Motion of a colloidal particle in an optical trap

Thermal position fluctuations of a colloidal particle in an optical trap are measured with microsecond resolution using back-focal-plane interferometry. The mean-square displacement $⟨\Delta x^2\left(t\right)⟩$ and power spectral density are in excellent agreement with the theory for a Brownian particle in a harmonic potential that accounts for hydrodynamic memory effects. The motion of a particle is dominated at short times by memory effects and at longer times by the potential. We identify the time below which the particle’s motion is not influenced by the potential, and find it to be approximately ${\tau }_k∕20$, where ${\tau }_k$ is the relaxation time of the restoring force of the potential. This allows us to exclude the existence of free diffusive motion, $⟨\Delta x^2\left(t\right)⟩\propto t$, even for a sphere with a radius as small as $0.27\mu \mathrm{m}$ in a potential as weak as $1.5\mu \mathrm{N}∕\mathrm{m}$. As the physics of Brownian motion can be used to calibrate an optical trap, we show that neglecting memory effects leads to an underestimation of more than 10% in the detector sensitivity and the trap stiffness for an experiment with a micrometer-sized particle and a sampling frequency above $200\mathrm{kHz}$. Furthermore, these calibration errors increase in a nontrivial fashion with particle size, trap stiffness, and sampling frequency. Finally, we present a method to evaluate calibration errors caused by memory effects for typical optical trapping experiments.

Figure 1

Measured (a) $⟨\Delta x^2\left(t\right)⟩$ and (b) $P\left(f\right)$ for a Brownian particle of radius $0.5\mu \mathrm{m}$ in a harmonic potential $k_1=64\mu \mathrm{N}∕\mathrm{m}$. Data for $P\left(f\right)$ are blocked in ten bins per decade. The dashed line indicates free diffusive Brownian motion [Eqs. (8, 16)]. Data are calibrated using the value for detector sensitivity from Table .

Figure 2

(a) Equation (9) (dotted lines) fitted to the measured $X\left(t\right)$ for a polystyrene sphere $(a=0.5\mu \mathrm{m})$ in the optical trap at various spring constants ($k_1=64\mu \mathrm{N}∕\mathrm{m}$ ●, $k_2=17\mu \mathrm{N}∕\mathrm{m}$ ◆, $k_3=3\mu \mathrm{N}∕\mathrm{m}$ ∎). (Only six data points per decade are shown for the sake of clarity.) (b) Measured $\tilde{X}\left(f\right)$ for the same time traces. Data are blocked in five bins per decade, and error bars indicate the standard deviation. Equation (17) (dotted lines) is fitted to the data. $X_{\mathit{\text{free}}}\left(t\right)$ [Eq. (10)] and ${\tilde{X}}_{\mathit{\text{free}}}\left(f\right)$ [Eq. (18)] for a free particle are indicated by a continuous line.

Figure 3

(a) Time ${\tau }_M$ at which $X\left(t\right)$ reaches its maximum versus ${\tau }_k$ ($a=0.27\mu \mathrm{m}$, ◻; $0.33\mu \mathrm{m}$, ▵; $0.50\mu \mathrm{m}$, 엯; $0.74\mu \mathrm{m}$, ▷; $0.99\mu \mathrm{m}$, ▿; $1.25\mu \mathrm{m}$, ◁) and theoretical prediction from Eq. (9) for each particle size (dashed lines). (Data and theory are given in grayscale to show the increase of ${\tau }_M$ with increasing $a$.) (b) ${\tau }_k∕{\tau }_M$ versus ${\tau }_k∕{\tau }_f$ (bottom axis) and versus $ak$ (top axis). The continuous line shows the theoretical prediction derived from Eq. (9). (c) Frequency ${\varphi }_M$ at which $\tilde{X}\left(f\right)$ reaches its maximum versus ${\varphi }_k$ and (d) ${\varphi }_M∕{\varphi }_k$ versus ${\varphi }_k∕{\varphi }_f$ (bottom axis) and versus $ak$ (top axis). The continuous line in (d) shows theoretical prediction from Eq. (17).

Figure 4

(a) ${\tau }_{\mathit{\text{free}}}$ plotted versus ${\tau }_k$ for polystyrene spheres of different radii ($a=0.27\mu \mathrm{m}$, ◻; $0.33\mu \mathrm{m}$, ▵; $0.50\mu \mathrm{m}$, 엯; $0.74\mu \mathrm{m}$, ▷; $0.99\mu \mathrm{m}$, ▿; $1.25\mu \mathrm{m}$, ◁). (b) ${\tau }_k∕{\tau }_{\mathit{\text{free}}}$ versus ${\tau }_k∕{\tau }_f$ (bottom axis) and versus $ak$ (top axis) The continuous line shows theoretical prediction (see text). Prediction for ${\tau }_k∕{\tau }_M$ from Eq. (21) is given by dashed line. (c) ${\varphi }_{\mathit{\text{free}}}$ plotted versus ${\varphi }_k$. (d) ${\varphi }_{\mathit{\text{free}}}∕{\varphi }_k$ versus ${\varphi }_k∕{\varphi }_f$ (bottom axis) and $ak$ (top axis). The continuous line in (d) shows theoretical prediction (see text). Prediction for ${\varphi }_M∕{\varphi }_k$ from Fig. 3d is given by dashed line.

Figure 5

(a) $X\left(t\right)$ for the sphere with $a=0.27\mu \mathrm{m}$ and $k=1.5\mu \mathrm{N}∕\mathrm{m}$, compared to the theory in Eq. (9) (dotted line). Theory for the free particle $X_{\mathit{\text{free}}}\left(t\right)$ is given by the continuous line [Eq. (10)], and time when $X_{\mathit{\text{free}}}\left(t\right)$ reaches the value of 0.98 is indicated by an arrow. (b) $\tilde{X}\left(f\right)$ for the same sphere compared to the theory in Eq. (17) (dotted line). The free particle case is given by a continuous line [Eq. (18)].

Figure 6

Underestimation error (a) ${\epsilon }_{\beta }$ for the detector sensitivity $\beta$ and (b) ${\epsilon }_k$ for the spring constant $k$, if inertial effects are neglected. Theories are fitted to the power spectrum in the fitting range between $1\mathrm{Hz}$ and $f_{\mathit{max}}=10\mathrm{kHz}$ (엯) and $100\mathrm{kHz}$ (◻). The error for the particle from Fig. 2 is highlighted as a full square.

Figure 7

(Color online) Contour plot of (a) ${\epsilon }_{\beta }$ and (b) ${\epsilon }_k$ interpolated from a grid of values (dots) for combinations of $ak$ (bottom axis) and $f_{\mathit{max}}∕{\varphi }_f$ (left axis). ${\epsilon }_{\beta }$ for $f_{\mathit{max}}=100\mathrm{kHz}$ from Fig. 6a is indicated by squares in (a). ${\epsilon }_k$ for the typical experiments employing optical trapping as listed in Table is indicated by squares in (b).