John C. Wheeler

Theory of critical micelle concentration for solutions of block copolymers

A simple microscopic model for micellar formation in mixtures of block copolymers and homopolymers is presented. The size, number of chains, and energy of formation of micelle are calculated. The fraction of copolymer chains aggregating into micelles is computed as a function of the overall copolymer content. A critical micelle concentration behavior is found for low copolymer contents even for weak incompatibilities of species. The similarity with an aborted phase transition is underlined.

’’Exactly soluble’’ two‐component lattice solution with upper and lower critical solution temperatures

A decorated lattice model of a two−component liquid solution is presented which has closed−loop coexistence curves with both upper and lower critical solution temperatures analogous to the behavior found in the nicotine + water and m−toluidine + glycerol systems. The model can be transformed exactly into the spin−1/2 Ising model for which exact results are known in two dimensions and reliable estimates are available in three dimensions. The model exhibits nonclassical critical exponents at both upper and lower critical solution temperatures and has coexistence curves in qualitative agreement with those for real systems. The coexistence curves exhibit characteristic features found in most systems with closed−loop coexistence curves.

Theory of lower critical solution points in aqueous mixtures

Two decorated lattice models of hydrogen bonded mixtures are presented that exhibit lower critical solution temperatures and closed‐loop coexistence curves and that account for the strongly asymmetric closed‐loop coexistence curves found in many aqueous mixtures. The models are extensions of an earlier decorated lattice model that produces only symmetric closed‐loop curves. They incorporate asymmetries in both directional and nondirectional energies as well as the possibility of multiple hydrogen bonding in water. The models are exactly soluble in terms of the spin‐1/2 Ising model, and exhibit nonclassical critical behavior at both upper and lower critical solution temperatures. The hydrogen bond energies in the models are in good agreement with those in real systems, and one of the models gives coexistence curves that are in reasonable quantitative agreement with most of the mixtures considered.

Directionality dependence of lattice models for solutions with closed loop coexistence curves

Two lattice solution models of systems which exhibit lower critical solution temperatures and closed‐loop coexistence curves are generalized to allow for arbitrary directionality in their interactions. It is found that in both models a substantial widening of the curves results from increasing the degree of directionality of the interaction between unlike molecules. One of these, an exactly soluble decorated latice model, produces reasonable quantitative agreement with the coexistence curves for the systems glycerol + guaiacol, glycerol + m‐toluidine, and glycerol + benzylethylamine. Hydrogen bond energies obtained from the generalized decorated lattice model are in reasonable agreement with those found in real systems.

On the density anomaly in sulfur at the polymerization transition

We show that the density anomaly in sulfur near the polymerization transition observed by Borst and co‐workers can be explained, at least in part, as a manifestation of nonclassical critical behavior. We combine the lattice solution model recently proposed to study equilibrium polymerization in sulfur with a phenomenological assumption to produce a dramatic improvement over earlier theories. Some discrepancies between the theory and experiment remain.

Some interesting new phase diagrams in hydrogen‐bonded liquid mixtures

A decorated lattice model of hydrogen‐bonded mixtures that exhibits closed‐loop coexistence curves and lower critical solution temperatures is shown to exhibit a low‐temperature phase separation with an upper critical solution point lying below the lower critical solution point of the closed‐loop curve. The model is exactly soluble in terms of the three‐dimensional spin–1/2 Ising model and therefore exhibits nonclassical critical behavior. The model also exhibits critical double points at which two coexistence curves merge at a common critical point and the critical exponent β is renormalized to 2β. For a special choice of the parameters the model exhibits a new type of phase diagram in the T–X plane in which the shape of the coexistence curve near the critical solution point is described by the exponent 3β. As a result the phase boundaries appear to approach the critical point nearly as straight lines. Coexistence curves are presented exhibiting the variety of behavior possible in the model, including cr...

Critical and tricritical phenomena in ‘‘living’’ polymers

The equilibrium polymerization of ‘‘living polymers’’ such as polytetrahydrofuran and poly‐α‐methylstyrene can be usefully described by the n=0 limit of the n vector model of magnetism. Nonclassical critical behavior is predicted near the ceiling temperature when the initiator concentration is sufficiently small. The small initiator concentration implies a small magnetic field in the n→0 vector model. The agreement of our theory with the limited experimental data available is good, and appears to confirm nonclassical critical behavior. Solutions of living polymers can be described by an annealed dilute n→0 vector model, and interesting new phase diagrams are predicted that include tricritical points analogous to those found in He3–He4 mixtures at very low temperature and in models of sulfur solutions and dilute ferromagnets. The relationship between the phase diagrams in these systems is elucidated.

Critical points and tricritical points in liquid sulfur solutions

Equilibrium polymerization in a solvent can be described by the n→0 limit of a dilute n‐vector model of magnetism in a small magnetic field. In the molecular field approximation the model becomes identical to the earlier theory of Scott for liquid sulfur solutions. The lower critical solution temperature in sulfur solutions is found to be intimately associated with a tricritical point in the magnetic model which accounts for the distinctive shape found by Scott for the high‐temperature phase separation curve. The connection with the magnetic model also establishes a relationship between the phase diagrams predicted by Scott for sulfur solutions and those predicted by Blume, Emery, and Griffiths for He3xnHe4 mixtures. Modern theory of critical and tricritical phenomena suggests that incorporation of nonclassical critical behavior in the dilute n→0 vector model may help to resolve certain discrepancies between Scott’s mean field theory and experimental coexistence curves for sulfur solutions.

Chemical reaction driven phase transitions and critical points

Two simple examples of model (mean‐field) equations of state for phase equilibrium in chemically reactive systems are examined for ‘‘unexpected’’ phase equilibria. They are, essentially, exactly soluble and give classical critical behavior. One of these leads to a lower critical solution temperature that results in closed‐loop coexistence curves similar to those seen in hydrogen‐bonding mixtures. The second leads to less familiar, but interesting phase diagrams that exhibit a phenomenon analogous to critical azeotropy. The same phenomena occur in two examples of lattice gas models the partition functions of which can be mapped exactly to that of the Ising model thus resulting in nonclassical critical behavior. These models demonstrate how a chemical reaction can provide a mechanism leading to interesting phase equilibria and critical phenomena.

Modified moments and continued fraction coefficients for the diatomic linear chain

A simple, stable, and rapid procedure for the calculation of modified moments and continued fraction coefficients for the diatomic linear chain is presented. The Chebyshev modified moments are obtained stably from a simple three‐term recursion relation. Stable determination of the continued fraction coefficients requires a more sophisticated approach, also based on three‐term recursion relations. This procedure also allows stable calculation of the continued fraction coefficients for more complex model densities with singularities within the bands. Examples are given and the corresponding behavior of the continued fraction coefficients is discussed.