John S. Biggins
John S. Biggins

Contact

TCM Group
Cavendish Laboratory
JJ Thomson Avenue
Cambridge CB3 0HE
United Kingdom

jsb56@cam.ac.uk
+44 (0) 1223 3 37360

I do very little of this research on my own. Some collaborators are mentioned below, and a more extensive list can be found on the links page. These days I also have a small research group who increasingly do the hard work!

# Elastic Instabilities in Soft Solids

When an elastic structure is pushed or pulled it responds by changing shape. For example, if you pull on the two ends of an elastic band it gets longer. For small forces the shape changes are typically simple and small, but if you apply enough force the structure sometimes responds by changing shape in dramatic and unexpected ways. We call these shape changes elastic instabilities. The simplest example of an elastic instability occurs when you push on a thin elastic rod. For small forces the rod simply gets shorter, but at a critical degree of compression the rod will buckle into a bent shape.

In a linear system doubling the force applied simply doubles the system's response. In elastic instabilities adding more force produces a completely different response, so they must be driven by non-linearities in the theory of elasticity. These non-linearities divide into two types, material non-linearities that stem from the constitutive law of a given material and geometric non-linearities that stem from the fact that large deformations do not commute. The former are specific to a given material but can occur at small strains while the latter are universal but only occur when the shape changes are geometrically large, so they tend to be restricted to soft solids.

 Right: Top view of a soft elastic layer sandwiched between two glass plates. The plates are pulled apart while maintaining adhesion to the layer, resulting in fingers of air invading the layer at its boundary. Experiment by Baudouin Saintyves.

I have worked on a new example of such a geometric elastic instability shown in the above video. We call it elastic fingering because it is a solid analogue of Saffman-Taylor fingering in viscous fluids. Understanding elastic fingering should help us understand the failure of polymeric glue joints.

 Sulcus on the surface of a bent rubber beam. Image from Evan Hohlfeld. Pattern of sulci on the surface of a slab of soft material subject to equal compression in both planar directions.

If any soft solid is subject to sufficient compression its free surface invaginates to form cusped furrows called sulci, shown above in in a bent rubber beam. This instability,discovered by Biot in the 1950s, is another example of a geometric elastic instability. Its mathematical structure remains poorly understood. I have worked on the pattern the sulci form when a surface is compressed in both in-plane directions rather than just one. An example of such a pattern is shown above. This is a natural situation in biology, where compression often derives from tissue growth, and in the swelling of gels. This work appeared on the cover of PRL.

# Mechanics of chains

Watch this video of a chain fountain by Steve Mould.

Why do you get a fountain? We have shown that, to get a fountain, when a link of the next link of chain from the is pulled into motion by the tension in the chain above it, it must also be pushed into motion by the pot. The existence of this new force will require us to re-examine some very old ideas about energy loss and chain pickup. We made an explanatory video

This work featured in several media outlets, including Nature and Science., Scientific American and, yes, the Daily Mail.

# Physical and Statistical Approaches to Biological Shape

The diversity of shapes found in biology is truly awe inspiring. Evolution has produced complex patterns and shapes, ranging in size from the molecular double spiral of DNA to the colossal branched structure of giant redwood trees. Biology is, to borrow the title of Richard Dawkins' latest book, "the greatest show on earth". However, to satisfactorily answer the question "why is it that shape?", which is surely one of the most fundamental questions about any biological object, one must understand much more than the mechanism of evolution. One must also understand how the shape actually grows, and why the shape in question is well suited to its environment. The answers to these biological questions often lie in physics and maths.

## Elastic instabilities that sculpt biological organs

When we see a complicated shape in biology - for example, the folds of the human brain - we typically think the development must be the result of a carefully regulated chemical process, with a Turing-like reaction diffusion instability at the heart of the pattern generation mechanism. This way of thinking fits easily into a biochemistry mindset dominated by genes and proteins, but sometimes things are simpler. I am interested in examples where the biological growth is very simple but the action of mechanical forces (i.e. elasticity) sculpts the tissue into the observed complex shape. My favorite hypothesis, hopefully soon to be convincingly published, is that the folds on the human brain are a simple mechanical consequence of the outside of the brain (the cerebral cortex) growing faster than the inside of the brain, and then being shaped by a sulcus forming compressive elastic instability akin to buckling and wrinkling. An example of the brain-like shapes this mechanism can produce is shown on the right.

