Thin Sheet Mechanics

Whenever one dimension of a solid becomes small compared to the others, we say that the object is thin.  Thin objects (films or sheets) deform in unique ways when confined, or in other words, stuffed into a space that is smaller than the sheets two longer sides.  If confinement is small, a sheet can simply bend to deal with the constraints.  This is mainly because bending (though rooted in stretching) is a much lower energy deformation than compressing the sheet.  However, if the confinement is increased further the sheet will eventually need to bend in an orthogonal direction to the first bend.  When this happens, a big energetic cost is incurred because the sheet must now stretch (its Gaussian curvature is forced to change).  This is a big energetic cost for the sheet, and it may cause the sheet to focus the stretching to a sharp point. 

While still continuum mechanics, the focusing is quite hard to understand and represents a weakness in our ability to model thin films.  What is more, the sharp point discussed above occurs in many applications involving thin sheets like origami, thin film laminating processes, encapsulation, and crumpling (the inevitable end point of highly confining a sheet).  It is an open problem understanding how sharp points and folds affect the strength of thin sheet structures.

Adhesion

Adhesion is a vast area of research, seamlessly connecting geckos to surface science in a truly interdisciplinary field.  Understanding the stickyness of two surfaces requires a deep understanding of the mechanics of both the adhering object and the apparatus (i.e. the gecko) used to test the adhesive forces.  Even though scientists and engineers have studied adhesion for many decades, there are many unsolved physical problems, and many open industrial niches.

Thin Polymer Films

Strange things can happen simply because a material is made small.  The most remarkable features of a thin polymer film include a glass transition that is vastly different than in a bulk sample.  The reason for these changes remain a significant challenge for polymer physics and its understanding has broad ranging implications.  An interesting way of phrasing the problem is that measurement of small things is quite difficult.  For example, there are many simple ways of measuring the modulus of a large piece of polystyrene (e.g. stretch it).  However, the same measurement on a nanometer thin sheet is incredibly difficult (and therefore prone to error) because thin sheets are sticky, fragile and dominated mechanically by the surfaces that they contact.  Each of these points represents important areas of current research.

Pattern Formation and Control

Patterns are everywhere in Nature, appearing on large lengthscales and small lengthscales, on the ground and in the sky and most certainly appear in biology.  What is a pattern though?  Have you ever swirled a glass of wine and noticed that small equally spaced droplets of fluid form on the inside of the glass?  Perhaps you have squeezed some skin together between your fingers and noticed equally spaced wrinkles appear.  Or maybe you have looked at a thin layer of oil heating in a pan, sprinkled some pepper, and noticed equally spaced 'cells' appear.  These are all examples of simple patterns, which we might formally define as a lengthscale appearing in a system (i.e. equally spaced 'things').  Ultimately, patterns appear because there are two different energies in a system each of which prefers a different lengthscale.  Since both energies cannot be simultaneously minimized, the system must find some lengthscale between the two intrinsic sizes.  For example, as you squeeze skin together between your fingers the top layer wants to bend - like a sheet of paper does.  The energy that occurs when an object is bent is minimized by the smallest possible curvature (or the biggest length) available.  Skin, however, is connected to the soft tissue below it, which must be stretched as the skin is bent.  Intuitively, stretching is minimized by the shortest possible distance (think of pulling on a spring).  Because the squeezed skin cannot have a large bend and a small stretch at once, it forms wrinkles of an intermediate size.  Finding and characterizing new patterns is both fundamentally exciting, and technologically useful - provided you can control the pattern.  Control is easy though if you understand the physics of the pattern.

Block Copolymer Physics

Block copolymers are made whenever polymers of two or more different varieties are joined together.  The simplest type of block copolymer is a diblock copolymer, made whenever two different polymers are covalently bound at one end.  Their behavior as a material is rich and full of fascinating physics.  For example, when cooled below a phase transition point diblocks spontaneously form nanoscopically ordered patterns, very similarly to certain surfactant or lipid systems.  The physics behind the structure has to do with the interplay between the chemical differences of the two blocks (think of oil and vinegar) and entropy loss (both because of localizing the junction point of the diblock to an interface, and because the polymer chains themselves get squeezed in the process).  While our physical understanding of diblocks has grown significantly in recent years, there remain many interesting and unsolved fundamental problems (fluctuations, confinement, non-equilibrium states ...).  Diblock copolymers are also hoped to be useful as lithographic masks of smaller dimension than can be reached by traditional means.  Because of this technological drive there are also many interesting applied problems in this field.