Monte Carlo Simulations of Colloidal Pair Potential Induced by Telechelic Polymers: Statistics of Loops and Bridges Vincent Testard, Julian Oberdisse,* and Christian Ligoure Laboratoire des Colloı ¨des, Verres et Nanomate´riaux, UMR 5587 CNRS/UM2, UniVersite´ Montpellier II, F-34095 Montpellier Cedex 5, France ReceiVed March 17, 2008; ReVised Manuscript ReceiVed July 3, 2008 ABSTRACT: A Monte Carlo study of the statistics of loop and bridge formation between colloidal particles, and in particular micelles, by telechelic polymers is presented. The experimental fact that the hydrophobic outer blocks of a triblock copolymer tend to stick into micelles in aqueous solution is mimicked by counting only polymer chains with both ends on the surface of the micelles. The long inner hydrophilic block is generated by a random walk procedure. It is excluded from the volume of the micelles, and it can form either a loop, with both stickers on the same micelle, or a bridge, with the stickers on two different micelles. In this paper, the pair potential is determined between two micelles as a function of distance and chain-to-micelle size ratio, for ideal and self-avoiding chains. In the latter case, the effect of many chains has also been explored. 1. Introduction Self-assembling networks are common in both natural and synthetic materials. They consist of self-assembled aggregates (nodes) that are reversibly connected by links with a finite lifetime as opposed to chemical gels where junctions are permanent. These physical gels exhibit two universal and independent features: a thermodynamic first-order phase separa- tion, which occurs at a low volume fraction between a dilute and concentrated solution even in the absence of any specific interaction, and a nonthermodynamic topological transition, where an infinite network spanning the entire volume of the system is formed. 1 Telechelic polymers are often used as model linkers because they are architecturally simple: they consist of a long solvophilic midblock with each end terminated by a solvophobic short chain (a sticker). The stickers incorporate into the solvophobic domains of the aggregates and can bridge them via their solvent-soluble midblock, resulting in an attractive interaction between the aggregates. The nature and morphologies of the aggregates forming the network are versatile: (i) telechelic polymers in binary solution 2 that self-assemble spontaneously into noninteracting flowerlike micelles at low concentrations and form three-dimensional networks above a threshold con- centration, 3 (ii) surfactant vesicles, 4,5 (iii) lyotropic lamellar phases, 6,7 (iv) wormlike micelles, 8-10 (v) spherical micelles, 7,11 and (vi) oil-in-water 12,13 or water-in-oil 14 microemulsion drop- lets. This last system (telechelic/microemulsion mixtures) is of particular fundamental interest. Indeed, the advantage of this system is that the parameters that control the thermodynamics and structure of the physical gel can be easily identified and independently controlled: the concentration of nodes (the droplets) and the number of polymers per droplet. This is in contrast with binary mixtures of telechelics, where the number of nodes formed by the associating chain ends cannot be controlled separately. Other advantages of this system are the spherical symmetry which allows, for instance, a simple structural analysis in the Fourier space from scattering experi- ments and the versatility of the control of the surface curvature. Because of the high solvophobic energy (≈10-20k B T), the chain ends are constrained to lie within the droplets and the number of dangling chains outside the droplets is statistically insignificant. Nevertheless, the stickers can detach from a droplet and switch between loops (with both ends inside the same droplet) and bridges (with the chain ends residing in different droplets). Stickers can also be exchanged between the droplets during droplet collisions. Phase behavior, structural properties on one hand, and elasticity in the gel region on the other hand will be controlled by the distribution of loops and bridges between the aggregates 15 and will depend on several parameters: the concentrations of droplets and linkers as well as the sizes of polymers and droplets. Modeling such a complicated self-assembly is a formidable challenge, because of the wide range of scales involved. Different complementary theoretical approaches to describing these reversible gels have been proposed. The mean field analytical model developed by Zilman et al. 16 allows for a generic qualitative prediction of the phase diagram of micro- emulsions with telechelic linkers: it predicts that the system undergoes a robust first-order phase separation into a dense highly connected phase in equilibrium with dilute droplets decorated by polymer loops as the number of polymers per drop is increased, despite the absence of any specific interaction between either the droplets or the polymer chains. On the other hand, a hybrid Monte Carlo/molecular dynamics numerical approach by Hurtado et al. 17 coarse grains the polymer descrip- tion and retains their effect only as links inducing a phenom- enological entropic interaction between the two droplets they connect. This model predicts the same type of phase diagram but also a gelation line determined by geometric percolation unrelated to phase separation, as observed experimentally by Filali et al. 14 It also allows the investigation of the gel dynamics, which is outside the scope of this paper. A quantitative description of the interactions induced by telechelic polymers between droplets needs a polymeric ap- proach that can predict the loop and bridge distribution as a function of the various relevant parameters, i.e., the mean droplet separation, the mean number of stickers per droplet, the droplet- to-polymer size ratio, and the quality of the solvent. Such an approach could be also useful for the quantitative analysis of the scattering experiments performed with these systems. 14,18-20 The first theoretical approach was proposed by Milner and Witten, who adapted self-consistent field calculations to tele- chelic polymer brushes between flat plates. 21 These authors found a weak attractive minimum in the free energy near a separation close to brush contact due to the increased entropy per bridge in the contact. This calculation was revisited by Meng * To whom correspondence should be addressed. E-mail: oberdisse@ lcvn.univ-montp2.fr. 7219 Macromolecules 2008, 41, 7219-7226 10.1021/ma8005813 CCC: $40.75  2008 American Chemical Society Published on Web 09/19/2008