Polymer-solvent interaction parameter fractionation

Fractional Precipitation of Cellulose Triacetate. The reported partial or non-fractionation of cellulose triacetate from chlorinated hydrocarbons or acetic acid may be explained in terms of the polymer-solvent Interaction parameter x (1-11) The x values for cellulose triacetate-tetrachloroethane and cellulose triacetate-chloroform systems are reported (10,21) as 0.29 and 0.34 respectively. The lower values of x for such systems will result in a smaller or negative heat of mixing (AHm) and therefore partial or non-fractionation of the polymer in question results. 

Xi and X2 are the mole fractions of solvent and polymer, respectively g is the polymer-solvent interaction parameter... 

Vs = molar volume of the solvent = volume fraction of polymer in the swollen gel % = polymer-solvent interaction parameter... 

Problem 3.15 A polymer solution was cooled very slowly until phase separation took place to give two phases in equilibrium. Analysis of the phases showed that the volume fractions of polymer in the two phases were 2 = 0.01 and < 2 = 0.89, respectively. Using the Flory-Huggins equation for A/ri [cf. Eq. (3.53)], together with the equilibrium condition = A/i", calculate an estimate of the polymer-solvent interaction parameter for the conditions of phase separation. 

FIGURE 6.17 Solubility of a homopolymer according to the Flory-Huggins theory. Variables are the excluded volume parameter ft (or the polymer-solvent interaction parameter y), the net volume fraction of polymer q>, and the polymer-to-solvent molecular volume ratio q. Solid lines denote binodal, the broken line spinodal decomposition. Critical points for decomposition (phase separation) are denoted by . See text. 

Here, >i is the volume fraction of the solvent in the swollen polymer, Oxx is the stress developed within the polymer, Dn is the mutual diffusion coefficient, % is the polymer-solvent interaction parameter, Vi is the molar volume of the solvent, T is the temperature, E is the modulus of die polymer and T is the viscosity of the polymer. Equation (1) is valid in the region between x s R and x = S. 

Here, the second virial coefficient (excluded-volume parameter), vb = [1 -2x T)]<0, is negative because water is a poor solvent for the hydrophobic block B. The third virial coefficient, Wb, is positive, and x= T- e /d is the relative deviation from the theta temperature. At small deviations from the theta point, r < 1, the surface tension y and the polymer volume fraction (p are related as y/k T = (p. However, at larger deviations from the theta point, (p becomes comparable to unity and the latter relationship breaks down. Because in a typical experimental situation (p = 1, we treat (p and y as two independent parameters. Note that in a general case, surface tension y and width A of the core-corona interface depend on both the polymer-solvent interaction parameter Xbs T) for the core-forming block and the incompatibility Xab between monomers of blocks A and B. That is, y could depend on the concentration of monomers of the coronal block A near the core surface. We, however, neglect this (weak) dependence and assume that the surface tension y is not affected by conformations of the coronal blocks in a micelle. 

Chemical potentials inside the membrane are determined by the Flory-Rehner-Donnan equation of state for one-dimensional swelling (5), since the clamped membrane s area. A, is constant. The polymer-solvent interaction parameter, f, is a linear function of the polymer volume fraction, i.e. J + Xi as proposed by Hirotsu (/ 7). Transport of water into and out of the hydrogel membrane causes its thickness, I, to change at a rate that is proportional to the difference in chemical potential of water between the membrane and Cell I and Cell II. 

Kamide,K., Sugamiya,K. Role of concentration dependence of polymer/ solvent interaction parameter in the polymer fractionation by successive precipitational method. Makromol. Chem. 139,197 (1970). 

Where, is volume fraction of the polymer, is molar volume of solvent, X is the polymer solvent interaction parameter. Me is the crosslink density of polymer, pi is solvent density, P2 is polymer density, / is functionality of crosslink, Fequ is equilibrium volume of the hydrogel. 

Calculations using (8) show that the temperature Tp ep significantly affects the monomer concentration in the unfrozen phase. Figure 6a, b shows how the monomer concentration C in the unfrozen domains and the volume fraction of frozen solvent in the resulting cryogel P vary with the temperature Cprep. Calculations were for water as the solvent and for various polymer-solvent interaction parameters x... 

Flory and Huggins used a liquid lattice theory to express p — p as a function of the volume fraction Vj of polymer in the mixture, the polymer-solvent interaction parameter % and the ratio x of molar volumes of polymer Vj and solvent V. ... 

The first term in equation (4.11) is a mixing term depending only on the relative volumes of polymer and solvent, and on the polymer-solvent interaction parameter, Xs The final form of this term will therefore be analogous to the result derived by Flory, with the volume fraction of... 

Here tj>, is the volume fraction solvent and x is the polymer-solvent interaction parameter. Tm is the melting temperature of the diluted polymer and AHf is the heat of fusion per mole of repeating units, V, and Vj ate the molar volume of the solvent and of the polymer repeating unit respectively. 

Notation cp = volume fraction, = correlation length, p= C./6, v= excluded volume parameter, 1 - 2x, N= number of bonds per chain, x= Flory-Huggins polymer-solvent interaction parameter, Vc = cross-over from swollen to ideal. Subscripts cr = critical, / = ideal, d = dilute, s = semi-dilute, m = marginal regime, c = concentrated regime, ov= cross-over from dilute to semi-dilute, cp = cross-over from concentrated to phase separated, sm = cross-over from semi-dilute to marginal, me = cross-over from marginal to concentrated. 

Here, Tm is the melting point of the polymer in the solvent, is the equilibrium melting point of the polymer, AHl is the equilibrium heat of fusion (J/mol) of the repeating unit, V2 is the molar volume of the polymer, Vi is the molar volume of the solvent Vi is the volume fraction of the solvent, %i is the polymer-solvent interaction parameter, and R is the universal gas constant. This equation can be simphfied to the following form ... 

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