Kirkwood-Riseman approximation

Thus a simple power law behavior with an exponent of 1.5 would result if 0"= 1 [125]. Zimm and Kilb [128] made a first attempt to calculate g for star branched macromolecules on the basis of the Kirkwood-Riseman approximation for the hydrodynamic interaction. They came to the conclusion that... 

Although flvv for V 0 can be readily computed from Eq. (110), we use the Kirkwood-Riseman approximation of replacing Ow by the values asymptotically valid for large v and v, ... 

By invoking the Kirkwood-Riseman approximation of diagonality in aw, we obtain... 

Dynamic light scattering from dilute solutions provides the value of the diffusion coefficient, which can be converted to hydrodynamic radius J h,star of the star polymer. The ratio Rh/(Rg) characterizes the compactness of the macromolecule for the uniform hard sphere impenettable for the flow, it is Rb/Rg= (5/3) 1.29, whereas for the Gaussian coil, Rh/(Rg) = 3a- / /8 0.66. For ideal stars (without excluded-volume interactions), the ratio Rh/(Rg) can be derived within the Kirkwood-Riseman approximation, which gives the value of Rh/(Rg) 0.93. Reported experimental values of the Rh/(Rg) ratio for star polymers and starlike block copolymer micelles are usually found close... 

Motion of a Fluid Zimm adopted the Kirkwood-Riseman approximate form of the Oseen interaction formula to describe the force on the motion of a fluid ... 

Yamakawa and Fujii formulated So of a KP cylinder on the basis of the Oseen-Burgers method with the preaveraged (or nonflurtuating) Oseen HI tensor and the Kirkwood-Riseman approximation. Their expression (for 2L>2.278) is written in terms of Rh as... 

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. 

Barrett [1984] has further proposed TP expressions for and an, formulated within the Kirkwood-Riseman hydrodynamic theory in the nondraining limit and using approximate formulas, based on numerical simulation, for the requisite statistical averages, (/f >and Rf/), where Ry refers to the distance between the ith andyth chain segments ... 

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). 

The Kirkwood-Riseman theory allows for an approximate calculation of the hydrodynamic drag of arbitrary clusters of small objects (molecules, particles) with undefined shape. It is based on the fact that, for low Reynolds-numbers, the velocity perturbations originating from the primary particles superpose linearly. The hydrodynamic force on a primary particle j results from the total perturbation field around j (yj) and from the undisturbed flow u°°. To first approximation, it can be assumed that the presence of j does not afl ect y, i.e. the spatial extension of particle j is neglected. Moreover, the perturbation of the velocity field at the place of j can be related to the perturbation forces acting on the other particles ... 

Due to its simplicity, the Kirkwood-Riseman theory has been widely used in the literature for estimation of the hydrodynamic diameter of colloidal aggregates (Chen et al. 1984 Hess et al. 1986 Wiltzius 1987 Naumann and Bunz 1991 Lattuada et al. 2003 Sandkiihler et al. 2005a). However, this is an approximate approach which considers the hydrodynamic interaction only within a first-order correction of the unperturbed force and which neglects the finite size and the shape of the primary particles. Hence, the Kirkwood-Riseman theory is best applicable for very porous aggregates and is expected to fail for very compact ones (de la Torre and Bloomfield 1977 Binder et al. 2006). Attempts to consider the size and shape of the primary particles within this framework lead to comparably cumbersome expressions of the hydrodynamic drag (de la Torre and Bloomfield 1977) and were obviously not able to compete with other theories or numerical methods. 

The extremely simple expressions for intrinsic viscosity at zero gradient and sedimentation coefficient containing in the denominator the square root of molecular weight agree very well with experimental data of theta solutions in the whole range from fully drained to impermeable coil. They turned out to be very good approximations of the more complicated expressions derived by Kirkwood-Riseman and Debye-Bueche ° and hence were used for the calculation of intrinsic viscosity of polydisperse samples. 

Tests of the validity of the Kirkwood-Riseman picture, inquiring directly if diffusing objects actually have cross-diffusion tensors that match their supposed hydrodynamic interactions, have recently been accomplished Crocker used videomicroscopy and optical tweezers to study the correlated Brownian motions of a pair of 0.9 xm polystyrene spheres, thereby determining their cross-diffusion ten-sors(3). Crocker found that the diffusion tensors are accurately described by the hydrodynamic interaction tensors, exactly as Kirkwood and Riseman had assumed. An optical trap experiment by Meiners and Quake observed the motions of two Brownian particles, further confirming the validity of the Oseen approximation for hydrodynamic interactions(4). 

The hydrodynamic scaling model is an extension of the Kirkwood-Riseman model for polymer dynamics(l). The original model considered a single polymer molecule. It effectively treats a polymer coil as a bag of beads. For their collective coordinates, the beads have three center-of-mass translations, three rotations around the center of mass, and unspecified other coordinates. The use of rotation coordinates causes the Kirkwood-Riseman model to differ from the Rouse and Zimm models(2,3). The other collective coordinates of the Kirkwood-Riseman model are lumped as internal coordinates whose fluctuations are in first approximation ignored. The beads are linked end-to-end, the links serving to estabhsh and maintain the coil s bead density and radius of gyration. However, the spring constant of the finks only affects the time evolution of the internal coordinates it has no effect on translation or rotation of the coil as a whole. 

When a coil moves with respect to the solvent, each bead sets up a wake, a fluid flow described in first approximation by the Oseen tensor. The fluid flow velocity near each bead is perturbed by the wakes established by all the other beads, so the fluid flow created by all the beads must be computed in a self-consistent manner. To find concentration dependences, an extended Kirkwood-Riseman model is applied to several polymer chains. The extended model leads to a power series in polymer concentration. A process is then needed to take the power series to large concentration. The original calculation used a self-similarity argument to compute the concentration dependence of Dj(4). The retardation of motion of one polymer... 

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... 

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 second average is not easy. Kirkwood and Riseman introduced the following approximation... 

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. 

The first term Sij/C is the local Rouse term the Oseen tensor describes the hydrodynamic interactions. A frequently used approximation, introduced first by Kirkwood and Riseman, averages the Oseen tensor over the configurations of the polymer chain in its equilibrium Gaussian state. This replaces the Oseen tensor by the preaveraged Oseen tensor... 

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