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Brownian Motion of an Ellipsoid

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Science  27 Oct 2006:
Vol. 314, Issue 5799, pp. 626-630
DOI: 10.1126/science.1130146

Abstract

We studied the Brownian motion of isolated ellipsoidal particles in water confined to two dimensions and elucidated the effects of coupling between rotational and translational motion. By using digital video microscopy, we quantified the crossover from short-time anisotropic to long-time isotropic diffusion and directly measured probability distributions functions for displacements. We confirmed and interpreted our measurements by using Langevin theory and numerical simulations. Our theory and observations provide insights into fundamental diffusive processes, which are potentially useful for understanding transport in membranes and for understanding the motions of anisotropic macromolecules.

Brownian motion (1), wherein small particles suspended in a fluid undergo continuous random displacements, has fascinated scientists since before it was first investigated by the botanist Robert Brown in the early 19th century. The origin of this mysterious motion was largely unexplained until Einstein's famous 1905 paper (2) that established a relation between the diffusion coefficient of a Brownian particle and its friction coefficient. One year later (3), Einstein extended the concept of Brownian dynamics to rotational and other degrees of freedom. The subsequent study of Brownian motion and its generalizations has had a profound impact on physics, mathematics, chemistry, and biology (4). Because direct detection of translational Brownian motion is relatively easy, many experiments elucidating the ideas of translational diffusion have been carried out. On the other hand, the direct visualization of rotational Brownian motion has not been an easy task, and fundamental concepts about motions of anisotropic macromolecules remain untested. For this contribution, we used digital video microscopy to study the Brownian motion of an isolated ellipsoid in suspension and thus directly observed the coupling effects between rotational and translational motion.

Particle anisotropy leads to dissipative coupling of translational to rotational motion and to physics first explored by F. Perrin (5, 6). A uniaxial anisotropic particle is characterized by two translational hydrodynamic friction coefficients, γa and γb, respectively, for motion parallel and perpendicular to its long axis. If a particle's rotation is prohibited, it will diffuse independently in directions parallel and perpendicular to its long axis with respective diffusion constants of Dα = kBTα for α either a or b, where kB is Boltzmann's constant and T is the temperature. In general, γa is less than γb (7), and consequently Da is greater than Db. If rotation is allowed, rotational diffusion, characterized in two dimensions by a single diffusion coefficient, Dθ, and associated diffusion time, τθ = 1/(2Dθ), washes out directional memory and leads to a crossover from anisotropic diffusion at short times to isotropic diffusion at times much longer than τθ. Figure 1, A and B, presents numerical simulations (8) that illustrate this behavior. Our experiments, which were restricted to two dimensions (2D), provide explicit verification of this behavior and some of its extensions. In addition, we show that a fundamental property of systems with dissipatively coupled translation and rotation is the existence of non-Gaussian probability density functions (PDFs) for displacements in the lab frame.

Fig. 1.

(A and B) 10,000-step 2D random walk trajectories for an ellipsoid with Da =0.99 and Db =0.01 during τθ and 100τθ, respectively. τθ is the time for the ellipsoid to diffuse 1 rad. (A Inset) 10,000-step trajectory for a sphere with Da = Db = 0.5 during τθ. The initial positions are represented by a green ellipse and a sphere. At step times long compared with τθ, the coarse-grained 100-step black trajectory in(B) is similar to that of the random walk of a spherical particle in fig. S1. (C) Representation of an ellipsoid in the x-y lab frame and the - body frame. The angle between two frames is θ(t). The displacement δx can be decomposed as (δx, δy) or (δ, δ). (D) True interference color in the reflection mode of the microscope. (E) Ellipsoid image in the transmission mode. (F) A typical 20-s experimental trajectory with step 1/30 s. Da/Db = 4.07. Orientations are further labeled with a rainbow color scale. For example, the purple parts of the trajectory reflect a higher mobility along the y direction, and the red parts reflect a higher mobility along the x direction.

