Pushing the limit: investigation of hydrodynamic forces on a trapped particle kicked by a laser pulse

Naja Villadsen; Daniel Ø. Andreasen; Jesper Hagelskjær; Jan Thøgersen; Alberto Imparato; Søren Rud Keiding

doi:10.1364/OE.23.013141

1. Introduction

Since the introduction of the optical tweezers in 1970 [1] lasers have been used extensively to push, hold and manipulate nano- and microscopic objects through the transfer of photon momentum. This has led to a plethora of applications, particularly in biophysics where the forces governing the interactions between biological objects and the achievable optical forces are comparable. However, a limitation to the precision by which we can determine the position and momentum of the trapped object, is the omnipresence of thermal fluctuations giving rise to Brownian motion [2]. In itself the study of Brownian motion is a very active research field: Recent advances in experimental techniques have led to the predicted observation of the transition from purely diffusive motion at long time scales $(x_{rms} (t) \propto \sqrt{t})$ [3,4] to ballistic motion at short timescales (x_rms(t) ∝ t). However, probing the short time dynamics remains a challenge: The deviation from normal diffusion, for a 5 μm radius particle in water, becomes observable only with sub-nanometer spatial resolution and with microsecond time resolution [5–14]. In this work we introduce a different experimental approach to investigate the dynamics of trapped particles. We show that by introducing a controlled repetitive perturbation, in the actual case a train of nanosecond pulses kicking the trapped object from the side, we can investigate the short time behavior of the trapped particle by considering only its average trajectory, and thus avoiding the calculation of higher order moments and the influence of the Brownian fluctuations [15]. In active microrheology, liquids are similarily studied either by actively exciting the liquid in which the particle is trapped or by modulating the trap strengths [16,17]. Also related is the technique localization microscopy [18] where increased spatial resolution is obtained by adding multiple images of individual fluorescent point sources. By bypassing the influence of the Brownian fluctuations our experiments give an unprecedented insight into the hydrodynamic forces acting on the trapped particle. Consider again the 5 μm sphere in water that has a position standard deviation of $\sqrt{2 D t} = 0.3 nm$ within 1 μs. Now consider replacing these faint molecular collisions underlying the Brownian motion with a short kick from a 225 nanosecond laser pulse. This will displace the sphere approximately 100 nm enabling a detailed experimental investigation of the dynamics of the sphere. The sudden impact of the laser pulse could also challenge the low Reynolds number restriction employed in deriving the hydrodynamic drag.

2. Theory

As a starting point for the discussion of the motion of the laser kicked sphere, we use the canonical Langevin description and write the total force acting on the probe particle as:

F_{res} (t) = F_{fric} (t) + F_{th} (t) + F_{ext} (t)

where F_fric is the frictional force, F_th is the stochastic thermal force, both originating from the environment of the particle, and F_ext is the additional external forces. We denote the particle position in a trajectory by y(t) and choose our origin such that ⟨y(0)⟩ = 0. The particle is laser-trapped in a harmonic potential, with strength κ, and kicked by a laser pulse at the beginning of each trajectory, as described below. The impulse force, I(t), resulting from the interaction between the laser pulse and the polystyrene bead, is modeled with a delta function having an amplitude, 2A, since the pulse duration is shorter than any other time scale in the system as discussed below. So we take I(t) = 2Aδ (t). The deterministic external force thus reads:

F_{ext} (t) = - κ y (t) + I (t)

The division of the forces from the environment into a friction and a stochastic force is made, such that the stochastic force has zero mean and the standard two time correlation is given by the fluctuation-dissipation theorem [19]. Choosing the simplest model possible for the friction, that is the Stokes friction, corresponding to a Reynolds number well below one and a steady flow [20]:

F_{fric} = - γ \dot{y} (t), γ = 6 π η R

our Langevin Eq. describes a Markovian model with a white noise thermal force, i.e. ⟨F_th(t)F_th(t′)⟩ = 2k_BTγδ (t−t′)

