How ill-defined constituents produce well-defined nanoparticles: Effect of polymer dispersity on the uniformity of copolymeric micelles

We investigate the effect of polymer length dispersity on the properties of self-assembled micelles in solution by self-consistent field calculations. Polydispersity stabilizes micelles by raising the free energy barriers of micelle formation and dissolution. Most importantly, it significantly reduces the size fluctuations of micelles: Block copolymers of moderate polydispersity form more uniform particles than their monodisperse counterparts. We attribute this to the fact that the packing of the solvophobic monomers in the core can be optimized if the constituent polymers have different length.

Figure 1

Left: Spatial density profiles of solvophobic A segments (full lines) and solvophilic B segments (dashed lines) in micelles made of A:B copolymers with $f=0.5$, $F_m/k_{B}T\approx 0$ for different bulk polydispersity indices Đ as indicated. Inset shows the corresponding shift of the polymer volume fraction ${\overline{\mathrm{\Phi }}}_p$ in the bulk. Right: Chain length distribution of copolymers in the bulk (dashed) and in the micelle (solid) for different values of Đ. A constant offset has been added for better visibility.

Figure 2

Left: Micelle free energy as a function of micelle radius for the same systems as in Fig. 1. Symbols show SCF results, solid lines a fit to the polynomial ${\sum }_{n=1}^4{a}_n{x}^n$. Right: Corresponding micelle size dispersity ${\sigma }_m$ as a function of copolymer polydispersity index Đ for different values of the scaling parameter $\overline{C}$. Cartoons illustrate the proposed stabilizing mechanism: Polydisperse solvophobic blocks can pack more efficiently in the core of the micelle.

Figure 3

(a) Density profiles of A and B segments (full/dashed lines) for micelles with micelle free energy $\mathrm{\Delta }{F}_m=0$ in systems where only the solvophobic (left) or solvophilic (right) chain block is polydisperse. The inset shows the corresponding bulk polymer volume fractions ${\overline{\mathrm{\Phi }}}_P$. (b) Height $\mathrm{\Delta }{F}_m$ of the free energy barrier for micelle dissolution. (c) Width ${\sigma }_m$ of micelle size distribution at $\overline{C}=4$ vs Đ for all systems considered in the present work. Here Đ refers to the dispersity in the corresponding polydisperse part of the chain (A or B or total).

Figure 4

Contributions to the micelle free energies from the interaction energy (black circles), the polymer entropy (red squares), the chemical work associated with the polymers (blue diamonds), and the sum of solvent entropy and solvent chemical work (green triangles) for the systems considered in this work. Full lines/full symbols refer to most probable micelles; dashed lines/open symbols to critical micelles. Inset shows the difference between the internal energy per polymer segment in the micelle system and in the bulk (in arbitrary units).