Micrometer-sized PMMA (polymethyl methacrylate) uniaxial ellipsoids (9) were under strong quasi–two-dimensional confinement in a thin glass cell. The choice of 2D rather than 3D for these studies substantially simplified the experimental imaging tasks as well as the data acquisition time and storage requirements. The choice also ensured that the measured effects would be large by virtue of the much larger friction anisotropy in 2D compared with 3D. The local cell thickness was ∼1 μm. It was measured to within 0.1 μm resolution by comparing the Michel-Levy chart (10) to the reflected interference colors produced by the two inner surfaces under white light illumination on the microscope (Fig. 1D). To avoid interactions between ellipsoids, we made the solution very dilute. The Brownian motion of a single ellipsoid in water was recorded by a charge-coupled device (CCD) camera on a videotape at 30 frame/s. From the image analyses, we obtained data sets consisting of a particle's center-of-mass positions x(tn) = [x(tn), y(tn)] in the lab frame and its orientation angles θ(tn) relative to the x axis at times tn = n(1/30) s, as shown in Fig. 1F. The orientational resolution is 1°, and spatial resolutions are 0.5 pixel = 40 nm along the particle's short axis and only 0.8 pixel along its long axis because of the superimposed small tumbling motion. We define each 1/30-s time interval as a step. During the nth step, the particle's position changes by δx(tn) = x(tn)–x(tn–1) and its angle by δθ(tn) = θ(tn)–θ(tn–1). From the data set, we extract an ensemble of particle trajectories starting at different times τ0 and ending a time t later. The total positional and angular displacements in these trajectories are, respectively, Δx(t) = x(t + τ0)–x(τ0) and Δθ(t) = θ(t + τ0)–θ(τ0).

We first consider the statistical properties of θ(t), which, as pointed out by Perrin (6), are independent of translational motions. We measured data from a 30-min trajectory of a 2.4 μm–by–0.3 μm–by–0.3 μm ellipsoid confined in an 846-nm-thick cell (Fig. 2A). The inset shows that the mean-square angular displacement 〈[Δθ(t)]2 〉 equals 2Dθt, where the average 〈 〉 is over all trajectories with different starting times τ0. The mean-square angular displacement has diffusive behavior over the entire range of observable times with a rotational time of τθ = 1/(2Dθ) = 3.1 s. Over the time scales we can observe, this diffusive behavior is independent of θ0. The PDF for Δθ(t) was measured to be Gaussian with variance 2Dθt, and the angles θ(t) were measured to be uniformly distributed in [0,2π].

Fig. 2.

(A) MSDs along a, b, x, and y axes. (Inset) Angular MSD. All curves have diffusive behavior (∝ t), and corresponding diffusion coefficients D = MSD/(2t) shown in the figure are from best fits. (B) Diffusion coefficients D in the lab frame. The initial orientation of each trajectory was chosen to be along the x axis (θ0 = 0), so that Dxx and Dyy change from Da and Db to , respectively, over time interval τθ. Symbols, experiment; error bars ∝ t. π Solid curves, Eq. 2 when θ0 = 0. (C) Mixed correlations of translational displacements and orientation. Symbols, experiment. Error bars ∝ πt. Solid curves, theoretical results from Eq. 3 for n = 2. Dashed curves, reference uncorrelated averages 〈Δx2 〉 〈cos2θ 〉/t, 〈Δy2 〉 〈cos2θ 〉/t, and 2 〈ΔxΔy 〉 〈sin2θ 〉/t = 0.

We now turn to the statistics of translational motion whose full understanding is facilitated by the consideration of decomposing the displacement δxn into its components δxni relative to the body frame or δxni relative to the fixed lab frame. As shown in Fig. 1C, the two are related via a rotation, δ xni = Rijδxnj, where the Einstein summation convention on repeated indices is understood and Math is the rotation matrix with θn = [θ(tn–1) + θ(tn)]/2. In practice, choosing θn = θ(tn–1) or θn = θ(tn) has little effect on our results because θ barely changes during 1/30 s. We can construct total body-frame displacements by summing over displacements in each step, Math, and from (tn) we can construct body-frame displacements for trajectories of duration t at starting time τ0 via Δx̃(t)= x̃(t + τ0)–x̃(τ0).

