5422
Macromolecules 2008, 41, 5422-5432
Transport Properties of Rodlike Particles Marc L. Mansfield*,† and Jack F. Douglas*,‡ Department of Chemistry and Chemical Biology, SteVens Institute of Technology, Hoboken, New Jersey 07030, and Polymers DiVision, National Institute of Standards and Technology, Gaithersburg, Maryland 20899 ReceiVed December 20, 2007; ReVised Manuscript ReceiVed April 23, 2008
ABSTRACT: The transport properties of rodlike particles have been of interest since the origins of polymer science. Indeed, Staudinger initially believed that polymers had exclusiVely this structure, although Onsager soon discounted this possibility through direct computation of the transport properties of slender (ellipsoidal) particles and comparison with early polymer solution measurements. History is apparently repeating itself in the study of carbon nanotubes, which are also often assumed to be generally rodlike. Computations of the transport properties of rods have often assumed such structures could be described by ellipsoids or have relied on slender body calculations of uncertain accuracy. To clarify the situation, we apply a path-integration method to compute the intrinsic viscosity [η] and friction coefficient f of right rectangular cylinders, cylinders with rounded ends, and other fiberlike structures having a variety of cross-sections. We then determine convenient approximants that describe these basic properties over a wide aspect ratio range and for all these cross-sectional shapes. Our computations are validated against independent boundary element computations by Aragon in the case of right rectangular cylinders. Finally, we discuss the apparent mass-scaling exponent of the intrinsic viscosity (Mark-Houwink exponent) for rodlike structures since this quantity is often reported in the polymer science literature.
1. Introduction Recent work on the characterization of carbon nanotubes and other similar structures has generated renewed interest in the transport properties of rodlike particles.1–12 Many of these studies have simply assumed that these structures are rodlike, ignoring observational evidence for finite persistence lengths, or other sources of flexibility. This paper is not concerned with the validity of the rodlike picture for any particular type of particle but instead with the validity of theoretical expressions of the transport properties of rods that are currently in use for the characterization of various nanoparticle structures. Theoretical work on slender-body hydrodynamics has a long history, beginning with early contributions of Jeffery,13 Onsager,14 and Perrin15 on ellipsoids of revolution. Other contributions on ellipsoids of revolution,16–20 cylinders,19,21–34 helices,35,36 tori,24,36–38 triaxial ellipsoids,39 and many general treatments23,40–49 have also appeared. Most theoretical results for cylinders and other nonellipsoidal shapes involve calculations whose accuracy is uncertain. We decided that it would be useful to compare the results of various approximations that have been proposed for rodlike bodies (listed in Appendix 1) against the predictions of a recently developed numerical path-integration method50–57 that gives precise results for particles of arbitrary shape. This comparison also includes boundary-element calculations provided by Aragon,58 which constitute an independent validation of our computations. We find that many of the expressions in active use have limited accuracy.59 Therefore, in place of these, we provide approximants that reproduce our numerical results for both the intrinsic viscosity and the friction coefficient, where the uncertainties in the approximants are specified. These results should be helpful both in the reliable characterization of rodlike particles, and also with providing an appreciation of the inherent limitations of slender-body * Corresponding authors. E-mail: (M.L.M.)
[email protected]; (J.F.D.)
[email protected]. † Department of Chemistry and Chemical Biology, Stevens Institute of Technology. ‡ National Institute of Standards and Technology.
theory, upon which much of polymer hydrodynamics is based. Similar calculations on the wormlike chain model and on random coils will be reported in separate publications.60,61 Let L, d, and r ) d/2 represent the length, diameter, and radius of any rodlike body. For any given property, we normally expect asymptotic scaling behavior in the “slender-body” limit, L . d. However, the expansion parameter of slender-body theory is ln(L/d), and asymptotic behavior is only achieved at aspect ratios around e20 ≈ 109. Evidently, rodlike structures should exhibit the properties of semiflexible (or wormlike) polymers before they enter the true asymptotic scaling regime described by slender rods. Therefore, reliable approximations of the transport properties require an accurate determination of the corrections to the asymptotic predictions for rodlike scaling. Our computational scheme is a numerical path integration that we have described in previous publications.53–57 The method involves uncertainties of about 1% or less for the hydrodynamic radius, and recently has been improved to give accuracies of about 1.5% or less for the intrinsic viscosity.62 In order to better understand the influence of end effects on slender bodies, we have examined results for two popular models of rods, cylinders and blunt-ended cylinders (Figure 1). The hydrodynamic radius, Rh, is the proportionality constant between the friction coefficient and the solvent viscosity, f ) 6πηRh. Through the Einstein relation, D ) kT/f, it is also related to the diffusivity. The intrinsic viscosity, [η] is the leading virial coefficient for the solution viscosity when the solute concentration is measured in volume fraction units: η ) η0(1 + [η]φ + ...)
