Calculation of the intrinsic viscosity

We shall now calculate the stress using the linearization approximation. As was shown in Section 3.7.3, under the velocity field v(r, t) = K(t) r, each bead has an additional velocity K t) R (see eqn (3.120)). This gives the velocity f(r) Xp in normal coordinates. Thus the Langevin equation for Xp now becomes 

To calculate Xp Xpp) it is convenient to rewrite this into an equivalent Smoluchowski equation.  

If we multiply both sides of eqn (4.140) by Xp Xpp and integrate over all the normal coordinates, we get, i er integration by parts 

This equation can be solved for arbitrary K pit) (see Section 7.6.3), but here we consider a special case of the shear flow given by eqn (4.115). To calculate the shear stress a yit), we have to solve 

For small k, (X y) may be replaced by the equilibrium value kaTIkp. Hence eqn (4.142) becomes 

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In 1959, Zimm and Kilb (34) made some calculations of the intrinsic viscosities of certain branched polymer molecules, taking into account the hydrodynamic interaction between portions of the polymer chain, using a modification of the Rouse procedure. They carried out these difficult calculations for a quite restricted range of models, obtaining numerical results for equalarmed stars with 3, 4, and 8 branches, and for one modified star with 2 long branches and 8 shorter branches. They found that their numerical results for this set of structures could be approximately represented by ... 

Calculation of the Intrinsic Viscosity During the Cellulolysis. The kinematic viscosity (v) is related to the outflow time (t) by... 

For calculation of the intrinsic viscosity [rj] of PSD solutions in tetra-hydrofuran Mark-Kuhn-Houwink equation fractal variant (the Eq. (10)) was used, where the constant c(a) is accepted equal to 2.91. and values, received experimentally, are adduced in Table 24 and value for PSD was calculated as follows. As it is known through Ref [1], for flexible-chain polymers, having a l, Huggins constant can be determined according to the Eq. (174), from which it follows that the condition cc l is fulfilled for the values k 0.55. The Table 24 data showed that the last... 

The Solomon-Ciuta equation allows for the calculation of the intrinsic viscosity from a single concentration ... 

Theoretical calculation of the intrinsic viscosity of regular stars goes back to 1959 [78]. In the free-draining case, the Rouse model for linear polymers applied to stars yields [79]... 

Calculations. For determination of the intrinsic viscosity [ti] the prepared pectins were solved in an 0.1 M phosphate buffer with pH 6.0. The relative viscosity was determined by a glass. Ubbelhode viscometer at 25 0.1 °C. The flow time of solvent (L) was 81.8 seconds. At least six pectin solutions with different concentrations were measured in a way that their flow times (ts) comply the order 1.2to[Pg.528]

Returning to Staudinger s derivation, it must be revised on two grounds. First the kinematics of motion is three rather than two-dimensional and the hydrodynamic volume spherical rather than cylindrical, i.e. 3. The detailed calculation for thin ellipsoidal particles (13) shows an approximate proportionality of the intrinsic viscosity with M1 7, a considerable difference from eq. (1) for large M. 

Since the intrinsic viscosity depends not only on the size of the macromolecule but also on its shape, on the solvent, and on the temperature, there is no simple relationship for the direct calculation of molecular weights from viscosity measurements. However, the Mark-Houwink-Kuhn equation gives a general description of how the molecular weight can be calculated from the intrinsic viscosity ... 

The limiting line for Cri = 0 has been calculated from the data plotted in Fig. 22. The dependence of the intrinsic viscosity of the polymer (and consequently of its molecular weight) on the alkylaluminum concentration... 

On the other hand, the slope of the lines plotted in Fig, 38, from the value of which the mean life of the chains is calculated, may be greatly effected even by small errors in the determination of the intrinsic viscosity of polymers obtained by shorter time tests. The resulting data are mostly concerned by eventual errors. 

The enzymic activity was proportional to the change of the inverse of the numerical average molecular weight per time unit. The linear relationship between Mv and Mn allowed a simple calculation of the activity from the knowledge of the intrinsic viscosity at zero shear rate as a function of the reaction time. A practical example is given. 

The value of the intrinsic viscosity of the sample calculated in this fashion, [77], can then be compared to that obtained experimentally for the whole sample, [77]. But you need a value of A to do these calculations. So what you actually do is first calculate the intrinsic viscosity for the sample as if it were linear, i.e., A = 0. Then, if the cal-... 

Fig. 5. Shearing stress dependence of the intrinsic viscosities of three a-helical polypeptides. The lines are theoretical curves (broken lines to the right of the arrows being extrapolated theoretical curves) 1, ijiiQ/T = 0.76 3, riaO/T = 0.054 2, calculated on the basis of three parts curve 1 and one part curve 3. Reproduced from Yang (1959).

Molecular Weight. The viscosity average molecular wei t of the homopolymer and copolymers was determined from single point determinations of viscosity of a 0.25% solution in Fisher purified cyclohexane at 30.0 + 0.01° C. The molecular wei t is calculated from the intrinsic viscosity, [q], using the relationship ( ... 

Comparison of Chain Dimensions. Knowledge of the intrinsic viscosities and molecular weights of a series of PMDA/DAPE polyamic acids allows estimation of the unperturbed chain dimensions. Comparison may then be made with the cured polyimide, with results obtained by other workers, and with calculated values. The expressions in the literature using values for [n] obtained in good solvents generally involve extrapolation of [t)]/M toM = 0, where excluded volume effects are presumed to be minimal (12-14). Although this method is not strictly valid, it is useful for comparison purposes when direct measurement of the dimensions is impractical. The data in Table I for the polyamic acid in distilled NMP and in the poorer mixed solvent NMP/dioxane yield (see Figure... 

Materials. Two samples of polyacrylic acid, PAA-1 (4) and PAA-2, were obtained as gifts from S. C. Johnson Company. Their degrees of polymerization were 3090 and 4420, respectively, as calculated from the intrinsic viscosities of their sodium salts in 0.102V NaBr at 15°C using the appropriate relation of Takahashi and Nagasawa (5). The samples of the hydrolyzed copolymers of maleic anhydride witn ethylene (HEMA) and with methyl vinyl ether (HVMEMA) have been described previously (4). 

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