Mean-square displacements (MSDs) in the body frame and in the lab frame were averaged over all trajectories with different initial angle θ0 ≡ θ(τ0) (Fig. 2A). They are all diffusive with 〈[Δ (t)]2 〉 = 2Dat, 〈[Δ(t)]2 〉 = 2Dbt, and 〈[Δx(t)]2 〉 = 〈[Δy(t)]2 〉 = (Da + Db)t ≡ 2 Dt. The full average 〈 〉 for any observable A can be viewed as an ensemble average of trajectories at fixed θ0 followed by a second average over all Math.

A particle with a given initial angle will diffuse more rapidly along its long axis than along its short axis. As time progresses, however, memory of its initial direction is lost, and diffusion becomes isotropic. Thus, averages over trajectories at fixed θ0 should exhibit a crossover from early-time anisotropic to late-time isotropic diffusion. Our measurements with θ0 = 0 of the time-dependent diffusion co-efficients Dxx(t) = 〈[Δx(t)]20/(2t) and Math provide direct verification of this crossover (Fig. 2B): At t << τθ, Dxx equals Da and Dyy equals Db, whereas for t >> τθ, Dxx equals Dyy equals .

The anisotropic-to-isotropic crossover was calculated in 3D by Perrin (6) and is mentioned in qualitative terms in a 3D simulation (11). We calculate the properties of this transition in 2D within the Langevin formalism and compare them with experiment. Because our time scales are much larger than the momentum relaxation times of a micrometer-sized particle in water (Irotm/γ ∼ 10–7 s), we can ignore inertial terms. The Langevin equations for displacement and angle in the lab frame in the presence of external forces described by a Hamiltonian H are Math(1a) Math(1b) where i = x,y for 2D and Γij = γ –1ij is the mobility tensor, which can be expressed in terms of the unit vector n(t) ≡ n[θ(t)] specifying the direction of the local anisotropy axis as Γij(t) = Γbδij + ΔΓni(t)nj(t) = Γδij + ΔΓMij[θ(t)]/2, where Γ = (Γa + Γb)/2, ΔΓ = Γa–Γb, and Mij(θ) = Math. χθ(t) and χ (t) random noise sources with zero mean and respective variances, 〈χθ(tθ(t′) 〉 = 2kBTΓθδ(tt′) = 2Dθδ(tt′) and 〈χi(tj(t′) 〉θ(t) =2kBTΓij[θ(t)]δ(tt′), dictated by the Einstein relation or equivalently by the requirement that thermal equilibrium be reached at long times. We retain H in Eq. 1 even though the external forces are zero in our experiments to emphasize that the mobilities Γij and Γθ relating velocity and angular velocity to force and torque, respectively, determine the variances of the random noise sources. χθ(t) obeys Gaussian statistics at all times, as does χi(t) for a fixed angle θ(t). The average 〈Aθ0 of any measurable quantity is equivalent to the average of A over both χi(t) and χθ(t) at fixed θ0.

Because there are no external forces in our experiments, we can set ∂H/∂x =0 and ∂H/∂θ =0. Equation 1b for θ(t) is simply the Langevin equation for 1D diffusion. It yields a time-independent diffusion coefficient Dθ = 〈[Δθ(t)]2 〉/(2t), a Gaussian PDF for Δθ(t) with variance 2Dθt, and consequently 〈cosnΔθ(t) 〉 = Re 〈einΔθ(t) 〉 = cosnθ0en2Dθt. From this we can calculate (8) the time-dependent displacement diffusion tensor for fixed θ0: Math(2) Math where ΔDDaDb and Math. Dxx(t,0) and Dyy(t,0) quantitatively match experimental results for θ0 = 0 as shown in Fig. 2B, with Da, Db,and Dθ equal to their values obtained from Fig. 2A. The average of Dij(t, θ0) in Eq. 2 over initial angles θ0 yields xx = yy = , in agreement with the MSDs of x and y in Fig. 2A. The 3D counterpart, xx = yy = zz =(Da + Db + Dc)/3, is widely used in dynamic light scattering (12).

Unlike spheres, anisotropic particles have anisotropic friction coefficients that are responsible for the coupling of translation and rotation. This coupling leads to nontrivial mixed correlation functions such as Math(3) Math where Math. Equation 3 is obtained from our Langevin formalism (8). Experimental results agree well with these theoretical predictions and deviate from the theoretical dashed curves obtained assuming translational and rotational motion are decoupled (Fig. 2C).

