Abstract
The introduction of improved steam engines at the end of the 18th century marked the start of the industrial revolution and the birth of classical thermodynamics. Currently, there is great interest in miniaturizing heat engines, but so far traditional heat engines operating with the expansion and compression of gas have not reached length scales shorter than one millimeter. Here, a micrometersized piston steam engine is implemented in an optical tweezer. The piston is a single colloidal microparticle that is driven by explosive vapourization of the surrounding liquid (cavitation bubbles) and by optical forces at a rate between a few tens of Hertz and one kiloHertz. The operation of the engine allows to exert impulsive forces with optical tweezers and induce streaming in the liquid, similar to the effect of transducers when driven at acoustic and ultrasound frequencies.
Introduction
In the last two decades heat engines have been miniaturized using different fabrication and control methods to explore new ways to power small devices and periodically displace microscopic objects^{1,2,3}. Environmental fluctuations can have a marked effect in the operation of microscopic heat engines. Recently, it was shown that a micrometersized heat engine can operate with a classical Stirling cycle between two heat baths (T_{C}=22 °C, T_{H}=90 °C) in an environment where the engine dynamics are dominated by Brownian fluctuations^{3}. The reported mean value of the extracted work w per cycle is 0.11k_{B}T_{C}=4.5 × 10^{−22} J, with maximum mean power w/τ for τ=7 s, corresponding to a rate of 0.14 Hz.
Here a micrometersized piston steam engine is implemented in an optical tweezer, where a single microparticle is driven by microscopic explosions resulting from explosive vapourization and optical forces. The mean work per cycle exceeds the amount generated by the stochastic heat engine^{3} by six to eight orders of magnitude, while the frequency and maximum displacements are at least two orders of magnitude larger.
Results
Engine cycle
In each cycle a microscopic volume of water is superheated above the normal boiling or saturation temperature, the superheated liquid is unstable and relaxes to an equilibrium saturated state with the nucleation of a vapour cavitation bubble (explosive vapourization). Figure 1a shows the phase diagram projected in pressure p and temperature T that is commonly used to describe superheating^{4}. The two continuous lines are the binodal and the spinodal. The binodal is the saturation curve T_{sat}(p) where the p/T_{sat} pairs can be calculated from the steam table for the saturated water line. The spinodal represents the limit at which the liquid can be superheated T_{sp}(p). The dotted line depicts the isobaric heating path at atmospheric pressure p_{0} in the engine cycle (Fig. 1a, 1→3), where the liquid is superheated reaching a temperature T_{cav} (T_{sat}(p_{0})<T_{cav}≤T_{sp}(p_{0})). The relaxation to the equilibrium saturated state at the binodal p/T_{sat} (Fig. 1a, 3→4) results in a large pressure rise and the explosive expansion of the saturated vapour (cavitation bubble).
The relaxation of a superheated liquid involves density fluctuations that eventually grow creating an exploding vapour bubble as described by the classical nucleation theory (homogeneous/heterogeneous)^{5} and spinodal decomposition at the superheat limit (T_{sp})^{6}. Experiments that have used current pulses to heat thin films and filaments immersed in water have reported temperatures T_{cav} at p_{0} around 302 °C (refs 7, 8), which correspond to pressure rises of 8.7 MPa (at the binodal). However, in general, reported values of T_{cav} are scattered in a wide range of temperatures^{9,10,11,12,13} showing that the results are very sensitive to the heating rate, surface properties of the heater and contaminants in the liquid. In this experiment T_{cav} is not measured, but the relevant parameters about the explosion are derived from the maximum bubble size.
Figure 1b shows the microparticle (piston) trajectory in the optical tweezer at different steps of the cycle. The microsphere absorbs at the trapping laser wavelength (975 nm) and it is heated by linear absorption. The liquid (water) is transparent at the laser wavelength.
The cycle starts with the particle below the geometrical focus or waist of the beam (step 1 in Fig. 1a,b), where the temperature is below the saturation temperature at p_{0}. Then, the particle is attracted and pushed towards the waist by optical gradient and scattering forces. As the particle gets closer to the geometrical focus (1→2) the intensity increases, raising the surface temperature of the bead and the steepness of the trapping potential. The temperature increase is larger at the bead surface that is closer to the focal volume of the trapping beam. Hence, the liquid in contact with that surface reaches the highest temperatures. Closer to the focus, the displacement of the microparticle (2→3) is sufficiently fast that the liquid in contact with the top surface is suddenly superheated reaching T_{cav} at p_{0} (3). The superheated liquid relaxes (3→4) by nucleating a vapour bubble that later collapses under the hydrostatic pressure of the surrounding liquid. The microsphere is driven away by the bubble expansion close to the starting position (1) in the subsaturated water region where the cycle starts again.