## Cell Division and Specialization

For a fertilized egg to turn into a fully-fledged organism requires many iterations of cell division and specialization, and this process continues later in life in stem-cell maintained tissues such as skin. Failure of these processes can result in cancer. Biologists have expended a great deal of effort trying to understand these fundamental processes from a "bottom up" perspective, attempting to identify regulatory gene networks and other signaling mechanisms. This approach has enjoyed some success but is consistently frustrated by the extreme complexity of the low level mechanisms. Condensed matter physics is full of sets of systems that exhibit very similar macroscopic behavior despite different microscopic details, and such behaviors are often best described by a ``top-down" approach, not asking what behavior a given set of microscopic rules give, but what constraints on the microscopic rules different behaviors require. I am attempting to use this philosophy to identify the macroscopically important features of cell division and differentiation rules.

This work is in collaboration with Ben Simons and several experimental biologists. Our approach is rooted in ``clone-fate'' data sets. These are from experiments in which a mother cell is marked, and then the number of its surviving daughter cells --- which form the ``clone'' --- is recorded at later times. At them moment I am working with data from a brain cancer tissue and a mutated skin system in which cell differentiation is severely inhibited. I am also working on analogous data (provided by Shankar Srinivas) from very early term mouse embryos (8-50 cells), attempting to ascertain when and why the very first cell specialisations are made.

# Liquid Crystal Elastomers

The exotic elasticity of liquid crystal elastomers was the focus of my PhD thesis, which was supervised by Mark Warner.

Liquid crystal elastomers (LCEs) are remarkable materials that combine the mobile orientational order of liquid crystals with the extreme stretchiness of rubber. LCEs are mixtures of rigid liquid-crystal rods and long flexible polymer chains that have been chemically cross-linked to form a rubber network. When an LCE is hot the rods point in random directions and the material behaves like a traditional isotropic rubber, but when the elastomer is cooled, the rods align, and the elastomer stretches by up to several times its own length in the alignment direction. A video of this happening from Eugene Terentjev's lab is shown on the right. Since the hot state is isotropic, on cooling any direction can be chosen for the rods to align in, and hence there are many equivalent cold states each with a different alignment and a different stretch with respect to the hot state. Stretches that turn one such aligned cold state into another different but equivalent one cannot cost energy --- they are ``soft'' deformations. Soft deformations are common in fluids --- we usually say the fluid is flowing --- but very unusual in a solid. Indeed we can fairly say the resultant material is somewhere between a solid and a liquid. Much more information about LCE's can be found at www.lcelastomer.org.uk.

My research into LCEs has focused on two main topics. Firstly I have studied the limitations of the above symmetry argument for completely soft deformations of monodomain LCEs. The essential issue is that, if the hot state is truly isotropic, then, on cooling, the liquid crystal rods will choose different directions to align at different points in the sample, so the cold aligned state will be have a polydomain structure and there will be no overall deformation. The rods can be made to all align in the same direction by imprinting a direction on the isotropic state, producing a monodomain elastomer which does change length considerably on cooling. However, in this case the hot state is not truly isotropic so the argument for the existence of soft modes is compromised. I have used phenomenological arguments to show that the resulting monodomain will still show qualitatively soft behavior, and, at some stretches, the modulus for additional extension will vanish altogether. I have also shown that chiral nematic monodomains will exhibit strain-induced electrical polarization, unlike nematic liquids or conventional elastomers.

My second focus has been the polydomain LCEs that form in the absence of imprinting. I have shown that, if these systems are prepared correctly, they will exhibit elasticity that is softer than the more fashionable monodomains. Since these samples are also much easier to produce, this could have technological implications. I collaborated closely with Kaushik Bhattacharya on this work.

 Maintained by John S. Biggins