Transforming Eq. 1a into the body frame at ∂H/∂x = 0, we obtain Math(4) The probability distribution of x̃i(t), which can be calculated directly from its definition and the properties of χi(t), is a Gaussian with zero mean and variance Math, where Math is a θ(t)-independent diagonal matrix with components Math and Math. Thus, 〈(Δ i)2 〉 equals 2Dit, where Di = (Da, Db), in agreement with the experimental data in Fig. 2A. Because x̃i is Gaussian, the PDF for body-frame displacements Δ i (t) is Gaussian at all times: Math(5) where σ 2i(t)=2Dit. Our measurements confirm this behavior in fig. S1. For our quasi-2D sample, the ellipsoid's friction and diffusion tensors are different at different heights within the cell (13). Therefore, the PDF of Δi should be an average of Gaussian PDFs with different variances. However, the interference color from the ellipsoid changed very little throughout the course of our experiment; from this result we estimate that the ellipsoid remains within 50 nm of the midplane of the cell and that the non-Gaussian effects are too small to be observable as is confirmed by our measurements.

Although the statistics of displacements in the body frame are Gaussian, those in the lab frame are not because of coupling between translation and rotation (14). Prager (15) calculated the non-Gaussian concentration for averaged initial angles in a particular geometry in three dimensions. The lab-frame noise, χi(t)= R –1ij[θ(t)] x̃i(t), is a nonlinear function of the independent noises χθ(t) and x̃i(t). Thus, although its probability distribution is Gaussian for fixed θ(t) and thus fixed χθ(t), its distribution averaged over χθ(t) is non-Gaussian, as is that for Δxi(t). At short times, the lab- and body-frame displacements are equal, and the PDF for Δxi(t) is Gaussian because that for Δ i(t) is. Directional information is lost at times greater than τθ. Therefore, at long times, Δx(t) is a sum of displacements from ∼tθ statistically independent steps, and the central limit theorem implies that its PDF is Gaussian. Thus at fixed θ0, we expect deviation from Gaussian behavior to vanish at t = 0 and t = ∞ and to reach a maximum at times of order τθ.

The simplest manifestations of non-Gaussian behavior are the nonzero values of the fourth- or higher-order cumulants of lab-frame displacements, which can be calculated (8) from our Langevin theory. For example, the fourth cumulant of Δx(t) for fixed initial orientation is Math(6) Math Math This function vanishes as t2+s2, where s2 > 0, as t → 0 and grows linearly in t as t → ∞. The non-Gaussian parameter, Math(7) Math vanishes with ts2 for t → 0 and as t–1 as t → ∞. The angle-averaged non-Gaussian parameter, Math(8) Math(9) where C(4)(t) = 〈[Δx(t)]4 〉–3 〈[Δx(t)]22, approaches a constant as t →0 and vanishes as t–1 as t → ∞. Because statistics in the body frame are Gaussian, the body-frame non-Gaussian parameter pb(t) is zero.

Equations 7 and 8 are confirmed numerically in Fig. 3A. Experimental measurements of both p(t0) and (t) have poor statistics at large t because their errors grow as t3/2. Nevertheless, we were able to extrapolate to the t → 0 limit of p (t) in seven samples with different aspect ratios and to confirm Eq. 9 in the Fig. 3A inset. Figure 3A confirms our qualitative expectations about the non-Gaussian parameter p in different frames and for different types of averages, specifically: (i) In the ensemble with fixed θ0, the non-Gaussian parameter vanishes for t << τθ and t >> τθ and reaches a maximum when t ∼ τθ;(ii) in the ensemble that averages over θ0, the non-Gaussian parameter is a maximum at t = 0 and vanishes for t >> τθ; and (iii) larger Da/Db causes larger non-Gaussian effects.

Fig. 3.

