P. D. Gujrati

Universal equation of state for an interacting multicomponent mixture of polymers

We present a closed form universal equation of state for an interacting multicomponent mixture of polymers of any architecture and dispersity. The equation is obtained by solving the model on a Bethe lattice and goes beyond the random mixing approximation. The latter property endows our theory with features that are consistent with real systems. The equation of state, though an approximate one, is thermodynamically consistent and is valid even in the incompressible limit. The predictions of the equation are consistent with simulations and experiments, as discussed.

A BINARY MIXTURE OF MONODISPERSE POLYMERS OF FIXED ARCHITECTURES, AND THE CRITICAL AND THE THETA STATES

We study the complete phase diagram for a model of a binary mixture of two interacting polymer species A and A′, each of fixed architecture (dendrimer, star, linear, or regularly branched polymer, brush, etc.) and size given by the number M (or M′) of monomers in it, on a lattice of coordination number q. For M′=1, the model describes a solution. Branchings, if any, are regular in these architectures. This feature alone makes these polymers different from polymers with random branchings studied in the preceding paper [J. Chem. Phys. 108, 5089 (1998)]. There exists a theta point regardless of the fixed architecture, which is not the case for random branchings. We identify this point as a tricritical point T at which one of the two sizes M and M′ diverges. Two critical lines C and C′ meet at T. The criticality along C corresponds to the criticality of an infinitely large polymer of any fixed architecture, not necessarily linear. This polymer is a fractal object. We identify the relevant order parameter and ...

LATTICE THEORY OF A MULTICOMPONENT MIXTURE OF MONODISPERSE POLYMERS OF FIXED ARCHITECTURES

We present a lattice theory for a multicomponent mixture of p distinct polymeric species, each of a prescribed architecture but without any cycles and s monomeric species along with a solvent species, the latter playing the role of a reference species whose amount is controlled not by any activity but by the sum rule of fixed amount of material. The theory is an extension of our previous work on a binary mixture of polymers in bulk or a general mixture next to a surface. The model allows for nearest-neighbor interactions between unlike species. The chemical bondings are allowed to be between monomers (of the same species) that are nearest-neighbor. The resulting theory is obtained by solving the model on a Bethe lattice. The theory has a very simple structure and supersedes random mixing approximation to which it reduces in a special limit, the random mixing approximation limit, see text. We study the behavior of a ternary system numerically and compare it with that of a binary system. We also compare the...

Polydisperse solution of randomly branched homopolymers, inversion symmetry and critical and theta states

We discuss the phase behavior of a model of a binary mixture of randomly branched homopolymers in a solution. The monomer–solvent interaction is determined by a Boltzmann weight w. The theory has been presented recently and is obtained by approximating the underlying lattice by a Bethe lattice of the same coordination number q. Of special interest is the class of randomly branched polymers with inversion symmetry (see the text). This class includes linear polymers. The phase diagram for the special class of polymers is very simple. There is a line C of critical points in the dilute limit on which branched polymers become a critical object in a good solvent. This is an extension of the result due to de Gennes for linear chains in an athermal solution to the above class of branched polymers in any good solvent. The line C meets with another critical line C′ for phase separation in a poor solvent. We identify the theta point as a tricritical point as first suggested by de Gennes for linear chains only. The t...

New statistical mechanical treatment of systems near surfaces. I. Theory and principles

We present a new theoretical framework for a statistical mechanical and thermodynamic description of any general inhomogeneous system (not necessarily polymeric) in the presence of surfaces. The framework is an extension of a lattice theory recently developed for a homogeneous system and requires approximating the original lattice by a recursive lattice which, for simplicity, we take to be a modified tree structure (see Fig. 4), TM as described in the text. The tree is formed recursively by two basic elements, the main tree T and the surface tree T. The model is solved exactly using a recursion technique. The technique allows us to account for connectivity, architecture, excluded-volume effects, interactions, etc. exactly. The resulting description goes beyond the random-mixing approximation used in most mean-field theories. We consider a general model of a multicomponent system and its exact solution on the modified tree TM provides us with an approximate theory of the inhomogeneous system on the origin...

New statistical mechanical treatment of systems near surfaces. II. Polydisperse linear and branched polymers in an athermal solution

We apply a recently developed analytic but approximate method to study the behavior of polydisperse linear and branched polymers in an athermal solution and near various kinds of surfaces. We consider equilibrium polydispersity controlled by a set of activities. The method allows us to account for polymer connectivity and excluded-volume effects and goes beyond the random mixing approximation. The density profiles of various kinds exhibit oscillations for bulk densities φmb larger than some threshold bulk density φmT. The origin of these oscillations is related to the decreasing branch of the recursion function, as explained in the text. The correlation length ξ related to these oscillations increases as φmb increases. On the other hand, the correlation length ξ controlling the approach of various density profiles to their respective bulk values in the range φmb<φmT increases as φmb decreases. The free energy and the entropy are uniquely determined. Various surface properties are also easily determined. W...

Lattice theory of polymer solutions with endgroup effects

We present a unified lattice theory for a binary solution where endgroups are treated differently from middle groups. This is a simple example of a triblock and the present study provides a starting point for studying a general triblock system. We replace the original homogeneous lattice by a Bethe lattice of the same coordination number as the original lattice. The model is solved exactly on the Bethe lattice. The resulting solution goes beyond the random mixing approximation and provides us with an approximate theory of the model on the regular lattice. The contributions of endgroups on various thermodynamic properties of a binary solution are investigated in a quantitative way using the theory. In particular, our theory predicts that contributions to the energy are more important than to the entropy.

New statistical mechanical treatment of systems near surfaces. IV. Surface and surface-induced capillary transitions in polymer solutions

We apply a recently developed analytic but approximate method to study surface and surface-induced capillary transitions in a solution of polydisperse linear and/or branched polymers confined between two infinite but identical surfaces. We use an equilibrium polymerization model where various densities in the system are controlled by the corresponding activities. The bulk region, i.e., the central region between the surfaces may or may not be in the bulk equilibrium state (see the text). We find a line of first-order transitions, commonly known as a prewetting transition line, passing through the phase separation point in the bulk equilibrium state, at which the surface undergoes a first-order transition. For a certain range around the bulk equilibrium transition point, the bulk region undergoes a stable-metastable transition. This presents us with the possibility of being able to prepare the system in a bulk metastable state, no matter how far apart the two surfaces are. This range is found to be identic...

Size disparity and lower critical solution temperature: A critical investigation of free-volume disparity

By critically examining a simple model system of equilibrium polymerization that is athermal in the traditional sense, we demonstrate that many of the consequences of the free-volume disparity induced by size disparity are inconsistent with its current understanding. Despite its traditional name, the model is not truly athermal because of the compressibility. The resulting energetics endows the model system with a very rich and complex behavior. The analytical results that are obtained in a mean-field approximation show how and when an upper critical solution temperature, a lower critical solution temperature (LCST) and an immiscibility loop may occur. We suggest that it is the difference in the thermal volume-expansion coefficients rather than the difference in free volumes of the coexisting phases (and not of the components) that plays a central role in determining the phenomenon of LCST and may be used to provide for its quantitative characterization. Too much or too little of free volume disfavors LCS...

MULTICOMPONENT SYSTEM WITH POLYDISPERSE SPECIES

We consider a multicomponent system containing polydisperse species of polymers produced in equilibrium polymerization. We present a general theory of such a system by solving the model on a Bethe lattice. We show that the resulting free energy has certain universal features regardless of the number of components, their architecture, and dispersity.