To account for the behavior at intermediate time scales between the diffusive and ballistic regimes, additional terms must be added to the Stokes friction to describe the flow around the trapped particle [6,7,10,11]. One then obtains the Stokes-Boussinesq friction force (also often referred to as the Basset force):

\begin{array}{l} F_{fric} = - γ (\dot{y} (t) + R \sqrt{\frac{ρ}{^{η}}} \frac{1}{\sqrt{π}} \int_{0}^{t} d t^{'} \frac{\ddot{y} (t^{'})}{\sqrt{t - t^{'}}}) \\ - \frac{1}{2} \frac{4}{3} π ρ R^{3} \ddot{y} (t) . \end{array}

This friction force accounts for the wobbly motion of the Brownian particle by adding a term taking into account the memory effect of the fluid and a mass correction term [21].

Requiring that the Stokes-Boussinesq friction force is related to the stochastic force by the fluctuation-dissipation theorem [10, suppl. mat.], we obtain the two time correlation for the stochastic force $〈 F_{th} (t) F_{th} (t^{'}) 〉 = - 3 k_{B} T R^{2} \sqrt{π η ρ} | t - t^{'} |^{- 3 / 2}$ . Thus, the noise field is no longer white.

Using the expressions for the forces (Eq. (2), (4)) the Langevin equation (Eq. (1)) can be rewritten as

\begin{array}{l} (τ_{p} + \frac{1}{9} τ_{f}) \ddot{y} (t) = - \dot{y} (t) - \sqrt{τ_{f}} \frac{1}{\sqrt{π}} \int_{0}^{t} d t^{'} \frac{\ddot{y} (t^{'})}{\sqrt{t - t^{'}}} \\ - \frac{1}{τ_{κ}} y (t) + 2 \bar{A} δ (t) + ξ_{th} (t) \end{array}

where we have introduced the momentum relaxation time, τ_p = m/γ, the fluid vorticity relaxation time, τ_f = R²ρ/η and the trap relaxation time, τ_κ = γ/κ. Furthermore the pulse amplitude

\bar{A} = A / γ

and the stochastic force ξ_th = F_th/γ have been rescaled. For the present experiments the time scales are approximately τ_p = 6μs, τ_f = 25μs, and τ_κ = 100ms.

Equation (5) becomes much more tractable in Laplace space. By Laplace transforming $(\tilde{f} (s) = \int_{0}^{\infty} d t e^{- s t} f (t))$ and solving for the Laplace transformed trajectory, we obtain

\begin{array}{l} \tilde{y} (s) = \tilde{G} (s) (\bar{A} + ζ_{t h}) \\ \tilde{G} (s) = {((τ_{p} + \frac{1}{9} τ_{f}) s^{2} + \sqrt{τ_{f}} s^{3} + s^{2} + τ_{κ}^{- 1})}^{- 1} \end{array}

where an implicit average over the initial position and velocity have been performed. The ensemble averaged trajectory is given by

〈 \tilde{y} (s) 〉 = \tilde{G} (s) \bar{A}

_, which is a purely deterministic prediction unaffected by any thermal motion. It is evident that without kicking the particle (A = 0) the average position ⟨y(t)⟩ of the particle would be stationary, therefore we would have had to resort to higher moments to investigate the system dynamics, as in, e.g. [6,7,10,11,22].

The average trajectory in direct space is now found using known formulas for the inverse Laplace transform [23, Eq. 2.1.18, 2.1.27, 2.2.23]

\begin{array}{l} 〈 y (t) 〉 = \frac{\bar{A}}{τ_{p} + \frac{1}{9} τ_{f}} (\frac{r_{1} e^{r_{1}^{2} t} erfc (- r_{1} \sqrt{t})}{(r_{1} - r_{2}) (r_{1} - r_{3}) (r_{1} - r_{4})} \\ + \frac{r_{2} e^{r_{2}^{2} t} erfc (- r_{2} \sqrt{t})}{(r_{2} - r_{1}) (r_{2} - r_{3}) (r_{2} - r_{4})} \\ + \frac{r_{3} e^{r_{3}^{2} t} erfc (- r_{3} \sqrt{t})}{(r_{3} - r_{1}) (r_{3} - r_{2}) (r_{3} - r_{4})} \\ + \frac{r_{4} e^{r_{4}^{2} t} erfc (- r_{4} \sqrt{t})}{(r_{4} - r_{1}) (r_{4} - r_{2}) (r_{4} - r_{3})}) \end{array}

where r₁, r₂, r₃ and r₄ are the roots of

{(\tilde{G})}^{- 1}

when interpreted as a 4th order polynomial in

\sqrt{s}

.

