Brownian Motion of Stiff Filaments in a Crowded Environment

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Science  24 Dec 2010:
Vol. 330, Issue 6012, pp. 1804-1807
DOI: 10.1126/science.1197321


The thermal motion of stiff filaments in a crowded environment is highly constrained and anisotropic; it underlies the behavior of such disparate systems as polymer materials, nanocomposites, and the cell cytoskeleton. Despite decades of theoretical study, the fundamental dynamics of such systems remains a mystery. Using near-infrared video microscopy, we studied the thermal diffusion of individual single-walled carbon nanotubes (SWNTs) confined in porous agarose networks. We found that even a small bending flexibility of SWNTs strongly enhances their motion: The rotational diffusion constant is proportional to the filament-bending compliance and is independent of the network pore size. The interplay between crowding and thermal bending implies that the notion of a filament’s stiffness depends on its confinement. Moreover, the mobility of SWNTs and other inclusions can be controlled by tailoring their stiffness.

Crowding greatly constrains the transversal mobility of a filament and causes anisotropic diffusion, which is limited to the filament axial direction. In the case of polymer solutions or melts, understanding the motion of a single polymer chain confined by the meshwork of its neighbors was key to a number of advances in polymer science. In their seminal work, de Gennes, Doi, and Edwards (13) modeled the effect of crowding on polymer dynamics by introducing the concept of a confining tube, together with preferential motion along the polymer’s axis, known as reptation because of its resemblance to the slithering of a snake (Fig. 1A, inset). This model captured many bulk dynamical properties of flexible polymer melts and solutions (4), although direct experimental evidence validating this powerful theoretical intuition came over two decades later, when reptation of flexible and semiflexible filaments was observed directly by imaging fluorescently labeled DNA (5) and actin (6).

Fig. 1

(A) (x, y) center-of-mass trajectories of a SWNT reptating in 1.5 w/w % agarose gel and representative NIR images of the SWNT, illustrating the effect of flexibility on reorientation of SWNT in different pores (scale bar, 5 μm). (Inset) Schematic of a stiff filament in a fixed network: L is the length of the filament, λ is the deflection length, and ξ is the pore size of the network. (B) Individual SWNT emission spectrum with peak at 985 nm, implying a (6, 5) structure with a diameter of 0.76 nm and persistence length of 26 μm. (C) Angular MSD showing the subdiffusive-to-diffusive behavior which occurs at disengagement time τd. The line is the best fit to the data. Dr is calculated from the long time (diffusive behavior). (Inset) Representative image of a SWNT in x-y lab frame; the orientation angle θ is the angle between the x axis and the major axis of the best-fitted ellipse to the shape of the SWNT.

In contrast, little is known about the thermal motion of stiff filaments such as carbon nanotubes, biopolymers, and stiff fibers in a network. In particular, the role of the bending stiffness of such inclusions remains controversial, with long-standing conflicting theoretical predictions (711). Doi predicted that rotational diffusion is independent of stiffness (7), whereas Odijk concluded that such diffusion should be enhanced by flexibility (9) and Sato concluded the opposite (11). Bulk experiments by means of birefringence and dichroism (1214) have also given conflicting results, mainly because polydispersity, aggregation, attractive forces, and strong coupling between translational and rotational diffusivities of the filaments all complicate the interpretation of the results. We directly visualized single-walled carbon nanotubes (SWNTs) reptating in a gel and established that flexibility substantially speeds up diffusion of stiff filaments under confinement, which is in accord with Odijk’s theory (9). We found that the rotational diffusion constant grows linearly with the bending flexibility and, counterintuitively, is independent of degree of crowding.

A natural measure of the stiffness of a filament is its persistence length, Lp = κ/kBT, which measures its thermal bending by Brownian forces. Here, κ is the bending stiffness, T is the temperature, and kB is Boltzmann’s constant. Doi (7) postulated that as long as the rods are stiff (L < Lp), flexibility does not affect diffusion, and that such a stiff filament of length L confined in a tube of diameter ξ << L would explore an angle θ ≈ ξ/L in the reptation time τrep = L2/D needed to diffuse a length L. This yields a rotational diffusivity DrDoi= θ2rep = kBTξ2L5, where D ~ kBTL is the translational diffusivity of an isolated filament in a solvent of viscosity η. In contrast, Odijk (9) argued that whenever the amplitude of the thermal undulations u = (L3/Lp)1/2 exceeds the pore diameter, confinement results in the independent deflection of segments of length λ = (Lpξ2)1/3, of which there are L/λ (15). As the filament reptates a distance λ, the ends of the filament reorient by an angle δθ ~ ξ/λ. After a reptation time, this results in a mean-square angular deflection of the filament of θ2 ~ ξ2L3 and an angular diffusivity of DrOdijk = kBTL2Lp. Doi’s theory is recovered for filaments shorter than λ, a length typically much shorter than the persistence length, at which the flexibility becomes irrelevant. Otherwise, Odijk theory predicts that rotational diffusion speeds up by a factor DrOdijk/DrDoi = (L/λ)3, for example, by three orders of magnitude for a 10–μm-long, stiff (Lp = 100 μm) filament moving through 100-nm pores. (9)

