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- 1. Rheol Acta (2009) 48:589–596 DOI 10.1007/s00397-009-0361-0 ORIGINAL CONTRIBUTION Rotational and translational diffusivities of germanium nanowires Bennett D. Marshall · Virginia A. Davis · Doh C. Lee · Brian A. Korgel Received: 11 September 2008 / Accepted: 11 April 2009 / Published online: 6 May 2009 © Springer-Verlag 2009 Abstract Understanding the rheological behavior of ological behavior of most nanocylinder dispersions. dilute dispersions of cylindrical nanomaterials in ﬂuids The extent to which these materials obey established is the ﬁrst step towards the development of rheological rheological models is an interesting scientiﬁc ques- models for these materials. Individual particle tracking tion because of the combination of nanocylinders’ was used to quantify the rotational and translational nanoscale diameters, nano- to microscale lengths, and diffusivities of high-aspect-ratio germanium nanowires extraordinary persistence lengths. As was the case for in alcohol solvents at room temperature. In spite of rod-like polymers (Donald and Windle 1992), rheolog- their long lengths and high aspect ratios, the rods were ical understanding of nanocylinder dispersions is not found to undergo Brownian motion. This work repre- only scientiﬁcally interesting but of signiﬁcant practical sents the ﬁrst time that the effects of solvent viscos- importance for applications development. The unique ity and conﬁnement have been directly measured and physical properties resulting from nanocylinders’ one the results compared to proposed theoretical models. dimensionality and potential for quantum conﬁne- Using viscosity as a single adjustable parameter in the ment have created signiﬁcant interest in their potential Kirkwood model for Brownian rods was found to be a use as building blocks for next-generation electronic, facile and versatile way of predicting the diffusivities of optoelectronic, photovoltaic, and structural materials nanowires across a broad range of length scales. (Lu et al. 2005; Murphy et al. 2005). These applica- tions generally require controlling the location, orien- Keywords Brownian dynamics · tation, and spacing of the nanocylinder building blocks Tracking · Suspension across volumes much larger than the nanocylinders themselves. Bottom-up assembly of ﬂuid-dispersed nanocylinders has proven to be one of the most pro- Introduction mising methodologies for producing materials with controlled spacing and orientation. However, optimiz- In spite of the signiﬁcant interest in cylindrical nano- ing these methodologies requires understanding the materials, little is known about the fundamental rhe- rheological characteristics of the dispersions. One of the most fundamental questions that must be answered is which, if any, classical models for the B. D. Marshall · V. A. Davis (B) Brownian motion of rods can be applied to dilute Department of Chemical Engineering, nanocylinder dispersions. Understanding the Brownian Auburn University, Auburn, AL 36849, USA e-mail: davisva@auburn.edu motion of dispersions of anisotropic rods is important because of the effect Brownian motion has on rheolog- D. C. Lee · B. A. Korgel ical response and dispersion processing (Mukhija and Department of Chemical Engineering, Texas Materials Solomon 2007). There are a growing number of stud- Institute and Center for Nano- and Molecular Science and Technology, The University of Texas at Austin, ies on single-walled and multiwalled carbon nanotube Austin, TX 78712, USA rheology (Chatterjee and Khrishnamoorti 2007; Davis
- 2. 590 Rheol Acta (2009) 48:589–596 et al. 2004; Duggal and Pasquali 2006; Fry et al. 2005; solid synthesis technique (Hanrath and Korgel 2004). Hobbie and Fry 2006, 2007; Parra-Vasquez et al. 2007; Rheologically dilute germanium nanowire dispersions Song 2006; Yang et al. 2006). However, the fundamen- with volume fraction φ ∼ 1 × 10−6 were prepared in tal rheological behaviors of dispersions of most inor- methanol and 1-propanol. The solvents were ob- ganic nanocylinders have yet to be characterized. Until tained from Sigma Aldrich and used as received. In recently, this was in part due to the limited availability spite of the large density difference between ger- of these materials. Additionally, the long lengths and manium (ρGe = 5.33 g/cm3 ) and the solvents (ρm = high aspect ratios of these materials limit the utility 0.792 g/cm3 and ρp = 0.804 g/cm3 ), the dilute disper- of