Polymer thermodynamic modelling

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-... 

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. 

Another quasi-thermodynamical model of ion transport in polymers is based on the concept of minimum configurational entropy required for rearrangement of the polymer, giving practically identical o—T and D — T dependences as the preceding model. 

DA Hoagland. Unified thermodynamic model for polymer separations produced by size exclusion chromatography, hydrodynamic chromatography, and gel electrophoresis. ACS Symp Ser 635 173-188, 1996. 

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 ... 

Gutowski, W. (1988). A thermodynamic model of the matrix-reinforcement interface Experimental verification. In Interfaces in Polymer. Ceramic and Metal Matrix Composites (Proc. ICCI-II) (H. Ishida ed.), Elsevier, New York, pp. 735-746. 

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. 

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. 

Polymer systems containing copolymers call for a further extension of the thermodynamic model. The interaction function for statistical copolymers was originally derived by Simha and Branson [34], discussed by Stockmayer [35] et al., and experimentally verified by Glbckner and Lohmann [36]. 

In Section I we introduce the gas-polymer-matrix model for gas sorption and transport in polymers (10, LI), which is based on the experimental evidence that even permanent gases interact with the polymeric chains, resulting in changes in the solubility and diffusion coefficients. Just as the dynamic properties of the matrix depend on gas-polymer-matrix composition, the matrix model predicts that the solubility and diffusion coefficients depend on gas concentration in the polymer. We present a mathematical description of the sorption and transport of gases in polymers (10, 11) that is based on the thermodynamic analysis of solubility (12), on the statistical mechanical model of diffusion (13), and on the theory of corresponding states (14). In Section II we use the matrix model to analyze the sorption, permeability and time-lag data for carbon dioxide in polycarbonate, and compare this analysis with the dual-mode model analysis (15). In Section III we comment on the physical implication of the gas-polymer-matrix model. 

LATTICE BASED MOLECULAR THERMODYNAMIC MODEL OF POLYMER SYSTEMS... 

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... 

Many polymer blends or block polymer melts separate microscopically into complex meso-scale structures. It is a challenge to predict the multiscale structure of polymer systems including phase diagram, morphology evolution of micro-phase separation, density and composition profiles, and molecular conformations in the interfacial region between different phases. The formation mechanism of micro-phase structures for polymer blends or block copolymers essentially roots in a delicate balance between entropic and enthalpic contributions to the Helmholtz energy. Therefore, it is the key to establish a molecular thermodynamic model of the Helmholtz energy considered for those complex meso-scale structures. In this paper, we introduced a theoretical method based on a lattice model developed in this laboratory to study the multi-scale structure of polymer systems. First, a molecular thermodynamic model for uniform polymer system is presented. This model can... 

To establish the molecular thermodynamic model for uniform systems based on concepts from statistical mechanics, an effective method by combining statistical mechanics and molecular simulation has been recommended (Hu and Liu, 2006). Here, the role of molecular simulation is not limited to be a standard to test the reliability of models. More directly, a few simulation results are used to determine the analytical form and the corresponding coefficients of the models. It retains the rigor of statistical mechanics, while mathematical difficulties are avoided by using simulation results. The method is characterized by two steps (1) based on a statistical-mechanical derivation, an analytical expression is obtained first. The expression may contain unknown functions or coefficients because of mathematical difficulty or sometimes because of the introduced simplifications. (2) The form of the unknown functions or unknown coefficients is then determined by simulation results. For the adsorption of polymers at interfaces, simulation was used to test the validity of the weighting function of the WDA in DFT. For the meso-structure of a diblock copolymer melt confined in curved surfaces, we found from MC simulation that some more complex structures exist. From the information provided by simulation, these complex structures were approximated as a combination of simple structures. Then, the Helmholtz energy of these complex structures can be calculated by summing those of the different simple structures. 

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. 

The static and dynamic properties of polymer-layered silicate nanocomposites are discussed, in the context of polymers in confined spaces and polymer brushes. A wide range of experimental techniques as applied to these systems are reviewed, and the salient results from these are compared with a mean field thermodynamic model and non-equilibrium molecular dynamics simulations. 

Fig. 6. The change of entropy per area versus the change in gallery height, for the polymer and the surfactant (octadecylammonium) functionalized surface based on the thermodynamic model presented in [26].

Hogfeldt, E. 1988. Application of a simple thermodynamic model to various ion exchange data. React. Polym. 7 81-87. 

Chapter 2 is an in depth discussion of the various theories important to phase equilibria in general and polymer thermodynamics specifically. First a review of phase equilibria is provided followed by more specific discussions of the thermodynamic models that are important to polymer solution thermodynamics. The chapter concludes with an analysis of the behavior of liquid-liquid systems and how their phase equilibrium can be correlated. 

THERMODYNAMIC MODELS OF POLYMER SOLUBILITY AND PHASE SEPARATION... 

In summary, IGC is an experimentally attractive method for obtaining polymer-polymer interaction parameters in polymer blends at temperatures above Tm for a crystalline blend, and above Tg for an amorphous blend. This technique yields interaction parameters that are generally consistent with data obtained with other techniques such as vapor sorption, melting point depression, neutron scattering, and small-angle X-ray scattering (40). Advances in IGC of polymer blends will require increased experimental precision in order to improve the consistency of the data, as well as refinements of thermodynamic models to allow better interpretation of interactions in ternary solutions. 

Thermodynamic Model for Phase Equilibrium between Polymer Solution and 6/W MlcroemulslonsT figures 6 and / show that when phase separation first occurs, most of the water is in the microemulsion. With an increase in salinity, however, much of the water shifts to the polymer solution. Thus, a concentrated polymer solution becomes dilute on increasing salinity. The objective of this model is to determine the partitioning of water between the microemulsion and the polymer-containing excess brine solution which are in equilibrium. For the sake of simplicity, it is assumed that there is no polymer in the microemulsion phase, and also no microemulsion drops in the polymer solution. The model is illustrated in Figure 12. The model also assumes that the value of the interaction parameter (x) or the volume of the polymer does not change with salinity. 

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