Spontaneous Formation of Complex Micelles from a Homogeneous Solution

We present an extensive computer simulation study of structure formation in amphiphilic block copolymer solutions after a quench from a homogeneous state. By using a mesoscopic field-based simulation method, we are able to access time scales in the range of a second. A “phase diagram” of final structures is mapped out as a function of the concentration and solvent philicity of the copolymers. A rich spectrum of structures is observed, ranging from spherical and rodlike micelles and vesicles to toroidal and net-cage micelles. The dynamical pathways leading to these structures are analyzed in detail, and possible ways to control the structures are discussed briefly.

Figure 1

Pathways of spontaneous vesicle formation in copolymer solutions. (a) PC: Micelle coalescence (a1),(a2), bilayer formation (a3), and bending (a4); (b) PG: micelle growth (b1),(b2), internal reorganization to semivesicle (b3), and swelling into vesicle (b4). The parameters of the simulations are ${\chi }_{BS}=0.128$ and ${\Phi }_p=0.2$, (a) ${\Phi }_P=0.15$ (b). Times $t$ are given in units of $T={10}^4{\tau }_0$. Here and throughout the Letter, structure snapshots show isodensity surfaces of $A$ blocks at ${\Phi }_A=0.625$.

Figure 2

“Phase diagram” of final structures after a sudden quench from an initially homogeneous copolymer solution, for a range of solvent-philicities ${\chi }_{BS}$ and copolymer volume fractions ${\Phi }_P$. The final structures in the regions V1/V2 correspond to vesicles (f), RS1/RS2 to mixtures of rod and sphere micelles (d),(e), T1 to ring micelles (c), and T2 to toroidal micelles (a),(b). In the region R0, the solution stayed homogeneous. The dotted lines are guides for the eye. The dashed line separates two dynamical regimes where the structure formation proceeds along different pathways: Micelle coalescence (pathway PC) in the regions RS1 and V1, micelle growth (pathway PG) in the regions RS2, V2, T1, and T2. The solid line shows the function ${\chi }^{*}=1.3-11.{\Phi }_P$ (see text and Fig. 4 for explanation).

Figure 3

Formation of toroidal structures at ${\chi }_{BS}=-0.512$: (a) rings (${\Phi }_P=0.2$), (b) toroidal micelles (${\Phi }_P=0.18$), (c) cage micelles (${\Phi }_P=0.17$). Times $t$ are given in units of $T={10}^4{\tau }_0$.

Figure 4

Inverse incubation time $1/{\tau }^{*}$ as a function of ${\chi }_{BS}$ for different copolymer volume fractions ${\chi }^{*}$ as indicated. The values ${\chi }_{BS}$ have been shifted by ${\chi }^{*}=1.3–11{\Phi }_P$. The solid line corresponds to the function $y=5.3\times {10}^{-6}{x}^{1.25}$, and the dashed line shows $y\propto x^{-1}$ for comparison. Inset: Evolution of the order parameter $\sigma$ with time at ${\chi }_{BS}=0$, ${\Phi }_P=0.17$. The incubation time is defined as the time where the tangent of $\sigma (t)$ at the inflection point and the base line $\sigma \equiv 0$ intersect each other.