Theory of block copolymer micelles in solution

Abstract

A simple model of AB diblock copolymer micelles in solution is presented. A spherical shape is assumed, with a completely uniform inner core formed from the insoluble B blocks and a uniform outer shell composed of the soluble A blocks. The interaction parameters, as well as the molecular weight, composition, and overall volume fraction of the copolymers, are assumed to be given. All energetic and entropic contributions to the free energy can be written down simply, with the exception of the interfacial tension y of the asymmetric interphase. An approximation for y is developed, and the free energy is minimized to obtain the equilibrium size of the micelle. Good agreement with the small-angle X-ray scattering data on the polystyrene/ polybutadiene / n-heptane system is obtained. Numerical results, as well as scaling arguments, indicate that the size of the micelle, even for different block copolymer compositions, is characterized by power law functions of the total molecular weight of the block copolymer.

Towards a structural characterization of charge-driven polymer micelles

“…5c, d). The observed increase in R c upon increasing N corona is in contradiction with all known micellization theories [15][16][17][18][19][20], and likely to result from the fact that R = R c , but in fact contains contributions from both core and corona scattering, with the latter contribution becoming increasingly important as N corona increases. To verify this hypothesis unambiguously is a hard task for the present system (even if contrast-variation techniques would be applied), wherein both core and corona suffer from low contrast in microscopy and scattering experiments due to the high solvent volume fraction.…”

Hydrodynamic studies on micellar solutions of styrene–butadiene block copolymers in selective solvents

“…Usually, the micelle size and aggregation number for the same block copolymer in n-heptane was higher compared to those in DMF. We could not observe very high micgle size and aggregation number for TX 006 (% PS 48, M,, total 375 000) in heptane as observed by Stacy and Kraus (5) and as predicted from the theoretical results of Noolandi and Hong (24). We have checked this point and results from solution of copolymers prepared at different times gave the same value.…”

“…The micelle formation in selective solvents (pure and mixed solvents) for each block has been studied using several techniques viz. light scattering (4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14), viscosity (4,7,9,(13)(14)(15), osmometry (12,15,16), ultramicroscopy (1 9), electron microscopy (19)(20)(21), gel permeation chromatography (20), nmr (22,23), small angle X-ray scattering (17,18,24,25), photon correlation spectroscopy (19,26), and sedimentation measurements (10,15).…”

Formation of Block Copolymer Micelles in Solution: A Monte Carlo Study of Chain Length Dependence

“…Since at equilibrium the free energy per molecule of a phase is a minimum, n j is found from From eq 2.10 one then finds the critical micelle concentration So far, we just have recalled the standard treatment of micellization for any amphiphilic molecules; the aspects specific for block copolymers enter in the treatments of the chemical potential in eq 2.12. The quantity µ°n/n -µ°1 represents the change in the reference state free energy when a single block copolymer molecule in solution is transferred into a micelle of aggregation number n. The various theories 9,11,14,18,19,[23][24][25][26][27] differ in their assumptions about this expression, however, although they all deal with micelles of spherical shape ( Figure 1). Here we assume that in the micelles the A-monomers in the core are fully segregated from the B-monomers in the corona, and follow de Gennes 14 assuming that the A-blocks are stretched with an endto-end distance equal to the core radius R (Figure 1).…”

“…[1][2][3][4][5][6][7] At the same time, the theory of micelle formation within the framework of statistical thermodynamics is a challenging and longstanding problem which still is not fully understood. Particularly interesting in this context are block copolymers 9,11,14,18,19,[23][24][25][26][27][28][29] A f B 1-f in solution, because both the composition f ) N A /N and the chain length N()N A + N B ) of the total macromolecule can be varied over a wide range, 28,[31][32][33][34][35][36] without affecting intermolecular forces significantly. Thus, choosing f and N as control parameters in experiments, one would perform a stringent test of the theories which make interesting predictions how the linear dimensions of the micelles and the number of chains per micelle depend on these parameters.…”