Non-draining limit

The calculation of g for Gaussian uniform star chains was carried out by Zimm and Kilb (ZK) [83]. They used a modified version of the dynamic Rouse theory including preaveraged HI (in the non-draining limit) that considers the particular connectivity of units consistently with the star architecture. This ap-... 

Fig. 3.1. Reduced stored free energy Fs vs. reduced shear rate py for three samples of anionic polystyrene in monobromo-benzene at 25° C 70). Data of these samples are gathered in Table 3.2. The following concentrations given in g/100 cm8, are used Taps. No. 5 ( -) 0.50, (- ) 0.35, (J) 0.20, (1) 0.10 Taps. No. 17 (o-) 1.00, (-O) 0.50, () 0.35, (i) 0.20 Taps. No. 15 (4) 1.00, (It) 0.633, (. ) 0.40. "F" indicates free-draining, N non-draining limit according to Zimm (SO). For the meaning of the broken lines see Section 5.3.2...

As mentioned above, in the non-draining limit, both the viscosity and friction... 

Zimm [116] in 1980 reported Monte Carlo solutions to the non-preaveiaged Kirkwood-Riseman equations, giving in the non-draining limit... 

As mentioned repeatedly, according to the two-parameter theory, the expansion factor as> the penetration function and the hydrodynamic expansion factors a, and at in the non-draining limit should become universal functions of a single variable z. These non-dimensional quantities are experimentally determinable without any assumption. Thus, the validity of the two-parameter theory can be tested directly by looking at whether a single curve independent of polymer rind solvent condition (solvent species and temperature) is obtciined or not when any of them is plotted against the other. Such tests were made by many authors (for example, see Ref. [2] and [119]). Here we refer to a recent one by Miyaki and Fujita [49] (and also Miyaki [44]), who used the following criteria A and B. 

Their approximation has already been discussed in Section 1.10 of Chapter 2, and, according to the estimate of Barrett shown there, the error introduced is as much as 10%. In this way, Weill and des Cloizeaux obtained for the friction expansion coefficient a = R i/Rii in the non-draining limit (h —> oo)... 

The intrinsic viscosity [//(r)] and the friction coefficient f r)g of an unperturbed ring polymer were first calculated by Bloomfield and Zimm [69] and by Fukatsu and Kurata [70] using the Kirkwood-Riseman theory with the preaveraged Oseen interaction tensor. In the non-draining limit their calculations yield... 

First-order calculations of [7 (r)j and /(r) at the non-draining limit were made by Norisuye and Fujita [84] and then by Shimada and Yamakawa [85],... 

DLS is probably the most frequently used technique to determine the size of microgel particles. Miaogels are strongly swollen by the solvent and usually the non-draining limit is reached, that is, the solvent is immobilized inside the microgel. Consequently, DLS is not sensitive to the internal stmcture of the miaogels and the hydrodynamic radius that is obtained... 

The predictions of the Zimm model for a Gaussian chain have a very simple interpretation the Zimm limit is a non-draining limit, the solvent flow does not penetrate into the polymer chain and, as far as hydrodynamic properties are concerned, a polymer chain can be considered as a hard sphere. The diffusion constant is then given by Stokes law, and the intrinsic viscosity by the Einstein equation for dilute solutions of hard spheres, the only difference being in the numerical factors. 

In the non-draining limit, polymer chains behave as hard spheres and the dynamic exponent is z = 3 leading to a variation of the characteristic time with the molecular mass... 

In a non-draining limit, the scaling structure of the intrinsic viscosity is given by equation (30)... 

In the non-draining limit, which is a good approximation for dilute solutions, polymer coils behave as hard spheres. For hard spheres of radius R there are discrepancies between the values of the hydrodynamic interaction coefficients calculated by different authors. A critical comparison of most of the published approaches can be found in refs. 40 and 41. The most commonly assumed values are kD= 1.56[(47t/ )/3] and ks=6.55[(47rR )/3]. 

Although/ A in both the non-free-draining limit for low salt solutions and the free-draining limit, the terms appearing as prefactors are qualitatively different. 

For K 7 0, a part of Aqo cancels 1 jft exactly in the non-free-draining limit and the remainder is dependent on the structure factor of the polymer and the size exponent v. For large values of KRg, p becomes... 

In this way an extrapolation of data to zero concentration seems avoidable. This is in accordance with an early expectation of Peterlin and Signer (114). A refinement of this view, however, will be given below and in Chapter 4. Furthermore, a fair reduction with respect to the molecular weight is noticed. All measuring points of Fig. 3.1, which were obtained at / values smaller than two fall within the limits of the theory indicated by the lines "N and F (non-draining and free-... 

Solutions. Four sets of data indicate that the non-free draining limit of the Zimm theory (29) describes low to moderate frequency dependence of intrinsic complex modulus for linear polymers in 0-solvents. Figure 3.2 shows the result for PIB in benzene at 24° C, the 0-temperature for this system (2,92). In this figure, G R and GR are... 

For the bead-spring model, a/b has to be smaller than 1/2 to avoid the interpenetration of the neighboring spheres. The value of ft described by Eq. (3.1) satisfies this criterion and so is consistent with the model on which the theory is based However, it should be noted that this favorable result is obtained within the framework of the Zimm theory. The value of ft at the non-free draining limit is 1/4 for Gaussian chains but it is different from 1/4 for chains of other distribution. Moreover,... 

Table 1. Calculated values of g, g, h and g2 for regular star-branched polymers (non-frec draining limit)...

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