Pattern formation in Nature

Surfaces and sheets of condensed and liquid phases are often subject to forces that destabilize their homogenous state and lead to spontaneous formation of rich morphologies characterized by scales that range from few nanometers to centimeters or more. Emergence of such patterns has strong impact on mechanical, chemical and electromagnetic properties of objects, and has far reaching consequences in material science and engineering, as well as in the bio-world. Experimental and theoretical studies on pattern formation phenomena are driven by the desire to understand the fundamental mechanisms by which homogenous forces lead to the spontaneous formation of small-scale structures, as well as by the promise held by the application of useful theoretical ideas to the development of efficient and cheap surface-pattering technologies.

The work of my group in the last few years has addressed the spontaneous formation of patterns under equilibrium and non-equilibrium conditions. Most of our recent studies have addressed three general problems: The morphogenesis of complex morphologies in elastic sheets, non equilibrium surface patterning problems, and the geometry-induced forced of particles on soft interfaces. The description below is quite general. To get a better idea on the concrete problem we are studying and out style of work, please visit the Nuggets page in this website. To get an even better idea, you are invited to read our papers or write to me.

Morphogenesis in elastic sheets 

Elastic sheets exhibit highly complex morphologies that are often described as wrinkles, creases, crumples, folds, and blisters. A close inspection of a candy wrap, a human skin, or a stretched plastic bag, reveals that these patterns often coexist in the same object, ranging from microns or even nanometers, to human body scales. A curious, critically minded observer, is ultimately led to ask the typical child’s questions:

  • Why do films become folded upon confinement whereas a thick slab of an identical material does not fold?
  • Why does paper tend to crumple whereas rubber sheets would smoothly wrinkle?
  • Why do certain plant leaves have a buckled shape, whereas others are flat?

Answering these questions is important not only for satisfying our natural curiosity. The emergence of complex morphologies in natural and synthesized sheets affects their mechanical, optical and chemical properties, and is thus highly relevant subject for material science, engineering,  and biomechanics. Understanding how complicated patterns in sheets emerge spontaneously under featureless forces may inspire efficient methods for tailoring a desired surface pattern, or the self-assembly of structures at will from a homogenous piece of matter.

The difficulty in understanding this morphological diversity stems from the highly nonlinear nature and the geometric complexity of the problem when the sheet thickness becomes exceedingly small. This complexity requires the development of new concepts and methods, beyond traditionally used ones. This theoretical challenge and a sequence of experiments on nanomertically-thin (ultrathin) sheets, have motivated our studies on several ”simplest yet nontrivial” problems. Inspired by the classical methodology of pattern formation theory, we addressed highly symmetric systems (see Figure), in which wrinkling, crumpling, and hierarchical multi-scale structures emerge from spontaneous symmetry-breaking instabilities. These studies enabled us to identify a set of generic, dimensionless parameters, those govern the transitions between distinct types of patterns. Analogously to pressure and temperature, that govern theremodynamic transitions between various phases of matter, one may view the bendability, confinement, and softness, defined in our studies as fundamental parameters that span morphological phase diagrams in which the various types of deformation are recognized as distinct “morphological phases”. By focusing on the essential symmetry-breaking mechanisms associated with those phases, these studies provide a conceptual basis for exploring the generic conditions underlying the existence and co-existence of wrinkling, crumpling, creasing, folding, and blistering. Furthermore, the construction of this theoretical framework brings up new basic questions concerning the links between morphological transitions in elastic sheets and other pattern forming systems in nature.

Particles on Curved Surfaces

Liquid interfaces are ubiquitous in nature and industry, with examples including the surfaces of liquid droplets in emulsions or of gas bubbles in foams.  In many relevant cases, these interfaces are crowded with particles that strongly adhere to the interface and stabilize the droplets, thus forming ‘Pickering’ emulsions.  Industrial examples include silica or clay particles stabilizing petroleum droplets, inorganic nanoparticles coating droplets in ore separations, and fat crystals stabilizing droplets in foods.  Recent experiments have shown that liquid interfaces can be used as templates, on which to assemble particles into a variety of materials and devices. Examples include nanoparticle-stabilized water droplets in oil capsules with size-selective permeability (‘colloidosomes’), ultra-thin membranes that are fluorescent or magnetic, and even electronic devices such as transistors.  Despite these applications and the potential for many more, fundamental questions remain about how these adsorbed particles interact with one another.  Over distances of a few nm, van der Waals predominate.  At greater separations, repulsion can arise from charges dissociating in one or both liquids.  In other cases, experiments show that particles at tract one another over a range of several microns (i.e., a few particle diameters).  Debate over the exact origin of this long-range attraction persists, but an important fact is that those ideas address primarily with initially flat interfaces. When particles are bound to a geometrically curved interface another type of interaction might arise, which stems from the deformation of the interface due to the the adsorbed particles.  These geometry-inducedinteractions would be relevant for directing particle motion and assembling novel materials.

In collaboration with Dinsmore’s group we commenced recently a research program whose experimental and theoretical parts are aimed at understanding those interactions that arise from curvature of a liquid interface. As we learned,  the geometric effect associated with the induced deformation of the interface could give rise to strong long-range interactions.  These geometry-induced interactions would be relevant for directing particle motion and assembling novel materials.  They might also explain how particles pack on droplets that are distorted by a surface, by shear, or by flow.

 

Nonequlibrium Pattern formation