, a critical point phenomenon that is relatively rare among low molecular weight solutions. Soon after the discovery of the universality of LCST behavior in polymer solutions, Flory and c o -~o r k e r s~-~ developed a new theory of solutions which incorporated the "equation of state" properties of the pure components. This new theory of solutions, hereafter referred to as the Flory theory, demonstrated that mixture thermodynamic properties depend on the thermodynamic properties of the pure components and that LCST behavior is related to the dissimilarity of the equation of state of properties of polymer and solvent. P a t t e r~o d -~ has also shown that LCST behavior is associated with differences in polymer/solvent properties by using the general corresponding states theory of Prigogine and collaborators.1° Classical polymer solution theory, i.e., Flory-Huggins theory," which ignores the equation of state properties of the pure components, completely fails to describe the LCST behavior.More recently, a new equation of state theory of pure and their solutions14 has been formulated by the present authors. This theory has been characterized as an Ising or lattice fluid theory (hereafter referred to as the lattice fluid (LF) theory). Both the Flory and LF theories require three equation of state parameters for each pure component. For mixtures, both reduce to the FloryHuggins theory'l at very low temperatures.Our general objective in the present paper is to survey the applicability of LF theory to polymer solutions. Pure Lattice Fluid PropertiesAs its name suggests, LF theory is founded on a lattice model description of a fluid. An example of such a system is shown in (4)and t* is the interaction per mer and L'* is the close-packed mer volume.At equilibrium the chemical potential is at a minimum and satisfies the following equation of state:In general there are three solutions to the equation of state. The solutions a t the lowest and highest values of 2, yield minimum values in the chemical potential eq 1, while the intermediate value of p produces a maximum in the free energy. The high-density minimum (few vacant sites) corresponds to a liquid phase while the low-density minimum corresponds to a gas or vapor phase (most sites are empty). Typically near the triple point, reduced liquid densities are between 0.7 and 0.9 and gas densities between 0.001 and 0.005. At a given pressure there will be a unique temperature at which the two minima are equal. This temperature and pressure are the s a t u r a t i o n temperature and pressure and the locus of all such T,P points defines the saturation or coexistence line where liquid and vapor are in equilibrium.As the saturation temperature and pressure increase, the difference in densities between liquid and vapor phase diminishes until a temperature and pressure are reached where the densities of the two phases are equal. This unique point in the T,P plane is the liquid-vapor critical point (TC,PJ. For the lattice fluid, the critical point in 1978 American Chemical Soc...

A molecular theory of classical fluids based on a well-defined statistical mechanical model is presented. Since the model fluid reduces to the classical lattice gas in one special case, it can be best characterized as an Ising or lattice fluid. The model fluid undergoes a liquid-vapor transition. Only three molecular parameters are required to describe a fluid; these parameters have been determined and tabulated for several fluids. The molecular weight dependence of the critical point and boiling point of a homologous series of fluids such as the normal alkanes is correctly predicted.' The equation of state does not satisfy a simple corresponding states principle, although polymeric liquids of sufficiently high molecular weight do satisfy a corresponding states principle. The Ising fluid better correlates experimental saturated vapor pressures and liquid-vapor densities than the van der Waals or related theories. When applied to polymeric liquids i,t correlates experimental density data as well as less tractable equations derived from cell theories. The basic simplicity and structure of the theory readily suggests a generalization to mixtures.

A unified molecular theory of liquid and gaseous mixtures based on a lattice model description of a fluid is formulated. Pure fluids are completely characterized by three molecular parameters which are known for many fluids. Characterization of a binary mixture requires a knowledge of the pure fluid parameters and an interaction energy; interaction energies have been determined for 18 representative binary mixtures containing nonpolar and polar components. Thermodynamic properties of ternary and higher order mixtures are completely defined in terms of the pure fluid parameters and the binary interaction energies. Quantitative prediction of heats of mixing of a ternary hydrocarbon mixture is demonstrated. Volume changes on mixing are calculable from heats of mixing; good agreement is obtained with experiment for hydrocarbon mixtures. Thermodynamic stability criteria for liquid-liquid (L-L) and liquid-vapor (L-V) phase transitions are shown to be interdependent. The common appearance of a lower critical solution temperature (LCST) in polymer solutions is explained as well as the observation that LCST's usually occur above the normal boiling point of the solvent. Four basic types of L-L phase diagrams are predicted which include upper critical solution temperatures, LCST's, and closed immiscibility loops. Predicted pressure and molecular weight dependences of GST's are in at least qualitative agreement with experiment. Use of the geometric mean law to estimate interaction energies and to predict mixture miscibility limits is evaluated. Rules for predicting polymer/polymer mixture miscibility are outlined. The second virial coefficient of the chemical potential is evaluated; its temperature dependence compares favorably with experiment. Predicted L-V phase diagrams include those with azeotropes and the unusual critical point phenomena of retrograde condensation. Simultaneous description of L-L and L-V equilibria is demonstrated for nonpolar/polar mixtures.

