Viscosity Kirkwood-Riseman theory

As in the case of the Rouse relaxation time, it is possible to express x2 in term of the intrinsic viscosity either by using Eq. (24) or the Kirkwood-Riseman theory [24, 47] ... 

The estimation of f from Stokes law when the bead is similar in size to a solvent molecule represents a dubious application of a classical equation derived for a continuous medium to a molecular phenomenon. The value used for f above could be considerably in error. Hence the real test of whether or not it is justifiable to neglect the second term in Eq. (19) is to be sought in experiment. It should be remarked also that the Kirkwood-Riseman theory, including their theory of viscosity to be discussed below, has been developed on the assumption that the hydrodynamics of the molecule, like its thermodynamic interactions, are equivalent to those of a cloud distribution of independent beads. A better approximation to the actual molecule would consist of a cylinder of roughly uniform cross section bent irregularly into a random, tortuous configuration. The accuracy with which the cloud model represents the behavior of the real polymer chain can be decided at present only from analysis of experimental data. 

Fig. 12. Viscosity-molecular weight relations in polystyrene. The three dotted curves show the effect of partial draining according to the Kirkwood-Riseman theory, Eqs. (50), (51) and (62), with three different choices of drainage parameter...

To investigate further the absence of the draining effect, three dotted curves are also given in Fig. 12. These correspond to the intrinsic viscosities in the theta solvent predicted by the Kirkwood-Riseman theory 139) according to Eqs. (50) and (51)1 with three different choices of the drainage parameter X. It is convenient to express the friction constant of a chain segment in the form of the Stokes law,... 

The authors of Ref. [53] have shown, that frictional properties of fractal clusters can be different essentially for the usual results for compact (Euclidean) structures. It is known through Ref. [54], that the polymer melt structure can be presented as a macromolecular coils sets, which are fractal objects. Therefore, the authors [55] proposed general structural treatment of polymer melt viscosity within the framework of fractal analysis, using the model [53]. Within the framework of the indicated model the derivations for translational friction coefficient f(N) of clusters from N particles in three-dimensional Euclidean space were received, calculated according to Kirkwood-Riseman theory in the presence of hydrodynami-cal interaction between the cluster particles. The fundamental relationship of this theory is the following equation [53] ... 

Based on the Kirkwood-Riseman theory, if polymer chains are nondraining, it follows [Kirkwood and Riseman, 1948 Auer and Gardner, 1955] that intrinsic viscosity data can be related to the radius of gyration, Rg, of flexible polymers. This can be expressed in an equation of the form [Flory and Fox, 1951]... 

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... 

The theoretical prediction of these properties for branched molecules has to take into account the peculiar aspects of these chains. It is possible to obtain these properties as the low gradient Hmits of non-equilibrium averages, calculated from dynamic models. The basic approach to the dynamics of flexible chains is given by the Rouse or the Rouse-Zimm theories [12,13,15,21]. How-ever,both the friction coefficient and the intrinsic viscosity can also be evaluated from equilibrium averages that involve the forces acting on each one of the units. This description is known as the Kirkwood-Riseman (KR) theory [15,71 ]. Thus, the translational friction coefficient, fl, relates the force applied to the center of masses of the molecule and its velocity... 

Several theoretical tentatives have been proposed to explain the empirical equations between [r ] and M. The effects of hydrodynamic interactions between the elements of a Gaussian chain were taken into account by Kirkwood and Riseman [46] in their theory of intrinsic viscosity describing the permeability of the polymer coil. Later, it was found that the Kirdwood - Riseman treatment contained errors which led to overestimate of hydrodynamic radii Rv Flory [47] has pointed out that most polymer chains with an appreciable molecular weight approximate the behavior of impermeable coils, and this leads to a great simplification in the interpretation of intrinsic viscosity. Substituting for the polymer coil a hydrodynamically equivalent sphere with a molar volume Ve, it was possible to obtain... 

The viscosities of many binary liquid systems display minima as functions of composition at constant temperature, so that negative values of D are also possible. Yajnik and his coworkers (265 ) long ago observed that very frequently an extremum in the isothermal vapor pressure-composition curve is accompanied by an extremum of the opposite sense in the viscosity-concentration curve. Data are apparently not available for solutions of very low-molecular-weight paraffins in carbon tetrachloride, but minima are found for the viscosities of solutions of CC14 with ethyl iodide, ethyl acetate and acetone, so that a minimum appears quite probable for mixtures of small aliphatic hydrocarbons with carbon tetrachloride. If this were true, the downward trend of the Meyer-Van der Wyk data on C17—C31 paraffins, earlier discussed in connection with the polyethylene plots of Fig. 14, would be understood. It will be recognized that such a trend is also precisely what is to be expected from the draining effect of the hydrodynamic theories of Debye and Bueche (79), Brinkman (45 ) and Kirkwood and Riseman (139). However, the absence of such a trend in the case of polyethylene... 

The theory of non-Newtonian viscosity for ellipsoidal particles was first explicitly stated by Kuhn and Kuhn (1945), using Peterlin s distribution function (Peterlin, 1938) and Jeffery s hydrodynamic treatment (Jeffery, 1922-1923) [Eq. (10)]. More elegant treatments have recently been developed by Saito (1951), using the same ellipsoidal model, and also by Kirkwood and his co-workers (Kirkwood, 1949 Kirkwood and Auer, 1951 Kirkwood and Plock, 1956 Riseman and Kirkwood, 1956) for rodlike particles. The equivalence of the three theories has also been demonstrated by Saito and Sugita (1952). The general solution of Eq. (10) for the viscosity increment, v, can be expressed in the form... 

The theory of Zimm (7) uses the assumptions of the Rouse theory and in addition considers hydrodynamic interactions between the moving submolecules and the solvent. The theory also makes use of the method formulated by Kirkwood and Riseman for the evaluation of the viscosity of dilute polymer solutions. A parameter h = where r is the... 

INTRINSIC VISCOSITY OF SHORT CHAINS. I. EXTENSION OF THE THEORY OF KIRKWOOD AND RISEMAN. 

The Zimm theory includes an alternative treatment in which frictional resistance to motion of the beads in the bead-spring chain is dominated by the viscous drag from other beads in the same chain (dominant hydrodynamic interaction. Fig. 9-5-11). The interaction is treated approximately as in the theory of Kirkwood and Riseman for the intrinsic viscosity of dilute polymer solutions, by use of the equilibrium-averaged Oseen tensor for the influence of the motion of one bead on another the average distances between pairs of beads are supposed to correspond to those in a 0-solvent. 

Fox and Flory then applied the work of Kirkwood and Riseman to the problem of intrinsic viscosities in polymer solutions. They were able to use the single chain expansion theory of Flory and Krigbaum to derive a simple expression for the intrinsic viscosity in the high molecular weight Umit ... 

Recently Riseman and Kirkwood extended a treatment previously developed for coiled macromoleculcs to the calculation of the viscosity of solutions of rod-like particles in strong BnowNian motion. Although their method of calculation is beyond reproach, it is difficult to compare the results cither with other theories or with experiments, They use as a model a series of pcarls " fixed at equal distances on a stiff rod ... 

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