Viscosity expansion factor

SAN resins show considerable resistance to solvents and are insoluble in carbon tetrachloride, ethyl alcohol, gasoline, and hydrocarbon solvents. They are swelled by solvents such as ben2ene, ether, and toluene. Polar solvents such as acetone, chloroform, dioxane, methyl ethyl ketone, and pyridine will dissolve SAN (14). The interactions of various solvents and SAN copolymers containing up to 52% acrylonitrile have been studied along with their thermodynamic parameters, ie, the second virial coefficient, free-energy parameter, expansion factor, and intrinsic viscosity (15). 

In order to achieve a quantitative separation of the factors responsible for the temperature coefficient of the intrinsic viscosity, K should first be established as a function of temperature by carrying out measurements in -solvents having s covering the temperature range of interest. The expansion factor may then be obtained from the intrinsic viscosity measured at the temperature T in the given solvent. If Cm occurring in Eq. (10) were independent of the temperature, — should then plot linearly with 1/T, However,... 

Fig. 144.—The treatment of expansion factor-temperature data obtained from intrinsic viscosities of polyisobutylene fractions in three pure solvents and in ethyl-benzene-diphenyl ether mixtures. Data for fractions having molecular weights Xl6 of 1.88, 1.46, and 0.180 are represented by O,, and Q, respectively. (Fox and Flory. 2)...

Diffusion and sedimentation measurements on dilute solutions of flexible chain molecules could be used to determine the molecular extension or the expansion factor a. However, the same information may be obtained with greater precision and with far less labor from viscosity measurements alone. For anisometric particles such as are common among proteins, on the other hand, sedimentation velocity measurements used in conjunction with the intrinsic viscosity may yield important information on the effective particle size and shape. ... 

Thus if we know [tj] and [rj]e as a function of molecular weight we can plot the chain expansion factor as a function of concentration. A plot for polybutadiene from the work of Graessley is shown in Figure 5.21 and uses Equation (5.81) to describe the relationship between concentration and intrinsic viscosity. 

Fig. 17. A log-log plot of expansion factor, (obtained from intrinsic viscosity calculations) vs reduced excluded volume, z, for MC data of 12 arms stars with N=25-109 units (symbols as in Fig. 11 ). The solid lines and figures represent slopes corresponding to the predicted asymptotic behaviors for the EV and sub-theta regimes. Reprinted with permission from [143]. Copyright (1992) American Chemical Society...

This value of kn is actually low by an order of magnitude for dilute suspensions of charged spheres of radius Rg. This is due to the neglect of interchain correlations for c < c in the structure factor used in the derivation of Eqs. (295)-(298). If the repulsive interaction between polyelectrolyte chains dominates, as expected in salt-free solutions, the virial expansion for viscosity may be valid over considerable range of concentrations where the average distance between chains scales as. This virial series may be approxi-... 

Ratio of a dimensional characteristic of a macromolecule in a given solvent at a given temperature to the same dimensional characteristic in the theta state at the same temperature. The most frequently used expansion factors are expansion factor of the mean-square end-to-end distance, Ur = (/o) expansion factor of the radius oj gyration, as = (/0) relative viscosity, = ([ /]/[ /]o), where [ ] and [ /]o are the intrinsic viscosity in a given solvent and in the theta state at the same temperature, respectively. 

In principle, intrinsic viscosities used for estimating branching should be measured under conditions where the expansion factor a is unity, but as indicated in Section 6, it is not easy to identify such conditions. Some authors, e.g. Moore and Millns (40) have measured [tf at the theta-temperature of the corresponding linear polymer, but it is doubtful whether a is unity at that temperature for either linear or branched polymer, if the theories of Casassa or of Candau et al. are valid. If a were the same for both linear and branched polymers under the same conditions g would be unaffected and g could be measured at any convenient temperature some authors have presented data suggesting that g is nearly the same in good and poor solvents, e.g. Hama (42) and Graessley (477), but other authors, e.g. Berry (43) have found g to vary. The best that can be done at present would appear to be to measure g at the theta-temperature on the assumption that this ratio will be less temperature-sensitive than either intrinsic viscosity, and that even if this temperature is not the correct one it will be near it. Errors in estimates of branching due to this effect are likely to be much less serious than those due to the use of an incorrect relation between g and g0. 

