Thermodynamic model, polymer solution

In Chap. 8 we discuss the thermodynamics of polymer solutions, specifically with respect to phase separation and osmotic pressure. We shall devote considerable attention to statistical models to describe both the entropy and the enthalpy of mixtures. Of particular interest is the idea that the thermodynamic... 

Thermodynamic modelling of solution phases lies at the core of the CALPHAD method. Only rarely do calculations involve purely stoichiometric compounds. The calculation of a complex system which may have literally 100 different stoichiometric substances usually has a phase such as the gas which is a mixture of many components, and in a complex metallic system with 10 or 11 alloying elements it is not unusual for all of the phases to involve solubility of the various elements. Solution phases will be defined here as any phase in which there is solubility of more than one component and within this chapter are broken down to four types (1) random substitutional, (2) sublattice, (3) ionic and (4) aqueous. Others types of solution phase, such as exist in polymers or complex organic systems, can also be modelled, but these four represent the major types which are currently available in CALPHAD software programmes. 

In this chapter, subsequent to an introduction to devolatilization equipment, we review the thermodynamics of polymer solution equilibrium, which determines the maximum amount of volatiles that can be separated under a given set of processing conditions the phenomena associated with diffusion and diffusivity of small molecules in polymeric melts, which affects the rate of mass transfer the phenomena and mechanisms involving devolatilization and their modeling and the detailed and complex morphologies within the growing bubbles created during devolatilization of melts. 

We have given this discussion at the outset and in some detail, because the question when and which model is a central question for any MC simulation of polymers Actually a large variety of models for macromolecules exists, differing in the details of how the coarse-graining is done. Actually, simple lattice models like the SAW have been central to the formulation of the first theories for the thermodynamics of polymer solutions and polymer blends, such as the Flory-Hu ns the-ory2-14,15,39 extensions, although excluded volume in... 

Another model describing the thermodynamics of polymer solutions was introduced by Flory and Huggins [5,6]. In the Flory-Huggins theory, the non-ideality of the polymer solutions due to the large difference in molecular size between the solute and solvent is taken into account. Based on this theory, the Gibbs free energies of binary and ternary systems are expressed in Equations 15.7 and 15.8, respectively [7] as follows ... 

In many process design applications like polymerization and plasticization, specific knowledge of the thermodynamics of polymer systems can be very useful. For example, non-ideal solution behavior strongly governs the diffusion phenomena observed for polymer melts and concentrated solutions. Hence, accurate modeling of... 

It is an arduous task to develop thermodynamic models or empirical equations that accurately predict solvent activities in polymer solutions. Even so, since Flory developed the well-known equation of state for polymer solutions, much work has been conducted in this area [50-52]. Consequently, extensive experimental data have been published in the literature by various researchers on different binary polymer-solvent sys-... 

Spin orbitals, 258, 277, 279 Square well potential, in calculation of thermodynamic quantities of clathrates, 33 Stability of clathrates, 18 Stark effect, 378 Stark patterns, 377 Statistical mechanics base, clathrates, 5 Statistical model of solutions, 134 Statistical theory for clathrates, 10 Steam + quartz system, 99 Stereoregular polymers, 165 Stereospecificity, 166, 169 Steric hindrance, 376, 391 Steric repulsion, 75, 389, 390 Styrene methyl methacrylate polymer, 150... 

The formation mechanism of structure of the crosslinked copolymer in the presence of solvents described on the basis of the Flory-Huggins theory of polymer solutions has been considered by Dusek [1,2]. In accordance with the proposed thermodynamic model [3], the main factors affecting phase separation in the course of heterophase crosslinking polymerization are the thermodynamic quality of the solvent determined by Huggins constant x for the polymer-solvent system and the quantity of the crosslinking agent introduced (polyvinyl comonomers). The theory makes it possible to determine the critical degree of copolymerization at which phase separation takes place. The study of this phenomenon is complex also because the comonomers act as diluents. 

Flory-Huggins model for polymer solutions, based on statistical thermodynamics, is often used for illustrating the behavior of polymer blends [6,7]. The expression for the free energy change... 

Our model predicts destabilization of colloidal dispersions at low polymer concentration and restabilisation in (very) concentrated polymer solutions. This restabilisation is not a kinetic effect, but is governed by equilibrium thermodynamics, the dispersed phase being the situation of lowest free energy at high polymer concentration. Restabilisation is a consequence of the fact that the depletion thickness is, in concentrated polymer solutions, (much) lower than the radius of gyration, leading to a weaker attraction. 

