Modeling size-controlled assembly of polymeric nanoparticles in interdigital micromixers

Abstract

We apply mean field continuum theories to model the assembly of particles in the co-solvent method, to which we refer as size-controlled assembly, with the objective to explain nanoparticle size dependencies on solvent mixing speeds.

Our investigation starts at considering a Cahn-Hilliard equation with a Flory-Huggins-de Gennes free energy functional restricted to homopolymers. Upon modeling solvent mixing by a time dependent interaction parameter, structure formation during spinodal decomposition is analyzed. The qualitative agreement of our simulated data to both recently published Molecular Dynamics simulations and experiments indicates that size-controlled assembly can, on principle, be described by relaxation dynamics within a mean field approximation, and suggests a response of molecular organization to solvent mixing in the very early stages of phase separation to eventually determine final particle sizes. In contrast to Molecular Dynamics simulations, the Cahn-Hilliard model is able to simulate realistic mixing times and enables a perturbation approximation. The perturbation approximation does not only give an analytical interpretation to the underlying physical mechanism of size-control as a competition between molecular repulsion and interfacial tension of diffuse interfaces, but also yields a general theoretical scaling behavior that is reflected in experiments and Molecular Dynamics simulations. After introducing the notion of effective two-component models, we combine the computational efficiency of models based on time dependent interaction parameters with a more realistic description of solvent mixing by relative chemical potentials of solvents. This novel description is then shown to agree with incompressible three-component dynamics in dilute solutions that correspond to experimental conditions.

Size-controlled assembly of amphiphilic diblock-copolymers is studied by inserting time dependent interaction parameters into an External Potential Dynamics model with a free energy functional from the Self Consistent Field Theory. A satisfactory analysis of particle size distributions requires the development of a new numerical integration scheme to deal with stiffness instabilities at high compressive moduli, which accelerates simulations by a factor of up to 100. Subsequent simulations indicate that neither the fundamental qualitative characteristics of particle size dependencies on mixing speeds nor the physical mechanism behind the size-control are significantly affected by copolymer architecture. Experimentally observed transitions of particle morphologies are also reproduced qualitatively. To conclude, an effective two-component model with a revised description of solvent mixing for copolymers is proposed. Based on the findings in the present work, we consider it a suitable starting point for quantitative studies of size-controlled copolymer assembly.