Had we used the standard Stokes friction force instead of the Stokes-Boussinesq friction, the average trajectory would simply read

〈 y (t) 〉 = \frac{\bar{A}}{ϑ} (e^{(ϑ - 1) \frac{t}{2 τ_{p}}} - e^{- (ϑ + 1) \frac{t}{2 τ_{p}}})

\approx \bar{A} e^{- \frac{t}{τ_{κ}}} for τ_{κ} ≫ τ_{p}

where

ϑ = \sqrt{1 - 4 τ_{p} / τ_{κ}}

. For both the predictions of the average trajectory, Eqs. (7) and (8), it is again essential to underline that similarly to their Laplace transforms, they represent deterministic predictions. By kicking the particle we found a simple way of the circumventing the question of the correlation of the noise field. Instead it becomes possible to directly test the relation between the hydrodynamics of the system and the friction force used in the theoretical description.

3. Experiment and results

The setup used for testing the theoretical predictions by trapping and kicking is based on a counter propagating optical trap, shown in Fig. 1. Within the trap individual micrometer particles were trapped in water using (5–10) mW laser light at 1065 nm (for further information about the trap see [24]). The trapping laser beams propagated along the ±z-axis and the position of the trapped particle was detected orthogonal to these beams, along the x- and y-axis. Detection was conducted using a very low noise 635 nm diode laser (Coherent, Lablaser 635nm 5mW C ULN) and a dual axis position sensitive detector, PSD, (Pacific Silicon Sensor, DL100-7-PCB3). The voltage from the PSD, corresponding to the x- and y-position, was acquired by a DAQ-card (National Instruments, NI PCI-6154, 16 bit) with a sampling rate, f_s = 250kHz. The positions were normalized with the intensity of the detection laser measured simultaneously with a photodiode.

Fig. 1 The counter propagating optical trap geometry, with the two 1065 nm trapping beams; the 635 nm detection beam; and the 532 nm kicking beam. O1, O2 and OA are objectives.

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The trap was calibrated using a polystyrene bead (R = 5.0±0.1μm, Fluka, 72986) trapped in water and held in the center of the trapping potential for 5 minutes. The temperature was 20.4 °C and controlled within 0.2 °C. Futhermore, we do not expect that the optical trap heated the particle more than 0.1 °C. By Fourier transforming the particle position we obtain the power spectral densities, S(f), shown in Fig. 2(a). The frequency dependent power spectral density is equivalent to the Fourier transform of the time dependent second moment of the particle position and it is used to obtain the voltage to nanometer calibration according to [26]. In Fig. 2(a) three regimes can be distinguish: Below 2 Hz, corresponding to the corner frequency of the trapped particle, the power spectral density is flat. In the intermediate region from 2 Hz to 10 kHz the power spectral density is diffusive and proportional to f⁻². Finally, from 10 kHz to 0.125 MHz, the power spectral density is constant due to the 2 nm spatial resolution of the detection system giving rise to a noise floor, S_f, where the particle fluctuations are smaller than the spatial resolution of the system. Similarly, we have investigated the time resolution of the detection system. The nominal resolutions of both the PSD and the DAQ-card are 4 μs, but in order to extract the detailed dynamics of the kicked particle, the exact time response of the system was measured. This measurement was conducted by using a 100 fs laser pulse at 637 nm to excite the PSD. Thereby we obtained the response function of the entire system, at the detection wavelength, as seen in Fig. 2(b). The response of the system consisting of both the PSD and the DAQ-card is characterized by a 4 μs rise time. The subsequent decay is dominated by a fast Gaussian decay time of 8.3 μs responsible for 77% of the signal and a residual slower exponential component with a decay time of 25.9 μs. We believe that such a direct measurement of the time dependent response function of the system provides an intuitive and transparent alternative to the often complex and intricate schemes used to account for the system response in the frequency domain. The regions of the power spectrum density at low and intermediate frequencies are fitted with the power spectrum based on Stokes (Eq. (3)) and Stokes-Boussinesq (Eq. (4)) friction. These fits, based on the analytical expression for the power spectral density given in reference [6] are shown in Fig. 2(a) as a red and green line, respectively. It is evident from these fits that even higher spatial resolution is needed in order to for example distinguish between Stokes and Stokes-Boussinesq friction when the analysis is based on measurements of the power spectral density. Following the procedure in [9] the noise floor was substracted from the power spectral density, showing a better agreement with the extended theory of Stokes-Boussinesq.

