A mean-field analytical model is developed for end-grafted polymers in contact with a solution containing surfactants. At concentrations above the critical association concentration, the surfactants are capable of cooperative association in a micellar form with the polymer chains. For single-grafted coils it is found that the coverage of the polymer chain by micelles increases continuously with increasing adsorption strength or bulk surfactant concentration. The size of the coils can increase, decrease, or have a maximum with increasing micellar coverage, depending on the relative strength of the monomer-monomer, monomer-micelle, and micelle-micelle excluded volume interactions. The amount of surfactants adsorbed on densely grafted polymer chains (brushes) is found to increase either continuously or discontinuously with increasing adsorption strength or decreasing grafting density. The character of the transition is determined by the interplay of the adsorption energy and the excluded volume interactions. When the adsorbed amount decreases continuously with increasing grafting density, the brush height increases at small and large grafting densities and has a local maximum in the intermediate range. In the case of an abrupt transition in the adsorbed amount the coexistence of stretched and collapsed chains is predicted in the transition regime. Critical adsorption strengths and grafting densities are discussed. Where possible, the numerical results are compared to experimental data.
Currie, E. P. K., van der Gucht, J., Borisov, O. V., & Cohen Stuart, M. A. (1998). End-grafted polymers with surfactants: a theoretical model. Langmuir, 14, 5740-5750. https://doi.org/10.1021/la980168m