Onset of self-assembly in polymer-surfactant systems

– The onset of self-assembly in a dilute aqueous solution containing a flexible polymer and surfactant is theoretically studied. Focusing on the effect of the surfactant on polymer conformation and using a conjecture of local instability (partial collapse) of the polymer at the onset of self-assembly, we obtain several simple predictions: (i) polymer–surfactant self-assembly always starts at a lower concentration (cac) than the one required for surfactant–surfactant self-assembly (cmc); (ii) in charged systems the cac increases with salt concentration and is almost independent of polymer charge; (iii) in weakly interacting systems the cac remains roughly proportional to the cmc over a wide range of cmc values. The special case of amphiphilic side-chain polymers strongly supports our basic conjecture. A similarity is found between the partial collapse induced by the surfactant and the effect of impurities on the critical behaviour of magnetic systems (the Harris criterion). Aqueous solutions containing polymers and surfactants have been the subject of extensive research in the past few decades [1]. The possibility to achieve polymer–surfactant aggregation at very low surfactant concentration offers a delicate control over the properties of the mixture, a feature being used in numerous applications [2]. The joint self-assembly of polymers and surfactants usually starts at a well-defined surfactant concentration, the “critical aggregation concentration” (cac). One of the most consistent experimental observations in polymer–surfactant systems is that the cac is always lower than the “critical micellar concentration” (cmc) of the polymer-free surfactant solution. Consequently, polymer–surfactant systems are commonly divided into two categories [3]: (i) polyelectrolyte and oppositely charged ionic surfactant, where the cac can become several orders of magnitude lower than the cmc; (ii) neutral polymer and ionic surfactant, where the cac is lower than, but comparable to the cmc. Less common are systems containing a polyelectrolyte and a non-ionic surfactant [4], which can be included in the second category as their cac is comparable to the cmc. Systems where both species are neutral exhibit very weak or negligible effect [5, 6]. Typeset using EURO-LaTEX 2 EUROPHYSICS LETTERS Several theories have been suggested for polymer–surfactant aggregation. [7]-[12]. They include various generalizations of micellization theories [13] in the presence of a polymer chain. However, most of those models practically ignore the polymeric degrees of freedom, treating the chains merely as “substrate” for binding of surfactant monomers or aggregates. This approach may be justified for rigid polymers such as DNA or strong polyelectrolytes in the absence of salt. It is somewhat more questionable in view of many experiments involving flexible polymers, which indicate strong conformational changes accompanying the self-assembly [5]. This Letter takes a different point of view and focuses on the effect of the surfactant on the statistics of a flexible polymer below and at the onset of self-assembly. We assert that in a flexible polymer–surfactant system the cac is associated with local instability of the polymer chain. This description is reminiscent of de Gennes and Brochard’s treatment of a polymer in a binary mixture of good solvents close to its critical point [14]. Similar to the latter scenario, the polymer is predicted to undergo partial collapse [14] at the cac, which marks the onset of aggregation. This approach allows us to distinguish and explain some common, “universal” features in the vast experimental literature which has accumulated on polymer–surfactant systems. (See, e.g., ref. [12] for a more microscopic approach.) The theory is restricted to the onset of binding (cac) and its vicinity. We assume a dilute polymer regime where many-chain effects can be neglected. The problem of self-assembly and phase behaviour of more concentrated polymer–surfactant systems is very interesting and important [5], but lies outside the scope of the current Letter. Consider a dilute solution of polymer and surfactant whose local concentrations are denoted by c and φ, respectively. The polymer is assumed to be flexible and in a good solvent. The surfactant is both below its cmc and cac. The free energy density can be divided into three terms accounting for the polymer contribution, the surfactant one, and the polymer–surfactant interaction, f(c, φ) = fp(c) + fs(φ) + fps(c, φ). (1) Since the concentrations of both species are low and we are interested only in the onset of binding, the leading quadratic term in the expansion of fps(c, φ) is sufficient, fps(c, φ) = −wcφ, where w ≡ −∂fps/∂c∂φ is a parameter characterizing the attraction strength. In the absence of polymer the surfactant concentration has the value φ = φb which minimizes fs(φ). Let us consider a small perturbation in local surfactant concentration, φ = φb + δφ. Since the solution is both below its cac and cmc, f can be expanded in small δφ to yield f = fp(c) + fs(φb)− wcφb − wcδφ+ 1