Huggins function

In both of these equations c is the polymer concentration in g/dl. The Kraemer function is based on the relative viscosity, which is a ratio of the solution viscosity [r ) and the viscosity of the pure solvent The Huggins function uses the specific viscosity, ri p, which is defined in terms of the relative viscosity as follows ... 

As this subchapter is devoted to solvent activities, only the monodisperse case will be taken into account here. However, the user has to be aware of the fact that most LLE-data were measured with polydisperse polymers. How to handle LLE-results of polydisperse polymers is the task of continuous thermodynamics, Refs. Nevertheless, also solutions of monodisperse polymers or copolymers show a strong dependence of LLE on molar mass of the polymer, or on chemical composition of a copolymer. The strong dependence on molar mass can be explained in principle within the simple Flory-Huggins %-function approach, please see Equation [4.4.61]. 

A numerical calculation needs knowledge of the solvent activity of die corresponding homopolymer solution at the same equilibrium concentration (here characterized by the value of the Flory-Huggins %-function) and the assumption of a deformation model that provides values of the factors A and B. There is an extensive literature for statistical thermodynamic models which provide, for example, Flory A = 1 and B = 0.5 Hermans A = 1 and B = 1 James and Guttf or Edwards and Freed A = 0.5 and B = 0. A detailed explanation was given recently by Heinrich et al. ... 

The simple Flory-Huggins %-function, combined with the solubility parameter approach may be used for a first rough guess about solvent activities of polymer solutions, if no experimental data are available. Nothing more should be expected. This also holds true for any calculations with the UNIFAC-fv or other group-contribution models. For a quantitative representation of solvent activities of polymer solutions, more sophisticated models have to be applied. The choice of a dedicated model, however, may depend, even today, on the nature of the polymer-solvent system and its physical properties (polar or non-polar, association or donor-acceptor interactions, subcritical or supercritical solvents, etc.), on the ranges of temperature, pressure and concentration one is interested in, on the question whether a special solution, special mixture, special application is to be handled or a more universal application is to be foxmd or a software tool is to be developed, on munerical simplicity or, on the other hand, on numerical stability and physically meaningftd roots of the non-linear equation systems to be solved. Finally, it may depend on the experience of the user (and sometimes it still seems to be a matter of taste). 

In many applications the phase stmcture as a function of the temperature is of interest. The discussion of this issue requires the knowledge of the temperature dependence of the Flory-Huggins parameter x (T). If the interactions... 

Figure 8.1 The entropy of mixing (in units of R) as a function of mole fraction solute for ideal mixing and for the Flory-Huggins lattice model with n = 50, 100, and 500. Values are calculated in Example 8.1.

In this section and the last, we have examined the lattice model of the Flory-Huggins theory for general expressions relating AHj and ASj to the composition of the mixture. The separate components can therefore be put together to give an expression for AGj as a function of temperature and composition ... 

The toughness of interfaces between immiscible amorphous polymers without any coupling agent has been the subject of a number of recent studies [15-18]. The width of a polymer/polymer interface is known to be controlled by the Flory-Huggins interaction parameter x between the two polymers. The value of x between a random copolymer and a homopolymer can be adjusted by changing the copolymer composition, so the main experimental protocol has been to measure the interface toughness between a copolymer and a homopolymer as a function of copolymer composition. In addition, the interface width has been measured by neutron reflection. Four different experimental systems have been used, all containing styrene. Schnell et al. studied PS joined to random copolymers of styrene with bromostyrene and styrene with paramethyl styrene [17,18]. Benkoski et al. joined polystyrene to a random copolymer of styrene with vinyl pyridine (PS/PS-r-PVP) [16], whilst Brown joined PMMA to a random copolymer of styrene with methacrylate (PMMA/PS-r-PMMA) [15]. The results of the latter study are shown in Fig. 9. 

Both Flory and Huggins derived statistical mechanical expressions for aS . Their expressions are still among the best available. For this reason, Prigogine and his co-workers concentrated their efforts on revising the statistical mechanical configurational partition function which leads, among other things, toAH. ... 