many established ensemble techniques, such as light sions were kinetically stable for a period of several scattering. However, recent developments in scalable days. Transmission electron microscopy images of the synthesis methods for inorganic nanocylinders have as-synthesized germanium nanowires showed that the improved availability, and advances in single-particle diameters ranged from 7 to 25 nm. Since dynamics visualization tools have allowed the characterization are weakly dependent on the length-to-diameter ra- of the rheological behavior of individual rods in a tio of the as-dispersed rods, diameter measurements polydisperse sample. were repeated for quenched dilute dispersions us- In this research, the rotational and translational ing atomic force microscopy (AFM). For both the diffusion coefﬁcients for dilute dispersions of ger- methanol and 1-propanol dispersions, measurement manium (Ge) nanowires in alcohols were evaluated of 100 cylindrical objects yielded diameters ranging using single-particle tracking. The measured rota- from 7 to 60 nm, with an average diameter of 20 nm tional and translational diffusivities were compared (Fig. 1). Therefore, the dispersions were predominantly to the three approaches for predicting rotational and individual germanium nanowires, but some bundles translational diffusivity: (1) the Kirkwood equation were also present. The similarity between the diameter (Doi and Edwards 1986; Kirkwood and Plock 1956; distributions indicates that both solvents had the same Parra-Vasquez et al. 2007); (2) the Kirkwood equa- tendency for bundle formation in the dilute concen- tion using viscosity as an adjustable parameter to ob- tration regime. However, at higher concentrations tain an apparent, or best ﬁt, viscosity (Duggal and of φ ∼ 10−3 , 1-propanol was found to provide better Pasquali 2006; Li and Tang 2004); and (3) and a con- kinetic stability. ﬁnement model (Duggal and Pasquali 2006; Li and All imaging was performed using transmitted bright- Tang 2004). To the authors’ knowledge, this investi- ﬁeld microscopy on a Nikon 80 i optical microscope gation is the ﬁrst time the diffusivities of individual with a 60× NA 1.49 objective and 2× additional mag- Ge nanowires have been measured. Moreover, this niﬁcation in front of the camera. Since the smallest is the ﬁrst investigation where both the poten- germanium nanowires were 2 μm in length, it was tial for wall effects and solvent viscosity effects not necessary to use any ﬂuorescent tags or a ﬂuores- were experimentally measured and compared to the- cent light source. The rods were imaged using Nikon ory. Germanium nanowires were chosen because the combination of their long lengths, high aspect ratios, and rigidity are distinct from previously in- vestigated materials including single-walled carbon 25 nanotubes (SWNTs), F-actin, polyphenylene tereph- 22 22 thalate, microtubules, and low-aspect-ratio selenium rods (Donald and Windle 1992; Gittes et al. 1993; 20 Maeda and Maeda 2007). In addition, germanium Frequency nanowires are a particularly important material for 15 14 13 12 the development of new electronic applications; they have a lower band gap and higher electron and hole 10 9 mobilities than silicon (Wang and Dai 2006). 5 5 5 4 1 1 0 Experimental 5 10 15 20 25 30 35 40 45 50 55 60 Diameter (+/- 2.5 nm) Germanium nanowires with hexene-treated surfaces Fig. 1 Distribution of germanium nanowire diameters after were produced by the supercritical ﬂuid–liquid– dispersion in methanol obtained by AFM characterization
- 3. Rheol Acta (2009) 48:589–596 591 Elements software, and tracking was performed with Theory Image Pro M.D.A 6.0 software. Sequences of images of rods were obtained at 18.5 fps, giving a time interval The rotational diffusivity for a Brownian rod is given by of t = 0.054 s. the Einstein equation. Two different sample geometries were used in order to evaluate whether the walls surrounding the sample kB T were creating additional drag on the cylinders’ rota- Dr = , (1) ξr tion parallel, or nearly parallel, to the walls. Samples of physically conﬁned rods were prepared by placing where kB is Boltzmann’s constant and T is tempera- 45 μL of sample on a glass slide and then quickly ture in Kelvin. According to Kirkwood theory (Doi covering the drop with a glass coverslip and sealing and Edwards 1986; Kirkwood and Plock 1956), the the edges with vacuum grease. The path length H rotational drag coefﬁcient ξ r for a rigid rod