Probing Fluid Flow Using the Force Measurement Capability of Optical Trapping

Transcription

1 Probing Fluid Flow Using the Force Measurement Capability of Optical Trapping Namsoon Eom 1, Rossen Sedev 1, Bruce Wedding 2 and Jason Connor 1 1 Ian Wark Research Institute University of South Australia, Mawson Lakes, 5095, South Australia, Australia 2 School of Engineering, University of South Australia, Mawson Lakes, 5095, South Australia, Australia jason.connor@unisa.edu.au Abstract Interest in microfluidics has rapidly expanded over the past decade particularly for use as miniature chemical reactors. Microfluidic channels are now complex as they combine several functions on a single chip. Fluid flow details are extremely important but few methods are suitable to probe fluid flow because of the confined geometry. We use optical trapping of a small dielectric particle to probe the fluid flow. A highly focused laser beam attracts particles suspended in a liquid to its focal point. Thus particles can be trapped and positioned in 3D. By measuring the displacement of the trapped particle from its equilibrium position one can estimate the external force causing the displacement. The trap stiffness (spring constant) of the optical trap is very low making it an extremely sensitive force measuring device. Rather than using the optical trap to position and release a particle for independent velocimetry measurement, we map the fluid flow by directly measuring the hydrodynamic force acting on a trapped particle. The flow rate of a dilute aqueous electrolyte flowing through a plastic microchannel (w h l = 5 mm 0.4 mm 50 mm) was mapped successfully by monitoring the force acting on a small (1 µm dia.) silica particle. The fluid velocity profile obtained experimentally was in very good agreement with the theoretical prediction. Our flow mapping approach is relatively easy and reliable and can be used in any low-opacity dilute as well as concentrated suspensions flowing in microchannels of various geometries. Keywords-optical trapping; microfluidics; fluid flow mapping; trap stiffness; viscous friction I. INTRODUCTION Probing the properties of a viscous fluid flow on the microscopic scale is important both fundamentally and for practical applications. Currently, the most commonly used technique in measuring fluid flow is particle image velocimetry (PIV) [1]. Although PIV provides an accurate three dimensional fluid velocity profile in microfluidic channels, a practical issue is that the numerous microspheres used to track the flow may affect the functioning of the device and the flow itself at high concentrations. Recently optical trapping has been introduced as an alternative technique for investigating fluid flow profiles in microfluidics due to its highly localised nature and highly localized sensing capability [2-4]. Optical trapping, first developed by Ashkin in the mideighties [5], is a tool that uses the momentum of photons to manipulate microscopic particles in three dimensions. In an optical trapping system a collimated laser beam is tightly focused by a high numerical aperture objective lens and the focal spot attracts dielectric particles to its centre. Although constrained, the trapped particle can be displaced from its equilibrium position due to thermal fluctuations or external forces such as viscous friction. By measuring the displacement of a trapped particle, which is proportional to the force acting on the probe particle, and after calibrating the trap stiffness (spring constant), the optical trap becomes a highly sensitive force measurement tool. Previously, optical trapping has been used to measure simultaneously fluid velocity and viscosity [3], the velocity vectors of flow fields in three dimensions [2], and to map fluid velocity and shear stress in microfluidic systems [4]. These fluid flow measurements combined optical trapping with video microscopy [2, 4] or with an acousto-optic deflector [3]. In most fluid velocity measurement the optical trap was used to capture and then release a probe particle. The velocity of the particle was then obtained from video microscopy. In this scenario the force measurement capability of the optical trap is not utilized. Although a quadrant photo detector (QPD) [6] has been used to measure the displacement of a particle due to fluid flow, the stiffness of the optical trap was not obtained in the measurements. Using the stiffness of the optical trap in fluid velocity measurements has been avoided to date mainly due to lack of confidence in its correct calibration. However, trap stiffness calibration using the power spectral density (PSD) [7] and the equipartition theorem [8] methods is relatively easy and fast and it can be obtained when the QPD is calibrated. In this paper we confirm that the stiffness of the optical trap can be successfully used for flow mapping by showing agreement between experimental measurements and theoretical calculations of fluid flow in a rectangular channel. This method can be used for flow mapping without using an expensive highspeed video camera.

