Exponent good solvent

The dynamic scaling argument supposes that when the geometrical parameters of the chain (i.e. N and lp) are changed from N into N/A and lp into lpV, any physical quantity (A), either static or dynamic, related to the molecular size will be transformed into XXA. The parameter v is the exponent in Eq. (9) and is equal to 1/2 in 0-solvent and 3/5 in good solvent. 

Relationships between dilute solution viscosity and MW have been determined for many hyperbranched systems and the Mark-Houwink constant typically varies between 0.5 and 0.2, depending on the DB. In contrast, the exponent is typically in the region of 0.6-0.8 for linear homopolymers in a good solvent with a random coil conformation. The contraction factors [84], g=< g >branched/ <-Rg >iinear. =[ l]branched/[ l]iinear. are another Way of cxprcssing the compact structure of branched polymers. Experimentally, g is computed from the intrinsic viscosity ratio at constant MW. The contraction factor can be expressed as the averaged value over the MWD or as a continuous fraction of MW. 

The plateau adsorbances at constant molecular weight increased linearly with the square root of NaCl concentration. For the same NaCl concentration the adsorbance was nearly independent of the molecular weight. The thickness of the adsorbed layer was approximately proportional to the square root of the molecular weight for the Theta solvent (4.17 M NaCl). For good solvents of lower NaCl concentrations the exponent of the molecular weight dependence of the thickness was less than 0.5. At the same adsorbance and molecular weight the cube of the expansion factor at, defined by the ratio of the thicknesses for good solvent and for Theta solvent, was proportional to the inverse square root of NaCl concentration. 

The experimental data corresponding to one labeled arm in stars of f=12 (good solvent) [68] shows, as expected, Kratky plot ordinates that increase monotonously with q. However, the plateau is only obtained with an apparent critical exponent of 2/3 (i.e., greater than the theoretical value, v-3/5). This seems an indication of the arm stretching effect, though the scaling and RG theoretical predictions describe this effect only in terms of a pre-exponential factor [11,42]. 

These low exponents seem to suggest poor solvent behavior. However, the second virial coefficients are clearly positive and still fairly large and prove good solvent behavior. Surprisingly the molar mass dependencies of and Rj of unfractionated samples are almost indistinguishable from those of their hnear analogues. 

The exponent can vary from v=0.33 for hard spheres up to v=1.0 for rigid rods. For linear chains v=0.5 refers to unperturbed coil dimensions in -solvents and v=0.588 [6] to good solvent conditions. Equation (37) maybe re-writ-ten by expressing the molar mass as a function of the radius of gyration, i.e.. 

As pointed out in [6] the extended coil conformation in good solvents leads to different exponents for p=air N p, and Tp=a ri N tyik Tand D=a k T/ (rfN ) with the Flory exponent v and the numerical prefactors aj also dependent on the conformation. 

These exponents in Eqs. (4.9) and (4.10) correspond to vF = 3/5 and 1/2 in Eq. (4.5), respectively. In other words, the solvent quality changes from a good solvent to near a solvent by the introduction of crosslinks. This conclusion agrees with the work by Cohen et al. [82] where they studied the crosslinking density dependence of the solvent quality. 

Figure 5.6 shows the same data plotted as a function of cM0 6S to test the low concentration reduction scheme based on c rf] with a typical value of the Mark-Houwink exponent for good solvents. The data have been shifted vertically to achieve superposition at high molecular weights. It is clear that the cM variable produces a better superposition of data at all molecular weights and concentrations. The apparent variation in the values of cM at the intersections in Fig. 5.4 (Table 5.1) is largely due to a lack of data to define the limiting behavior at low molecular weights at some concentrations. The intersection on the superposed plot in the composite Fig. 5.5 is cM = 30000, giving Mc = 30600 for undiluted polystyrene (q = 0.98 at T = 217° C, in good agreement with the value 31200 reported by Berry and Fox (16).

Here F(q) is a function of radius of gyration and composition of the block copolymer. This equation should be compared with eqn 2.11 for block copolymer melts. The effective chi parameter in semidilute solution is X N = %abiV0(1+ )/(,v l), where yAB is the chi parameter for the block copolymer, v is the Flory exponent (v = 0.588 in good solvents) and z = 0.22 (Fredrickson and Leibler 1989 Olvera de la Cruz 1989). The function F q) has a minimum, and hence S(<7)-1 has a maximum, at q = q, which is independent of % and thus temperature. Empirically, is found to be inversely proportional to temperature... 

Experimentally, good solvent conditions have been observed [22,23,27,28, 34,35]. On the other hand, none has been reported for the prediction of the theta condition, y = 101, whereas the prediction of poor solvent conditions giving rise to y > 3 has been reported. These all have y < 20 except for two they are poly(methyl acrylate) at lower temperatures [34] and poly(dimethyl siloxane) [24]. Others have failed to reproduce them since. A caveat needs to be raised with these results. Since the semi-dilute regime is so narrow in r before the collapse state sets in whereby the power exponent is commonly deduced for a r range less than one full decade hence, the r scaling is at best qualitative in the static characterization. 

Real chains in good solvents have the same universal features as self-avoiding walks on a lattice. These features are described by two "critical" exponents y and v. The first is related to chain entropy, the second to chain size a real chain has a size that is much larger than that of an ideal chain (Nv instead of N1/2, where v 3/5 in good solvents) in good solvents the conformation of the chain is "swollen". 

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