Equations polymer thermodynamics

The molar activity coefficient y, shown in Equations 16.20 and 16.21, is not always a suitable quantity in polymer thermodynamics. Molar activity coefficients can reach very low values, especially at very high polymer molecular weights. More convenient is the weight fraction activity coefficient (Qj), which is the ratio of the activity to the weight fraction ... 

Since 1980 polymer thermodynamics has been developed considerably and, to date, models are available that are suitable for at least satisfactory calculations of VLE and, qualitatively, also for LLE. Some of these methods are models for the activity coefficient, which are modifications of the FH equation. These modifications use a similar to FH but better combinatorial/free-volume expression and a local-composition-type energetic term such as those found in the UNIQUAC and UNIFAC models. Models like the UNIFAC-FV and the Entropic-FV are discussed in Section 16.4. 

The free-volume (FV) concept has a special importance in polymer thermodynamics. FV is the volume allocated to the molecules for movement when their own volume is substracted. Patterson,in his excellent review on FV, offers a qualitative description of the relationship between FV and polymer solubility. Elbro demonstrated, using a simple defiiution for the FV (Equation 16.46), that the FV percentages of solvents and polymers are different. It is exactly these differences in FV (or expansivities), which were ignored in early theories like the famous FH equation, hi the typical case, the FV percentage of solvents is greater (40 to 50%) than that of polymers (30 to 40%). There are two exceptions to this rule water and Water has lower FV than other solvents and closer to that of most... 

This remarkably simple expression is the famous Floiy-Hug ns equation for the Gibbs free energy of mixing, which has been the cornerstone of polymer thermodynamics for more than five decades. 

Doi first proposed the generalized dynamic equations for the concentrated solution of rod-like polymers. Such constitutive equations can be derived from the molecular theory developed by Doi and Edwards (1986). The basis for the molecular theory is the Smoluchowski equation or Fokker-Planck equation in thermodynamics with the mean field approximation of molecular interaction. 

Sadowski, G., 2011. Modeling of polymer phase equilibria using equations of state. In Enders, S. and Wolf, B.A., eds. Polymer Thermodynamics Liquid Polymer-Containing Mixtures. Advances in Polymer Science, Vol. 238, p. 389, Springer-Verlag, Berlin, Heidelberg. 

It is perhaps advisable to reiterate here that all of the thermodynamic analysis of SANS data is based on a mean-field theory, essentially that of Flory and Huggins, and this is known to be inadequate. It may be that, if a proper account were taken of the different expansibilities of the polymer mixture components, as in equation-of-state or lattice-fluid theories of polymer thermodynamics, then the dependencies on microstructure and a g-dependent % might disappear. 

In writing Eq. (8.33), we have divided by Nj , since the polymer molecules are interchangeable. Equation (8.33) gives the thermodynamic probabilty for the system according to this model, since there is only one way to place the solvent molecules once the polymer molecules have been positioned on the lattice. 

Many of these features are interrelated. Finely divided soHds such as talc [14807-96-6] are excellent barriers to mechanical interlocking and interdiffusion. They also reduce the area of contact over which short-range intermolecular forces can interact. Because compatibiUty of different polymers is the exception rather than the rule, preformed sheets of a different polymer usually prevent interdiffusion and are an effective way of controlling adhesion, provided no new strong interfacial interactions are thereby introduced. Surface tension and thermodynamic work of adhesion are interrelated, as shown in equations 1, 2, and 3, and are a direct consequence of the intermolecular forces that also control adsorption and chemical reactivity. 

The diffusion coefficient, sometimes called the diffusivity, is the kinetic term that describes the speed of movement. The solubiHty coefficient, which should not be called the solubiHty, is the thermodynamic term that describes the amount of permeant that will dissolve ia the polymer. The solubiHty coefficient is a reciprocal Henry s Law coefficient as shown ia equation 3. 

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-... 

Lack of termination in a polymerization process has another important consequence. Propagation is represented by the reaction Pn+M -> Pn+1 and the principle of microscopic reversibility demands that the reverse reaction should also proceed, i.e., Pn+1 -> Pn+M. Since there is no termination, the system must eventually attain an equilibrium state in which the equilibrium concentration of the monomer is given by the equation Pn- -M Pn+1 Hence the equilibrium constant, and all other thermodynamic functions characterizing the system monomer-polymer, are determined by simple measurements of the equilibrium concentration of monomer at various temperatures. 

From the foregoing discussion it will be clear that the stoichiometry of the oxidation [n in Eq. (1)] has no thermodynamic significance. It should not be used in the Nemst equation to describe the potential dependence of the equilibrium shown in Eq. (1). It is therefore better to describe n as the degree of oxidation of the polymer (i.e., the average number of holes per monomer unit), n is a potential-dependent parameter,... 

The estimation of f from Stokes law when the bead is similar in size to a solvent molecule represents a dubious application of a classical equation derived for a continuous medium to a molecular phenomenon. The value used for f above could be considerably in error. Hence the real test of whether or not it is justifiable to neglect the second term in Eq. (19) is to be sought in experiment. It should be remarked also that the Kirkwood-Riseman theory, including their theory of viscosity to be discussed below, has been developed on the assumption that the hydrodynamics of the molecule, like its thermodynamic interactions, are equivalent to those of a cloud distribution of independent beads. A better approximation to the actual molecule would consist of a cylinder of roughly uniform cross section bent irregularly into a random, tortuous configuration. The accuracy with which the cloud model represents the behavior of the real polymer chain can be decided at present only from analysis of experimental data. 

At the initial stage of bulk copolymerization the reaction system represents the diluted solution of macromolecules in monomers. Every radical here is an individual microreactor with boundaries permeable to monomer molecules, whose concentrations in this microreactor are governed by the thermodynamic equilibrium whereas the polymer chain propagation is kinetically controlled. The evolution of the composition of a macroradical X under the increase of its length Z is described by the set of equations ... 

The structure of hydrogels that do not contain ionic moieties can be analyzed by the Flory Rehner theory (Flory and Rehner 1943a). This combination of thermodynamic and elasticity theories states that a cross-linked polymer gel which is immersed in a fluid and allowed to reach equilibrium with its surroundings is subject only to two opposing forces, the thermodynamic force of mixing and the retractive force of the polymer chains. At equilibrium, these two forces are equal. Equation (1) describes the physical situation in terms of the Gibbs free energy. 

Volume approximation (when the surface contribution to the free energy of a globule is neglected) works the better the farther the system is from the point of the coil-to-globule transition. In the framework of this approximation, it coincides with the -point, whereas under the theoretical consideration where the surface layer is taken into account, a gap appears separating these two points. The less is the length of polymer chain l, the more pronounced is this gap. Hence, the condition, imposed on the thermodynamic and stoichiometric parameters of the system by the equation of the -point,... 

As pointed out in the foregoing, there are two specific peculiarities qualitatively distinguishing these systems from the classical ones. These peculiarities are intramolecular chemical inhomogeneity of polymer chains and the dependence of the composition of macromolecules X on their length l. Experimental data for several nonclassical systems indicate that at a fixed monomer mixture composition x° and temperature such dependence of X on l is of universal character for any concentration of initiator and chain transfer agent [63,72,76]. This function X(l), within the context of the theory proposed here, is obtainable from the solution of kinetic equations (Eq. 62), supplemented by thermodynamic equations (Eq. 63). For heavily swollen globules, when vector-function F(X) can be presented in explicit analytical form... 

---