2 II. BACKGROUND THEORY When an external force is applied to a micrometre-sized particle held in an optical trap, the particle is displaced from the equilibrium point (bottom) of the trap by an amount proportional to the applied force. It has been shown [9-11] that the radiation pressure leads to a Hookean restoring force acting on the particle. That is, (1) where F is the restoring force, k is the trap stiffness, and x is the displacement of the particle from the equilibrium point in the trap. Current theory for the computation of the trapping force for particular objects and trapping geometries is unreliable [12], thus forces must be determined empirically. In this paper, we have used the power spectral method and the equipartition theorem methods. The equipartition theorem states that the kinetic energy of a particle is entirely determined by the local temperature. As the trapped particle is in equilibrium with the surrounding medium, the potential energy of the trap can be equated to the thermal energy and the equipartition theorem gives where k B is the Boltzmann constant, T is the absolute temperature, and < x² > is the mean square displacement from the equilibrium position. Thus the stiffness of the trap can be determined by measuring the positional variance of the trapped particle. The power spectral method is based on the fact that the trapped particle behaves like an overdamped oscillator because it is subjected to Brownian motion. The power spectral density (PSD) of the mean square displacement < x² > is described by a Lorentzian function [12, 13] (2), (3) where f C is the corner (cutoff) frequency at which the power spectral density drops by 3dB. The corner frequency is directly related to the spring constant of the trap by where γ is the drag coefficient. In the optical trap, the viscous drag is generated by the motion of the particle with respect to the surrounding liquid and is balanced by the optical trapping force that prevents the release of the particle. The beam focus with the trapped particle can be moved by beam steering, or the fluid can be pumped past the trapped particle at increasing velocity until the particle escapes. The drag force that the particle experiences is given by the Stokes law: (4) (5) where v is the velocity of the particle with respect to liquid, r is the radius of spherical particle, and η is the viscosity of the liquid. III. EXPERIMENTAL METHODS A. Experimental Setup and Sample Preparation A Thorlabs optical trapping kit (OTKB/M) and a force measurement module (OTKBFM) with a single mode laser diode (975 nm) and a Nikon 100 Oil Immersion Objective (NA 1.25, working distance 0.23 mm) were used. Even though this is a commercial trapping system it still requires careful beam alignment during the assembly to achieve 3D trapping. A USB oscilloscope (Picoscope 3224, 12 bits, 10 MHz) was additionally connected to the piezo stage controller and the QPD reader for data acquisition. A lock-in amplifier (Stanford Research, SR850) was used to feed an input signal to the piezo driver in order to move the stage linearly or sinusoidally as required for the trap stiffness calibration. The laser power was set to 11.4 mw for the 3D trapping results presented here and a vibration isolation system (Thorlabs, PTT600600) was also used to minimize errors during the measurements. A plastic flow cell (Ibidi µ-slide I 0.4 Luer) was used for the flow mapping. The channel volume was 100 µl (w h l = 5 mm 0.4 mm 50 mm) and the thickness of the bottom channel wall (wall nearest objective lens) was 0.17 mm. The probing particles used were 1 µm in diameter nonfunctionalised fused silica beads (Bangs Laboratories, catalogue code SS04N/9857) suspended in deionised water. The coefficient of variation (standard deviation divided by the mean value) of the bead size was 15% as listed by the manufacturer. The 1 µm silica beads were chosen as larger particles may perturb the fluid flow [4]. The microchannel was connected to a 10 ml plastic syringe with an 18 Gauge syringe needle using silicon tubing. Once the channel and the tubing were filled with the liquid suspension containing particles, the syringe was connected to a motion controller (Oriel Encoder Mike Controller). Water was injected into the microchannel with a constant velocity (volume flow rate of m 3 /s). Injecting fluid into the microchannel without trapping air bubbles was critical as bubbles can cause significant errors in the measurement. B. Calibration of Trap Stiffness and Flow Mapping The power spectral method and the equipartition theorem were used to calibrate the trap stiffness. The trap stiffness, k, was calibrated at each position where the probing particle was placed under quiescent conditions. Once the trap stiffness was calibrated, liquid was allowed to flow through the microchannel and the displacement of the trapped particle, x, parallel to the flow path was measured using the QPD. As it is difficult to set the QPD voltage to zero with a trapped particle due to its random motion, the average particle position was used to define the equilibrium position and subtracted from of the signal to obtain the displacement. The displacement was measured for 1 second at a sampling frequency of 50 khz.

