes, renormalizing both bending rigidity and elastic moduli so that in particular they become stiffer to bending than their bare bending rigidity would suggest. We use molecular dynamics simulations to examine how the mechanical behavior of thermalized two-dimensional clamped ribbons (cantilevers) depends on their precise topology and geometry. We find that a simple slit smooths roughness as measured by the variance of height fluctuations. This counterintuitive effect may be due to the counter-posed coupling of the lips of the slit to twist in the intact regions of the ribbon.

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arXiv2017

Non-Hookean statistical mechanics of clamped graphene ribbons

Thermally fluctuating sheets and ribbons provide an intriguing forum in which to investigate strong violations of Hooke's Law: large distance elastic parameters are in fact not constant, but instead depend on the macroscopic dimensions. Inspired by recent experiments on free-standing graphene cantilevers, we combine the statistical mechanics of thin elastic plates and large-scale numerical simulations to investigate the thermal renormalization of the bending rigidity of graphene ribbons clamped at one end. For ribbons of dimensions $W\times L$ (with $L\geq W$), the macroscopic bending rigidity $κ_R$ determined from cantilever deformations is independent of the width when $W<\ell_\textrm{th}$, where $\ell_\textrm{th}$ is a thermal length scale, as expected. When $W>\ell_\textrm{th}$, however, this thermally renormalized bending rigidity begins to systematically increase, in agreement with the scaling theory, although in our simulations we were not quite able to reach the system sizes necessary to determine the fully developed power law dependence on $W$. When the ribbon length $L > \ell_p$, where $\ell_p$ is the $W$-dependent thermally renormalized ribbon persistence length, we observe a scaling collapse and the beginnings of large scale random walk behavior.

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arXiv2017

Shapes and singularities in triatic liquid crystal vesicles

Determining the equilibrium configuration and shape of curved two-dimensional films with (generalized) liquid crystalline (LC) order is a difficult infinite dimensional problem of direct relevance to the study of generalized polymersomes, soft matter and the fascinating problem of understanding the origin and formation of shape (morphogenesis). The symmetry of the free energy of the LC film being considered and the topology of the surface to be determined often requires that the equilibrium configuration possesses singular structures in the form of topological defects such as disclinations for nematic films. The precise number and type of defect plays a fundamental role in restricting the space of possible equilibrium shapes. Flexible closed vesicles with spherical topology and nematic or smectic order, for example, inevitably possess four elementary strength $+1/2$ disclination defects positioned at the four vertices of a tetrahedral shell. Here we address the problem of determining the equilibrium shape of flexible vesicles with generalized LC order. The order parameter in these cases is an element of $S^1/Z_p$, for any positive integer $p$. We will focus on the case $p =3$, known as triatic LCs. We construct the appropriate order parameter for triatics and find the associated free energy. We then describe the structure of the elementary defects of strength $+1/3$ in flat space. Finally, we prove that sufficiently floppy triatic vesicles with the topology of the 2-sphere equilibrate to octahedral shells with strength $+1/3$ defects at each of the six vertices, independently of scale.

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arXiv2017

Thermal stiffening of clamped elastic ribbons

We use molecular dynamics to study the vibrations of a thermally fluctuating two-dimensional elastic membrane clamped at both ends. We directly extract the eigenmodes from resonant peaks in the frequency domain of the time-dependent height and measure the dependence of the corresponding eigenfrequencies on the microscopic bending rigidity of the membrane, taking care also of the subtle role of thermal contraction in generating a tension when the projected area is fixed. At finite temperatures we show that the effective (macroscopic) bending rigidity tends to a constant as the bare bending rigidity vanishes, consistent with theoretical arguments that the large-scale bending rigidity of the membrane arises from a strong thermal renormalization of the microscopic bending rigidity. Experimental realizations include covalently-bonded two-dimensional atomically thin membranes such as graphene and molybdenum disulfide or soft matter systems such as the spectrin skeleton of red blood cells or diblock copolymers.

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arXiv2017

Defect driven shapes in nematic droplets: analogies with cell division

Building on the striking similarity between the structure of the spindle during mitosis in living cells and nematic textures in confined liquid crystals, we use a continuum model of two-dimensional nematic liquid crystal droplets, to examine the physical aspects of cell division. The model investigates the interplay between bulk elasticity of the microtubule assembly, described as a nematic liquid crystal, and surface elasticity of the cell cortex, modelled as a bounding flexible membrane, in controlling cell shape and division. The centrosomes at the spindle poles correspond to the cores of the topological defects required to accommodate nematic order in a closed geometry. We map out the progression of both healthy bipolar and faulty multi-polar division as a function of an effective parameter that incorporates active processes and controls centrosome separation. A robust prediction, independent of energetic considerations, is that the transition from a single cell to daughters cells occurs at critical value of this parameter. Our model additionally suggests that microtubule anchoring at the cell cortex may play an important role for successful bipolar division. This can be tested experimentally by regulating microtubule anchoring.

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arXiv2016

Planar and curved droplet networks

We study how a two-dimensional square sheet of droplets evolves to the preferred triangular lattice. The rearrangement can be induced either by an impurity seed for flat sheets or by curving the sheet. We analyze the propagation of transformation fronts and the spatial reordering subsequent to the initial perturbation.

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arXiv2014

Effects of scars on crystalline shell stability under external pressure

We study how the stability of spherical crystalline shells under external pressure is influenced by the defect structure. In particular, we compare stability for shells with a minimal set of topologically-required defects to shells with extended defect arrays (grain boundary "scars" with non-vanishing net disclination charge). We perform Monte Carlo simulations to compare how shells with and without scars deform quasi-statically under external hydrostatic pressure. We find that the critical pressure at which shells collapse is lowered for scarred configurations that break icosahedral symmetry and raised for scars that preserve icosahedral symmetry. The particular shapes which arise from breaking of an initial icosahedrally-symmetric shell depend on the Föppl-von Kármán number.

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arXiv2014

Topology and Dynamics of Active Nematic Vesicles

Engineering synthetic materials that mimic the remarkable complexity of living organisms is a fundamental challenge in science and technology. We study the spatiotemporal patterns that emerge when an active nematicfilm of microtubules and molecular motors is encapsulated within a shape-changing lipid vesicle. Unlike in equilibrium systems, where defects are largely static structures, in active nematics defects move spontaneously and can be described as self-propelled particles. The combination of activity, topological constraints and vesicle deformability produces a myriad of dynamical states. We highlight two dynamical modes: a tunable periodic state that oscillates between two defect configurations, and shape-changing vesicles with streaming filopodia-like protrusions. These results demonstrate how biomimetic materials can be obtained when topological constraints are used to control the non-equilibrium dynamics of active matter.

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arXiv2014

Defect dynamics in active nematics

Topological defects are distinctive signatures of liquid crystals. They profoundly affect the viscoelastic behavior of the fluid by constraining the orientational structure in a way that inevitably requires global changes not achievable with any set of local deformations. In active nematic liquid crystals topological defects not only dictate the global structure of the director, but also act as local sources of motion, behaving as self-propelled particles. In this article we present a detailed analytical and numerical study of the mechanics of topological defects in active nematic liquid crystals.

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