Now let s discuss the pressure computations. The observed reactor pressure is a sum of the partial pressures of nitrogen and the styrene monomer vapor. The vapor pressure of the styrene vapor is an increasing function of temperature and decreasing function of conversion. This is explained by the Flory-Huggins relationship ( ). 

This channel has recently been observed in the 335-352 nm region of the Huggins band.49 Such a channel is not inconsistent with previously determined branching ratios in the Hartley band. These determinations typically rely on measuring the amount of 0(3P) as a function of time and inferring the channels present from the observed kinetics. Such a measurement accounts for the amount of 0(4D) versus 0(3P) formed but not which channels gives rise to these species. 

This expression accounts for the configurational entropy of an ideal binary mixture with identical molecular sizes, but not for that of a polymer solution, since polymer chains are large and flexible. For that case, more contributions arise from the chain conformational entropy, first considered by Meyer [19] and then derived by Huggins [20] and Flory [21]. In analogy with a nonreversing random walk on a lattice, the conformational contribution of polymer chains to the partition function is given by... 

Figure 14. The phase diagram of the gradient copolymer melt with the distribution functions g(x) = l — tanh(ciit(x —fo)) shown in the insert of this figure for ci = 3,/o = 0.5 (solid line), and/o — 0.3 (dashed line), x gives the position of ith monomer from the end of the chain in the units of the linear chain length. % is the Flory-Huggins interaction parameter, N is a polymerization index, and/ is the composition (/ = J0 g(x) dx). The Euler characteristic of the isotropic phase (I) is zero, and that of the hexagonal phase (H) is zero. For the bcc phase (B), XEuier = 4 per unit cell for the double gyroid phase (G), XEuier = -16 per unit cell and for the lamellar phases (LAM), XEuier = 0.

For the solubility of TPA in prepolymer, no data are available and the polymer-solvent interaction parameter X of the Flory-Huggins relationship is not accurately known. No experimental data are available for the vapour pressures of dimer or trimer. The published values for the diffusion coefficient of EG in solid and molten PET vary by orders of magnitude. For the diffusion of water, acetaldehyde and DEG in polymer, no reliable data are available. It is not even agreed upon if the mutual diffusion coefficients depend on the polymer molecular weight or on the melt viscosity, and if they are linear or exponential functions of temperature. Molecular modelling, accompanied by the rapid growth of computer performance, will hopefully help to solve this problem in the near future. The mass-transfer mechanisms for by-products in solid PET are not established, and the dependency of the solid-state polycondensation rate on crystallinity is still a matter of assumptions. 

For each sample, the Flory Huggins constants (k, Eq. 5.25) were also determined (by viscosity measurements) as function of sonication time and are given in Tab. 5.17. 

Ky is the Flory-Huggins interaction parameter between the i and j monomers. In Eq. 6.6, the matrices have a dimension (m) (m). We note that the s-depen-dence of the excluded volume matrix is solely determined by the contribution of the bare susceptibility yoo(Q> ) he invisible matrix component 0 . Finally, combining Eq. 6.6 with Eq. 6.1 the response function in the interacting system is given by ... 

Different models determine A in different ways. Nation exhibits a water-uptake isotherm as shown in Figure 7. The dashed line in the figure shows the effects of Schroeder s paradox, where there is a discontinuous jump in the value of A. Furthermore, the transport properties have different values and functional forms at that point. Most models used correlate A with the water-vapor activity, since it is an easily calculated quantity. An exception to this is the model of Siegel et al., ° which assumes a simple mass-transfer relationship. There are also models that model the isotherm either by Flory—Huggins theory" or equilibrium between water and hydrated protons in the membrane and water vapor... 

Disposing the Flory-Huggins modified equation, including the free entropy of mixing per total volume, AS , as a function of conversion and the enthalpy term expressed with the interaction parameter [66-68,72] ... 

Recent developments in the theory of polymer solutions have been reviewed by Berry and Casassa (32), and by Casassa (71). Casassa, who has contributed very largely to these developments, has adopted a statistical mechanical approach using molecular distribution functions, as first outlined by Zimm (72), rather than using a lattice model like that used by Flory, Huggins, and many later workers. 

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