(A) Non-Gaussian parameters as a function of t. Symbols, simulation in the lab frame for an ellipsoid with Da, Db,and Dθ from Fig. 2A. Error bars ∝ t3/2. Curves, theoretical predictions for p(t,0) and p(t,π/2) in Eq. 7 and (t) in Eq. 8. Dashed line, pb(t) = 0 in the body frame. (Inset) (t =0) for ellipsoids with different aspect ratios and confinements. Symbols, experiments; curve, theoretical prediction of Eq. 9. The double arrows in the figure and the inset indicate equivalent points for which PDFs are shown in (B). (B) Lab-frame PDFs for Δx(t) and Δy(t) at t = 0.1 s. Measured fΔx(x) (open circles) and fΔy(x) (open squares) π agree with the theory (solid dark curve) of Eq. 10 with no free parameter (Embedded Image, with Da from the fit of Fig. 2A). Dashed curve, best Gaussian fit; light curve, simulation.

It is clear from Eqs. 6 to 9 that non-Gaussian behavior originates in particle anisotropy and vanishes when ΔD vanishes. Thus, non-Gaussian effects for anisotropic particles diffusing in 3D with stick boundary conditions should be small because 1 < Da/Db = γba < 2 when 1 < a/b < ∞ (5, 7). For some small molecules, the slip boundary condition is more appropriate (16, 17) and γba diverges (11) even in 3D. Under quasi-2D conditions with stick boundary conditions, however, Da/Db increases with aspect ratio and finally saturates (13) at a value much larger than 2. In summary, non-Gaussian effects are strong when Da >> Db, i.e., for particles with a high aspect ratio confined in quasi-2D (in our case, Da/Db reaches about 4) or for some molecules with slip boundary conditions.

Lastly, we consider the lab-frame PDF for Δx(t). The expectation is that these PDFs at fixed θ0 will be non-Gaussian and exhibit maximum deviations from Gaussian behavior at times of order τθ. We have verified that this is the case within our statistical errors, but the deviations are very small. The lab-frame PDF averaged over θ0 shows more striking deviations from Gaussian behavior (8), particularly as t → 0: Math(10) where σ2(θ) = σ 2acos2θ + σ 2bsin2θ with σ 2i = 2Dit. The physical meaning of Eq. 10 is apparent. When t → 0, the orientation θ does not change during the displacement. Those Δx with the same θ0 follow a Gaussian distribution with σ = σ(θ0) because the hydrodynamic drag co-efficient γ(θ0) is a constant. Averaging Gaussian PDFs with different θ0 over [0,2π] yields a non-Gaussian PDF as shown in Eq. 10.

Experimental angle-averaged PDFs of lab-frame displacement are shown (Fig. 3B) at time intervals of t = 0.1 s. The system's isotropy is confirmed by fΔx(x) = fΔy(x). Interestingly, there are more tiny and large steps and fewer middlesized steps than there are in a Gaussian distribution (fig. S2). This PDF agrees with Eq. 10 very well with no free parameters. We measured the PDFs of 15 samples with different aspect ratios at different confinements. All agreed with Eq. 10. When Da/Db < 2.5, the measured non-Gaussian PDF becomes indistinguishable from a Gaussian distribution.

The most common experimental probes typically measure only second moments, from which diffusion coefficients can be extracted, that provide no information about non-Gaussian behavior. For example, dynamic light scattering (12, 18) and nuclear magnetic resonance (NMR) (19) measure θ0-averaged translational diffusion coefficients of anisotropic constituents; and NMR (19), fluorescence depolarization (20), electric birefringence (21), dichroism (22), and depolarized dynamic light scattering (19) measure rotational diffusion coefficients. In principle, some of these probes, light scattering in particular, could provide a measure of Math and higher moments, but we are aware of any such measurements. Certainly, the non-Gaussian effects would be very small, especially for particles in 3D where Da/Db < 2.

Our observations using digital video microscopy of the Brownian motion of an isolated ellipsoid in two-dimensions provide exquisitely detailed information about the diffusive properties of anisotropic objects and the subtle interplay between orientational and translational motions. Besides providing us with new insights about a fundamental phenomenon, these observations and underlying theory are potentially useful for research on diffusion of anisotropic molecules in membranes (16), on the hydrodynamics and kinetics of ensembles of anisotropic particles, and on anisotropic molecules that experience slip boundary conditions and thus have a large ratio of γa to γb.

Supporting Online Material

www.sciencemag.org/cgi/content/full/314/5799/626/DC1

Materials and Methods

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Figs. S1 to S5

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