SWNTs are the ideal system to study confined dynamics of stiff filaments. SWNTs are slender (typical diameters of d ≈ 0.7 to 1.2 nm), sufficiently long to be visualized through optical microscopy (L ≈ 3 to 15 μm), and share many dynamical characteristics with polymers (17, 18). SWNTs are considered stiff because their persistence length ranges from 20 to 150 μm and scales with their diameter cubed, Lp ~ d3, similar to the bending stiffness of a macroscopic hollow pipe (19). Individual semiconducting SWNTs can be visualized directly because of their bright near-infrared (NIR) luminescence, and their diameter can be determined simultaneously spectroscopically (20). We image the quasi-two-dimensional dynamics of these individual SWNTs in agarose gel (21), a permanent network with pores ξ ≈ 0.1 to 1 μm [depending on agarose concentration (22, 23)], which mimics the reptation ansatz of a filament moving in a fixed network of frozen obstacles (13). The diameter (hence the persistence length) of each SWNT was determined from its emission spectrum (19, 20). By means of image analysis, we extracted frame-by-frame each SWNT’s center-of-mass position ri = [xi, yi] in the lab coordinates and its orientation θi relative to the x axis (i represents the frame number spaced by 30 ms acquisition time). Figure 1A depicts the center-of-mass trajectory of a 4.5-μm-long (6,5) SWNT [deduced from its emission spectrum (Fig. 1B)], with a 0.76 nm diameter and Lp = 26 μm (19) in a 1.5% w/w agarose gel (ξ ≈ 0.2 μm); this figure and the accompanying video (24) show unequivocally snake-like motion. NIR fluorescence snapshots demonstrate that flexibility substantially affects reorientation of the SWNT in a new confining tube. At first, the SWNT slides back and forth partially out of the confining tube. By bending slightly, the end of the SWNT has more freedom to explore various paths while translating along its length, even though most of the SWNT is still caged and thus restricted to a certain orientation. Eventually, the SWNT completely slides out of the original confining tube and reorients in another tube.

We quantify rotational motion by the statistics of the angle θi. A typical time-evolution of the mean-square angular displacement (MSAD), Δθ2, is shown in Fig. 1C. At short times, the SWNT’s angular diffusion is subdiffusive (Δθ2τn, n << 1), reflecting the confinement in the initial tube. At longer times, the SWNT diffuses out of the initial tube, and the mean angular displacement behaves diffusively Δθ2=2Drτ (25), yielding the value of the rotational diffusivity Dr.

We measured the rotational diffusivity of 35 SWNTs with different lengths (2 to 10 μm) and persistence lengths (26 to 60 μm), reptating in agarose gels of several concentrations (hence pore sizes). We collapsed the rotational diffusivity on a master curve (Fig. 2) by plotting the normalized rotational diffusivities Dr/DrOdijk = DrηL2Lp/kBT versus normalized length L/λ. In such a plot, Doi’s theory predicts a power law with scaling exponent –3 [(L/λ)–3] across the whole range of normalized length (Fig. 2, dashed line), whereas Odijk’s theory predicts a plateau at ~1 for L > λ (Fig. 2, solid line). The data show that when L ≥ λ, flexibility does not affect mobility (which is in agreement with both Doi and Odijk), whereas for L > λ flexibility clearly speeds up long-time diffusion, which follows Odijk’s scaling. Therefore, the effective rigidity of a filament depends on its degree of confinement. Whereas in the absence of confinement Brownian filaments can be considered essentially as rigid when L < Lp, confined filaments (ξ < L) behave as rigid when L > λ.

Fig. 2

Normalized rotational diffusivity of 35 SWNTs with different length and persistence length [denoted by different symbols (table S1)], reptating in different concentrations of agarose gel versus normalized length by deflection length. Doi’s theory is shown by a dashed line and predicts a power law with scaling exponent –3 across the whole range of normalized length. Odijk’s theory is denoted by a solid line and predicts a plateau at ~1 for L > λ.

We next turned to the short-time subdiffusive dynamics of the MSAD (Fig. 1C). To cross over from short time subdiffusive behavior to long-time diffusive motion, a filament must diffuse by a length λ out of its initial confining tube. This occurs on a time scale known as the disengagement time τd, which is the time scale a SWNT needs to reptate by a deflection length and is determined from the free parallel diffusion constant of the center of the mass, τd = λ2/D ~ ηλ2L/kBT (9). At times shorter than τd, the SWNT wiggles “freely” inside its initial confining tube, with minimal angular reorientation (θ < ξ/L, hence the subdiffusive behavior of MSAD in Fig. 1C). At times longer than τd, the SWNT slides out of the initial confining tube and starts exploring the other accessible tubes. Show in Fig. 3 are the disengagement times normalized to λ2 obtained for 11 (6,5) SWNTs by fitting the MSAD with Δθ2=θ02+2Drτ and setting τd=θ02/2Dr. We found that the measured τd normalized by deflection length λ2 scales linearly with length L, confirming Odijk’s prediction (9) for short-time translational diffusion (26) and showing that flexibility speeds up disengagement.

