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.
A statistical thermodynamic theory is used to derive the mean-field equations for the fundamental probability distribution functions characterizing a system of two immiscible homopolymers (A and B), diluted with solvent, in the presence of a diblock copolymer (AB). The equations are solved numerically in a "computer experiment" and the various contributions to the free energy and interfacial tension are evaluated to determine their relative importance. For a symmetric diblock copolymer and homopolymers of infinite molecular weight (as well as a symmetric solvent), we find that the reduction in interfacial tension, 7, with increasing copolymer molecular weight and concentration arises mainly from the energetically preferred orientation of the blocks at the interface into their respective compatible homopolymers. The main counterbalancing term in the expression for 7 is the entropy loss of the copolymer which localizes at the interface. The loss of conformational or "turning-back" entropy of both copolymers and homopolymers at the interface is shown to contribute little to Ay. Keeping only the two main terms in Ay, we find an exponential dependence on the parameter (: , where Zc is the degree of polymerization of the symmetric copolymer, is the interaction parameter between A and B, and ? is the volume fraction of homopolymer.
A theory for inhomogeneous multicomponent polymer systems developed earlier by the authors is simplified for the case where the inhomogeneity is weak, and this theory is used to study the phase diagrams of a mixture of block copolymers, homopolymers, and solvents. The free energy of the system is expressed as a functional of the deviations in the concentration profiles from their homogeneous values, and a perturbation expansion up to fourth order in the fluctuations is carried out. For a mixture of a block copolymer and a nonselective solvent a simple expression is derived for the inhomogeneous free energy term, and some typical phase diagrams are calculated. The existence of eutectic points, similar to those studied in metallurgy, is demonstrated. Phase diagrams for block copolymer-homopolymer systems are also discussed, and homopolymer-induced mesophase formation is predicted. The periodicity of the lamellar structure with varying homopolymer concentration is calculated, and the density profiles near demixing are shown.
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