in a dilute of these conﬁned samples (Fig. 2) was measured us- solution is ing the z-motors of the microscope and a Heidenhein z-encoder with 50-nm accuracy. Measurement at mul- tiple locations on the sample yielded an average π ηL3 ξr = , (2) value of H = 1.66 ± 0.28 μm. Since the rods ranged 3 (ln (L/b ) + γ ) in length from 2 to 10 μm in length, rod rotation was largely conﬁned to the focal plane. All rods in this where η is the solvent viscosity, L is the rod length, geometry were observed for 185 frames over 10 s. b is the diameter for the string of beads considered Samples were also prepared in rectangular Vitrocom to be comprising the rods, and γ r is a correction for glass capillary tubes with internal dimensions of 2 mm the cylindrical geometry of the rod. For an inﬁnitely in width, 50 mm in length, and H = 100 μm. Capillary long cylinder, γ r = −0.447 (Broersma 1981). This equa- action was used to draw the dispersion into the tube; tion applies to a free rod with no additional drag the ends of the tube were sealed to prevent the concen- from outside forces or walls. Similarly, the translational tration and ﬂow effects that would accompany solvent diffusivities are given by equations of the same form: evaporation. Solvent viscosities were measured at 23◦ C using an kB T Anton Paar Physica MCR 301 rotational rheometer in a Dt = , (3) ξt sealed Couette geometry with Peltier temperature con- trol. In each test, the viscosity was measured at shear where ξ t is the translational drag coefﬁcient given by rates ranging from 1 to 100 s−1 , and the procedure was (Doi and Edwards 1986) repeated at least twice per solvent. Solvent viscosities were 0.55 ± 0.01 and 2.2 ± 0.01 mPa s for methanol and 1-propanol, respectively. 8π η L ξt = , (4) 3 ln (L/b ) + 2γ// + γ⊥ and γ // = −0.114 and γ⊥ = 0.886 (Broersma 1981). In the case of a conﬁned rod, an additional drag force Camera and may be imposed by the walls of the vessel (Jeffrey and Imaging Software Onishi 1981). Hunt et al. (1994) derived an expression Objective for the drag coefﬁcient per unit length of an inﬁnitely long rod moving parallel to a wall wall x H dx θ c// = 2π η cosh−1 (h/r) , (5) h y focal plane wall where h is the distance between the cylinder axis and the wall and r is the cylinder’s radius. The drag coef- Fig. 2 Diagram of a rod conﬁned between two glass walls. H is ﬁcient for an inﬁnitely long rod moving perpendicular the gap between the slide and coverslip or between the walls of the Vitrocom cell. θ , or the amount of tilt, represents the angle to a wall is given as c⊥ = 2 c// . Using these drag co- between the ﬁlament axis x and the wall, and y is the distance efﬁcients, Li and Tang (2004) derived the expressions between the rod center and the bottom wall given by Eqs. 6–8 for the rotational diffusivity of a
- 4. 592 Rheol Acta (2009) 48:589–596 cylinder conﬁned between two plates with separation H, rotating through an angle θ (Fig. 2) H /2 sin −1 (2y/ L) dy dθ Dr (y, θ)dθ 0 − sin −1 (2y/ L) Dr = (6) H /2 sin −1 (2y/ L) dy dθ 0 −sin −1 (2y/ L) Fig. 3 Binary sequence of a 3.2-μm-long germanium nanowire undergoing rotational diffusion, the time interval t = 0.054 s. Scale bar is 2 μm kT Dr (y, θ) = (7) ξr (y, θ) mean squared displacement of the orientation of the L/2 nanowire projected onto the focal plane 4π η x2 cos2 θ ξr (y, θ) = −1 ϕ 2 = 2Dr t (10) cosh ((y + x sin θ)/r) −L/2 Similarly, the two-dimensional translational diffusion 4 π ηx cos θ 2 2 coefﬁcient Dt was determined from the measured MSD + dx (8) cosh−1 ((H − y − x sin θ )/r) of the position of the center of mass of the nanowire r. Equation 6 gives the rotational diffusivity averaged r2 = 4Dt t (11) over all states. Dr (y, θ) and ξ r (y, θ) are the rotational Figure 4 shows the linearity of plots of < ϕ 2 > diffusivity and rotational drag coefﬁcient as a function and < r2 > against t for a nanowire 3.37 μm in of cylinder position and orientation. Li and Tang also length. Based on the analysis of internal averaging by derived a similar set of equations for the translational Saxton (1997) and the methodology used by Li and diffusivity averaged over all allowed states of a cylinder Tang (2004), the maximum time lag for calculating conﬁned between two glass plates (Eq. 9). diffusivities from linear ﬁts of the MSDs was limited to sin−1 (2y/ L) one quarter of the total number of time steps. Plotting H /2 dy Dt (y, θ) dθ the resulting values of Dr for rods in methanol with 0 − sin−1 (2y/ L) H = 1.66 