Long-term influence of fluid inertia on the diffusion of a Brownian particle

We experimentally measure the effects of fluid inertia on the diffusion of a Brownian particle at very long time scales. In previous experiments, the use of standard optical tweezers introduced a cutoff in the free diffusion of the particle, which limited the measurement of these effects to times comparable with the relaxation time of the fluid inertia, i.e., a few milliseconds. Here, by using blinking optical tweezers, we detect these inertial effects on time scales several orders longer up to a few seconds. The measured mean square displacement of a freely diffusing Brownian particle in a liquid shows a deviation from the Einstein-Smoluchowsky theory that diverges with time. These results are consistent with a generalized theory that takes into account not only the particle inertia but also the inertia of the surrounding fluid.

Figure 1

(a) Error in the estimation of the $\Delta \langle x^2\rangle \left(\tau \right)$ in the case of a melamine particle $\left(d=8.1\mu \text{m}\right)$ taking into account its mass (dashed line) and taking into account also the fluid inertia (solid line). While the former quickly reaches a plateau, the latter diverges as a function of $\tau$. (b) Error in the estimation of the diffusion coefficient $D$ for the same particle immersed in the various fluids used in this work taking into account the presence of the fluid inertia [Eq. (5)], which decays polynomially with ${\tau }^{-1/2}$. When only the particle inertia is taken into account, the error on the estimation of $D$ decays exponentially with characteristic time ${\tau }_{\mathrm{m}}$ so that the error is negligible on the plotted scales.

Figure 2

(a) The setup consists of a blinking optical tweezer combined with a fast digital video acquisition system. (b) Image and trajectory of a particle freely diffusing after being released from the trap. (c) Example of the $x$ component of the trajectory acquired while the particle is trapped and released repeatedly by the optical tweezers. The gray-shaded areas represent the periods of time when the trapping laser is on.

Figure 3

Errors in the determination of the $\Delta \langle x^2\rangle \left(\tau \right)$ (a) and of the diffusion coefficient $D$ (b) for different combinations of particles and fluids: $3.1 \mu \text{m}$ Licristar bead in acetone (green squares and dotted line), $5.0 \mu \text{m}$ polystyrene bead in methanol (brown circles and solid line), $4.8 \mu \text{m}$ silica bead in methanol (yellow up triangles and dashed line), and $3.1 \mu \text{m}$ Licristar bead in water (blue right triangles and dashed-dotted line). The experimental errors are similar to those reported in Fig. 4 and are not shown for clarity.

Figure 4

The polynomial behavior of the error in the estimation of (a) the $\Delta \langle x^2\rangle \left(\tau \right)$ and (b) of the diffusion coefficient $D$ for an $8.1\mu \mathrm{m}$ melamine particle up to a few seconds. The fitting to the theoretical model (dashed lines) is used as a guide for the eyes. In fact, the presence of a boundary introduces a change in the polynomial order of the deviation at short time scales that was not included in the model.