Solution It is apparent from the units of b] that solute concentration has been expressed in g/cm3. Dividing this concentration by the density of the unsolvated protein converts the concentration to dry volume fraction units. Since the concentration appears as a reciprocal in the definition of [17], we must multiply bl by p2 to obtain (lAA Mb/r/o) - 1]. For this protein the latter is given by (3.36)(1.34) = 4.50. If the particles were unsolvated, this quantity would equal 2.5 since the molecules are stated to be spherical. Hence the ratio 4.50/2.50 = 1.80 gives the volume expansion factor, which equals [1 + (mhb/m2)(p2/p )]. Therefore (m, tblm2) = 0.80(1.00/ 1.34) = 0.6O. The intrinsic viscosity reveals the solvation of these particles to be 0.60 g HaO per gram of protein. ... 

The relation (1 10) leads to a number of interesting consequences. In a theta solvent, in which the shape of the chain is described by the random flight model, is proportional to M2, so that the intrinsic viscosity should be proportional to M /2. And this prediction has been applied and verified. In solvent media better than 0-solvents, the theory of Flory [11,46] predicts that the linear expansion factor a increases for any polymer - homologous series with chain length. Thus the exponent v in the empirical equation should be larger than 0.50. 

Fig. 8. Variations of viscosity parameter with expansion factor. Experimental data of Schulz (229) (circles) and Krigbaum and Carpenter (146) (crosses) Theoretical curves according to Eqs. (55) and (56) for homogeneous polymers (broken curve and chain curve) and for broad fractions (dotted curve and full curve)...

The factor 1.7 arises from the viscosity dependence of 0.7 and the thermal expansion factor of 1.0. With constant inlet and outlet pressures, pjp is constant, and the outlet flow rate then changes with the 1.7 power of the temperature ... 

Now in some cases a direct study of a polymer under 0 conditions is not feasible or, more frequently, not desired. The need to work under 0 conditions when determining unperturbed chain dimensions might be circumvented if one could rely on theories connecting accurately measurable quantities such as intrinsic viscosity, second virial coefficient etc. obtained in good solvents, with the chain expansion factor. 

These data were obtained by estimating the chain expansion factor a from viscosity and osmotic second virial coefficient measurements, through the combined use of equations (10) and (11). These results tend to confirm the main predictions of the theory presented by the above mentioned authors both as regards the steep decrease in with... 

It has been established that Ko normally is independent of the solvent and the molecular weight of the polymer, though often dependent to some extent on the temperature. It is therefore possible to deduce values for the expansion factor in good solvents from intrinsic viscosities measured in them. From Eqs. (3.181) and (3.184) the linear expansion factor a.jj, which is a measure of long range interactions and pertains to hydrodynamic chain dimensions, is thus given by... 

Problem 3.24 The intrinsic viscosity of polystyrene of molecular weight 3.2x10 in toluene at 30°C was determined to be 0.846 dlVg. In a theta solvent (cyclohexane at 34°C) the same polymer had an intrinsic viscosity of 0.464 dL/g. Calculate (a) unperturbed end-to-end distance of the polymer molecule, (b) end-to-end distance of the polymer in toluene solution at 30°C, and (c) volume expansion factor in toluene solution. (3> = 2.5x10 mol )... 

From Equation 12.71, the intrinsic viscosity depends on the molecular weight as a result of the factor and also through the dependence of the expansion factor on molecular weight. By choosing a theta-solvent or 0 temperature, the influence of the molecular expansion due to intramolecular interactions can be eliminated. Under these conditions, a = 1, and the intrinsic viscosity depends only on the molecular weight. Thus Equation 12.71 is reduced to ... 

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