Entropic effects, shape, and size of mixed micelles formed by copolymers with complex architectures

The entropic effects in the comicellization behavior of amphiphilic $AB$ copolymers differing in the chain size of solvophilic $A$ parts were studied by means of molecular dynamics simulations. In particular, mixtures of miktoarm star copolymers differing in the molecular weight of solvophilic arms were investigated. We found that the critical micelle concentration values show a positive deviation from the analytical predictions of the molecular theory of comicellization for chemically identical copolymers. This can be attributed to the effective interactions between copolymers originated from the arm size asymmetry. The effective interactions induce a very small decrease in the aggregation number of preferential micelles triggering the nonrandom mixing between the solvophilic moieties in the corona. Additionally, in order to specify how the chain architecture affects the size distribution and the shape of mixed micelles we studied star-shaped, H-shaped, and homo-linked-rings–linear mixtures. In the first case the individual constituents form micelles with preferential and wide aggregation numbers and in the latter case the individual constituents form wormlike and spherical micelles.

Figure 1

Cartoon representation of (a) miktoarm star copolymer ${\left(A_{16}\right)}_4{B}_{30}$, (b) H-shaped copolymer (${\left(A_{30}\right)}_2{B}_{30}{\left(A_{30}\right)}_2$, (c) linear diblock copolymers $A_{15}{B}_{30}$, and (d) homo-linked-rings copolymer $A_{15}{B}_{30}$.

Figure 2

Tracer autocorrelation function $C\left(t\right)$ for different copolymer mixtures.

Figure 3

Plots of the total free copolymer concentration against the total copolymer concentration $\left[X\right]$ for the simulated binary mixtures of (a) star ${\left(A_{16}\right)}_4{B}_{30}$–star ${\left(A_8\right)}_4{B}_{30}$, (b) star ${\left(A_{60}\right)}_2{B}_{30}$–H-shaped ${\left(A_{30}\right)}_2{B}_{30}{\left(A_{30}\right)}_2$, and (c) linear $A_{16}{B}_{30}$–homo-linked-rings $A_{16}{B}_{30}$ at different molar fractions of two types of amphiphilic copolymers.

Figure 4

Plots of the cmc values of (a) star ${\left(A_{16}\right)}_4{B}_{30}$–star ${\left(A_8\right)}_4{B}_{30}$ mixtures with respect to molar fraction of star copolymer ${\left(A_{16}\right)}_4{B}_{30}$, (b) star ${\left(A_{60}\right)}_2{B}_{30}$–H-shaped ${\left(A_{30}\right)}_2{B}_{30}{\left(A_{30}\right)}_2$ mixture, (c) linear $A_{16}{B}_{30}$–homo-linked-rings $A_{16}{B}_{30}$. Within the same figure the analytical results obtained from Eq. (9) with unity activity coefficients are also shown.

Figure 5

Mass distributions of micelles in the mixtures (a) star ${\left(A_{16}\right)}_4{B}_{30}$–star ${\left(A_8\right)}_4{B}_{30}$, (b) star ${\left(A_{60}\right)}_2{B}_{30}$–H-shaped ${\left(A_{30}\right)}_2{B}_{30}{\left(A_{30}\right)}_2$ mixture, (c) linear $A_{16}{B}_{30}$–homo-linked-rings $A_{16}{B}_{30}$ as a function of the aggregation number $N_p$.

Figure 6

The most probable aggregation numbers of micelles $N_p$for star ${\left(A_{16}\right)}_4{B}_{30}$–star ${\left(A_8\right)}_4{B}_{30}$ mixtures.

Figure 7

Radial distribution functions of micelles with the most probable aggregation number for star ${\left(A_{16}\right)}_4{B}_{30}$–star ${\left(A_8\right)}_4{B}_{30}$ copolymer mixtures.

Figure 8

Snapshots of micelles: (a) $\left[X_{\left(A16\right)4B30}\right]=0.5$ with $N_p=12$, (b) $\left[X_{\left(A60\right)2B30}\right]=0.5$ with $N_p=13$, (c) $\left[X_{\mathrm{linked}-\mathrm{rings}}\right]=0.8$ with $N_p=24$, (d) $\left[X_{\mathrm{linked}-\mathrm{rings}}\right]=0.5$ with $N_p=36$, (e) $\left[X_{\mathrm{linked}-\mathrm{rings}}\right]=1$ with $N=36$, (f) $\left[X_{\mathrm{linked}-\mathrm{rings}}\right]=0.5$ with $N=50$.