Fig. 2 (a) shows the power spectrum, S(f), of the measured data (•) and the measured data substracted the noise floor (○) of a R = 5μm bead and the associated fits with the Stokes (- - -) and the Stokes-Boussinesq friction (—), respectively. A few noise peaks were excluded before the measured power spectrum were averaged in logarithmic blocks and fitted taking both blur and anti aliasing into account [25]. (b) shows the measured response function (•) of the detection system at 637 nm. It rises instantaneously and falls of as: $F_{respons} = A_{1} \exp (- t^{2} / (2 t_{1}^{2})) + A_{2} \exp (- t / t_{2})$ , where A₁ = 0.77, A₂ = 0.22, t₁ = 8.3μs, and t₂ = 25.9μs.

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A segment of the calibrated trace is shown in Fig. 3(a). The trace shows the fluctuations of the particle caused by thermal motions, resulting in a standard deviation of the particle position σ_y = 54nm. In addition the trap force constant, based on the corner frequency, f_c, κ_y = 2πγf_c = 1.22 ± 0.05pN/μm [27], the position resolution, $Δ y_{res} = \sqrt{S_{f} f_{s}} = 2 nm$ , based on the noise floor, S_f, and the sampling rate, f_s, [25] were obtained from the power spectrum as well.

Fig. 3 Particle trace (R = 5μm particle) in the direction of kicking, with y = 0 at the center of the trap. (a) shows the trace of the trapped particle without kicking. The particle fluctuates around the center of the trap with a standard deviation of σ_y = 54nm. (b) show three examples of traces when the particle is kicked by a laser pulse at t = 0 marked with short black lines. (c) shows the average trace corresponding to 1200 kick pulses. The fluctuations caused by Brownian motion are averaged out and the dynamics of the kicked particle is clearly visible. The standard deviation of the averaged trace is reduced by approximately $\sqrt{n}$ , where n corresponds to the 1200 events and is equal to $σ_{y}^{avg} = 1.5 nm$ .

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A pulsed frequency doubled Nd:YAG laser (Clark-MXR, Inc., ORC-1000, 532 nm) was used to kick the particle at the beginning of each trajectory. The pulse duration was measured to be 225 ns with a fast photodiode (Thorlabs DET10A). This is faster than the sampling time equal to 4 μs and the characteristic times scales of the particle motion (τ_p,τ_f, τ_κ), thereby justifying the use of a delta kick in modelling the data. The laser was focused on the trapped particle from below by an objective (Oplympus, LMPL20XIR), see Fig. 1. This way, the particle was kicked by the laser pulses aligned orthogonally to the trapping beams, resulting in a displacement along the y-axis. Depending on the alignment a displacement can also be induced in the x- and z-axis. Due to Brownian motion, the position of the trapped particle will fluctuate with respect to the beam waist of the kicking laser. However, the beam waist of the kicking laser (15 μm) is significantly larger than the amplitude of the Brownian motion (0.1 μm) and this ensures a constant amplitude and direction of each kicking pulse. The laser was triggered by the DAQ-card, responsible for PSD data acquisition. The timing was checked by detecting a small amount of the kicking light on the PSD, and reassuring that consecutive pulses always were separated by the same number of measurement points. For 2 Hz kicking and 250 kHz sampling rate the kicking pulses hit the bead exactly every 125000th sampling point.

Subsequently, the kicking light was blocked from the detector, and the trapped particle was kicked repeatedly – 1200 times at 2 Hz.