3 For the power spectral method a particle was trapped at each position in the channel where the fluid velocity was to be measured. As the trapped particle undergoes random thermal motion, the QPD output fluctuates. The signal exhibits white noise characteristics due to the Brownian motion of the trapped particle. The thermal motion of a trapped particle was again followed for 1 s at a sampling frequency of 50 khz at different distances from the channel wall. The experimentally found corner frequency f c and the theoretically calculated drag coefficient γ were used to determine the trap stiffness using (4). For the equipartition theorem method the trap stiffness was determined using the same QPD data collected for the power spectral calculation. The QPD voltage as a function of time of Brownian motion of the trapped particle was converted into displacement using the QPD calibration factor (nm/v). The trap stiffness was then calculated using (2). An advantage of the equipartition theorem method is that it does not depend on the viscosity of the surrounding liquid, the particle s shape or its distance from the wall [14]. The fluid velocity was measured immediately after measuring the trap stiffness using the same particle. Water was injected into the microchannel at a constant velocity and to avoid interference from the sides of the channel, the position of the particle in the z-direction (See Fig. 1) was 2.5 mm. where γ is the drag coefficient, 6πηr, r is the radius of the particle, and h is the distance between the centre of the particle and the surface. The drag coefficient increases by a factor of 3 when the particle is virtually touching the wall and decays rapidly until it is essentially the same as the bulk value at about 10 radii. The difference between the trap stiffness corrected by Faxen s law and that of uncorrected was most prominent at 1 µm away from the surface as expected and is shown in Fig. 2. It was found that the trap stiffness increases until about 5 µm away from the surface and decreases beyond 15 µm. The stiffness dependence on the distance to the wall [17] has been investigated and our result shown in Fig. 2 is consistent with the earlier measurement in [17]. The trap stiffness variation away from the wall is mainly due to spherical aberrations [18, 19] caused by the refractive index mismatch at the plasticwater interface. Note that when using a water immersion objective [19], the trap stiffness is essentially independent from the distance to the wall. (7) The inertial force on the probing particle was negligible because the flow was laminar as is typical for microfluidic channels under these conditions. Thus, the external force acting on the particle, F, was purely due to the viscous friction exerted by the fluid on the particle and (1) and (5) can be equated as. (6) Every parameter was known or measured experimentally except the local fluid velocity, v, which can be subsequently calculated using (6). Figure 1. Dimensions of the microchannel and direction of the fluid flow. IV. RESULTS AND DISCUSSSION A. Trap stiffness The trap stiffness, k, was determined for each position firstly using the power spectral method and is shown in Fig. 2. The effect of the proximity of the probe particle to the channel surface can be large when the distance from the surface is comparable to the particle radius [12]. Thus Faxen s law [15, 16] was applied in determining the drag coefficient in very close proximity of the wall: y z Figure 2. Trap stiffness determined using the power spectrum method. Faxen s correction is important in close proximity to the surface and negligible at distances greater than 20 times the radius of particle. Although our measurements show a similar dependence on the distance to those in [17], the magnitude of the change is different. A notable difference between the two experimental setups was the thickness of coverslip used: it was 3.4 times smaller in [17]. Overall changes in trap stiffness could be explained by the intensity gradient reduction and local laser heating. Whether the thickness of cover slip influences the local heating is not clear at this stage. Close to the surface, the spherical aberration due to the refractive index mismatch is negligible, however, this may cause local heating leading to a more intense thermal motion of the probing particle. It was previously investigated [20] that the trap stiffness determined by the power spectral method can be overestimated by 2% when a laser power of 100 mw is used if the local viscosity change due to heating is not taken into consideration. In our experiment, therefore, the uncertainty associated with the laser heating was regarded as negligible. Trap stiffness determined by using the equipartition theorem and is shown in Fig. 3. The trap stiffness determined by the equipartition theorem is of the same order of magnitude