μm shows that, in spite of the nanowires’ Dt = (9) exceptionally long lengths and high aspect ratios, Dr H /2 sin−1 (2y/ L) dy dθ scales with L3 (Fig. 5) in accordance with Kirkwood 0 − sin−1 (2y/ L) 1.6 Translation Results and discussion Rotation In the geometry where H = 1.66 ± 0.28 μm, the nano- 1.2 wires generally stayed in the focal plane; rotation and translation were nearly parallel to the walls formed by the slide and coverslip. This is similar to observations 〈 Δr2 〉 0.8 by Li and Tang (2004) for F-actin and Duggal and Pasquali (2006) for SWNTs. Figure 3 shows the rota- tional motion of a 3.2-μm-long germanium nanowire 0.4 dispersed in methanol. The image has been converted to binary to enhance contrast. 〈 Δϕ2 〉 The mean square displacements (MSDs) of the 0 rods < ϕ 2 > and < r2 > were calculated using in- 0 0.2 0.4 0.6 ternal averaging; all pairs of points were included for Δt(s) t = n tmin , where tmin = 0.054 s and n = 1, 2, 3, 4. . . (Saxton 1997). The one-dimensional rotational Fig. 4 Plot of mean square displacement (MSD) versus t for diffusion coefﬁcient Dr was determined from the translation and rotation of a 3.37-μm-long germanium nanowire
- 5. Rheol Acta (2009) 48:589–596 593 10 2 H = 1.68 μm 1.6 H = 100 mm Kirkwood η = 0.55 mPa s 1 Confinement Model 1.2 Kirkwood η = 0.83 mPa s 0.8 0.1 0.4 0 0.01 0 2 4 6 8 10 1 10 Fig. 6 Comparison of rotational diffusivities Dr with H = Fig. 5 Rotational diffusivities of germanium nanowires, where 1.68 μm (open circles) and H = 100 mm (solid squares). The H = 1.66 μm. The slope of the best-ﬁt line is −2.9 and R2 = 0.93 diffusivity error bars are the error in the linear ﬁts and the length error is given by 2D// t (Li and Tang 2004). There is no statisti- cally meaningful difference between the data sets. The Kirkwood model slightly overpredicts the data. The conﬁnement model theory for dilute Brownian rods. The rod lengths were slightly underpredicts the data. Using an apparent viscosity of 0.83 mPas in the Kirkwood model provides the most accurate ﬁt corrected for diffraction by assuming that the differ- ence between the observed rod length and actual rod length was equal to the difference between the ob- served rod diameter and the average rod diameter model slightly overpredicts the data while Li and Tang’s measured by AFM. The error length measurement was conﬁnement model, which was developed to account estimated as the root mean squared displacement due for drag, slightly underpredicts the values of Dr . The √ to Brownian motion 2Dll t, where Dll is the transla- best ﬁt for the data is obtained by adjusting the solvent tional diffusion coefﬁcient along the nanowire (Li and viscosity to 0.83 mPa s in the Kirkwood equation. This Tang 2004). The error bars are larger for the smaller apparent, or best-ﬁt, viscosity is 51% higher than the rods; the small gap conﬁned the motion of all the rods, measured value. Similarly, Duggal and Pasquali (2006) the shorter rods move faster and could geometrically found that the apparent viscosity required to obtain achieve higher values of θ. The error bars for Dr are the best ﬁt for SWNT rotational diffusivity was 40% based on the deviation from the linear ﬁt. higher than the measured bulk solvent viscosity. In In order to determine if the rods were experiencing addition, Li and Tang (2004) observed that the best ﬁt drag from the cell walls, samples were placed in the for the rotational diffusivity of F-actin was based on an Vitrocom tubes with H = 100 μm. To maximize the apparent viscosity almost three times higher than the distance between the rods and the walls, the focal measured value for the bulk solvent. plane was chosen as the midpoint of the tube. In this geometry, the cylinders were free to rotate in three di- mensions; obtaining one-dimensional diffusivities com- 1.6 parable to those obtained at H = 1.6 μm, required measuring the orientation of nanowires rotating nearly methanol 1.2 1-propanol parallel to the focal plane. This restriction reduced the time that nanowires could be observed to between 0.8 3.25 and 10 s, or 60 to 185 frames. Longer nanowires could typically be observed for longer times due to their lower diffusivities. Figure 6 shows that the measured 0.4 values of Dr for both H = 1.6 μm and H = 100 μm are comparable. In fact, ANOVA analysis of all data 0 sets using 1/L3 as a covariate conﬁrmed that the one- 0 2 4 6 8 10 dimensional rotational diffusivities were not affected by the sample geometry. This strongly suggests that, Fig. 7 Rotational diffusivities as a function of length for ger- even for H = 1.66 μm, the rods did not experience any manium nanowires in methanol and 1-propanol. Methanol and drag from the walls. Interestingly, when the measured 1-propanol diffusivity data are represented by open circles and methanol viscosity of 0.55 mPa s is used, the Kirkwood solid diamonds, respectively
