Theory of inhomogeneous multicomponent polymer systems

Abstract

A new theory of inhomogeneous multicomponent systems is developed, starting from the functional integral representation of the partition function. The theory can be used to determine the interfacial properties and microdomain structures of a combination of homopolymers, block copolymers, monomers, and solvents.For the case of no polymers present, our theory reduces to that of Cahn and Hilliard for small molecules.In our approach, the general free energy functional is optimized by the saddle-function method, subject to constraints of no volume change with mixing and constancy of the individual particle numbers, to obtain new equations for the mean fields acting on the polymers. We are thus able to restrict density fluctuations in the system, without introducing an "ad hoc" term involving a vanishing compressibility into the free energy. In addition, by using the gradient expansion for the interaction energy, we demonstrate explicitly the absence of all "kinetic" terms involving the (inverse) degree of polymerization from the mean-field equations, when we choose to introduce the local, homogeneous Flory-Huggins free energy for reference. The mathematical formalism is worked out in detail for the multicomponent homopolymer-solvent system, and it is shown how to extend the theory to include block copolymers. Finally, in order to give a definite example, the variation of the interfacial properties with molecular weight and interaction parameter is worked out numerically for the simplest case of a homopolymer-solvent system. The numerical method is discussed in detail. Applications to more complicated systems will be given in the future.

Spatially Localized Structures in Diblock Copolymer Mixtures

“…Another approach which has become widely utilized is the self-consistent (or mean-) field approximation of the Gaussian chain ensemble with Flory-type interaction [17,19,23,25]. In this theory, there are two nondimensional parameters: the volume fraction of A monomers f , and the interaction strength χN , which inversely corresponds to temperature.…”

The structure and phase transitions in polymer blends, diblock copolymers and liquid crystalline polymers: The Landau‐Ginzburg approach

“…Again, this can be expressed in a scaling form 27) with p fixed. Therefore, the presence of the Lifshitz tricritical behavior makes itself known by the fact that, in a fit of the structure function at long wavelengths, terms of order q 4 cannot be neglected with respect to terms of order q 2 .…”

“…A Landau-Ginzburg model of the phase transitions in diblock copolymer system was first formulated by Leibler 22) and later refined by Fredrickson et al 20,23) . Matsen and Schick 24,25) applied the self consistent field theory 26,27) to the phase transitions in this system in the weak segregation limit 28) , where the domain boundaries between the A rich and B rich domains are wide. The opposite case is the strong segregation regime with very sharp and narrow domain boundaries; here we consider only the former case.…”

Fundamentals of Phase Separation in Polymer Blend Thin Films

“…We see, from Eq. (3.27), that the constraint is enforced by λ acting as a constant chemical potential across the film [48,49] (the functional derivative δF [φ(z)] /δφ is proportional to the chemical potential. This is discussed in depth in section 4.3).…”