Statistical thermodynamics of polymer crystallization

Therefore, the partition function of polymer solutions can be expressed as 

At the right-hand side of (10.8), the first five terms come from Floiy s semiflexibility treatment (8.55), the sixth term comes from the mean-field estimation for the pair interactions of parallel bonds (10.7), and the last term comes from the mean-field estimation for the mixing interactions between the chain units and the solvent molecules (8.21). According to the Boltzmann s relation F = —kTlnZ, the free energy of the solution system can be obtained as 

In the practical systems, the mixing free energy change is estimated with the reference to the amorphous bulk phase of the polymer, so 

From (10.9), one can obtain the expression of the mixing free energy consistent with the Floiy-Huggins equation, as given by 

Although the mixing interaction parameter exhibits the same formulas with the Flory-Huggins parameter, it contains contributions from both the mixing energy and the parallel packing energy. 

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Simha, R., Jain, R. K., Statistical thermodynamics of polymer crystal and melt. Journal of Polymer Science, Polymer Physics Edition, 16(8), pp. 1471-1489 (1978). 

Figure 1 also shows that plasticized polyvinyl chloride begins to flow at a lower temperature. This is to be expected in view of the fact that equilibrium melting temperature of polymer crystals is depressed by monomeric diluents. A statistical thermodynamic treatment by Flory (13), showed that this effect depends on the nature of the polymer, concentration of the diluent, and the degree of polymer-diluent interaction in the following manner ... 

We made a brief introduction about our current theoretical models of thermodynamics and kinetics of polymer crystallization. We first introduced basic thermodynamic concepts, including the melting point, the phase diagram, the metastable state, and the mesophase. The mean-field statistical thermodynamics based on a... 

Flory PJ (1941) Thermodynamics of high polymer solutions. J Chem Phys 9(8) 660 Flory PJ (1942) Thermodynamics of high polymer solutions. J Chem Phys 10(1) 51-61 Flory PJ (1953) Principles of polymer chemistry. Cornell University Press, Ithaca Flory PJ (1955) Theory of crystallization in copolymers. Trans Faraday Soc 51 848-857 Flory PJ (1956) Statistical thermodynamics of semi-flexible chain molecules. Proc R Soc Lond A Math Phys Sci 234(1196) 60-73... 

It is shown that the problem of formulating the statistical thermodynamics of networks can be resolved by the use of ideas from quantum statistical mechanics by appropriate generalization. Examples are given in terms of gaussian and liquid crystal polymer networks. 

Construction of theoretical phase diagrams similar to the diagram in Fig. 2.17 should facilitate the search for major proaches to the analysis of polymer-solvent systems in which equilibria involving isotropic, liquid-crystalline, crystal solvate, and truly crystalline phases are complexly associated. This is valid, since, as indicated in the classic course of statistical thermodynamics of van der Waals and Konstamm [45], physicists and chemists do not need the precise quantitative dependence for the concrete case as much as to establish general types and then to study whether the qualitative differences of these types coincide with the experimentally found types. ... 

Sato H, Fujikake H, Lino Y, Kawakita M, Kikuchi H (2002) Flexible grayscale ferroelectric liquid crystal device containing polymer walls and networks. Jpn J Appl Phys 41 5302-5306 Sato H, Fujikake H, Kikuchi H, Kurita T (2003) Rollable polymer stabilized ferroelectric liquid crystal device using thin plastic substrates. Opt Rev 10(5) 352-356 Schrader DM, Jean YC (1988) Positron and positronium chemistry. Elsevier, Amsterdam Shinkawa K, Takahashi H, Fume H (2008) Ferroelectric liquid crystal cell with phase separated composite organic film. Ferroelectrics 364 107-112 Simha R, Somcynsky T (1969) On the statistical thermodynamics of spherical and chain molecule fluids. Macromolecules 2 342-350... 

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 making models, we must respect the principles of equilibrium statistical mechanics, but cannot wholly rely on them, since we have every reason to believe that in polymer crystallization we have only a frustrated approach towards thermodynamic equilibrium. Most of the models are in some way or another founded on their authors conceptions of the nature of the process of high-polymer crystallization from the melt. That is a process on which direct detailed information is hard to get, and some imaginative extrapolation from what one knows about related problems is almost unavoidable. 

The principles of polymer fractionation by solubility or crystallization in solution have been extensively reviewed on the basis of Hory-Huggins statistical thermodynamic treatment [58,59], which accounts for melting point depression by the presence of solvents. For random copolymers the classical Flory equation [60] applies ... 

In Fig. 11.3, we made a comparison between the binodals obtained from dynamic Monte Carlo simulations (data points) and from mean-field statistical thermodynamics (solid lines). First, one can see that even with zero mixing interactions B = 0, due to the contribution of Ep, the binodal curve is still located above the liquid-solid coexistence curve (dashed lines). This result implies that the phase separation of polymer blends occurs prior to the crystallization on cooUng. This is exactly the component-selective crystallizability-driven phase separation, as discussed above. Second, one can see that, far away from the liquid-solid coexistence curves (dashed lines), the simulated binodals (data points) are well consistent... 

Fig. 2.2. Two-phase model for a cylindrical polymer crystal, (a) Loops and tails are explicitly considered as an amorphous fraction in thermodynamic equilibrium with the crystalline fraction. The height of the amorphous layers is denotes by h, keeping the notation for m as the length of the crystalline stems. Both length scales are considered in units of statistical segments, (b) Illustration of the thermodynamic equilibrium system. Segments can be exchanged between two phases and the temperature is considered to be lower than equilibrium melting temperature...

Some transitions that are only known for macromolecules, however, will not be mentioned at all since they are covered elsewhere in this Encyclopedia (see, eg. Gel Point). Also we shall not be concerned here with the transformations from the molten state to the solid state of polymeric materials, since this is the subject of separate treatments (see Crystallization Kinetics Glass Transition Viscoelasticity). Unlike other materials, polymers in the solid state rarely reach full thermal equilibrium. Of course, all amorphous materials can be considered as frozen fluids (see Glass Transition) Rather perfect crystals exist for metals, oxides, semiconductors etc, whereas polymers typically are semicrystalline, where amorphous regions alternate with crystalline lamellae, and the detailed structure and properties are history-dependent (see Semicrystalline Polymers). Such out-of-equilibrium aspects are out of the scope of the present article, which rather emphasizes general facts of the statistical thermodynamics (qv) of phase transitions and their applications to polymers in fluid phases. 

Chapter 1 introduces basic elements of polymer physics (interactions and force fields for describing polymer systems, conformational statistics of polymer chains, Flory mixing thermodynamics. Rouse, Zimm, and reptation dynamics, glass transition, and crystallization). It provides a brief overview of equilibrium and nonequilibrium statistical mechanics (quantum and classical descriptions of material systems, dynamics, ergodicity, Liouville equation, equilibrium statistical ensembles and connections between them, calculation of pressure and chemical potential, fluctuation... 

When individual, isolated molecules exist in helical, or other ordered forms, environmental changes, either in the temperature or solvent composition, can disrupt the ordered structure and transform the chain to a statistical coil. This conformational change takes place within a small range of an intensive thermodynamic variable and is indicative of a highly cooperative process. This reversible intramolecular order-disorder transformation is popularly called the helix-coil transition. It is an elementary, one-dimensional, manifestation of polymer melting and crystallization. 

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