The Flory Parameter

The tendency of differing blocks to microseparate from each other is quantified by Flory s chi parameter /, introduced in Chapter 2. An increasing, positive value of x implies an increasing tendency for the two chemically dissimilar species to segregate from each other. As discussed in Section 2.3.1.2, for a blend of two different homopolymers (A and B) of equal degree of polymerization Na — Ag at a 50/50 composition, the Flory-Huggins theory predicts that phase separation should occur at a critical value of Xc = For block 

Typically, x can be increased by lowering the temperature. The temperature-dependence of X is often represented as 

The temperature dependence of x varies considerably from one pair of monomers to another. For example, for the pair styrene/isoprene, one estimate of x is (Lin et al. 19941 

For the pair styrene/isoprene, various other dependences of x or temperature have been reported in the literature, which give values of x that at 100°C differ by almost a factor of two, from 0.06 to around 0.14 (see Fig. 13-3) (Lin et al. 1994 Han et al. 1995). These estimates of x ot obtained by direct calorimetric measurements (the energetic effects are too minute to measure), but by fitting the predictions of thermodynamic theories (such as that of Leibler or Fredrickson and Helfand see below) to x-ray or neutron scattering data for diblock copolymers or blends. The values of / thereby obtained are only as accurate as the theories to which the data are fitted. 

Furthermore, in defining /, one must deal with the complication that the monomer volumes of the two blocks usually differ, and so one must arbitrarily choose a reference volume, V, of a reference chain segment. This could, for example, be the volume of one 

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Xaf. the Flory parameter of pair interaction between units Ma and M ... 

In these formulas the letter X stands for the average copolymer composition, while of denotes the dispersion of the SCD quantitatively characterizing its width. The second of these statistical characteristics is extremely significant for the thermodynamics of the melt of a heteropolymer specimen, being in a simple way AHmix = RT jof connected with the specific enthalpy of mixing Affmix per mole of monomeric units. Here T is the absolute temperature, R represents the gas constant, whereas / denotes the Flory /-parameter whose values are available from the literature for many pairs of monomeric units (see, for example, [7]). 

The Flory parameter is in essence analogous to the activity coefficient of the HOC in the DOM phase and consists of two components %u and 5Cs> which are the enthalpic and entropic contributions, respectively ... 

Where V2/V1 (r) is the ratio of the molar volume of the polymer to that of the solvent and x is the Flory parameter which depends primarily on the intermolec-ular forces between solute and solvent. According to the original formulation, this parameter is zero for athermal mixtures. However, subsequent work has shown that both the excess entropy and the excess enthalpy contribute to x ... 

Monte Carlo calculations based on the lower bound method were applied to estimate the Flory parameter for asymmetric stars, whereas, for the symmetric stars, experimental results on three- and four-arm homopolymer stars were used. The obtained values of gg for different macroscopic states together with those obtained from RG and MC calculations are presented in Table 2. p is the ratio of the lengths of the B and A arms (p=NB/NA). 

In practice, the Flory-Huggins theory fails to predict many features of polymers solutions, either qualitatively or quantitatively, but remains widely used because of its simplicity. The Flory parameter x, assumed to be constant, often increases with interaction-energy scaled on kT, often exhibits a more complicated temperature dependence than 1/T (Flory, 1970). Such behavior stems from energetic effects, such as directional polar... 

Since the Flory interaction parameter, x> was derived by considering only interaction energies between the molecules, it should not contain any entropic contributions and Equation (2D-9) should yield the correct value for the Flory-/ parameter. Unfortunately, x contains not only enthalpic contributions from interaction energies, but also entropic contributions. The solubility parameter includes only interaction energies and by the definition of regular solutions does not include any excess entropy contributions. Blanks and Prausnitz (1964) point out that the Flory / parameter is best calculated from... 

Mantovani F, Grassi M, Colombo I, Lapasin R. A combination of vapor sorption and dynamic laser light scattering methods for the determination of the Flory parameter chi and the crosslink density of a powdered polymeric gel. Fluid Phase Equilib 2000 167(1) 63— 81. 

So while not perfect, the value of the Flory parameter chosen, 0.475, gives a reasonably good description of the cyclohexane-polyisobutylene system. 