Figure 3(b), shows three examples of traces of the kicked particle. The particle displacement is composed of the displacement due to kicking at t = 0, and the inherent thermal fluctuations of the particle. Consequently, as the two contributions are comparable in amplitude, the kick is hardly distinguishable from the Brownian fluctuation and each relaxation curve looks very different. However, by averaging the 1200 synchronized kicking events, the thermal fluctuations are completely eliminated and the underlying dynamics stands out clearly. This is evident in Fig. 3(c). In this experiment the motion consists of a very fast rise starting at t = 0 followed by a exponential decay. The slow relaxation is well described by the Stokes friction as the memory-effects related to the fluid vorticity relaxation time are short-time effects. The slow decay is thus given by the long-time approximation of Eq. (8b): ⟨y(t)⟩ ∝ exp(−t/τ_κ), with a trap relaxation time of τ_κ = 72 ± 5ms. This is in agreement with τ_κ = 76±5ms calculated from the corner frequency of the power spectrum [15]. Similar agreement is observed when changing the size of the trapped particles (R=10.0 ± 0.2 μm and R=2.50 ± 0.05 μm), and changing the corner frequency and the kicking power.

The remaining position standard deviation $(σ_{y}^{avg})$ after elimination of the thermal fluctuations, was obtained by subtracting the exponential decay given in Eq. (8b) from the averaged trace at t > 0.1s and was found to be $σ_{y}^{avg} = 1.5 nm$ . This reduced standard deviation is caused by the coherent averaging bypassing the thermal fluctuations and agrees with the estimate given by $σ_{y}^{avg} \approx σ_{y} / \sqrt{n} = 2.0 nm$ , where n corresponds to the 1200 events. The same enhancement reduces the noise floor resulting in an improved resolution of: $Δ y_{res}^{avg} \approx Δ y_{res} / \sqrt{n} = 0.09 nm$ . So, in other words, by coherently adding the 1200 traces of the particle position we obtain a simultaneous time- and spatial resolution corresponding to 4 μs and 0.1 nm. This enables investigation of the details of the motion of the kicked particle with a signal to noise ratio exceeding 10³.

We now focus on the short time dynamics by zooming in on the first half millisecond of the kick averaged trajectory, shown in Fig. 3(c). With the improved spatial resolution and precise knowledge of the time response of the detection system, we are able to compare the short time dynamics of the measured average trajectory and the predicted trajectory using either the Stokes or the Stokes-Boussinesq friction. The dynamics of the displacement, caused by the kick from the laser pulse is shown in Fig. 4. The force from the kick accelerates the particle and imposes an initial velocity on the particle. The friction of the liquid will reduce the velocity and eventually the particle is dragged back to the origin by the trapping force, see Fig. 3(c).

Fig. 4 (a) is a zoom of the first 0.5 ms of Fig. 3(c) (•) with the scaled Stokes (- - -) and Stokes-Boussinesq (—) averaged position. The calculated positions have been convolved with the time response function of the detector. (b) is the experimental traces (normalized to equal amplitude) at 0.56(○), 0.78(∇), 1(+), and times the maximal kicking power. The error bars in both (a) and (b) are based on the averaged position standard deviation, $σ_{y}^{avg} = 1.5 nm$ .

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The rise time of the particle is expected to be longer than the value predicted by the Stokes friction Eq. (3) because of the additional terms of Eq. (4). In order to compare the theoretical prediction, of Eq. (8a) and (7) with the measurements, the theoretical curves were convolved using the response function of the detection system shown in Fig. 2(b), and normalized to the maximum displacement of the measurement. The normalization corresponds to a fit of the kick interaction amplitude $\bar{A}$ to the experimental result and is shown in Fig. 4(a). It is clearly seen that the Stokes prediction rises too fast compared to the measured displacement. The Stokes-Boussinesq prediction, however, rises slower, and is observed to be in good agreement with experimental data.

The same agreement is observed when changing the the size of the particle. Figure 5(a) shows the normalized rise of particles with radius 2.5 μm, 5 μm, and 10 μm. As expected from Eq. (4) the particles rises slower with increasing radius due to the additional drag. Changing the solvent to heavy water, that has a higher density and a higher viscosity compared to normal water, the agreement still holds (data not shown).

