Free Energy Change of Micelle Formation for Sodium Dodecyl Sulfate from a Dispersed State in Solution to Complete Micelles along Its Aggregation Pathways Evaluated by Chemical Species Model Combined with Molecular Dynamics Calculations

Surfactant molecules, when dispersed in solution, have been shown to spontaneously form aggregates. Our previous studies on molecular dynamics (MD) calculations have shown that ionic sodium dodecyl sulfate molecules quickly aggregated even when the aggregation number is small. The aggregation rate, however, decreased for larger aggregation numbers. In addition, studies have shown that micelle formation was not completed even after a 100 ns-long MD run (Chem. Phys. Lett. 2016, 646, 36). Herein, we analyze the free energy change of micelle formation based on chemical species model combined with molecular dynamics calculations. First, the free energy landscape of the aggregation, ΔG_i+j^†, where two aggregates with sizes i and j associate to form the (i + j)-mer, was investigated using the free energy of micelle formation of the i-mer, G_i^†, which was obtained through MD calculations. The calculated ΔG_i+j^† was negative for all the aggregations where the sum of DS ions in the two aggregates was 60 or less. From the viewpoint of chemical equilibrium, aggregation to the stable micelle is desired. Further, the free energy profile along possible aggregation pathways was investigated, starting from small aggregates and ending with the complete thermodynamically stable micelles in solution. The free energy profiles, G(l, k), of the aggregates at l-th aggregation path and k-th state were evaluated by the formation free energy $\sum\limits_i {{n_i}\left( {l, k} \right)G_i^\dagger } $ and the free energy of mixing $\sum\limits_i {{n_i}(l, k){k_B}Tln({n_i}(l, k)/n(l, k))} $, where n_i(l, k) is the number of i-mer in the system at the l-th aggregation path and k-th state, with $n\left( {l, k} \right) = \sum\limits_i {{n_i}\left( {l, k} \right)} $. All the aggregation pathways were obtained from the initial state of 12 pentamers to the stable micelle with i = 60. All the calculated G(l, k) values monotonically decreased with increasing k. This indicates that there are no free energy barriers along the pathways. Hence, the slowdown is not due to the thermodynamic stability of the aggregates, but rather the kinetics that inhibit the association of the fragments. The time required for a collision between aggregates, one of the kinetic factors, was evaluated using the fast passage time, t_FPT. The calculated t_FPT was about 20 ns for the aggregates with N = 31. Therefore, if aggregation is a diffusion-controlled process, it should be completed within the 100 ns-simulation. However, aggregation does not occur due to the free energy barrier between the aggregates, that is, the repulsive force acting on them. This may be caused by electrostatic repulsions produced by the overlap of the electric double layers, which are formed by the negative charge of the hydrophilic groups and counter sodium ions on the surface of the aggregates.