Mechanistic Origin of Compressed Exponential Time Dependence in Block Copolymer Micelle Fragmentation

Abstract Multiple studies of block copolymer micelle fragmentation have revealed a surprising compressed exponential time dependence of relaxation toward the equilibrium size, following exp[(–t/τ)n], with n ≈ 2. We present a simple two-step kinetic model that quantitatively explains the mechanistic origin of the apparent compressed exponential behavior. In this model, as-prepared micelles evolve toward steady-state micelles via metastable, “peanut-shaped” intermediates. Concentrations of the three states are obtained as a function of time by coupling simple rate laws. The initial and final sizes are known from experiments. The size of the intermediates is estimated by modelling them as prolate ellipsoids, leading to a radius about 2% larger than that of the as-prepared micelles. The two rate constants, k1 and k2, for the formation and decomposition of the intermediate state, respectively, are estimated by fitting experimental data, while providing the sizes of the three states as input. Fitting reveals that k1 and k2 need to be very similar (within a factor of about 2) to observe the apparent compressed exponential dynamics. Equivalent magnitudes of k1 and k2 suggest that the intermediate state is metastable with a finite lifetime and not a transition state, an inference that is supported by the appearance of the peanut-shaped intermediates in previously conducted in-situ TEM experiments. Calculations also indicate that the relative population of the intermediate state reaches a peak value of 20–40%, which should facilitate further imaging of intermediates via TEM. We are not aware of a current theory for block copolymer micelle fragmentation that proposes passage through a metastable intermediate, so this question certainly merits further examination.