Volume fraction polymer equilibrium

Figure 8.3 Volume fraction polymer in equilibrium phases for chains of different length, (a) Theoretical curves drawn for the indicated value of n, with the interaction parameter as the ordinate. Note that x increases downward. (Redrawn from Ref. 6.) (b) Experimental curves for the molecular weights indicated, with temperature as the ordinate. [Reprinted with permission from A. R. Shultz and P. J. Flory, J. Am. Chem. Soc. 74 4760 (1952), copyright 1952 by the American Chemical Society.]...

At least two different techniques are available to compress an emulsion at a given osmotic pressure H. One technique consists of introducing the emulsion into a semipermeable dialysis bag and to immerse it into a large reservoir filled with a stressing polymer solution. This latter sets the osmotic pressure H. The permeability of the dialysis membrane is such that only solvent molecules from the continuous phase and surfactant are exchanged across the membrane until the osmotic pressure in the emulsion becomes equal to that of the reservoir. The dialysis bag is then removed and the droplet volume fraction at equilibrium is measured. 

Here, u2,r is defined as the polymer volume fraction after crosslinking but before swelling (the relaxed polymer volume fraction) and u2-s is the polymer volume fraction after equilibrium swelling (swollen polymer volume fraction). 

Figure 8.7. A schematic volume fraction profile for a brush-like layer at an interface. 0s and 0OO are the surface volume fraction and equilibrium bulk volume fi action of the polymer, respectively.

Feed ratio of OC2H5 TEOS groups to OH chain ends. b Volume fraction of polymer present at swelling equilibrium in benzene at room temperature. c Elongation at initial upturn in modulus. d Ultimate strength as represented by the nominal stress at rupture. Energy required for rupture... 

They showed further that the limiting slope (RTA2) of the plot of the osmotic pressure-concentration ratio tz/c against the polymer concentration in a binary solvent mixture should be proportional to the value of the quantity on the left side of Eq. (17),f with V2 representing the volume fraction of solvent in the nonsolvent-solvent mixture which is in osmotic equilibrium with the solution. The composition of the liquid medium outside the polymer molecules in a dilute solution must likewise be given by V2. The composition of the solvent mixture within the domains of the polymer molecules may differ slightly from that outside owing to selective absorption of solvent in preference to the nonsolvent. This internal composition is not directly of concern here. If the solution is made sufficiently dilute, the external nonsolvent-solvent composition v2 = l—Vi) will be practically equal to the over-all solvent composition for the solution as a whole. Hence... 

Here % is the Flory-Huggins interaction parameter and ( ), is the penetrant volume fraction. In order to use Eqs. (26)—(28) for the prediction of D, one needs a great deal of data. However, much of it is readily available. For example, Vf and Vf can be estimated by equating them to equilibrium liquid volume at 0 K, and Ku/y and K22 - Tg2 can be computed from WLF constants which are available for a large number of polymers [31]. Kn/y and A n - Tg can be evaluated by using solvent viscosity-temperature data [28], The interaction parameters, %, can be determined experimentally and, for many polymer-penetrant systems, are available in the literature. 

Volume fraction of polymer at equilibrium swelling in CgHg. 

The crucial question is at what value of <)> is the attraction high enough to induce phase separation De Hek and Vrij (6) assume that the critical flocculation concentration is equivalent to the phase separation condition defined by the spinodal point. From the pair potential between two hard spheres in a polymer solution they calculate the second virial coefficient B2 for the particles, and derive from the spinodal condition that if B2 = 1/2 (where is the volume fraction of particles in the dispersion) phase separation occurs. For a system in thermodynamic equilibrium, two phases coexist if the chemical potential of the hard spheres is the same in the dispersion and in the floe phase (i.e., the binodal condition). 

The chemical potential of the polymer is affected by "impurities" such as solvents or copolymerized units. For an equilibrium condition in the presence of water as the diluent, the melting temperature of starch (Tm) would be lower because p in the presence of diluent is less than pi). For the starch-water system at equilibrium, the difference between the chemical potentials of the crystalline phase and the phase in the standard state (pure polymer at the same temperature and pressure) must be equal to the decrease in chemical potential of the polymer unit in solution relative to the same standard state (Flory, 1953). By considering the free energy of fusion per repeating unit and volume fraction of water (diluent), the... 

One of the most common techniques for determining x parameters for polymer-solvent systems is the vapor pressure method.(10) In this approach, the uncrosslinked polymer is exposed to solvent vapor of known pressure, p. The polymer absorbs solvent until equilibrium is established, x is related to p and V2, the volume fraction of polymer at equilibrium, by the Flory-Huggins equation (ll)... 

For comparison, a telechelic sulfonated polystyrene with a functionality f = 1.95 was prepared. In cyclohexane the material forms a gel independent of the concentration. At high concentrations the sample swells. When lower concentrations were prepared, separation to a gel and sol phase was observed. Thus, dilution in cyclohexane does not result in dissolution of the gel even at elevated temperatures. Given the high equilibrium constant determined for the association of the mono functional sample, the amount of polymer in the sol phase can be neglected. Hence, the volume fraction of polymer in the gel phase can be calculated from the volume ratio of the sol and gel phases and the total polymer concentration. The plot in Figure 9 shows that the polymer volume fraction in the gel is constant over a wide range of concentrations. 

The data discussed above are still too incomplete to draw quantitative conclusions, but they may be helpful in designing new experiments. We especially hope to gain additional insight from temperature dependent light scattering experiments on monofunctional samples, and from the equilibrium polymer volume fractions of the telechelic polymers in the gel in different solvents. 

Swelling equilibrium is defined as the point at which the activity of the solvent is unity, or p, = p , and the polymer volume fraction at this point is defined as o2jS (see Fig. lb). At equilibrium, Eq. (9) becomes,... 

In Eq. (1), a is the equilibrium stress (Nm 2) supported by the swollen specimen a is the stretched specimen length divided by the unstretched length (extension ratio) v2 is the volume fraction of dry protein and p is the density of dry protein. In the common case of tetrafunctional crosslinks, the concentration of network chains n (mol network chains/g polymer) is exactly one-half the concentration of crosslinks, so that n = 2c. The hypothesis that a specimen behaves as if it were an ideal rubber can be confirmed by observing a linear relation with zero intercept between a and the strain function (a — 1/a2) and by establishing a direct proportionality between a and the absolute temperature at constant value of the extension ratio, as stipulated by Eq. (1). 

Fig. 17. Equilibrium swelling and relative volume of the network phase after phase separation in crosslinking of existing chains. Initial volume fraction of the polymer 0.2, x = 0.45, P -> oo [DuSbk (49)]...

The conclusion that the free-volume fraction at Tg is not a universal parameter for linear polymers of differing molecular structure can be qualitatively confirmed by the following arguments71. Assume that at temperatures far below Tg polymeric chains are in a state of minimum energy of intramolecular interaction, Le. the fraction of higher-energy ( flexed ) bonds is zeroS4. On the other hand, let the equilibrium fraction of flexed bonds at T> Tg obey the Boltzmann statistics and be a function of Boltzmann s factor e/kT. Thus, the fraction of flexed bonds at Tg can be estimated from the familiar expression ... 

---