1. Optimal vein density in artificial and real leaves X. Noblin, L. Mahadevan, I. Coomaraswamy, D. Weitz, N. Holbrook and M. Zwieniecki,  Proceedings of the National Academy of Sciences (USA),  105, 9140, 2008.
    [View PDF] [Download PDF] The long evolution of vascular plants has resulted in a tremendous variety of natural networks responsible for the evaporatively driven transport of water. Nevertheless, little is known about the physical principles that constrain vascular architecture. Inspired by plant leaves, we used microfluidic devices consisting of simple parallel channel networks in a polymeric material layer, permeable to water, to study the mechanisms of and the limits to evaporationdriven flow. We show that the flow rate through our biomimetic leaves increases linearly with channel density (1/d) until the distance between channels (d) is comparable with the thickness of the polymer layer (), above which the flow rate saturates. A comparison with the plant vascular networks shows that the same optimization criterion can be used to describe the placement of veins in leaves. These scaling relations for evaporatively driven flow through simple networks reveal basic design principles for the engineering of evaporation–permeation-driven devices, and highlight the role of physical constraints on the biological design of leaves.
  2. Localized and extended deformations of elastic shells A. Vaziri and L. Mahadevan,  Proceedings of the National Academy of Sciences (USA),  105, 7913, 2008.
    [View PDF] [Download PDF] The dried raisin, the crushed soda can, and the collapsed bicycle inner tube exemplify the nonlinear mechanical response of naturally curved elastic surfaces with different intrinsic curvatures to a variety of different external loads. To understand the formation and evolution of these features in a minimal setting, we consider a simple assay: the response of curved surfaces to point indentation. We find that for surfaces with zero or positive Gauss curvature, a common feature of the response is the appearance of faceted structures that are organized in intricate localized patterns, with hysteretic transitions between multiple metastable states. In contrast, for surfaces with negative Gauss curvature the surface deforms nonlocally along characteristic lines that extend through the entire system. These different responses may be understood quantitatively by using numerical simulations and classified qualitatively by using simple geometric ideas. Our ideas have implications for the behavior of small-scale structures.
  3. Signal processing by HOG MAP kinase pathway P. Hersen, M. McClean, L. Mahadevan, and S. Ramanathan,  Proceedings of the National Academy of Sciences (USA),  105, 7165, 2008.
    [View PDF] [Download PDF] Signaling pathways relay information about changes in the external environment so that cells can respond appropriately. How much information a pathway can carry depends on its bandwidth. We designed a microfluidic device to reliably change the environment of single cells over a range of frequencies. Using this device, we measured the bandwidth of the Saccharomyces cerevisiae signaling pathway that responds to high osmolarity. This prototypical pathway, the HOG pathway, is shown to act as a low-pass filter, integrating the signal when it changes rapidly and following it faithfully when it changes more slowly. We study the dependence of the pathway’s bandwidth on its architecture. We measure previously unknown bounds on all of the in vivo reaction rates acting in this pathway. We find that the two-component Ssk1 branch of this pathway is capable of fast signal integration, whereas the kinase Ste11 branch is not. Our experimental techniques can be applied to other signaling pathways, allowing the measurement of their in vivo kinetics and the quantification of their information capacity.
  4. Limbless undulatory propulsion on land Z. Guo and L. Mahadevan,  Proceedings of the National Academy of Sciences (USA),  105, 3179, 2008.
    [View PDF] [Download PDF] We analyze the lateral undulatory motion of a natural or artificial snake or other slender organism that ‘‘swims’’ on land by propagating retrograde flexural waves. The governing equations for the planar lateral undulation of a thin filament that interacts frictionally with its environment lead to an incomplete system. Closures accounting for the forces generated by the internal muscles and the interaction of the filament with its environment lead to a nonlinear boundary value problem, which we solve using a combination of analytical and numerical methods. We find that the primary determinant of the shape of the organism is its interaction with the external environment, whereas the speed of the organism is determined primarily by the internal muscular forces, consistent with prior qualitative observations. Our model also allows us to pose and solve a variety of optimization problems such as those associated with maximum speed and mechanical efficiency, thus defining the performance envelope of this mode of locomotion.
  5. Equilibrium of an elastically confined liquid drop H-M. Kwon, H-Y. Kim, J. Puell and L. Mahadevan,  Journal of Applied Physics,  103, 093519, 2008.
    [View PDF] [Download PDF] When a liquid drop is confined between an elastic plate and a rigid substrate, it spreads spontaneously due to the effects of interfacial forces, eventually reaching an equilibrium shape determined by the balance between elastic and capillary effects. We provide an analytical theory for the static shape of the sheet and the extent of liquid spreading and show that our experiments are quantitatively consistent with the theory. The theory is relevant for the first step of painting when a brush is brought down on to canvas. More mundanely, it allows us to understand the stiction of microcantilevers to wafer substrates occurring in microelectromechanical fabrication processes
  6. A quantitative analysis of contractility in active cytoskeletal protein networks P. Bendix, G. Koenderink, D. Cuvelier, Z. Dogic, B. Koeleman, W. Brieher, C. Field, L. Mahadevan and D. Weitz,  Biophysical Journal,  94, 3126, 2008.
