Soft Matter

The most consequential fight was about whether large molecules exist. In 1920 the respectable position was that rubber, cellulose and proteins are aggregates — many small molecules held together by weak forces — because the available analytical methods could not weigh anything bigger and because colloidal behaviour was taken to indicate aggregation.

Hermann Staudinger insisted they are single covalently bonded molecules with thousands of atoms, and was told publicly, by senior chemists, to stop. His decisive experiment was to hydrogenate natural rubber. If the colloidal properties came from association at the double bonds, saturating every one of them should destroy those properties. The hydrogenated product was still a rubbery, high-viscosity, colloidal material. There were no association sites left to be responsible.

Accepting long chains made a quantitative theory possible, and it came from an unexpected direction: random walks. Werner Kuhn and then Paul Flory treated a chain as a sequence of freely jointed segments, and the mathematics of probability theory supplied the rest.

Model a polymer chain as $N$ rigid segments of length $b$, each free to point anywhere — a random walk. The end-to-end distance is not $Nb$ but

$$R = b\sqrt{N}.$$

For a chain of $N = 1000$ segments of $b = 0.25$ nm, the contour length is $Nb = 250$ nm while the typical end-to-end distance is

$$R = 0.25\sqrt{1000} = 7.9\ \mathrm{nm}.$$

The chain is coiled into a ball about thirty times smaller than its own length, for the same reason that a thousand coin flips rarely come out all heads: there are vastly more configurations with the ends close together than with them far apart.

That count is the elasticity. The number of configurations with end-to-end vector $R$ falls off as a Gaussian, so the entropy is

$$S(R) = k_B \ln \Omega(R) = \text{const} - \frac{3k_B R^{2}}{2Nb^{2}},$$

and since the internal energy barely changes — no bonds are being stretched — the free energy $F = U - TS$ gives a restoring force

$$f = \frac{\partial F}{\partial R} = -T\frac{\partial S}{\partial R} = \frac{3k_B T}{Nb^{2}}R.$$

A Hookean spring, with a stiffness proportional to $T$. The force exists because stretching reduces the number of available configurations, and for no other reason.

Two consequences are testable with a rubber band and your lip. First, the modulus rises with temperature: a crosslinked network with $n$ chains per unit volume has shear modulus $G = n k_B T$. Writing $n = \rho N_A/M_c$ with $M_c$ the mass between crosslinks, for natural rubber with $\rho = 950$ kg/m³ and $M_c = 5$ kg/mol at 300 K:

$$G = \frac{\rho R T}{M_c} = \frac{(950)(8.314)(300)}{5} = 4.7\times10^{5}\ \mathrm{Pa} \approx 0.5\ \mathrm{MPa},$$

which is the measured value, and five orders of magnitude below steel's 80 GPa. Heat a loaded rubber band and it contracts, pulling harder — the opposite of thermal expansion, and the direct signature of entropic elasticity.

Second, stretching must release heat. The work done goes into reducing entropy at constant internal energy, so $T\Delta S$ comes out as heat: stretch a thick rubber band quickly and press it to your lip, and it is noticeably warm; let it snap back and it cools. The demonstration takes five seconds and is a direct measurement of a thermodynamic identity.

Pierre-Gilles de Gennes later showed that the $\sqrt{N}$ law is only the ideal case. A real chain cannot pass through itself, and allowing for that excluded volume changes the exponent to roughly $N^{3/5}$ in three dimensions. His route to that result is the striking part: the polymer problem maps onto a magnet near its critical point in the limit where the number of spin components goes to zero, so the renormalisation group computes polymer exponents. The chemistry of the chain enters nowhere.

The newest part of the field breaks the thermodynamic frame rather than extending it. In ordinary soft matter, energy enters at the boundaries and the interior relaxes towards equilibrium. In active matter each unit consumes fuel and propels itself, so momentum is injected everywhere at once.

Tamás Vicsek's 1995 model is about as simple as a model can be: particles move at fixed speed and each turns to match the average heading of its neighbours, with some noise. Below a noise threshold, order appears — the whole crowd moves together, with no leader, no attraction, and in two dimensions, where equilibrium statistical mechanics forbids such long-range order. Bacteria in suspension, vibrated rods, synthetic swimmers and mixtures of cytoskeletal filaments with their motor proteins all show the predicted behaviour, including spontaneous flows and density fluctuations far larger than equilibrium allows.

This is where the thread rejoins biology. A cell's interior is a crosslinked network of semiflexible filaments driven by motors burning ATP, which is the definitional case of active matter, and treating it as such predicts the flows seen during division and the way a tissue behaves as a fluid over hours and a solid over seconds — the mechanics described from the other side under cell biology. Meanwhile the oldest question in the field remains the glass transition: cool a liquid fast and its viscosity rises by fourteen orders of magnitude while its structure stays liquid, and sixty years of argument have not settled whether anything is actually transitioning.