Futerko and Hsing presented a thermodynamic model for water vapor uptake in perfluorosulfonic acid membranes.The following expression was used for the membrane—internal water activity, a, which was borrowed from the standard Flory—Huggins theory of concentrated polymer solutions ... 

The distinct properties of liquid-crystalline polymer solutions arise mainly from extended conformations of the polymers. Thus it is reasonable to start theoretical considerations of liquid-crystalline polymers from those of straight rods. Long ago, Onsager [2] and Flory [3] worked out statistical thermodynamic theories for rodlike polymer solutions, which aimed at explaining the isotropic-liquid crystal phase behavior of liquid-crystalline polymer solutions. Dynamical properties of these systems have often been discussed by using the tube model theory for rodlike polymer solutions due originally to Doi and Edwards [4], This theory, the counterpart of Doi and Edward s tube model theory for flexible polymers, can intuitively explain the dynamic difference between rodlike and flexible polymers in concentrated systems [4]. 

In the present article, we focus on the scaled particle theory as the theoretical basis for interpreting the static solution properties of liquid-crystalline polymers. It is a statistical mechanical theory originally proposed to formulate the equation of state of hard sphere fluids [11], and has been applied to obtain approximate analytical expressions for the thermodynamic quantities of solutions of hard (sphero)cylinders [12-16] or wormlike hard spherocylinders [17, 18]. Its superiority to the Onsager theory lies in that it takes higher virial terms into account, and it is distinctive from the Flory theory in that it uses no artificial lattice model. We survey this theory for wormlike hard spherocylinders in Sect. 2, and compare its predictions with typical data of various static solution properties of liquid-crystalline polymers in Sects. 3-5. As is well known, the wormlike chain (or wormlike cylinder) is a simple yet adequate model for describing dilute solution properties of stiff or semiflexible polymers. 

Another important application of experimentally determined values of the osmotic second virial coefficient is in the estimation of the corresponding values of the Flory-Huggins interaction parameters x 12, X14 and X24. In practice, these parameters are commonly used within the framework of the Flory-Huggins lattice model approach to the thermodynamic description of solutions of polymer + solvent or polymer] + polymer2 + solvent (Flory, 1942 Huggins, 1942 Tanford, 1961 Zeman and Patterson, 1972 Hsu and Prausnitz, 1974 Johansson et al., 2000) ... 

If we turn from phenomenological thermodynamics to statistical thermodynamics, then we can interpret the second virial coefficient in terms of molecular parameters via a model. We pursue this approach for two different models, namely, the excluded-volume model for solute molecules with rigid structures and the Flory-Huggins model for polymer chains, in Section 3.4. 

I. C. Sanchez and C. G. Panayiotou, Equation of State Thermodynamics of Polymer and Related Solutions, chapt. 3 in "Models for Thermodynamic and Phase equilibria Calculations, S. L. Sandler ed., M. Dekker, New York, 1994. 

The miscibility behaviour of polymer systems has been studied extensively, and experimental data and thermodynamic models have been generated for (co)polymer solutions and for polymer blends. 

The above molecular thermodynamic model for polymer systems has been widely tested by comparing with simulation results (Yang et al., 2006a Xin et al., 2008a). Figure 8 shows the comparisons between predicted critical temperature and critical volume fraction for binary polymer solutions at different chain lengths of with the... 

Monolayers of micro- and nanoparticles at fluid/liquid interfaces can be described in a similar way as surfactants or polymers, easily studied via surface pressure/area isotherms. Such studies provide information on the properties of particles (dimensions, interfacial contact angles), the structure of interfacial layers, interactions between the particles as well as about relaxation processes within the layers. Such type of information is important for understanding how the particles stabilize (or destabilize) emulsions and foams. The performed analysis shows that for an adequate description of II-A dependencies for nanoparticle monolayers the significant difference in size of particles and solvent molecules has be taken into account. The corresponding equations can be obtained by using a thermodynamic model developed for two-dimensional solutions. The obtained equations provide a satisfactory agreement with experimental data of surface pressure isotherms in a wide range of particle sizes between 75 pm and 7.5 nm. Moreover, the model can predict the area per particle and per solvent molecule close to real values. Similar equations were applied also to protein monolayers at liquid interfaces. 

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