Sylvia Jeney is employed at the Swiss Federal Institute of Technology in Lausanne (EPFL) and is leading a research project on nanospectroscopic methods in living systems at the Swiss Nanoscience Institute. Being an experimental biophysicist by training, her research interests often cross the borders between physics and biology. Her work on Brownian motion has been published in Physical Review Letters, Nature and in Nature Physics.

Brownian Motion beyond Einstein

Sylvia Jeney, École polytechnique fédérale de Lausanne

 

Brownian Motion and Diffusion

The observation of the erratic motion of microscopic particles suspended in a liquid dates back to the invention of optical microscopy in the 17th century. This phenomenon was later named after the Scottish botanist Robert Brown who published in 1827 his observation of random and apparently inexhaustible motions in a wide range of suspensions. From then on, understanding the origin of Brownian motion was under debate [1], until 1905, when Einstein gave a convincing theoretical description. He assumed that the fluctuations of small-sized spherical particles floating in a fluid are caused by momentum transfer from thermally excited fluid molecules. He identified the mean square displacement (MSD) of the particle as the characteristic experimental observable of Brownian motion, and showed that it grows linearly with time, , thereby introducing the diffusion coefficient D = kBT/6πηR, where R is the particle’s radius, η the viscosity of the fluid, T its temperature, and kB the Boltzmann constant [2]. Attempts to measure velocities of a Brownian particle in experiments had failed so far because these velocities were too high and changed too rapidly.

In 1908, Langevin adapted Newton’s force balance equation to the problem of Brownian motion, by introducing a random thermal force Fth that arises from random fluctuations of the fluid molecules excited by the thermal energy kBT [3]. The particle’s motion is then on one hand driven by collisions with the fluid molecules, and on the other hand strongly damped by the Stokes friction force Ffr = -6πηRv of the surrounding viscous fluid, which is instantaneously linear with the particle’s velocity v [4].

Experimental verification of the theoretical framework, developed in parallel by Einstein, Smoluchowski and Langevin, was soon possible thanks to several technical breakthroughs. It was Langevin’s friend, Jean Perrin and his students who employed a combination of the both newly invented "Ultramicroscope" [5] and film camera to precisely determine the sizes of thousands of colloidal resin spheres and record each of their trajectories. Careful analysis confirmed that at equilibrium, i.e. times larger than τP = m/6πηR (m being the particle’s mass), particle motion was completely diffusive as predicted by Einstein. Perrin’s experiments related diffusion, an observable bulk phenomenon, to the non-observable fluctuations of the molecules constituting the fluid [6]. He was awarded the Nobel prize in 1926 for proving the existence of molecules and atoms.

Short-time dynamics of Brownian Motion

Throughout the 20th century the theory of Brownian motion was generalized, in particular for all times and at non-equilibrium. Therefore, the forces involved in the Langevin equation had to be defined more carefully. Fth(t) was associated to a Dirac delta distribution, hence a Gaussian white noise spectrum, which means that the force spectrum is constant over a wide range of frequencies [7, 8]. As a result for short times, the particle, kicked by the surrounding fluid molecules, moves with a mean velocity and motion is ballistic with . At long times, diffusive motion according to Einstein is recovered. The exponential transition from ballistic to diffusive motion occurs at the characteristic timescale τP (Fig. 1, left).

Uhlenbeck and Ornstein [9] further provided a solution of the Langevin equation for a harmonically bound Brownian sphere exposed to an external harmonic force Fext(t) = -Kx, with K the force constant. Motion becomes that of an overdamped harmonic oscillator reaching at long times the constant value . The transition from diffusive to confined motion is then characterized by τK = 6πηR/K (Fig. 1, right).

 

Fig 1: Characteristic time scales of a Brownian particle confined by a harmonic potential. On very short timescales (t << τP), the particle undergoes ballistic motion governed by its mass (left). On timescale τf, hydrodynamic backflow develops (centre; solid lines show the emerging fluid velocity field). Finally, for t ≥ τK , the harmonic potential sets in and confines particle diffusion (right).
Fig 2: MSD (a) and VAF (b) of a single resin sphere (m = 21 pg, R = 1.5 μm) in water (ρf = 1000 g/l) for K = 0,2 mN/m. Calibration of the particle’s trajectory and optical trap is achieved by fitting the theory (black continuous line) to the measured data (blue dots).
Fig 3: (a) Normalized power spectral density (PSD) of the thermal force. Deviations from Gaussian white noise are towards the blue end of the spectrum at frequencies that are much smaller than the collision rate of the solvent molecules, and reflect the color of thermal force. Inset: Corresponding fluid velocity field developing around the sphere. (b) Normalized PSD of the position fluctuations of the resin sphere. The blue line indicates exponential relaxation when overdamped motion, according to Uhlenbeck and Ornstein, is assumed. The black lines correspond to the full hydrodynamic theory.
Fig 4: (a) Theoretical PSD of a resin sphere in very strong traps (blue: K=1.02 mN/m, green: 2.04 mN/m, red: 4.07 mN/m). The dashed lines give a damped harmonic oscillator. The colored dots represent the corresponding PSD after modulating the trap strength at frequency fexc=2fpeak demonstrating the possibility of parametric excitation of the Brownian resonator. (b) A Brownian nanomechanical resonator sensing molecules within its fluid environment.

