Comparison of Approximate Methods for Calculating the Friction Coefficient and Intrinsic Viscosity of Nanoparticles and Macromolecules

Abstract

A number of methods for estimating the translational friction coefficient f and the intrinsic viscosity [η] of polymers and nanoparticles have been proposed. These methods range from first-principles “boundary-element” or “bead-model” solutions of the Stokes equation employing a precise description of particle shape, to coarse-grained descriptions of polymer structures and approximate computational methods at an intermediate level of description, and finally to phenomenological estimates that relate f to the surface area of the particle. Analytic treatments normally involve slender-body and various “preaveraging” approximations, etc., that render the calculation analytically tractable, but numerically uncertain. Powerful numerical path-integral methods have become available in recent years that allow the assessment of the accuracy of the various approximate methods. We compare several methods of computing f and [η] to determine their applicability to various classes of particle shapes.

Evaluation of the Kirkwood approximation for the diffusivity of channel-confined DNA chains in the de Gennes regime

“…To distinguish more quantitative between cases (ii) and (iii), Fig. 9 plots the mean span, hXi ¼ hmaxðR i;x Þ À minðR i;x Þi; (25) of the chain as a function of the inverse channel size. In the latter, R i,x is the x component of the position vector R i of bead i.…”

“…(2) and hereafter, we interpret the square of a vector as R 2 ¼ R Á R, not as a dyadic product RR. The difference between D (K) and D L has been addressed for flexible and semiflexible polymer chains in free solution, [18][19][20][21][22][23][24][25][26] with errors in the range of 1% to 25% for different approaches and different polymer models. For example, the error increases by increasing the chain size, 24 or by increasing the flexibility of the chain 26 or by reducing the solvent quality.…”

“…For example, the error increases by increasing the chain size, 24 or by increasing the flexibility of the chain 26 or by reducing the solvent quality. 23,25 Our goal here is to evaluate the ability of the Kirkwood approximation to model experimental data for DNA confinement, for example, the relaxation time data from Reisner et al 27 The total duration of these experiments is long compared to the autocorrelation time of the mean span of the chain, but is rarely longer than the time s for center-of-mass diffusion over the size of the chain. As we will see, there is a fast transient in hR 2 cm i that captures much of the difference between the Kirkwood diffusion coefficient and the long-time diffusion coefficient.…”

Newtonian Viscosity of Dilute, Semidilute, and Concentrated Polymer Solutions

“…The accuracy of each of the finite element methods increases as the number of finite elements, N, increases, but so does the computational time. The numerical path integration method claims an advantage [Mansfield et al, 2007] since, for such objects, the computational time scales as O(N), whereas for the other two methods, the computational time scales as O(N 3 ). Each of these methods enables the introduction of microscopic surface detail into the computation, and offers, for example, the possibility of obtaining a consistent picture of protein hydration and of determining from viscometric measurements whether the solution conformation of a protein differs from that in the crystalline state.…”

On the strength of glasses

“…We suspect this simplification is probably not too bad for small stresses and not too far from T K . This surmise is buttressed by the experience for the corresponding hydrodynamic problem of computing the intrinsic viscosity of complex shapes, a problem that has been extensively studied in polymer chemistry (40). In that problem the shape effects are quite modest.…”