Kirkwood-Riseman equation

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

For a large value of the degree of polymerization N (where N = M/Mq), the Kirkwood-Riseman equation of the intrinsic viscosity is reduced to... 

Random coils. Equation (9.53) gives the Kirkwood-Riseman expression for the friction factor of a random coil. In the free-draining limit, the segmental friction factor can, in turn, be evaluated from f. In the nondraining limit the radius of gyration can be determined. We have already discussed f in Chap. 2 and (rg ) in this chapter and again in Chapter 10, so we shall not examine the information provided by D for the random coil any further. 

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. 

Kirkwood and Riseman have developed a theory that allows for variable degrees of solvent drainage through the coil domain. We shall not go into this theory in any detail, except to note that it should reduce to Equation (87) in the free-draining limit and to the Einstein equation in the nondraining limit. The Kirkwood-Riseman theory can be written in the form... 

Yamakawa (38) and Imai (83) have published an alternative description based on a random coil model and the Kirkwood-Riseman theory (62) and obtained for theta-solvent conditions an equation equivalent to ... 

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]... 

Auer, R. L., and Gardner, C. S., Solution of the Kirkwood-Riseman integral equation in the asymptotic limit, J. Chem. Phys., 23,1546-1547 (1955). 

The Yamakawa-Fujii theory [2, 3] was developed by using the Kirkwood-Riseman formalism with the effect of chain thickness approximately taken into account. The following remarks may be in order. The Oseen interaction tensor was preaveraged. Force points were distributed along the centroid of the wormlike cylinder (not over the entire domain occupied by the cylinder). The no-slip hydrodynamic condition was approximated by equating the mean solvent velocity over each cross-section of the cylinder to the velocity of the cylinder at that cross-section (Burgers approximate boundary condition). 

It can also be shown from the preaveraged Kirkwood-Riseman-Zimm equation that the mean square displacement of a labeled monomer i is proportional to the 2/3 power of time for times shorter than rzimm. 

The results of Equations 7.46, 7.48, and 7.50 for the Zimm dynamics are entirely consistent with the universal laws expected in Section 7.2.1 and are fully supported by experimental data in dilute solutions. If the hydrodynamic interaction among segments is suppressed in the Kirkwood-Riseman-Zimm equation, then the problem reduces to the Rouse dynamics and all results of Section 7.2.2 are recovered. 

At zero wavevector =0, this formula is equivalent to the Kirkwood-Riseman formula (equation 27) D( = 0) is the center of mass diffusion constant Dq. 

Hie cylinder model gave no theoretical lalue of ]i/] for short flexible chains (AL<2.278 and Ad>0.1) owing to the nature of the kernel in the integral equation. To evaluate ]i/] for such chains, Yoshizaki et adopted the touched-bead model, that is, a discrete chain consisting of N beads of diameter db whose centers are located on the continuous KP or HW contour. These authors showed ](/] to be represented approximately by the sum of the solution of the Kirkwood-Riseman integral equation and the contribution of N beads as Einstein spheres. The result for the KP chain may be expressed in the form... 

The integral equation for < )(x) was originally derived by Kirkwood and Riseman [1]. By defining the Fourier transforms... 

Equation (3.60) has just the form of the well-known Flory-Fox equation (98). As is well-known, this equation holds for sufficiently high molecular weights. With decreasing molecular weights shielding is expected to become less effective [cf. also Kirkwood and Riseman ( 8)]. Apparently, the Flory-Fox parameter of eq. (3.60) reads... 

It is interesting to compare this parameter with the one obtained by Kirkwood and Riseman for the non-draining case, when these authors equated the centre of resistance with the centre of gravity. It reads... 

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... 

In these equations, e and es are the values of v and ve, respectively, in the free-draining case. The former quantity is tabulated by Kirkwood and Riseman (139) and the latter is easily found from their equation for the friction constant. 

After solving the integral equation (see Appendix), Kirkwood and Riseman found that... 

Equations (25) - (29) determine the simplest approach to the dynamics of a macromolecule, even so, it appears to be rather complex if the effects of excluded volume, hydrodynamic interaction, and internal viscosity are taken into account. Due to these effects, all the beads in the chain ought to be considered to interact with each other in a non-linear way. To tackle with the problem, this set of coupled non-linear equations is usually simplified. There exist the different simpler approaches originating in works of Kirkwood and Riseman [46], Rouse [2], Zimm [5], Cerf [4], Peterlin [6] to the dynamics of a bead-spring chain in the flow of viscous liquid. The linearization is usually achieved by using preliminary-averaged forms of the matrix of hydrodynamic resistance (hydrodynamic interaction) [5] and the matrix of the internal viscosity [4]. In the last case, to ensure the proper covariance properties when the coil is rotated as a whole, Eq. (29) must be modified and written thus... 

The general expectations embodied in Equations 7.12, 7.16, and 7.19 are borne out to be valid as shown by experiments in dilute solutions of uncharged polymers. Depending on the experimental conditions, the value of the size exponent changes and this change is directly manifest in D, rj, and t in terms of their dependencies on the molecular weight of the polymer and solvent conditions. In order to obtain the numerical prefactors for the above scaling laws and to understand the internal dynamics of the polymer molecules, it is necessary to build polymer models that explicitly account for the chain connectivity. The two basic models of polymer dynamics are the Rouse and Zimm models (Rouse 1953, Kirkwood and Riseman 1948, Zimm 1956), which are discussed next. 

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