Also Ffr(t) had to be refined, as the Stokes force only applies for steady flow at long times and for particles, which are much denser than their medium. However, in most experiments (also Perrin’s) neutrally buoyant particles are used implying that the particle has a density similar to the surrounding medium. When the particle receives momentum from the fluctuating fluid molecules, it displaces fluid in its immediate vicinity. The non-negligible inertia of entrained fluid acts back on the sphere. As a consequence, the fluid bears a memory on the particle’s past motion. The time needed by the perturbed fluid flow field to diffuse over one particle radius is then given by τf = R2ρf/η , with ρf the density of the fluid (Fig. 1, middle). The phenomenon is well known for macroscopic objects. For example, when a swimmer stops his movement abruptly, he can still feel the ongoing drag of the water.

Following Navier-Stokes hydrodynamics, the complete expression for Ffr(t) was already given in the late 19th century by Boussinesq [10] and Basset [11], but yielded a complicated Langevin equation, which was only solved in 1945 by Vladimirsky and Terletzky who provided the exact expression for the MSD [12]. However their contribution, published in Russian, remained largely ignored. In 1967, Alder and Wainwright discovered, in numerical simulations, that the particle’s velocity autocorrelation function (VAF, another characteristic observable of Brownian motion) displays a power-law decay [13] instead of an exponential relaxation, as expected for simple Stokes friction. These simulations led theoreticians in the 1970’s to reconsider the contribution of fluid mechanics to Brownian motion [14, 15 and many others]. In 1992, Clercx and Schram provided the full theoretical description of a harmonically bound Brownian sphere including hydrodynamic memory [16].

Indirect experiments using dynamic light scattering in colloidal suspensions confirmed that the diffusion of colloidal particles is influenced by fluid mechanics [17, 18 and many others]. However, to achieve a high enough resolution, averaging over an ensemble of different particles was necessary.

Direct observation of Brownian Motion at ballistic and hydrodynamic timescales

This year, about 100 years after Perrin’s pioneering experiments, again recent technical improvements and a careful consideration of theoretical predictions led to the experimental verification of the short-time behavior of a single micrometer sized Brownian particle. For such a particle, ballistic motion exists at timescales significantly faster than τP = 100 ns, and the corresponding average displacement is of the order of 1 Å.

This extraordinary high spatial and temporal resolution was achieved by optical trapping interferometry. A resin sphere (R = 1.5 µm) immersed in a fluid was held in the well-defined harmonic potential of a laser trap (also called optical tweezers). The interference pattern created between the laser light scattered by the fluctuating sphere and the non-scattered light was recorded on a high-bandwidth position detector [19]. The optical trap provided thereby the light source for illuminating the particle and, at the same time, ensured that it remained within the detector range [20]. Such configuration allowed reaching the early ballistic regime of Brownian motion, characterized by a t2-dependence in the MSD (Fig. 2a) [21]. Also the VAF became measurable, displaying an initial exponential decay determined by the particle’s mass and followed by the predicted, but up to now never measured, t-3/2 power-law decay [21, 22] (Fig. 2b).

Resonances and the Color of Thermal Noise

According to the fluctuation-dissipation theorem [23], a direct consequence of the hydrodynamic coupling between sphere and fluid is that Fth(t) is not only characterized by a delta-correlated white noise term, but has also a colored, frequency-dependent component. By increasing the trapping force Fext(t) and decreasing the viscosity of the fluid, i.e. reducing Ffr(t), it could be shown that the spectrum of the thermal force indeed grows with increasing frequencies (Fig. 3a) [24].

The colored thermal noise led to a resonance in the spectrum of the bead’s positional fluctuations (Fig. 3b). The appearance of such peak was up to now overlooked since hydrodynamic memory is commonly neglected and the paradigm of overdamped Brownian motion dating back to Uhlenbeck and Ornstein is assumed in optical trapping experiments. Interestingly, resonances disappeared when bringing the particle close to a hard surface. Due to the increased surface friction, the diffusion of the hydrodynamic backflow was hindered [supplementary information in 24, 25, 26].
Stronger and narrower resonances could be obtained, but yet only theoretically or in computer simulations, by increasing and modulating the trap strength through parametric excitation (Fig. 4a) [24]. It was found that the amplitude of the resonance is strongly sensitive to the stiffness of the harmonic potential, the boundary conditions at the fluid-particle interface, the fluid properties, as well as the size and mass of the particle.

A Brownian Nanoresonator

Inspired by microcantilever-based sensors [27], the particle-fluid-trap system has the potential to turn into a nano-mechanical resonator to characterize fluid properties, detect the presence of analytes and probe even deeper into the molecular world (Fig. 4b). Here also theoretical predictions and new instruments will pave the way for new exciting and challenging experiments on Brownian motion.

 

 

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[Released: January 2012]