    [View PDF] [Download PDF] Cells actively produce contractile forces for a variety of processes including cytokinesis and motility. Contractility is known to rely on myosin II motors which convert chemical energy from ATP hydrolysis into forces on actin filaments. However, the basic physical principles of cell contractility remain poorly understood. We reconstitute contractility in a simplified model system of purified F-actin, muscle myosin II motors, and a-actinin cross-linkers. We show that contractility occurs above a threshold motor concentration and within a window of cross-linker concentrations. We also quantify the pore size of the bundled networks and find contractility to occur at a critical distance between the bundles. We propose a simple mechanism of contraction based on myosin filaments pulling neighboring bundles together into an aggregated structure. Observations of this reconstituted system in both bulk and low-dimensional geometries show that the contracting gels pull on and deform their surface with a contractile force of ;1mN, or ;100 pN per F-actin bundle. Cytoplasmic extracts contracting in identical environments show a similar behavior and dependence on myosin as the reconstituted system. Our results suggest that cellular contractility can be sensitively regulated by tuning the (local) activity of molecular motors and the cross-linker density and binding affinity.
  7. Life and times of a cellular bleb G. Charras, M. Coughlin, T. Mitchison and L. Mahadevan,  Biophysical Journal,  94, 1836, 2008.
    [View PDF] [Download PDF] Blebs are spherical cellular protrusions that occur in many physiological situations. Two distinct phases make up the life of a bleb, each of which have their own biology and physics: expansion, which lasts ;30 s, and retraction, which lasts ;2 min. We investigate these phases using optical microscopy and simple theoretical concepts, seeking information on blebbing itself, and on cytomechanics in general. We show that bleb nucleation depends on pressure, membrane-cortex adhesion energy, and membrane tension, and test this experimentally. Bleb growth occurs through a combination of bulk flow of lipids and delamination from the cell cortex via the formation and propagation of tears. In extreme cases, this can give rise to a traveling wave around the cell periphery, known as ‘‘circus movement.’’ When growth stalls, an actin cortex reforms under the bleb membrane, and retraction starts, driven by myosin-II. Using flicker spectroscopy, we find that retracting blebs are fivefold more rigid than expanding blebs, an increase entirely explained by the properties of the newly formed cortical actin mesh. Finally, using artificially nucleated blebs as pressure sensors, we show that cells rounded up in mitosis possess a substantial intracellular pressure.
  8. Power-limited contraction dynamics of Vorticella convallaria: an ultrafast biological spring A. Upadhyaya, M. Baraban, J. Wong, P. Matsudaira, A. van Oudenaarden and L. Mahadevan,  Biophysical Journal , 94, 265, 2008.
    [View PDF] [Download PDF] Vorticella convallaria is one of the fastest and most powerful cellular machines. The cell body is attached to a substrate by a slender stalk containing a polymeric structure—the spasmoneme. Helical coiling of the stalk results from rapid contraction of the spasmoneme, an event mediated by calcium binding to a negatively charged polymeric backbone. We use high speed imaging to measure the contraction velocity as a function of the viscosity of the external environment and find that the maximum velocity scales inversely with the square root of the viscosity. This can be explained if the rate of contraction is ultimately limited by the power delivered by the actively contracting spasmoneme. Microscopically, this scenario would arise if the mechanochemical wave that propagates along the spasmoneme is faster than the rate at which the cell body can respond due to its large hydrodynamic resistance. We corroborate this by using beads as markers on the stalk and find that the contraction starts at the cell body and proceeds down the stalk at a speed that exceeds the velocity of the cell body.
  9. Sickle cell vaso-occlusion and rescue in a microfluidic device J. Higgins, D. Eddington, S. Bhatia and L. Mahadevan,  Proceedings of the National Academy of Sciences (USA) , 104, 20496, 2007.
    [View PDF] [Download PDF] The pathophysiology of sickle cell disease is complicated by the multiscale processes that link the molecular genotype to the organismal phenotype: hemoglobin polymerization occurring in milliseconds, microscopic cellular sickling in a few seconds or less [Eaton WA, Hofrichter J (1990) Adv Protein Chem 40:63–279], and macroscopic vessel occlusion over a time scale of minutes, the last of which is necessary for a crisis [Bunn HF (1997) N Engl J Med 337:762–769]. Using a minimal but robust artificial microfluidic environment, we show that it is possible to evoke, control, and inhibit the collective vasoocclusive or jamming event in sickle cell disease. We use a combination of geometric, physical, chemical, and biological means to quantify the phase space for the onset of a jamming event, as well as its dissolution, and find that oxygendependent sickle hemoglobin polymerization and melting alone are sufficient to recreate jamming and rescue. We further show that a key source of the heterogeneity in occlusion arises from the slow collective jamming of a confined, flowing suspension of soft cells that change their morphology and rheology relatively quickly. Finally, we quantify and investigate the effects of small-molecule inhibitors of polymerization and therapeutic red blood cell exchange on this dynamical process. Our experimental study integrates the dynamics of collective processes associated with occlusion at the molecular, polymer, cellular, and tissue level; lays the foundation for a quantitative understanding of the rate-limiting processes; and provides a potential tool for optimizing and individualizing treatment, and identifying new therapies.
  10. Settling and swimming of flexible fluid lubricated foils M. Argentina, J. Skotheim and L. Mahadevan,  Physical Review Letters , 99, 224053, 2007.
    [View PDF] [Download PDF] We study the dynamics of a flexible foil immersed in a fluid and moving close to a rigid wall. Lubrication theory allows us to derive equations of motion for the foil and thus examine the passive settling and the active swimming of a foil. This also allows us to partly answer the long-standing question in cartoon physics—can carpets fly? Our analysis suggests a region in parameter space where one may realize this dream and move the virtual towards reality.
  11. Mechanosensation and mechanical loads modulate the locomotory gait of swimming C. elegans J. Korta, D. Clark, C. Gabel, L. Mahadevan, and A. Samuel,  Journal of Experimental Biology , 210, 2383, 2007.