(1)
where η, η0, and φ are the solution viscosity, the solvent viscosity, and the volume fraction, respectively. Therefore, [η] is a dimensionless, scale-invariant functional of the shape of the particle. The quantity that is most frequently encountered in experimental work, the “practical” intrinsic viscosity, [η]P, is the viscosity virial coefficient when solute concentration is measured in mass/volume units. The two are related by the simple conversion:
10.1021/ma702839w CCC: $40.75 2008 American Chemical Society Published on Web 06/25/2008
Macromolecules, Vol. 41, No. 14, 2008
[η]P )
V[η] m
Transport Properties of Rodlike Particles 5423
(2)
for V and m the volume and mass of the particle, respectively. The product J ) V[η] ) m[η]P has units of volume and is a measure of the volume of the region over which Stokes flow is perturbed by the body. The quantity Vh )
2J 2 2 ) V[η] ) m[η]P 5 5 5
(3)
is known as the “hydrodynamic volume,” and equals the volume of an equivalent sphere, since for the sphere, we have the Einstein result [η] ) 5/2. We emphasize that the appropriate comparison, when replacing any complex particle with a simpler effective body, such as a helix with a cylinder or a globular molecule with a sphere, should be between the hydrodynamic volumes of the two bodies, eq 3, and not their intrinsic viscosities. The best way to see this is to rewrite eq 1 as η ) η0(1 + [η]
NV + ...) Vtot
(4)
where Vtot is the total volume of the solution, and N is the number of solute particles present. Replacing the actual solute particles, one for one, with some standard hypothetical particle will only have the same hydrodynamic effect if the product J ) V[η] is the same for the two particles. Therefore, a rodlike particle and its effective cylinder will have the same hydrodynamic volume, but since their actual volumes can differ substantially, then so also can their intrinsic viscosities.63 Likewise, it is obvious from Stokes’ law, f ) 6πηRh, that particles that are hydrodynamically equivalent with respect to friction must have the same hydrodynamic radii. In this paper, we use the terms “rod” or “rodlike” to represent straight, slender, nontapering structures. By this definition, both cylinders and blunt-ended cylinders are rodlike. A rigid helix is also rodlike: It has a uniformly periodic profile along its length and does not taper off toward the ends. Because of the availability of exact solutions for ellipsoids of revolution,13–16,18 and because of the expectation of universal scaling behavior at sufficiently high aspect ratios, there has been a repeated tendency over the years to use highly prolate ellipsoids of revolution as a model of rodlike particles. Another goal of this paper is to point out the pitfalls of such an approach, which was also observed by Haltner and Zimm19 based on direct measurements on macroscopic cylinders and ellipsoids. We will show that because ellipsoids taper toward either end, they do not have the same longitudinal Stokes force distribution as cylinders and other nontapering particles. (See also Gluckman et al.20) This means that there are fundamental differences between the hydrodynamic properties of slender circular cylinders and slender prolate ellipsoids. For example, there is no single prolate ellipsoid having the same length as a given rodlike particle that can simultaneously reproduce the hydrodynamic radius and the hydrodynamic volume. Of course, these discrepancies between cylinders and ellipsoids do eventually disappear in the slenderbody limit, but as already mentioned, this regime corresponds to aspect ratios of no physical interest. Furthermore, since the longitudinal mass distributions of ellipsoids and cylinders are not the same, a cylinder and an ellipsoid of the same total length do not have the same radius of gyration, Rg. Fortunately, with the formulas published here, as well as a few others of comparable accuracy,28,29,32,44,64,65 it is no longer necessary to model rodlike particles as ellipsoids.66 We also provide computations of the dimensionless ratios Rg/Rh, Rη/Rh, and Rg3/V[η] of rods as functions of aspect ratio, A ) L/d, where L and d are respectively the length and diameter of the rod. Each ratio combines the results of two experiments
Figure 1. Ratio of hydrodynamic radius to length, Rh/L, for several slender bodies as a function of aspect ratio, A. Also shown are our definitions of the dimensional parameters of both cylinders and blunt cylinders.
