| Molecules with directional interactions, such as hydrogen bonds, and molecules with combinations of different types of segments, such as copolymers, can have unusual and useful properties. Predicting the behavior of such systems is difficult; however, lattice theory can be used to provide insights into the behavior of such molecules, and can be used to model their behavior with reasonable accuracy. Aranovich and Donohue recently developed a lattice density functional theory for two-dimensional and three-dimensional mixtures. The resulting equations, while quite accurate and general, could not be integrated analytically to give expressions for the free energy. In this work, a new expression for the free energy has been derived from an internal energy model that is both accurate and integrable. Additionally, this theory is expressed in a general form such that each side of each type of component may have different interaction energies.; As many copolymers undergo a microphase separation a free energy model of this system must account for the fundamental differences between the segregation of copolymer segment types and the segregation of homopolymer chains. The entropic contribution to the free energy of a microphase separation is analogous to the entropic contribution to the free energy of de-mixing. In this work, a rederivation of the configurational degeneracy of Guggenheim's Random Mixing theory is used to calculate the configurational entropy of compressible multi-component mixtures containing copolymers. The microphase separation temperature of a diblock copolymer melt is compared to the liquid de-mixing temperature of an analogous homopolymer blend. This random mixing entropy model can be used as the infinite temperature limit in the derivation of a nonrandom mixing expression for the change in free energy upon the microphase separation of a copolymer. Monte Carlo simulation results of an equimolar AA and BB dimer blend, and an AB dimer melt demonstrate the accuracy of the lattice density functional theory and differences between the phase behavior of an equimolar AA and BB dimer blend, and an AB dimer melt. An extension of this approach for dimers to homopolymers and copolymers demonstrates the accuracy of the model and the limitations inherent in the underlying assumptions. |