    [View PDF] [Download PDF] Animals move through their environments by selecting gaits that are adapted to the physical nature of their surroundings. The nematode Caenorhabditis elegans swims through fluids or crawls on surfaces by propagating flexural waves along its slender body and offers a unique opportunity for detailed analysis of locomotory gait at multiple levels including kinematics, biomechanics and the molecular and physiological operation of sensory and motor systems. Here, we study the swimming gait of C. elegans in viscous fluids in the range 0.05–50·Pa s. We find that the spatial form of the swimming gait does not vary across this range of viscosities and that the temporal frequency of the swimming gait only decreases by about 20% with every 10-fold increase in viscosity. Thus, C. elegans swims in low gear, such that its musculature can deliver mechanical force and power nearly 1000-fold higher than it delivers when swimming in water. We find that mutations that disrupt mechanosensation, or the laser killing of specific touch receptor neurons, increase the temporal frequency of the undulating gait, revealing a novel effect of mechanosensory input in regulating the putative central pattern generator that produces locomotion. The adaptability of locomotory gait in C. elegans may be encoded in sensory and motor systems that allow the worm to respond to its own movement in different physical surroundings.
  12. Gravitational stability of suspensions of attractive colloidal particles. C. Kim, Y. Liu, A. Kuhnle, S. Hess, S. Viereck, T. Danner, L. Mahadevan, and D. Weitz,  Physical Review Letters , 99, 028303, 2007.
    [View PDF] [Download PDF] Colloidal suspensions are susceptible to gravitationally induced phase separation. This can be mitigated by the formation of a particle network caused by depletion attraction. The effectiveness of this network in supporting the buoyant weight of the suspension can be characterized by its compressional modulus. We measure the compressional modulus for emulsion networks induced by depletion attraction and present a model that quantitatively predicts their gravitational stability. We also determine the relationship between the strength of the depletion attraction and the magnitude of the compressional modulus.
  13. Force of an actin spring J.H. Shin, B.K. Tam, R.R. Brau, M.J. Lang, L. Mahadevan, and Paul Matsudaira,  Biophysical Journal , 92, 3729, 2007.
    [View PDF] [Download PDF] Cellular movements are produced by forces. Typically, cytoskeletal proteins such as microtubules and actin filaments generate forces via polymerization or in conjunction with molecular motors. However, the fertilization of a Limulus polyphemus egg involves a third type of actin-based cellular engine—a biological spring. During the acrosome reaction, a 60-mm long coiled and twisted bundle of actin filaments straightens and extends from a sperm cell, penetrating the vitelline layer surrounding the egg. A subtle overtwist of 0.2/subunit underlies the mechanochemical basis for the extension of this actin spring. Upon calcium activation, this conformational strain energy is converted to mechanical work, generating the force required to extend the bundle through the vitelline layer. In this article, we stall the extension of the acrosome bundle in agarose gels of different concentrations. From the stall forces, we estimate a maximum force of 2 nN and a puncturing pressure of 1.6 MPa. We show the maximum force of extension is three times larger than the force required to puncture the vitelline layer. Thus, the elastic strain energy stored in the acrosome bundle is more than sufficient to power the acrosome reaction through the egg envelope.
  14. The universal dynamics of cell spreading D. Cuvelier, M. Thery, Y-S. Chu, S. Dufour, J-P. Thiery, M. Bornens, P. Nassoy, and L. Mahadevan,  Current Biology  , 17, 694, 2007.
    [View PDF] [Download PDF] Cell adhesion and motility depend strongly on the interactions between cells and extracellular matrix (ECM) substrates. When plated onto artificial adhesive surfaces, cells first flatten and deform extensively as they spread. At the molecular level, the interaction of membrane-based integrins with the ECM has been shown to initiate a complex cascade of signaling events [1], which subsequently triggers cellular morphological changes and results in the generation of contractile forces [2]. Here, we focus on the early stages of cell spreading and probe their dynamics by quantitative visualization and biochemical manipulation with a variety of cell types and adhesive surfaces, adhesion receptors, and cytoskeleton-altering drugs. We find that the dynamics of adhesion follows a universal power-law behavior. This is in sharp contrast with the common belief that spreading is regulated by either the diffusion of adhesion receptors toward the growing adhesive patch [3–5] or by actin polymerization [6–8]. To explain this, we propose a simple quantitative and predictive theory that models cells as viscous adhesive cortical shells enclosing a less viscous interior. Thus, although cell spreading is driven by well-identified biomolecular interactions, it is dynamically limited by its mesoscopic structure and material properties.
  15. Nonlinear mechanics of soft fibrous networks A. Kabla and L. Mahadevan,  Journal of the Royal Society - Interface , 4, 99, 2007.
    [View PDF] [Download PDF] Mechanical networks of fibres arise on a range of scales in nature and technology, from the cytoskeleton of a cell to blood clots, from textiles and felts to skin and collageneous tissues. Their collective response is dependent on the individual response of the constituent filaments as well as density, topology and order in the network. Here, we use the example of a lowdensity synthetic felt of athermal filaments to study the generic features of the mechanical response of such networks including strain stiffening and large effective Poisson ratios. A simple microscopic model allows us to explain these features of our observations, and provides us with a baseline framework to understand active biomechanical networks.
