Application of computer simulation free-energy methods to compute the free energy of micellization as a function of micelle composition. 1. Theory

The widespread use of surfactant mixtures and surfactant/solubilizate mixtures in practical applications motivates the development of predictive theoretical approaches to improve fundamental understanding of the behavior of these complex self-assembling systems and to facilitate the design and optimization of new surfactant and surfactant/solubilizate mixtures. This paper is the first of two articles introducing a new computer simulation-free-energy/molecular thermodynamic (CS-FE/MT) model. The two articles explore the application of computer simulation free-energy methods to quantify the thermodynamics associated with mixed surfactant/cosurfactant and surfactant/solubilizate micelle formation in aqueous solution. In this paper (article 1 of the series), a theoretical approach is introduced to use computer simulation free-energy methods to compute the free-energy change associated with changing micelle composition (referred to as DeltaDeltaGi). In this approach, experimental critical micelle concentration (CMC) data, or a molecular thermodynamic model of micelle formation, is first used to evaluate the free energy associated with single (pure) surfactant micelle formation, g(form,single), in which the single surfactant micelle contains only surfactant A molecules. An iterative approach is proposed to combine the estimated value of gform,single with free-energy estimates of DeltaDeltaGi based on computer simulation to determine the optimal free energy of mixed micelle formation, the optimal micelle aggregation number and composition, and the optimal bulk solution composition. After introducing the CS-FE/MT modeling framework, a variety of free-energy methods are briefly reviewed, and the selection of the thermodynamic integration free-energy method is justified and selected to implement the CS-FE/MT model. An alchemical free-energy pathway is proposed to allow evaluation of the free-energy change associated with exchanging a surfactant A molecule with a surfactant/solubilizate B molecule through thermodynamic integration. In article 2 of this series, the implementation of the CS-FE/MT model to make DeltaDeltaGi free-energy predictions for several surfactant/solubilizate systems is discussed, and the predictions of the CS-FE/MT model are compared with the DeltaDeltaGi predictions of a molecular thermodynamic model fitted to relevant experimental data.