- 6. 594 Rheol Acta (2009) 48:589–596 2 There are several potential factors to consider. The ﬁrst Experimental possible origin of the apparent viscosity being higher 1.6 Kirkwood η = 0.55 mPa s than the measured bulk viscosity is that the two mea- Confinement Model surements were carried out at different temperatures. Kirkwood η = 1.06 mPa s 1.2 In this work, the bulk viscosities were measured at the room temperature of 23◦ C using Peltier temperature 0.8 control. While heating from the microscope lamp dur- ing particle tracking could have caused a slight change 0.4 in temperature, this would have resulted in increased temperature and a lower viscosity. Therefore, the ap- parent viscosity being higher than the measured bulk 0 0 2 4 6 8 10 viscosity is not the result of temperature effects. The second possibility is that the polydispersity of diameters in the sample accounts for the difference in Fig. 8 Translational diffusivities of germanium nanowires dis- the apparent and bulk viscosities. However, since Dr persed in methanol. As in the case of rotational diffusivity, adjusting viscosity in the Kirkwood equation to an apparent value is only weakly dependent on rod diameter, this is not provides the best ﬁt to the data. However, the apparent viscosity likely to be signiﬁcant. In fact, Li and Tang showed for translation is higher than the apparent viscosity for rotation that even a doubling of the hydrodynamic radius only resulted in a 15% change in the best-ﬁt (apparent) viscosity. To further explore the consistency of the observed The third possibility is that the difference between apparent viscosity, nanowire rotational diffusivities in the apparent and bulk viscosity is due to errors in length 1-propanol were also measured with H = 100 μm. The measurement. If this is the case, the apparent viscosity diffusivities show the expected decrease in rotational obtained by ﬁtting Dt should be closer to the bulk diffusivity in this higher-viscosity solvent (Fig. 7). In value than that obtained by ﬁtting Dr , since Dt is more both cases, the Kirkwood equation predicts diffusivi- weakly dependent on L. The translational diffusivities ties that are higher than the actual diffusivities. The of the germanium nanowires in methanol with H = apparent viscosity for the 1-propanol data is 3.2 mPa 1.66 μm are shown in Fig. 8. The best ﬁt was obtained s, compared to a measured viscosity of 2.2 mPa s. This using the Kirkwood model and a viscosity of 1.06 mPa 46% increase in apparent viscosity for 1-propanol is s, nearly two times the bulk value of 0.54 mPa s, and comparable to the 51% observed in methanol. In other approximately 30% higher than the apparent viscosity words, although the magnitudes of the apparent viscosi- obtained based on measurements of Dr . Li and Tang ties are different from the measured bulk viscosity, Dr (2004) observed that apparent viscosity based on mea- has the expected linear dependence on η described by surements of Dt was approximately 30% lower than the Kirkwood equation. This suggests that there is a that that for Dr , but still approximately 70% higher consistent physical reason for the difference between than the bulk viscosity. Since, in both cases, there was the apparent viscosity and the bulk solvent viscosity. a difference in the apparent viscosities obtained for Dt Although this difference between the apparent, or and Dr , it is likely that length measurement errors are best-ﬁt, viscosity and the measured bulk solvent viscos- a contributing factor. However, the quantitative effect ity has been previously reported (Duggal and Pasquali cannot be evaluated based on existing data. 2006; Li and Tang 2004), it has not previously been The fourth possibility is that rod ﬂexibility is causing conﬁrmed for the same type of nanocylinder in multi- the high apparent viscosities. In addition to violating ple solvents. Furthermore, the origin of the observed the model assumptions, rod ﬂexibility would contribute higher apparent viscosities is not well understood. to length measurement errors. Rod ﬂexibility can be Table 1 Dimensions and rigidities of brownian rods (Duggal and Pasquali 2006; Gittes et al. 1993; Li and Tang 2004; Ngo et al. 2006; Smith et al. 2008) L D Lp Lp /L μm nm μm F-Actin 2–5 ∼8 ∼ 18 4–12 SWNTs 2–5 ∼1 32–174 7–76 Germanium 2–9 7–25 2.8 × 103 − 4.7 × 105 3.1 × 102 − 2.4 × 105