  16. Persistence of a pinch in a pipe L. Mahadevan, A. Vaziri and M. Das,  Europhysics Letters , 77, 40003, 2007.
    [View PDF] [Download PDF] The response of low-dimensional solid objects combines geometry and physics in unusual ways, exemplified in structures of great utility such as a thin-walled tube that is ubiquitous in nature and technology. Here we provide a consequence of this confluence of geometry and physics in tubular structures: our analysis shows that the persistence of a localized pinch in an elastic pipe whose effect decays as an oscillatory exponential with a persistence length that diverges as the thickness of the tube vanishes, which we confirm using simulations and simple experiments. The result is more a consequence of geometry than material properties, and is thus equally applicable to carbon nanotubes as it is to oil pipelines.
  17. Curvature condensation and bifurcation in an elastic shell M. Das, A. Vaziri, A. Kudrolli, and L. Mahadevan,  Physical Review Letters,  98, 014301, 2007.
    [View PDF] [Download PDF] We study the formation and evolution of localized geometrical defects in an indented cylindrical elastic shell using a combination of experiment and numerical simulation. We find that as a symmetric localized indentation on a semicylindrical shell increases, there is a transition from a global mode of deformation to a localized one which leads to the condensation of curvature along a symmetric parabolic defect. This process introduces a soft mode in the system, converting a load-bearing structure into a hinged, kinematic mechanism. Further indentation leads to twinning wherein the parabolic defect bifurcates into two defects that move apart on either side of the line of symmetry. A qualitative theory captures the main features of the phenomena but leads to further questions about the mechanism of defect nucleation
  18. A dynamic fate map of the forebrain shows how vertebrate eyes form and explains two causes of cyclopia S.J. England, G.B. Blanchard, L. Mahadevan, and R.J. Adams,  Development , 133, 4613, 2006.
    [View PDF] [Download PDF] Mechanisms for shaping and folding sheets of cells during development are poorly understood. An example is the complex reorganisation of the forebrain neural plate during neurulation, which must fold a sheet into a tube while evaginating two eyes from a single contiguous domain within the neural plate. We, for the first time, track these cell rearrangements to show that forebrain morphogenesis differs significantly from prior hypotheses. We postulate a new model for forebrain neurulation and demonstrate how mutations affecting two signalling pathways can generate cyclopic phenotypes by disrupting normal cell movements or introducing new erroneous behaviours.
  19. Sensorimotor control during isothermal tracking in Caenorhabditis elegans L. Luo, D.A. Clark, D. Biron, L. Mahadevan, and A.D.T. Samuel,  The Journal of Experimental Biology  209, 4652, 2006.
    [View PDF] [Download PDF] In order to purposefully navigate their environments, animals rely on precise coordination between their sensory and motor systems. The integrated performance of circuits for sensorimotor control may be analyzed by quantifying an animal’s motile behavior in defined sensory environments. Here, we analyze the ability of the nematode C. elegans to crawl isothermally in spatial thermal gradients by quantifying the trajectories of individual worms responding to defined spatiotemporal thermal gradients. We show that sensorimotor control during isothermal tracking may be summarized as a strategy in which the worm changes the curvature of its propulsive undulations in response to temperature changes measured at its head. We show that a concise mathematical model for this strategy for sensorimotor control is consistent with the exquisite stability of the worm’s isothermal alignment in spatial thermal gradients as well as its more complex trajectories in spatiotemporal thermal gradients.
  20. Mechanics of Interfacial Composite Materials A.B. Subramaniam, M. Abkarian, L. Mahadevan, and H.A. Stone,  Langmuir , 22, 10204, 2006.
    [View PDF] [Download PDF] Recent experiments and simulations have demonstrated that particle-covered fluid/fluid interfaces can exist in stable nonspherical shapes as a result of the steric jamming of the interfacially trapped particles. The jamming confers the interface with solidlike properties. We provide an experimental and theoretical characterization of the mechanical properties of these armored objects, with attention given to the two-dimensional granular state of the interface. Small inhomogeneous stresses produce a plastic response, while homogeneous stresses produce a weak elastic response. Shear-driven particle-scale rearrangements explain the basic threshold needed to obtain the near-perfect plastic deformation that is observed. Furthermore, the inhomogeneous stress state of the interface is exhibited experimentally by using surfactants to destabilize the particles on the surface. Since the interfacially trapped particles retain their individual characteristics, armored interfaces can be recognized as a kind of composite material with distinct chemical,structural,
  21. Fall and rise of a viscoelastic filament A. Roy, L. Mahadevan and J-L Thiffeault,  Journal of Fluid Mechanics  , 563, 283, 2006.
    [View PDF] [Download PDF] When a viscoelastic fluid blob is stretched out into a thin horizontal filament, it sags and falls gradually under its own weight, forming a catenary-like structure that evolves dynamically. If the ends are brought together rapidly after stretching, the falling filament tends to straighten by rising. These two effects are strongly influenced by the elasticity of the fluid and yield qualitatively different behaviours from the case of a purely viscous filament analysed previously (Teichman & Mahadevan, J. Fluid Mech. vol. 478, 2003, p. 71). Starting from the bulk equations for the motion of a viscoelastic fluid, we derive a simplified equation for the dynamics of a viscoelastic filament and analyse this equation in some simple settings to explain our observations.
  22. Microtubules can bear enhanced compressive loads in living cells because of lateral reinforcements C. Brangwynne, F. Mackintosh, S. Kumar, N. Geisse, J. Talbot, L. Mahadevan, K. Parker, D. Ingber and D. Weitz,  Journal of Cell Biology  , 173, 733, 2006.