and carries scale-invariant information about the shape of the solute particle.67 The intrinsic-viscosity-mass scaling exponent, or the so-called Mark-Houwink exponent is also examined here since it is widely considered as a measure of particle shape. Finally, because many carbon nanotube and similar systems often have a distribution of lengths and diameters, we consider the effects of polydispersity on the intrinsic viscosity. Our computations rely on an interesting analogy between electrostatics and hydrodynamics.50–55 For example, the charge distribution over the surface of a charged conductor is approximately proportional to the distribution of Stokes force over the same surface in Stokes flow. The following approximate proportionalities also result from the hydrodynamic-electrostatic analogy:52 Rh ) qhC [η] )
qη〈R〉 V
(5) (6)
where qh and qη are constants of proportionality. Here, C is the electrostatic capacity, and 〈R〉 is the mean electrostatic polarizability, or one-third the trace of the polarizability tensor, of a perfect conductor having the same size and shape as the body. If the analogy were exact, the two terms qh and qη would be constant. In reality, they are weak, scale-invariant, dimensionless functionals of the particle shape. The functional qh lies in the range qh ) 1.00 ( 0.01
(7)
52
for all bodies. With such small variability, we usually assume that qh is unity and that Rh and C are interchangeable. The variability of qη is larger; for all shapes it lies in the range52 qη ) 0.79 ( 0.04
(8)
Given this analogy, in this paper we will shift between a hydrodynamic and an electrostatic description, as it suites us. Our path-integration technique is rigorous (to within sampling error) for the capacity and polarizability tensor and can be applied to objects of any shape. We then infer the hydrodynamic radius and the intrinsic viscosity using eq 5 to eq 8. In a more recent development,62 we use eq 5 to eq 7 along with an estimate for qη obtained from the eigenvalues of the polarizability tensor,
5424 Mansfield and Douglas
Macromolecules, Vol. 41, No. 14, 2008
Table 1. Volume and Gyration Radius of Model Rodlike
Particlesa
cylinders
blunt-ended cylinders
A-1
δ ) d/LB ) (A - 1)-1 V ) πLB3(δ2/4 + δ3/6) Rg2 ) (LB2/12)[(1 + 2δ + 3δ2 + 6/5δ3)/(1 + 8δ)] These formulas are exact for the models as defined in Figure 1.
δ ) d/L ) V ) πL3δ2/4 Rg2 ) (L2/12)[1 + 3δ2/2] a
which appears to give a more accurate approximation than eq 8. This latter approach requires additional testing, but initial results suggest an uncertainty of about 1.5% for the value of qη. Unless otherwise noted, the intrinsic viscosity results described here were obtained by the latter approach. 2. Formulas for the Volume, Gyration Radius, Hydrodynamic Radius, and Intrinsic Viscosity of Cylinders As already mentioned, we let L, r, and d ) 2r represent the length, radius, and diameter of a cylinder. Similarly, we let L ) LB + d, r, and d represent the total length, radius, and diameter of a blunt-ended cylinder, there being length contributions of LB and d, respectively, from the cylindrical portion and the two hemispherical caps; see Figure 1. The volumes and gyration radii of either cylinder model are given in Table 1. The aspect ratios of both types of cylinders are defined as the length-to-diameter ratio: L L ) (cylinder) (9) d 2r L LB + d A) ) (blunt cylinder) (10) d d The hydrodynamic radius and intrinsic viscosity of both models obey the following asymptotic forms as A f ∞:25,47 A)
Rh A f r ln A [η] f
8A2 45 ln A
(cylinder) (cylinder)
(prolate ellipsoid)
t ) (ln A)-1 for the expansion parameter of slender-body theory.
(15)
(11) (12)
Rh 4A ) A ln r e
(13)
4A2 (prolate ellipsoid) (14) 15 ln A Convergence to these asymptotes occurs only when ln A . 1, and eqs 11–14 are in significant error anywhere within the range of physically realizable aspect ratios. For example, we find that eqs 11 and 12 are in error by about 2% and 4%, respectively, even at A ) 109. Appendix 1 summarizes a few formulas from the literature that have been presented in attempts to improve on these asymptotic expressions. In this section, we present our own results as alternatives to the formulas in Appendix 1. As already mentioned, we computed the hydrodynamic radius and the intrinsic viscosity of cylinders and ellipsoids of revolution as a function of aspect ratio by our path-integral formalism, with results displayed in Figures 1 and 2. Because the logarithmic scales of these plots deemphasize the relative errors, we point out that the discrepancies in Figure 1 vary from 19% at A ) 1 to 5% at A ) 103. Similarly, in Figure 2 they vary from 13% to 34%. The resulting data were used to develop several Pade´ approximants. One set of these, which we will refer to as the “higher-accuracy” approximants, are extremely [η] f
accurate (typically four significant figures). However they are also quite cumbersome, and therefore are given in the supporting material accompanying this article.68 Below, we give several “lower-accuracy” approximants for the hydrodynamic radius and intrinsic viscosity. These are more succinct but still reproduce the raw integration data to better than about 0.5%. In the following, we have
Hydrodynamic Radius. The following formula agrees with the path integration results to about 0.4% or better for cylinders in the range 0 < t < 2 or 1.65 < A < ∞:
The analogous expressions for ellipsoids of revolution (major axis ) c, minor axis ) a, A ) c/a) are as follows:14,47 Rh A f a ln A
Figure 2. Intrinsic viscosity of several slender bodies as a function of aspect ratio.
-1
[ ( )]
[
×
1 - 0.782t + 0.691t1.67 + 0.622t1.77 + 0.418t2.16 1 - 0.677t + 1.601t2.07 + 0.178t2.26
]
(cylinders, 1.65 < A < ∞) (16) The analogous formula for blunt cylinders, valid to better than about 0.5% in the range 0