A model is presented to predict the depression of the glass transition temperature of a polymer in the presence of a liquid, gas, or supercritical fluid as a function of pressure. It is developed using lattice fluid theory and the Gibbs-Di Marzio criterion, which states that the entropy is zero at the glass transition. Four fundamental types of Tg versus pressure behavior are identified and interpreted as a function of three factors: the solubility of the compressed fluid in the polymer, the flexibility of the polymer molecule, and the critical temperature of the pure fluid. A new phenomenon is predicted where a liquid to glass transition occurs with increasing temperature, which we define as retrograde vitrification. This retrograde behavior is a consequence of the complex effects of temperature and pressure on sorption. For the limited data which are available for the polystyrene-COa and poly(methyl methacrylatel-COa systems, the predictions of the model are in good agreement with experiment.

0 = 10.5 for all values of n. Like the graft copolymers, the n-arm star diblock copolymers (each arm is a diblock copolymer of composition / containing N monomer units) do not have a critical point. At / = 0.5, ( ) equals 8.86, 7.07, 5.32, and 4.33 for = 2, 4, 10, and 30, respectively. At a spinodal point the static structure factor S(q) diverges at a finite wave vector q*. Near a critical point

I , , " I 271 (1976).R. Guillard and A. Englert, Biopolymers, 15, 1301 (1976). V. V. Zelinski, V. P. Kolobkov, and L. G. Pikulik, Opt. Spektrosk., 1, 560 (1956). (38) As to the active sphere radius r, ' for intermolecular electron transfer, there are some arguments. If one assumes a uniform distribution of energy acceptors around an energy donor, the condition that the total transfer efficiency calculated by the active sphere model is the same as that for the exact case, yields, ro6 Jrd4*r2 dr = 4*r2 drwhere r, denotes the Forster distance. Integration gives r, ' = 1.16rp The above equation means the radius of the active sphere should be a little larger than the Forster distance. Jabionski (A. Jabronski, Bull. Acad. Pol. Sci., Ser. Sci. Math., Astron. Phys., 6,663 (1958)) reported a similar calculation and suggested r, ' = 1.33r0. On the other hand, in the intramolecular case, the distribution of acceptors cannot be uniform and,. strictly speaking, there can be no simple way to define r,'. In fact, a comparison of static transfer efficiency for the active sphere model (broken line in Figure 3) with that for the exact case (solid line, ro = 24 A) indicates that r, ' should be smaller than ro for n < 12 but larger than ro for n < 12. However, since the difference between the two lines is not significant, it is sufficient to use the Forster distance as an active sphere radius in our approximate analysis. A small change in the active sphere radius did not change any conclusions in the text. The authors wish to thank the referee for informine us of this Doint.ABSTRACT: A mean field theory of chain dimensions is formulated which is very similar to the van der Waals theory of a simple fluid. In the limit of infinite chain length, the chain undergoes a Landau-type second-order phase transition. For finite chains, the transition is pseudo-second-order. At low temperatures, the chain is in a condensed or globular state, and the mean square gyration radius (S2) varies as r2/3 where r is proportional to chain length. At high temperatures, the chain is in a gaslike or coil state where (p) varies as r6/5. In the globular state, fluctuations in (S2) are very small, whereas they are very large in the coiled state. A characteristic feature of the theory is that ternary and higher order intramolecular interactions are approximated. At high temperatures, only binary interactions are important, but a t low temperatures, many of the higher order terms contribute. An important conclusion of this study is that a polymer chain does not obey ideal chain statistics a t the 8 temperature. Although the second virial coefficient vanishes at 8, the third virial coefficient does not; its presence is responsible for the perturbation of the chain statistics.For an infinite chain, 8 and the second-order phase-transition temperature are identical. For finite chains, the pseudo-second-order transition temperature is less than 8. When generalized to d dimensions, the theory yields at low temperatures ( S 2 ) d / 2r for all d and at high temperatur...

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We investigate the solvent density driven changes in polymer conformation and phase behavior that occur in a supercritical fluid, with a particular emphasis on conditions near the lower critical solution temperature ͑LCST͒ phase boundary. Using continuous space Monte Carlo simulations, the mean square end-to-end distance (R) and radius of gyration (R g ) are calculated for a single chain with 20 Lennard-Jones segments in a monomeric solvent over a broad range of densities and temperatures. The chains collapse as temperature increases at constant pressure, or as density decreases at constant temperature. A minimum in R and R g occurs at a temperature slightly above the coil-to-globule transition temperature ͑C-GTT͒, where the chain adopts a quasi-ideal conformation, defined by the balance of binary attractive and repulsive interactions. Expanded ensemble simulations of finite-concentration polymer-solvent mixtures reveal that the LCST phase boundary correlates well with the single chain C-GTT. At temperatures well above the LCST, the chain expands again suggesting an upper critical solution temperature ͑UCST͒ phase boundary above the LCST.