    [View PDF] [Download PDF] Cytoskeletal microtubules have been proposed to infl uence cell shape and mechanics based on their ability to resist large-scale compressive forces exerted by the surrounding contractile cytoskeleton. Consistent with this, cytoplasmic microtubules are often highly curved and appear buckled because of compressive loads. However, the results of in vitro studies suggest that microtubules should buckle at much larger length scales, withstanding only exceedingly small compressive forces. This discrepancy calls into question the structural role of microtubules, and highlights our lack of quantitative knowledge of the magnitude of the forces they experience and can withstand in living cells. We show that intracellular microtubules do bear large-scale compressive loads from a variety of physiological forces, but their buckling wavelength is reduced signifi cantly because of mechanical coupling to the surrounding elastic cytoskeleton. We quantitatively explain this behavior, and show that this coupling dramatically increases the compressive forces that microtubules can sustain, suggesting they can make a more signifi cant structural contribution to the mechanical behavior of the cell than previously thought possible
  23. Modeling DNA loops using the theory of elasticity A. Balaeff, L. Mahadevan and K. Schulten,  Physical Review E,  73, 031919, 2006.
    [View PDF] [Download PDF] An elastic rod model of a protein-bound DNA loop is adapted for application in multi-scale simulations of protein-DNA complexes. The classical Kirchhoff system of equations which describes the equilibrium structure of the elastic loop is modified to account for the intrinsic twist and curvature, anisotropic bending properties, and electrostatic charge of DNA. The effects of bending anisotropy and electrostatics are studied for the DNA loop clamped by the lac repressor protein. For two possible lengths of the loop, several topologically different conformations are predicted and extensively analyzed over the broad range of model parameters describing DNA bending and electrostatic properties. The scope and applications of the model in already accomplished and in future multi-scale studies of protein-DNA complexes are discussed.
  24. Dynamics of fracture in drying suspensions E. Dufresene, D. Stark, N. Greenblatt, J. Cheng, J. Hutchinson, L. Mahadevan and D. Weitz,  Langmuir , 22, 7144, 2006.
    [View PDF] [Download PDF] We investigate the dynamics of fracture in drying films of colloidal silica. Water loss quenches the nanoparticle dispersionsto form a liquid-saturated elastic network of particlesthat relieves drying-induced strain by cracking. These cracks display intriguing intermittent motion originating from the deformation of arrested crack tips and aging of the elastic network. The dynamics of a single crack exhibits a universal evolution, described by a balance of the driving elastic power with the sum of interfacial power and the viscous dissipation rate of flowing interstitial fluid.
  25. Dynamics of surfactant-driven fracture of particle rafts D. Vella, H-Y Kim, P. Aussillous and L. Mahadevan,  Physical Review Letters , 96, 178301, 2006.
    [View PDF] [Download PDF] We investigate the dynamic fracture of a close-packed monolayer of particles, or particle raft, floating at a liquid-gas interface induced by the localized addition of surfactant. Unusually for a two-dimensional solid, our experiments show that the speed of crack propagation here is not affected by the elastic properties of the raft. Instead it is controlled by the rate at which surfactant is advected to the crack tip by means of the induced Marangoni flows. Further, the velocity of propagation is not constant in time and the length of the crack scales as t 3=4. More broadly, this surfactant-induced rupture of interfacial rafts suggests ways to manipulate them for applications.
  26. Crack-front instability in a confined elastic film M. Adda Bedia and L. Mahadevan,  Proceedings of the Royal Society of London, series A , 462, 3233, 2006.
    [View PDF] [Download PDF] We study the undulatory instability of a straight crack front generated by peeling a flexible elastic plate from a thin elastomeric adhesive film. We show that there is a threshold for the onset of the instability that is dependent on the ratio of two lengthscales that arise naturally in the problem: the thickness of the film and an elastic length defined by the stiffness of the plate and that of the film. A linear stability analysis predicts that the wavelength of the instability scales linearly with the film thickness. Our results are qualitatively and quantitatively consistent with recent experiments, and show how crack fronts may lose stability due to a competition between bulk and surface effects in the presence of multiple length scales.
  27. A simple model for the dynamics of adhesive failure D. Vella and L. Mahadevan,  Langmuir , 22, 163, 2006.
    [View PDF] [Download PDF] We consider a microscopic model for the failure of soft adhesives in tension based on ideas of bond rupture under dynamic loading. By focusing on adhesive failure under loading at constant velocity, we demonstrate that bimodal curves of stress against strain may occur due to effects of finite polymer chain or bond length and characterize the loading conditions under which such bimodal behavior is observed. The results of this analysis are in qualitative agreement with experiments performed on unconfined adhesives in which failure does not occur by cavitation.
  28. Superficial wrinkles in stretched, drying gelatin films R. Rizzieri, L. Mahadevan, A. Vaziri and A. Donald,  Langmuir , 22, 3622, 2006.
    [View PDF] [Download PDF] When a thin film of initially hydrated gelatin is allowed to dry from the surface, superficial changes in the structure of the material affect the local mechanical properties of the drying region. If the film is simultaneously subjected to large strain deformation (above 20%), a periodic pattern of wrinkles appears on the surface of the gelatin along the length of the sample in the direction of the applied force. These wrinkles are uniformly distributed on the surface of the gelatin with a wavelength that is much smaller than the sample thickness, which changes with sample composition, aging time, and deformation rate. We investigate these patterns via in situ environmentalscanning electron microscopy (ESEM) and provide a theory for their origin.
