Some surfactant molecules in solution self-assemble and form long wormlike micelles which continously break and recombine.[1] Their mass distribution is, hence, in thermal equilibrium and they present an important example of the vaste class of systems termed equilibrium polymers (EP). Other examples include liquid sulfur and selenium, supramolecular aggregates of dyes, dipolar colloids, protein filaments, self-assembled stacks of discotic molecules. All these examples differ by the nature of the intermolecular forces involved in the self-assembling of the basic units, but they lead to a similar physical situation bearing much analogy with a traditional system of polydisperse flexibles polymers when their length becomes sufficiently large with respect to their persistence length.
The specificity and originality of these supramolecular polymers comes from the fact that these chains are continuously subject to scissions at random places along their contour and subject to end to end recombinations, leading to a dynamical equilibrium between different chain lengths species. These supramolecular polymers are typical soft matter systems and the chain length distribution which determines their properties, is very sensitive to external conditions (temperature, concentration, external fields, salt contents, etc...). In particular, micellar solutions exhibit fascinating rheological behaviour, such as shear-banding, shear-thickenning, Maxwell fluid behaviour or anomalous diffusion (Levy flight) which are ultimately due to the permanent scission-recombination processes.
Mesoscopic scale computer simulations can shed much light on these rich but complex systems, thanks to techniques borrowed from simulations of classical polymers. The advantage of numerical studies is their ability to make links either to experiments or to mean-field type theories, given that those systems are difficult to characterize by experiments and the results are not simple to interpret.
The free energy E of the (spherical) end cap of these micelles has been estimated to be of order 10 kBT. This energy penalty (together with the density of surfactants) determines essentially the static properties and fixes the ratio of the scission and recombination rates, ks and kr. Additionally, these rates are influenced by the barrier height B which has been estimated to be similar to the end cap energy. Both important energy scales have been sketched schematically as a function of a generic reaction coordinate q.
In our numerical studies we represent these micellar systems by coarse-grained effective potentials in terms of standard bead-spring models. The end cap energy becomes now an energy penalty for scission events, i.e., the creation of two unsaturated chain ends. This penalty (together with the monomer density) determines essentially the static properties and fixes the ratio of the scission and recombination rates, ks and kr.
The dynamical barrier is taken into account by means of an attempt frequency exp(-B/kBT). This garantees that the barrier may only determine the dynamical properties (through the effective chemical reaction constants) while leaving the static properties of the samples unchanged. If the attempt frequency is large, successive breakage and recombination events for a given chain can be assumed to be uncorrelated and the recombination of a newly created chain ends will be of standard mean-field type. On the other hand, the (return) probability that two newly created chain ends recombine immediately must be particulary important at high frequency. These highly correlated ``diffusion controlled" recombination events do not contribute do the effective macroscopic reaction rates which determine the dynamics of the system. It is therefore important to compute the true effective rates as a function of the attempt frequency.
Both energy scales E and B are assumed to be independent on the density and on the position of the bonded connected along the chain. In most of our studies we do not allow for the formation of closed loops.