Fig. 5 (a) shows the size dependent rise of the particles for radii of: 2.5 μm (∇), 5 μm (○), and 10 μm (Δ), respectively (b) shows the maximal displacement for a R = 5μm at different kick pulse intensities. The line is a linear fits given as a guide to the eye. The error bars in (a) and (b) corresponds to the appertaining position standard deviations of the traces

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According to Eq. (7) the force of the kick will only influence the amplitude of the displacement and not the dynamics. It is seen in Fig. 4(b) that this is indeed the case; the temporal dynamics of the particle is independent of the amplitude of the kick and thus independent of the initial velocity. That is, we observe no indication that the averaged trajectory is dependent on Reynolds number (at least within the range of Reynolds numbers considered here). We further note, that Fig. 4(b) together with Fig. 5(b) shows that the kicking strength only influence the motion through the maximal displacement of the particle and does so linearly, again in agreement with the description given in Eq. (7).

4. Summery

In summary, we have introduced a new technique enabling the study of the motion of a trapped particle bypassing the Brownian fluctuations. The particle is kicked by a short laser pulse with a force of approximately 15 nanoNewton giving it an acceleration corresponding to 10⁴g and a subsequent initial velocity of approximately 0.006 m/s or two orders of magnitude larger than typical thermal velocities. In spite of this seemingly drastic perturbation of the trapped particle, the temporal dynamics is still well described by the Stokes-Boussinesq friction model. With the improved sensitivity of the present method, we are able to confirm the results [5–13, 25] obtained from the recent analysis of the short time dynamics of Brownian motion. Due to our analysis, based on the more direct evaluation of the first moment of the displacement rather than of the second moment of the position or velocity, the spatio-temporal resolution of the system can be relaxed. Thereby making the technique applicable to other repetitive systems influenced by Brownian motions. With a different focusing geometry of the kicking and detection laser and increased displacements we have been able to observed small temporal dynamics deviating from the Stokes-Boussinesq friction model. However, these experiments were conducted at laser fluences close to the damage threshold of the trapped particles.

The Stokes-Boussinesq friction force is derived under several conditions [21, §24]. One of them being the assumption that the flow have a Reynolds number well below one. The Reynolds numbers is given as the product of a characteristic length and velocity divided by the kinematic viscosity. A crude estimate of the initial Reynold number can be obtained from the maximum displacement of the particle, y_max; Re ∼ 50y_max/τ_p. In the present work the maximum Reynold number is thus around 0.1. As shown in Fig. 4(b) we do not observe any changes in the temporal dynamics of the trapped particle when changing the Reynolds number and thus, can conclude that even with a relative high Reynolds number pertaining to our experiments, we do not challenge the assumptions of the Stokes-Boussinesq friction model.

There are other assumptions needed to derive the Stokes-Boussinesq friction, among those the assumption of incompressibility of the fluid. However, the density fluctuations in a compressible fluid decays with a time constant, τ_c = Rv_c, given by the speed of sound, v_c and the radius of the particle [28]. That is the density fluctuations will have vanished within 3 ns for our 5 μm particle. This is faster than the sampling time and therefore it will not have any influence on our measurements. However, an interesting extension of our experimental technique, replacing both the kicking- and tracking laser with femtosecond laser pulses, can increase the time resolution to femtoseconds. This will reveal an entirely unexplored time domain of dissipative motion where the viscoelastic properties of both the fluid and the trapped particle can be observed [28]. If we consider a trapped particle that can absorb photons from the kicking laser, the femtosecond technique could be used to observe how momentum transfer to individual molecules is gradually transferred into motion of the macroscopic particle.

Acknowledgments

We gratefully acknowledge the financial support from the Danish Natural Science Research Council.

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Pushing the limit: investigation of hydrodynamic forces on a trapped particle kicked by a laser pulse

Abstract

1. Introduction

2. Theory

3. Experiment and results

4. Summery

Acknowledgments

References and links

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Figures (5)

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