  29. Capillary rise between elastic sheets H-Y Kim and L. Mahadevan,  Journal of Fluid Mechanics , 548, 141, 2006.
    [View PDF] [Download PDF] When a paintbrush is dipped into a pot of paint and pulled out, surface tension forces cause the individual hairs in the brush to coalesce even as the brush becomes impregnated with paint. We study a simple model of this elastocapillary interaction in the context of the surface-tension-driven vertical rise of a liquid between two long flexible hydrophilic sheets that are held a small distance apart at one end. We provide an analytic theory for the static shapes of the sheets as well as the liquid rise height which is different from that of the classical law of Jurin, and show that our experiments are quantitatively consistent with the theory.
  30. Transitions to nematic states in homogeneous suspensions of high aspect ratio magnetic rods A. Gopinath, L. Mahadevan, and R.A. Armstrong,  Physics of Fluids , 18, 028102, 2006.
    [View PDF] [Download PDF] Isotropic-nematic and nematic-nematic transitions from a homogeneous suspension of high aspect ratio magnetic rods are studied for both Maier-Saupe and Onsager excluded volume potentials. Asymptotic analysis in the vicinity of critical points yields insight into the stability and type of polarized nematic states emanating from nonpolarized equilibrium states. This, in conjunction with recently published global numerical results, yields a unified picture of the bifurcation diagram and provides a convenient base state to study effects of external orienting fields.
  31. Non-spherical bubbles A. Balasubramaniam, M. Abkarian, L. Mahadevan and H.A. Stone,  Nature , 438, 930, 2005.
    [View PDF] [Download PDF]
  32. Powerful curves L. Mahadevan and T. Mitchison,  Nature , 435, 895, 2005.
    [View PDF] [Download PDF] A cell’s contents are organized by a scaffolding of microtubules. These long, thin polymers continuously grow and shrink, and the structures of two forms of the constituent protein provide clues to how this occurs.
  33. Solenoids and Plectonemes in stretched and twisted elastomeric filaments,  A. Ghatak and L. Mahadevan,  Physical Review Letters  , 95, 057801, 2005.
    [View PDF] [Download PDF] We study the behavior of a naturally straight highly extensible elastic filament subjected to large extensional and twisting strains. We find that two different phases can coexist for a range of parameter values: the plectoneme and the solenoid. A simple theory based on a neo-Hookean model for the material of the filament and accounting for the slender geometry suffices to explain these observations, and leads to a phase diagram that is consistent with observations. Extension and relaxation experiments on these phases show the presence of large hysteresis loops and sawtoothlike force-displacement curves which are different for the plectoneme and the solenoid.
  34. Soft lubrication: the elastohydrodynamics of conforming and non-conforming contacts J. Skotheim and L. Mahadevan,  Physics of Fluids , 17, 092101, 2005.
    [View PDF] [Download PDF] We study the lubrication of fluid-immersed soft interfaces and show that elastic deformation couples tangential and normal forces and thus generates lift. We consider materials that deform easily, due to either geometry e.g., a shell or constitutive properties e.g., a gel or a rubber, so that the effects of pressure and temperature on the fluid properties may be neglected. Four different system geometries are considered: a rigid cylinder moving parallel to a soft layer coating a rigid substrate; a soft cylinder moving parallel to a rigid substrate; a cylindrical shell moving parallel to a rigid substrate; and finally a cylindrical conforming journal bearing coated with a thin soft layer. In addition, for the particular case of a soft layer coating a rigid substrate, we consider both elastic and poroelastic material responses. For all these cases, we find the same generic behavior: there is an optimal combination of geometric and material parameters that maximizes the dimensionless normal force as a function of the softness parameter = hydrodynamic pressure/elastic stiffness = surface deflection/ gap thickness, which characterizes the fluid-induced deformation of the interface. The corresponding cases for a spherical slider are treated using scaling concepts.
  35. The ‘Cheerios Effect’ D. Vella and L. Mahadevan,  American Journal of Physics , 73, 817-825, 2005.
    [View PDF] [Download PDF] Objects that float at the interface between a liquid and a gas interact because of interfacial deformation and the effect of gravity. We highlight the crucial role of buoyancy in this interaction, which, for small particles, prevails over the capillary suction that often is assumed to be the dominant effect. We emphasize this point using a simple classroom demonstration, and then derive the physical conditions leading to mutual attraction or repulsion. We also quantify the force of interaction in particular instances and present a simple dynamical model of this interaction. The results obtained from this model are validated by comparison to experimental results for the mutual attraction of two identical spherical particles. We consider some of the applications of the effect that can be found in nature and the laboratory
  36. Gravitational collapse of colloidal gels S. Manley, J.M. Skotheim, L. Mahadevan, D. Weitz,  Physical Review Letters , 94, 218302, 2005 
    [View PDF] [Download PDF] We present a unified framework for understanding the compaction of colloidal gels under their own weight. The dynamics of the collapse are determined by the value of the gravitational stress g, as compared to the yield stress Y of the network. For g < Y, gels collapse poroelastically, and their rate of compression decays exponentially in time. For g > Y, the network eventually yields, leading to rapid settling. In both cases, the rate of collapse is backflow limited, while its overall magnitude is determined by a balance between gravitational stress and network elastic stress.
  37. Physical limits and design principles for plant and fungal movements J. Skotheim and L. Mahadevan,  Science,  308, 1308-10, 2005.
    [View PDF] [Download PDF]
  38. Non-equilibration of hydrostatic pressure in blebbing cells, G. Charras, J. Yarrow, M. Horton, L. Mahadevan and T. Mitchison,  Nature , 435, 365-69. 2005.