Fig. 3

(A) The disengagement time normalized by deflection length λ2 scales linearly with length L. (B) Time domains in translational MSDs parallel (Δs2) and perpendicular (Δn2) to the time-averaged reptation path (corresponding to the SWNT in Fig. 1A). τ < τd is the dynamics inside the tube. τd < τ < τr is the crossover region between anisotropic and isotropic diffusion. τ > τr is the isotropic diffusive long-time dynamics.

Because SWNTs explore orientation space by reptating in and out of pores, rotational and translational diffusion should be strongly coupled at time scales below the rotational diffusion time τr = 1/2Dr. Such coupling occurs even in the much simpler case of two-dimensional Brownian motion of an unconstrained ellipsoid and is well described in terms of Perrin-Smoluchowski theory (27). Theoretical calculations and simulations have recently shown that this same theory can capture such coupling in the motion of confined rigid rods (infinite Lp) (28). To investigate this coupling experimentally, we measured the time evolution of the center-of-mass mean square displacements (MSDs) parallel (Δs2) and perpendicular (Δn2) to the orientation of the reptation tube, averaged over the same time window (23). The parallel and perpendicular MSDs versus time (normalized by the rotational diffusion time) are shown in Fig. 3B.

At short times (τ < τd), SWNT diffusion is anisotropic—Δs2 >> Δn2; SWNTs diffuse much faster parallel than perpendicular to the tube axis. In this time regime, the dynamics of center of mass is dominated by the relaxation of thermally excited elastic bending modes of the SWNT, with relaxation times τrn~ηln4/κ, where ln is the mode wavelength (29). For a given time τ, long-wave modes (τrn>τ) are effectively “frozen,” whereas short-wave modes (τrn<τ) evolve and contribute to the amplitude of thermal undulations. At time τ, the longest (dominant) bending mode has a wavelength of l(τ) ~ (κτ/η)1/4. The mean square amplitude of the transverse fluctuations (Δu2) of this mode dominate the transverse diffusion of the center of mass and evolves with time (2931) as Δu2 ≈ Δn2 ~ l(τ)3/Lp ~ τ3/4, which is indeed the subdiffusive power law τ3/4 we measured (Fig. 3B). The same time dependence, τ3/4, is also expected for the mean square amplitude of the longitudinal fluctuations of the SWNT (29, 31); these dominate the mean square longitudinal displacement of the center of mass Δs2, as shown in Fig. 3B. At longer times, Δs2 crosses over to a linear diffusion regime, indicating that the SWNT has fully reptated along its length (Δs2 ~ τ). In this crossover regime, the transverse MSD Δn2 grows super-linearly with time because reptation occurs along a curved path—a motion that couples rotation and translation (Δn2 ~ DτΔθ2 ~ DDrτ2) (10, 28). Thus on intermediate time scales between disengagement and rotational diffusion times (τd and τr), translational diffusion perpendicular to the filament is also enhanced by flexibility. At times longer than rotational diffusion time τr, the SWNT loses memory of its initial orientation, and its diffusion becomes isotropic. On these time scales, translational diffusion is weakly reduced by flexibility (32).

By varying SWNT surface modifications (33), we can selectively tune the sensitivity of the carbon nanotubes to the different physical properties of the porous media for transport and sensing applications (such as a cellular crowded environment). The orientation dynamic behavior of SWNTs in a fixed network is a starting point to study the dynamics of concentrated solutions of SWNTs as well as SWNT composite materials. Our results indicate that the SWNT shapes are altered by the presence of the pores and that bent shapes can be very long lived. Rotational diffusion and coupling between translational and rotational motion of SWNTs can provide a useful counterpart to translational diffusion approaches in microrheology techniques and render the ability to probe different viscoelastic modes or local heterogeneity in complex fluids and biological media.

Supporting Online Material

Materials and Methods

Fig. S1

Table S1


Movie S1

References and Notes

  1. This characteristic length scale has been estimated from thermal fluctuations of flexible filaments in an ordered polymer background (16).
  2. Materials and methods are available as supporting material on Science Online.
  3. Rotational diffusion is characterized in two dimensions by a single diffusion coefficient, Dr, and associated diffusion time, τr = 1/2Dr.
  4. The errors introduced by the limited angular resolution in our measurements can affect the interpretation of the short time dynamics. The microscope angular resolution is ≈a/L, where a is the pixel size. Therefore, resolution limits our experiments below a resolution time of τresolutiona2Lpπη/2kBT. In the experimental conditions of Fig. 3A, τresolutiond = [(λ/L)(a/ξ)2]/4 ranges from 0.01 to 0.26; therefore, τresolution << τd and resolution does not substantially affect the measurement of the subdiffisive regime due to the short-time dynamics in the system.
  5. This work was supported by the NSF Center for Biological and Environmental Nanotechnology (EEC-0118007 and EEC-0647452), the Welch Foundation (grant C-1668), the Advanced Energy Consortium (, the Région Aquitaine, the Agence Nationale pour la Recherche (ANR PNANO), the European Research Council (grant n 232942), and the Foundation for Fundamental Research on Matter (FOM), which is part of the Netherlands Organisation for Scientific Research (NWO).
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