- 7. Rheol Acta (2009) 48:589–596 595 considered in terms of the ratio of the persistence the nanocylinders. For the ﬁrst time, measurements length Lp to the physical length L (Table 1). For F- were made in multiple solvents, and it was shown that actin, Lp /L is relatively low (Gittes et al. 1993; Li the apparent viscosity scaled with the bulk viscosity of and Tang 2004). For SWNTs, the persistence length is the solvents. These data provide conﬁrmation that the greater; the values shown in Table 1 were calculated diffusivities of high-aspect-ratio nanocylinders can be by Duggal and Pasquali (2006) based on the observed determined by adjusting the viscosity in the Kirkwood bending moments of the rods. This method was shown equation to obtain the best ﬁt, and that there is a to provide comparable results to Lp = κ/(kb T), where consistent physical origin to the difference between κ = CD3 /8 for hollow rods and C = 345 N/m based on the apparent viscosity and the bulk solvent viscosity. the in-plane stiffness of carbon nanotubes (Yakobson Length measurement errors are likely to contribute and Couchman 2004). For the majority of commercial to this difference, but future clariﬁcation of potential SWNTs, L < 500 nm, in which case, Lp /L is on the nanoscale size effects is needed. order 100 and they are clearly rigid rods (Lp >> L). The ﬁndings of this research provide fundamental However, in the case of the longer SWNTs imaged information on the rheological behavior of germanium by Duggal and Pasquali (2006), Lp /L is comparable to nanowires and a methodology for determining the ro- that of F-actin for the longest SWNTs, and signiﬁcantly tational and translational diffusivities of other systems. higher for shorter, larger-diameter SWNTs. In contrast, In addition, they provide a ﬁrst step in providing key the germanium nanowires are considerably more rigid. rheological information to help guide the bottom-up In fact, no bending moments could be observed for assembly of ﬂuid-dispersed nanowires into functional even the longest rods. The persistence lengths were devices. instead estimated using κ = ED4 /64 for a solid cylinder and a Young’s modulus E ∼ 100 GPa (Ngo et al. 2006; Acknowledgements This work was supported by the National Smith et al. 2008). The large range of Lp /L values is due Science Foundation Nanoscale Exploratory Research Program to the scaling of the persistence length with D4 and the NSF-CMMI-0707981 and the Auburn Undergraduate Research range of rod lengths. In all cases, Lp >> L, so ﬂexibility Fellowship Program. DCL and BCK acknowledge funding of this could not have contributed to the difference between research by the Robert A. Welch Foundation. The authors would also like to thank Matthew J. Kayatin for rotational rheology of the apparent and bulk viscosity. the solvents, Vinod Radhakrishnan and Shanthi Murali for AFM, The ﬁnal possible origin of the high apparent viscos- and Dr. Rajat Duggal for useful discussions. ity is that, in addition to length measurement errors, the Kirkwood equation’s assumption of hard colloidal rods may be inadequate for high-aspect-ratio cylinders with nanoscale diameters. Future research may eluci- References date contributions from potential size effects related to high surface-to-volume ratios, relative size of the rod Broersma S (1981) Viscous force and torque constants for a cylinder. J Chem Phys 74:6989–6990 diameter and the solvent molecules, or the relative size Chatterjee T, Khrishnamoorti R (2007) Dynamic consequences of the diameter and the Debye screening length (Maeda of the fractal network of nanotube-poly(ethylene oxide) and Maeda 2007). Nonetheless, using the apparent or nanocomposites. Phys Rev E 75:114307 best-ﬁt viscosity in the Kirkwood equation is a simple Davis VA et al (2004) Phase behavior and rheology of SWNTs in superacids. Macromolecules 37:154–160 method for obtaining the rotational and translational Doi M, Edwards SF (1986) The theory of polymer dynamics. diffusivities of high-aspect-ratio nanocylinders for a Oxford University Press, Oxford broad range of gap sizes. Donald AM, Windle AH (1992) Liquid crystalline polymers. 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