    [View PDF] [Download PDF] Current models for protrusive motility in animal cells focus on cytoskeleton-based mechanisms, where localized protrusion is driven by local regulation of actin biochemistry1–3. In plants and fungi, protrusion is driven primarily by hydrostatic pressure4–6. For hydrostatic pressure to drive localized protrusion in animal cells7,8, it would have to be locally regulated, but current models treating cytoplasm as an incompressible viscoelastic continuum9 or viscous liquid10 require that hydrostatic pressure equilibrates essentially instantaneously over the whole cell. Here, we use cell blebs as reporters of local pressure in the cytoplasm. When we locally perfuse blebbing cells with cortex-relaxing drugs to dissipate pressure on one side, blebbing continues on the untreated side, implying non-equilibration of pressure on scales of approximately 10 mm and 10 s. We can account for localization of pressure by considering the cytoplasm as a contractile, elastic network infiltrated by cytosol. Motion of the fluid relative to the network generates spatially heterogeneous transients in the pressure field, and can be described in the framework of poroelasticity11,12.
  39. Self-organized origami, L. Mahadevan and S. Rica,  Science , 307, 1740, 2005.
    [View PDF] [Download PDF] The controlled folding and unfolding of maps, space structures, wings, leaves, petals, and other foldable laminae is potentially complicated by the independence of individual folds; as their number increases, there is a combinatorial explosion in the number of folded possibilities. The artificially constructed Miura-ori (1) pattern, with a periodic array of geometrically and elastically coupled mountain and valley folds (Fig. 1A), circumvents this complication by allowing the entire structure to be folded or unfolded simultaneously. Making such a pattern is not easy, so it may be surprising to find an elegant natural counterpart that is a few hundred millennia old. In Fig. 1B, we show the different stages of the opening of a hornbeam leaf that starts life in its bud as a Miura-ori folded pattern (2). Similar structures arise in insect wings (3) and elsewhere in nature (4), suggesting that these origami patterns are a result of convergent design. This raises a question of mechanism: How might this spatial organization of folds be brought about?
  40. Self-similar nested wrinkling patterns in skins K. Efimenko, M. Rackaitis, E. Manias, A. Vaziri, L. Mahadevan and J. Genzer,  Nature — Materials , 4, 293-97, 2005.
    [View PDF] [Download PDF] Stiff thin fi lms on soft substrates are both ancient and commonplace in nature; for instance, animal skin comprises a stiff epidermis attached to a soft dermis. Although more recent and rare, artifi cial skins are increasingly used in a broad range of applications, including fl exible electronics1 , tunable diff raction gratings2,3, force spectroscopy in cells4 , modern metrology methods5 , and other devices6–8. Here we show that model elastomeric artifi cial skins wrinkle in a hierarchical pattern consisting of self-similar buckles extending over fi ve orders of magnitude in length scale, ranging from a few nanometres to a few millimetres. We provide a mechanism for the formation of this hierarchical wrinkling pattern, and quantify our experimental fi ndings with both computations and a simple scaling theory. Th is allows us to harness the substrates for applications. In particular, we show how to use the multigeneration-wrinkled substrate for separating particles based on their size, while simultaneously forming linear chains of monodisperse particles.
  41. How the Venus Flytrap snaps Y. Forterre, J. Skotheim, J. Dumais and L. Mahadevan,  Nature,  433, 421-25, 2005.
    [View PDF] [Download PDF] The rapid closure of the Venus flytrap (Dionaea muscipula) leaf in about 100 ms is one of the fastest movements in the plant kingdom. This led Darwin to describe the plant as “one of the most wonderful in the world”1 . The trap closure is initiated by the mechanical stimulation of trigger hairs. Previous studies2–7 have focused on the biochemical response of the trigger hairs to stimuli and quantified the propagation of action potentials in the leaves. Here we complement these studies by considering the post-stimulation mechanical aspects of Venus flytrap closure. Using high-speed video imaging, non-invasive microscopy techniques and a simple theoretical model, we show that the fast closure of the trap results from a snap-buckling instability, the onset of which is controlled actively by the plant. Our study identifies an ingenious solution to scaling up movements in non-muscular engines and provides a general framework for understanding nastic motion in plants.
  42. Hydrodynamical models for the chaotic dripping faucet, P. Coullet, L. Mahadevan and C. Riera,  Journal of Fluid Mechanics,  526, 1-17, 2005.
    [View PDF] [Download PDF] We give a hydrodynamical explanation for the chaotic behaviour of a dripping faucet using the results of the stability analysis of a static pendant drop and a proper orthogonal decomposition (POD) of the complete dynamics. We find that the only relevant modes are the two classical normal forms associated with a saddle– node–Andronov bifurcation and a Shilnikov homoclinic bifurcation. This allows us to construct a hierarchy of reduced-order models including maps and ordinary differential equations which are able to qualitatively explain prior experiments and numerical simulations of the governing partial differential equations and provide an explanation for the complexity in dripping. We also provide a new mechanical analogue for the dripping faucet and a simple rationale for the transition from dripping to jetting modes in the flow from a faucet.
  43. Confined elastic developable surfaces: cylinders, cones and the elastica E. Cerda and L. Mahadevan,  Proceedings of the Royal Society of London (A),  461, 671-700, 2005.
    [View PDF] [Download PDF] We consider two of the simplest problems associated with the packing of a naturally flat thin elastic sheet. Both problems involve packing the sheet into a hollow cylinder; the first considers the partial contact of a cylindrically curved sheet with a cylindrical surface, while the second considers the partial contact of a conically curved sheet with the edge of a cylindrical surface. In each case, we solve the free-boundary problems to determine the shape, response and stability of the confined surfaces. In particular, we show that an exact description of both the cylindrical and conical structures is given by solutions of the Elastica equation, allowing us to present a unified description of a large class of elastic developable surfaces. This includes what is possibly the simplest example of strain localization, occurring at a point and forming one of the constituent elements of a crumpled elastic sheet.
