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  Instabilities and morphologies of elastic membranes

 

Thin solid sheets exhibit a rich variety of patterns when subjected to a featureless distribution of forces or geometric constraints. This property can be demonstrated through two simple yet representative examples: First, force a flat piece of paper into your fist and watch its crumpled shape - a disordered network of ridges and peaks in which the sheet is strongly distorted, leaving the rest of it intact. Second, hold a thin rectangular rubber sheet between your palms and pull apart two opposite edges - the response is now characterized by the spontaneous emergence of a parallel array of wrinkles. A curious, critically-minded observer, is ultimately led to ask the typical child's questions: Why do papers crumple, whereas rubber sheets are smoothly (and reversibly) distorted under similar confinement conditions? Why do thin rubber sheets develop wrinkles upon stretching, whereas a thick slab of identical material would not wrinkle? Why does a torn edge of a ripped plastic sheet appears wavy, whereas a torn piece of paper remains flat? 

Adopting the language and concepts used to describe pattern formation in continuous media, these puzzles translate into questions, such as:  what are the fundamental instabilities by which the continuous symmetries of an elastic membrane are broken? what are the characteristic features of patterns that emanate from these instabilities? Finally, one may ask - is it possible to classify the rich variety of observed membrane patterns in a "phase diagram", whose axis correspond to a small set of universal, morphologically-relevant parameters?

 

 

In a recent work we addressed these questions by focusing on a simple system: homogenous membrane, of highly-symmetric (rectangular) shape, that is bent due to a uniaxial compression (see Fig. 1) along direction y.  A central result of our analysis is the prediction of a novel series of elastic "period fissioning" instabilities, that govern the pattern when sufficiently large tension is exerted along the perpendicular direction (x), if boundary conditions (BC) forbid the formation of an "ideal" 1d wrinkling shape. This result explains an experimental observation that motivated this study: a repetitive, step-like increase of wrinkles periodicity, near an edge (x=0) of an ultrathin uniaxially- compressed membrane that is floating on liquid, and subjected to tension and geometric frustration due to strong capillary forces (see Fig. 2). Furthermore, our analysis led us to conjecture the existence of a surprisingly rich phase diagram for this problem (see Fig. 3), in which all patterns are classified according to two morphologically-relevant parameters: A mechanical one (e) - the compression-to-tension ratio, and a geometric one (n) – the wavelength contrast along the tension direction. The predicted phases consists of hierarchical patterns of distinct types:  Smooth "period fissioning" cascades, with n generations of wrinkles (Fn), "Irregular" shapes, composed of k generations of sharp folds (Ik), and "Mixed" patterns, in which n generations of smooth wrinkles "coexist" with k generations of sharp folds (Mn,k).

 

Fundamental questions that we now study, concern the nature of morphological phases and phase transitions in elastic membranes, and possible links to similar phenomena in other energy-dominated branched patterns (such as the near-edge morphology of type-I superconductors). Furthermore, we explore the possibility that similar instability mechanism underlies the hierarchical wrinkling patterns that appear on certain types of plant leaves. Finally, we explore the possibility that the conjectured phase diagram may be a cornerstone for a high-dimensional phase space of membrane morphologies, whose additional axis represent new universal parameters, that are not reflected in the simple conditions assumed in Fig. 1, but may be dominant in many natural and technological environments.

Publications:

1) J.Huang, B. Davidovitch, C. Santangelo, T. R. Russell, and N. Menon , “Smooth cascades of wrinkles at the edge of ultrathin floating sheets” http://xxx.lanl.gov/abs/0901.2892   

2) B. Davidovitch, “Period fissioning and other instabilities of stressed elastic membranes”

http://xxx.lanl.gov/abs/0901.2719

 

                             

Patterns on eroded surfaces

 

Uniform ion beam sputter erosion of a solid surface often causes a spontaneously-arising pattern in the surface topography, that can take the form of one-dimensional ripples or a two-dimensional ordered or disordered array of dots with typical length scales of 10-1000 nm (Fig. 4,5). The promise held by this phenomenon to surface-based technologies is obvious: imagine a device which produces a desired surface pattern, whose typical scales could be 10 or 100 nm, by tuning macroscopic parameters such as angle or energy of ion beam, or surface temperature. Developing such device requires, however, a quantitative theory that enables one to predict what type of surface pattern emerges for a given set of system parameters.

Despite significant progress over the last couple of decades, this goal has not been accomplished yet. An important step towards developing such theory was made by Bradley and Harper (BH, 1988), who incorporated Sigmund’s “ellipsoidal” surface response function (to ion impact), into a continuum approach, that enabled the derivation of an linear equation that describes the surface dynamics on scales much larger than ion penetration depth. Generalizations of BH equation into the nonlinear regime, led to certain predictions concerning the type of the evolving patterns. Many of these predictions, however, as well as several predictions of the linear BH equation, have been contrasted by various experiments, in which ion energy and beam angle are controlled. For example, ion-sputtered Si surfaces (at ~0.5-2 KeV) are homogenously eroded and do not become unstable at certain ranges of beam angle, in contradiction to a central prediction of BH theory.

 

In a recent theoretical paper, we suggested a generalized form of the linear dynamics, that is necessary to explain these observation. An important conclusion of this theory, which was later on substantiated by experimental work, is that the type of transitions between stable and unstable dynamics (as ion energy or beam angle vary), may indicate on the dominance of physical mechanisms that are fundamentally different than the local, erosive response envisioned by Sigmund. Current work on this problem is focused on theoretical studies of various phenomena in the nonlinear regime, such as the evolution of localized structures.   