The SEC columns were calibrated using narrow molecular weight distribution water soluble polymer standards sodium polystyrene sulfonate, NaPSS (Pressure Chemical Company, Pittsburgh, PA) and dextrans (Pharmacia Fine Chemicals, Piscataway, NJ). The universal calibration curve proposed by Grubisic et al. (13) was found not to be applicable for these polymer standards. The alternative calibration procedure of Coll and Prusinowski (J ) which incorporates excluded volume effects in the numerical value of the Flory parameter was found to be applicable for this system ( ). The calibration curve for the two-column network system described above is shown in Figure 3. The Kyyy parameter used on the abscissa of Figure 3 is a normalized elution volume parameter defined by Equation 1,... 

Depending on the Flory parameter x, there is a particular special temperature T = 6 at X = 1/2, which corresponds to an exact cancellation between steric repulsion and van der Waals attraction between monomers, and thus the chains are nearly ideal. This temperature is known as the collapse temperature or theta temperature [15]. Equation (15.10) for the free energy of mixing is an expression that finds wide use in physical chemistry. A quantitative understanding of hydrophobie eollapse is required to understand the initial stage of protein folding, as proteins are often a finite chain consisting of a 50-300 amino acid residue linear chain, which in many aspects resembles a heteropolymer. 

The usual one is to take into account only tte active chains and jum tions. The difunctional junctions are thus not iiKluded. Therefore the molecular wdght between crosslinks can be different from M and its distribution is unknown. Nevertheless one gains more knowledge about the functionality of the considered juiKtims, three in an imperfect trifunctional network, a mixture of three and four in a tetrafunctional one. The Flory parameter v. and the Graessley parameter h are not defined. One can take an average over the distribution of molecular weights between crosslinks and the different functionalities. 

Table 7. Values of the Flory parameter x for imperfect trifunctkxial networks = 0), using e p i-mental data from Ref. S2...

The aim of this paper is to clarify current discussions about rubber elasticity particularly with regard to entanglement contributions. [A similar review convering some of the same topics has recmtly been prqjared by Eichinger After a brief review of the existing molecular theories, it is shown that the Flory parameters can be determined by a new method of fitting the stress-strain curves obtained for perfect end-linked PDMS nrtworks. For networks assumed to be imperfect, the branching... 

The Flory % parameter has foimd widespread use in the discussion of any two-component mixture, including polymer mixtures. The Gibbs energy of mixing can then be written as ... 

The van t Hoff Law is recovered in the limit of infinite dilution, since second virial coefficient is zero when % = 1/2. The identification of the Flory temperature as the point where the Flory % parameter is 1/2 helps to guide the discussion of the behavior of the osmotic pressure as a function of %. 

The volume fraction of the network at swelling equilibrium depends on the number density of elastically effective chains in the dry network and the solvent quality as measured by the Flory % parameter. In general, the volume... 

Solution ProportiGS. The amide group (—CONH2) in poly(acrylamide) provides for its solubility in water and in a few other polar solvents such as glycerol, ethylene glycol, and formamide. We can acquire a sense of poly(acrylamide) s affinity for water by examining a few characteristic parameters. Theta ( ) conditions for a polymer delineate a particular combination of solvent and temperature at which the polymer acts in an ideal manner (22), ie, the chains behave as random coils. The 0 temperature of poly(acrylamide) in water has been determined to be -8°C (23). Thus, water at 25°C is a solvent of intermediate quality for poly(acrylamide). Aqueous methanol (40 vol%), however, is a solvent for poly(acrylamide) at 25°C (24). The Flory / parameter, which is a measure of the relative affinity between the polymer segments with each other vs with the solvent, is 0.5 under conditions. The Flory / parameter of poly(acrylamide) has been determined to be 0.48 in water at 30°C (25). These and other properties of poly(acrylamide) in solution are collected in Table 2. 

V (mVkg) is the specific volume per segment m (kg/mol) is the molecular weight per segment Na is the Avogadro number X is the Flory parameter... 

Ivin and Leonard 9 extended the thermodynamic treatment of the polymer-monomer equilibrium to the nonideal systems, accounting for the polymer-monomer interaction described by the Flory parameter Xmp bulk process, the following expression was obtained for the free energy change upon the conversion of one mole of pure monomer into one base-mole of amorphous polymer (AG ) ... 

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