  44. Using the peel test to measure the work of adhesion in a confined elastic film A. Ghatak, L. Mahadevan and M. Chaudhury,  Langmuir,  21, 1277-81, 2005.
    [View PDF] [Download PDF] We present a theoretical and experimental study of a method for the determination of the adhesion strength between a thin elastomeric film bonded to a rigid substrate and a flexible plate in a geometry common in the peel test. In particular, we characterize the work of adhesion in terms of the length of an equilibrium crack, generated by a spacer of known thickness wedged between the flexible plate and adhesive film, and the elastic and geometric properties of the film and the plate. We treat both the limit of perfect bonding and that of perfect slippage at the interface of the adhesive film and the flexible plate. A series of experiments allow us to verify the theory quantitatively and thus validate our method, which ought to be of value in many technological situations.
  45. Buckling of drying droplets of colloidal suspensions, N. Tsapsi, E. Dufresne, S. Sinha, C. Riera, J. Hutchinson, L. Mahadevan, D. Weitz,  Physical Review Letters , 94, 018302, 2005.
    [View PDF] [Download PDF] Minute concentrations of suspended particles can dramatically alter the behavior of a drying droplet. After a period of isotropic shrinkage, similar to droplets of a pure liquid, these droplets suddenly buckle like an elastic shell. While linear elasticity is able to describe the morphology of the buckled droplets, it fails to predict the onset of buckling. Instead, we find that buckling is coincident with a stress-induced fluid to solid transition in a shell of particles at a droplet’s surface, occurring when attractive capillary forces overcome stabilizing electrostatic forces between particles.
  46. Spontaneous folding of planar surfaces into three-dimensional objects by magnetic self-assembly, M. Boncheva, S. Andreev, L. Mahadevan, A. Winkelman, D. Reichman, M. Prentiss, S. Whitesides and G. Whitesides,  Proceedings of the National Academy of Sciences (USA),  102, 3924-29, 2005.
    [View PDF] [Download PDF] This report describes the spontaneous folding of flat elastomeric sheets, patterned with magnetic dipoles, into free-standing, 3D objects that are the topological equivalents of spherical shells. The path of the self-assembly is determined by a competition between mechanical and magnetic interactions. The potential of this strategy for the fabrication of 3D electronic devices is demonstrated by generating a simple electrical circuit surrounding a spherical cavity.
  47. Fluid-flow induced flutter of a flag M. Argentina and L. Mahadevan,  Proceedings of the National Academy of Sciences (USA), 102, 1829-34, 2005.
    [View PDF] [Download PDF] We give an explanation for the onset of fluid-flow-induced flutter in a flag. Our theory accounts for the various physical mechanisms at work: the finite length and the small but finite bending stiffness of the flag, the unsteadiness of the flow, the added mass effect, and vortex shedding from the trailing edge. Our analysis allows us to predict a critical speed for the onset of flapping as well as the frequency of flapping. We find that in a particular limit corresponding to a low-density fluid flowing over a soft high-density flag, the flapping instability is akin to a resonance between the mode of oscillation of a rigid pivoted airfoil in a flow and a hinged-free elastic plate vibrating in its lowest mode.
  48. Photo-induced deformation of beams, plates and films Warner, M. and L. Mahadevan,  Physical Review Letters , 92, 134302, 2004.
    [View PDF] [Download PDF] Photoresponsive solids such as nematic photoelastomers can undergo large deformations induced by light absorbed into rodlike molecules which bend and disrupt liquid crystal order. Significant variation of photoabsorption through the solid leads to nonuniform elastic deformations such as bending of beams and plates and pitting of layers. Such effects are also found in the presence of inhomogeneous thermal or swelling fields in solids or gels. We analyze the small deflection limit of these problems and show that beams made of these materials can have two elastically neutral planes, and that plates of these materials have a typical saddle shape. We also give a scaling analysis of the elasticity of photoinduced mounds and pits and speculate on their applications.
  49. Soft lubrication Skotheim, J. and L. Mahadevan,  Physical Review Letters , 92, 245509, 2004.
    [View PDF] [Download PDF] We consider some basic principles of fluid-induced lubrication at soft interfaces. In particular, we quantify how a soft substrate changes the geometry of and the forces between surfaces sliding past each other. By considering the model problem of a symmetric nonconforming contact moving tangentially to a thin elastic layer, we determine the normal force in the small and large deflection limit, and show that there is an optimal combination of material and geometric properties which maximizes the normal force. Our results can be generalized to a variety of other geometries which show the same qualitative behavior. Thus, they are relevant in the elastohydrodynamic lubrication of soft elastic and poroelastic gels and shells, and in the context of biolubrication in cartilaginous joints.
  50. Peeling, healing and bursting in lubricated elastic sheets, Hosoi, A. and L. Mahadevan,  Physical Review Letters , 93, 137802, 2004.
    [View PDF] [Download PDF] We consider the dynamics of an elastic sheet lubricated by the flow of a thin layer of fluid that separates it from a rigid wall. By considering long wavelength deformations of the sheet, we derive an evolution equation for its motion, accounting for the effects of elastic bending, viscous lubrication, and body forces. We then analyze various steady and unsteady problems for the sheet, such as peeling, healing, levitating, and bursting, using a combination of numerical simulation and dimensional analysis. On the macroscale, we corroborate our theory with a simple experiment, and, on the microscale, we analyze an oscillatory valve that can transform a continuous stream of fluid into a series of discrete pulses.
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