 

   

 

Publications:

1) B. Davidovitch, M.J. Aziz, and M.P. Brenner, “On the stability of ion-sputtered surfaces”, Phys.Rev. B. (2007), Phys. Rev. B 76, 205420 (2007). Also:

http://xxx.lanl.gov/abs/0706.2662

2) C.S. Madi, B. Davidovitch, H.B. Goeoge, S.A. Norris, M.P. Brenner and M.J. Aziz, “Multiple bifurcation types and the linear dynamics of ion-sputterd surfaces”, Phys. Rev. Lett. 101, 246102  (2008).

3) B. Davidovitch, M.P. Brenner and M.J. Aziz, J. Cond, Matter (in press)

 

 

 

 

Dynamics of charged interfaces: Moving boundary approximation

 

Fronts propagation - the motion of narrow zones that separate two “bulk” regions that are physically distinct from each other – is ubiquitous in nature. A classical example is the solidification of supercooled liquid, that often gives rise to fascinating branched patterns such as ice dendrites. This could be described as the dynamics of the solid-liquid interfacial zone (front), from the solid (bulk phase A) into the liquid (bulk phase B). The emerging convoluted pattern is then associated with the (Mullins-Sekerka) linear instability of the planar front dynamics with respect to sinusoidal undulation at a certain range of wavelengths, and to the nonlinear coupling between these unstable modes. The bulk regions separated by the front zone, should not necessarily be “classical” (equilibrium) thermodynamic phases. An example is the dynamics of negative ionization fronts (“streamers”), formed by negative charge carrier that are accelerated (in neutral gas) under high electrostatic field, and ionize the gas molecules through collisions. The “microscopic” dynamics often leads to a “front” in which the negative charge is concentrated, that separates a neutral region (ahead of the front), from a fully ionized one behind it. One may thus hope that the evolving streamer patterns could be described by starting from an appropriate continuum dynamics, that gives rise to a steadily propagating charged front, and analyzing its stability with respect to large scale undulations. Such moving boundary approach was recently initiated, and effective equations that describe the nonlinear front dynamics were derived. We are currently exploring various aspects concerning the relevant dispersion relations, and the adequacy of this approach for describing the spatio-temporal behavior of fully developed charged fronts.  

 

 

Publications:

1) F. Brau, B. Davidovitch, U. Ebert, “Moving boundary approximation for curved steamer ionization fronts: Solvability analysis”, Phys. Rev. E 78, 056212  (2008). Also:

http://xxx.lanl.gov/abs/0807.4614

2) F. Brau, A. Luque, B. Davidovitch, U. Ebert, “Moving boundary approximation for curved steamer ionization fronts: Numerical tests”, http://xxx.lanl.gov/abs/0901.1916

 

 

Shear banding in complex fluids

 

When sheared out of equilibrium, the response of many types of complex fluids is highly non-Newtonian, and nonlinear effects can prevail even for non-inertial flow rates. A striking and technologically important phenomenon, common to flows in many types of complex fluids, is the formation of shear bands characterized by different strain rates throughout the sheared sample. The number, direction, and size of these bands are determined by the rate and magnitude of the applied force in a rather nontrivial way, and are typically attributed to the nonlinear rheology of the material. A variety of experimental techniques have been used to characterize shear banded flows: birefringence, NMR, ultrasound velocity imaging, and more recently, confocal microscopy. Understanding the physical mechanisms for formation, structure, and stability of shear banded flows will enable controlling the effect in technological applications, and moreover, may reveal key principles underlying the nonequilibrium behavior of complex fluids.

A system in which shear banding has been recently observed is a partially ordered colloidal suspension: hexagonal, densely-packed layers of colloids, that are randomly stacked on each other. Transitions from homogenous to banded flow profiles are observed upon varying oscillations amplitude and frequency. Studying this system is valuable not only for understanding flow in colloidal suspensions, but also due to the simplicity it offers in addressing the fundamental questions raised above concerning the mechanisms underlying shear banding phenomena. Colloidal trajectories can be directly visualized at high precision by confocal microscopy techniques, thus enabling exact characterization of the spatio-temporal structures inside and between shear bands.

One measurement, which revealed a purely harmonic response of the colloidal trajectories under oscillatory shear, led us to realize that despite the appearance of nonhomogenous flow, the response is nearly linear throughout the whole system. We proposed a phenomenological nonstandard model for the dynamics, in which shear banding is related to the existence of stable and metastable phases of the partially-ordered suspension, and thus does not assume any nonlinear local rheology. While analysis of experimental data shows good agreement with the predictions of this phenomenological model, thus providing good evidence that shear banding does not necessarily indicate local nonlinear rheology, fundamental questions concerning the selection mechanisms of number and shear rates in bands, and the transition between confined and unconfined behavior, are beyond the scope of this model, and requires the development of a "first principles" model. A simple, yet nontrivial model, is currently being studied, that will enable us deeper understanding of these puzzles. 

 

 

 

 

Publications:

1)  I. Cohen, B. Davidovitch, A. Schofield, M.P. Brenner, and D.A. Weitz, “Slip, yield, and bands in colloidal crystals under oscillatory shear”,  Phys. Rev. Lett. 97, 215502 (2006).   

 

 

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Benny Davidovitch (bdavidov@physics.umass.edu)