High-resolution detection of Brownian motion for quantitative optical tweezers experiments

We have developed an in situ method to calibrate optical tweezers experiments and simultaneously measure the size of the trapped particle or the viscosity of the surrounding fluid. The positional fluctuations of the trapped particle are recorded with a high-bandwidth photodetector. We compute the mean-square displacement, as well as the velocity autocorrelation function of the sphere, and compare it to the theory of Brownian motion including hydrodynamic memory effects. A careful measurement and analysis of the time scales characterizing the dynamics of the harmonically bound sphere fluctuating in a viscous medium directly yields all relevant parameters. Finally, we test the method for different optical trap strengths, with different bead sizes and in different fluids, and we find excellent agreement with the values provided by the manufacturers. The proposed approach overcomes the most commonly encountered limitations in precision when analyzing the power spectrum of position fluctuations in the region around the corner frequency. These low frequencies are usually prone to errors due to drift, limitations in the detection, and trap linearity as well as short acquisition times resulting in poor statistics. Furthermore, the strategy can be generalized to Brownian motion in more complex environments, provided the adequate theories are available.

Figure 1

(a) A lin-log representation of the uncalibrated MSD of an optically trapped melamine resin microsphere in water ($R=1.4$μm, ${\rho }_p=1510$ g/l, ${\rho }_f=1000$ g/l, and ${\tau }_p/{\tau }_f=0.45$) yielding ${\langle \Delta x^2\left(\infty \right)\rangle }^{\mathsf{V}}=1.54{\mathrm{mV}}^2$ and $t_{1/2}=327$µs. The inset shows a log-log representation of the same MSD normalized by its plateau value. (b) A lin-log representation of the corresponding uncalibrated VACF, with $t_0=39.3$ µs and ${\langle v\left(t_{\min }\right)v\left(0\right)\rangle }^{\mathsf{V}}=-2.64\times {10}^6{\mathrm{mV}}^2/{\mathrm{s}}^2$. The inset shows a log-log representation of ${|\langle v\left(t\right)v\left(0\right)\rangle }^{\mathsf{V}}|$. The amplitude of the $t^{-3/2}$ long-time tail reads $B^{\mathsf{V}}=0.65{\mathrm{mV}}^2/{\mathrm{s}}^{1/2}$. Data were blocked in ten bins per decade. Error bars give the standard error on the mean from blocking. (c) Top: Theoretical ratio $t_0/{\tau }_f$ as a function of ${\tau }_K/{\tau }_f$ for a resin sphere (${\rho }_p=1510$ g/l) in water (${\rho }_f=1000$ g/l). The trap relaxation time increases from left to right, while the trap stiffness decreases. Bottom: Corresponding theoretical values of $t_{1/2}/t_0$. Each value necessary to calibrate the data shown as an example is highlighted by an arrow.

Figure 2

(a) Calibrated MSD and (b) VACF of a single resin sphere (${\rho }_p=1510$g/l and $R=1.4\mu \mathrm{m}$) in water (${\rho }_f=1000$g/l, $\eta =0.98\mathrm{cP}$, $T=21{}^{\circ }\mathrm{C}$, ${\tau }_p/{\tau }_f=0.45$, and ${\tau }_f=2\mu s$) for increasing $K$. Data were blocked in ten bins per decade. Error bars give the standard error on the mean from blocking. (c) Overview of the parameters obtained from calibrating the data.

Figure 3

(a) Calibrated MSD and (b) VACF of single resin particles (${\rho }_p=1510\mathrm{g}/\mathrm{l}$) with different sizes in water (${\rho }_f=1000\mathrm{g}/\mathrm{l}$, $\eta =0.98\mathrm{cP}$, and ${\tau }_p/{\tau }_f=0.45$). Data were blocked in ten bins per decade. Error bars give the standard error on the mean from blocking. (c) Overview of the parameters obtained from calibrating the data.

Figure 4

(a) Calibrated MSD and (b) VACF of resin particles in acetone (${\rho }_f=790\mathrm{g}/\mathrm{l}$), water (${\rho }_f=1000\mathrm{g}/\mathrm{l}$), and glycerol (${\rho }_f=1066\mathrm{g}/\mathrm{l}$). Data were blocked in ten bins per decade. Error bars give the standard error on the mean from blocking. (c) Overview of the parameters obtained from calibrating the data.