4 as the one determined by power spectrum method. In both cases, the trap stiffness dependence on the distance from the surface can be clearly seen. The temperature at the trap spot could not be measured due to its inaccessibility. We measured the temperature at the surface of the flow cell and estimated the error to be 0.5 C. In the experiment described in [21], the upper limit for the local temperature increase due to laser heating was also estimated to be 0.5 C. Laser-induced temperature increase in water solution in optical traps [20] has been known to be relatively small (~ 0.8 C). The average value of trap stiffness determined by the equipartition theorem and power spectrum methods was used in the determination of the fluid velocity. Trap stiffness determined by using the equipartition theorem and is shown in Fig. 3. The trap stiffness determined by the equipartition theorem is of the same order of magnitude as the one determined by power spectrum method. In both cases, the trap stiffness dependence on the distance from the surface can be clearly seen. The temperature at the trap spot could not be measured due to its inaccessibility. We measured the temperature at the surface of the flow cell and estimated the error to be 0.5 C. In the experiment described in [21], the upper limit for the local temperature increase due to laser heating was also estimated to be 0.5 C. Laser-induced temperature increase in water solution in optical traps [20] has been known to be relatively small (~ 0.8 C). The average value of trap stiffness determined by the equipartition theorem and power spectrum methods was used in the determination of the fluid velocity. where v x (z,y) is the velocity of the fluid in the x direction (see Fig. 1) at a position (z,y) in the microchannel, p is the pressure difference between the channel inlet and outlet, η is the viscosity of the fluid, and L, W, and H are the length, width, and height of the channel, respectively. The pressure difference is found using the equation where Q is the volumetric flow rate in m 3 /s and a is dimensionless parameter [22] defined as (8) (9). (10) The volume flow rate in our experiments was m 3 /s and the theoretical fluid velocity across the microchannel was calculated using MATLAB and shown in Fig. 4. Note that the velocity profile in the height or y axis appears parabolic at any position, and the influence of channel walls extends approximately 1 mm into the fluid. Figure 3. Trap stiffness determined using the equipartition theorem. The feature showing the dependence of trap stiffness on distance is consistent with the result obtained using the power spectral method. B. Theoretical Fluid Flow in a Microchannel The fluid velocity in a rectangular microchannel was determined using an analytical solution of the Navier-Stokes equations for a pressure driven flow with a no-slip boundary condition at the interface between the fluid and the microchannel surface [22]. The fluid velocity is given by Figure 4. Theoretical fluid flow profile in the microchannel. The fluid velocity profile in the height axis is parabolic at any position, whereas uniformity is obtained once approx. 1 mm from channel walls. In the experiment, the working distance of the objective lens (~230 μm) and the thickness of the bottom wall of flow cell (170 μm) restricted the trapping of the probe particle to a maximum of approximately 50 μm from the surface. Thus the experimentally measured fluid velocity located in the region indicated by the arrow. Thus the experimentally measured fluid velocity located in the region indicated by the arrow. C. Experimental Determination of Fluid Flow The displacement of a trapped particle from the equilibrium position due to fluid flow is shown in Fig. 5. It appears to be parabolic in shape, however, it does not reflect directly the fluid velocity profile because the trap stiffness is also a function of the distance from the wall. In order to calculate the

5 drag force on the probing particle, the displacement was multiplied by the trap stiffness determined at the same position Figure 5. Displacement of the trapped particle from the equilibrium position due to drag force. A parabolic trendline has be superimposed over the measured data. Fig. 6 shows the experimentally measured fluid velocity (data points) and theoretical velocity (solid line). Although the theoretical velocity appears to be almost linear it is in fact a portion of a parabola that represents the velocity profile that is closest to the wall as indicated in Fig. 4. The fluid velocity determined by optical trapping is in good agreement with the theoretical prediction. The magnitude of the measured drag force indicates that the sensitivity of the optical trapping system built for this research is the subpiconewton range. measurement (calibration) had to be done at every single point where the velocity was measured, however this is a rapid process. Although this method could be used in concentrated solutions it is preferable to use dilute suspensions as it is difficult to keep a single particle at the trap if there are many nearby. Even though the trap successfully holds a single particle, other particles passing above the trapped particle in front of the QPD detection system may affect the measurement. In this case the signal was normally quite noticeable. Similar artefacts could be detected in the QPD signal when two particles are trapped. When this was the case the measurement was discarded. However it may be possible to exploit these effects to further understand multiple particle interaction. Ideally, the same particle should be used as the probe for the whole velocity profile. However, in our experience this is difficult if the particle trap is very weak. It would be difficult to measure a wide range of velocity in the one experiment since, in order to measure a high velocity the trap should be strong hence laser power increased, but then it could be too strong for low fluid velocity close to the surface, i.e. insensitive to smaller forces. If the laser power is too weak with a high fluid velocity, the trap wouldn t be able to hold the particle. CONCLUSIONS We used the stiffness of an optical trap to measure the fluid velocity in a rectangular microfluidic channel. The quadrant photodiode detector (QPD) was calibrated and used for detecting the position of the micron sized trapped particle with nanometre resolution. The trap stiffness was determined using two alternative methods: the power spectrum method and the equipartition theorem method. The fluid velocity obtained experimentally using optical trapping is in agreement with the standard hydrodynamic calculation, and the flow mapping technique developed in this work the optical trap is used for both particle manipulation and sensitive force measurement. Figure 6. Fluid velocity and drag force determined using the experimentally measured displacement of the trapped particle and trap stiffness (data points) is compared with the theoretical curve shown as a solid line. Due to the relatively thick bottom channel wall (0.17 mm) compared to the working distance (0.23 mm) of the objective lens, measuring the fluid velocity was limited to 50 µm nearest the surface. As a result, the experimentally determined fluid velocity profile showed only a small fraction of the full parabolic profile that was determined theoretically. Using a micro channel with a thinner bottom wall would enable full velocity mapping over the channel. Our experiment, nevertheless, verified that force measurement capability of optically trapped particles can be used to determine the fluid velocity. An advantage of using this technique is that it is relatively unobtrusive and straightforward to map the channel fluid velocities. One of pitfalls was that the trap stiffness ACKNOWLEDGMENT This research was supported under Australian Research Council s Discovery Projects funding scheme (project number DP ), the Department of Innovation, Industry, Science and Research (Australian Government) through the Australia India Strategic Research Fund, and Re-LIEF grant from the University of South Australia. REFERENCES [1] C. D. Meinhardt, S. T. Wereley, and G. Santiago, "PIV measurements of a microchannel flow," Exp. Fluids, vol. 27, pp , [2] H. Mushfique, et al., "3D mapping of microfluidic flow in laboratoryon-a-chip structures using optical tweezers," Anal. Chem, vol. 80, pp , [3] B. A. Nemet and M. C. Golomb, "Microscopic flow measurements with optically trapped microprobes," Opt. Lett, vol. 27, pp , [4] J. Wu, D. Day, and M. Gu, "Shear stress mapping in microfluidic devices by optical tweezers," J. Opt. Soc. Am, vol. 18, pp , [5] A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, "Observation of a single beam gradient force optical trap for dielectric particles," Opt. Lett, vol. 11, pp , 1986.

6 [6] G. Knöner, S. Parkin, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Characterization of optically driven fluid stress fields with optical tweezers," Phys. Rev, vol. E 72, pp , [7] F. Gittes and C. F. Schmidt, "Signals and noise in micromechanical measurements," Methods Cell Biol, vol. 55, pp , [8] [8] K. Visscher and S. M. Block, "Versatile optical traps with feedback control," Meth. Enzymol, vol. 298, pp , [9] N. Malagnino, G. Pesce, A. Sasso, and E. Arimondo, "Measurements of trapping efficiency and stiffness in optical tweezers," Opt. Comm, vol. 214, pp , [10] M. Michihata, T. Hayashi, and Y. Takaya, "Measurement of axial and transverse trapping stiffness of optical tweezers in air using a radically polarized beam," Appl. Opt, vol. 48, pp , [11] F. Gittes and C. F. Schmidt, "Interference model for back-focal-plane displacement detection in optical tweezers," Opt.Lett, vol. 23, pp. 7-9, [12] K. Svoboda and S. M. Block, "Biological applications of optical forces," Annu. Rev. Biophys. Biomol. Struct., vol. 23, pp , [13] K. Berg-Sørensen and H. Flyvbjerg, "Power spectrum analysis for optical tweezers," Rev. Sci. Instrum, vol. 75, pp , [14] O. Marti and K. Hübner, Force measurement with optical tweezers, in Handbook of nanotechnology, B. Bhushan, Ed. Springer Verlag. 2010, pp [15] J. Happel and H. Brenner, Low reynolds number hydrodynamics, Leyden, the Netherlands: Noordhoff International Publishing,1973. [16] H. Faxen, Influence of walls on the drag of a small sphere moving in a viscous fluid, Ph.D.thesis. Uppsala University,1921 [17] A. Buosciolo, G. Pesce, and A. Sasso, "New calibration method for position detector for simultaneous measurements of force constants and local viscosity in optical tweezers," Opt. Comm, vol. 230, pp , [18] A. C. Dogariu and R. Rajagopalan, "Optical traps as force transducers: The effects of focusing the trapping beam through a dielectric interface," Langmuir, vol. 16, pp , [19] K. C. Vermeulen, G. J. L. Wuite, G. J. M. Stienen, and C. F. Schmidt, "Optical trap stiffness in the presence and absence of spherical aberrations," J. Opt. Soc. Am, vol. 45, pp , [20] E. J. G. Peterman, F. Gittes, and C. F. Schmidt, "Laser-Indused heating in optical traps," Biophys. J, vol. 84, pp , [21] S. F. Tolić-Nørrelykke, et al., "Calibration of optical tweezers with positional detection in the back focal plane," Rev. Sci. Instrum, vol. 77, pp , [22] F. M. White, Viscous fluid flow, New York: McGraw-Hill 1974.