Experimental implementation of the engine
The experiments are done in a standard optical tweezer setup. The colloid solution is a dilute aqueous suspension of monodisperse microparticles (1.05 or 3.16 μm diameter 2R_{p}, Promag Bangs) that absorb at the trapping laser wavelength (975 nm). The laser is focused by a × 100 (1.25 NA, oil) microscope objective. The dynamics are detected with a high speed video recorder at frame rates of 300,000 frames per second (fps), with transverse/axial resolutions of 50/100 nm and 100/400 nm for 3 and 1μm particles, respectively. The videos have a duration between 0.15 and 0.3 s. The algorithm to extract the three dimensional position is described in the Methods section.
The engine works for optical powers P above a threshold ~46 mW, where the heating rate at some point in the trajectory is sufficiently large to superheat the surface of the particle that is closer to the waist of the trapping beam, inducing explosive vapourization. At low P up to 24 mW, the particle can be trapped just like in regular optical tweezers, as the intensity is not sufficiently large to increase the temperature to the saturation value. For larger powers between 25 and 45 mW, it may be possible to achieve the saturation temperature and vapour may appear at the surface of the particles resulting in ejection from the trap. Experiments were done at P between 52 and 84 mW. However, the results mainly depend in the particle size R_{p}.
Figure 2a shows selected frames from a single cycle of a 3μm particle with P=58 mW. The first three frames (from −33.3 ms to −167.7 μs) show the particle approaching the focus from below, becoming less fuzzy. The point of closest approach to the waist is reached at t=−3.3 μs and the surface closer to the waist has been superheated. At t=0 μs explosive vapourization is captured in one frame, where a blurred image of the explosion shows the bubble pushing the particle below the focus of the microscope objective. In the next frame t=3.3 μs, the bubble has vanished (collapsed by the hydrostatic pressure of the surrounding liquid) and the particle is close to the starting position where the cycle restarts. No vapour is observed at any frames before the explosion. The full microparticle trajectory for the cycle depicted in Fig. 2a is shown in Fig. 2b, where the positions (X(t), Y(t), Z(t)) are measured relative to the location of the geometrical focus. As the particle gets closer to the focus and the optical gradient force increases, the width of the Brownian fluctuations decreases (Fig. 2b); this is similar to compression in the stochastic heat engine^{3}. Figure 2c shows a few cycles in the XY plane where it is easier to observe the direction of the engine that is constantly changing due to fluctuations that affect the trajectory. In Fig. 2d, the particle distance measured relative to the position of the geometrical focus is plotted for a 3μm and a 1μm particle (P=58 mW, see Supplementary Movies 1 and 2).
For each cycle, the maximum displacement induced by the explosion is defined as ΔD (Fig. 2d), which is measured for 52 and 382 cycles at different P for the 3 and 1μm particles, respectively. ΔD_{3 μm}=7.6±1 μm and ΔD_{1 μm}=2.3±0.6 μm. The distance of closest approach for a given cycle is D_{min} (defined as the minimum distance between the bead center and the geometrical focus), which is about 5 μm for the 3μm particles and 2.8 μm for the 1μm beads. The measurements are summarized in Table 1.
The measured cycle frequency f is in the following intervals f_{3 μm}=(23, 48) Hz and f_{1 μm}=(357, 1,025) Hz. The microparticle trajectory has a fast motion during the explosion and then a slow return with optical forces. Hence, the cycle frequency essentially depends on the maximum displacement ΔD and the mean speed during the slow return. The speed can be estimated by assuming that the optical force is constant and balanced by the drag, which is given by Stokes law F=6πμR_{p}v (μ is the dynamic viscosity and v the velocity). In this way f=v/ΔD∝1/(R_{p}ΔD). The expected frequency ratio f_{1 μm}/f_{3 μm} can be estimated assuming that the optical force is the same for both sizes. In this way f_{1 μm}/f_{3 μm}~ΔD_{3 μm}R_{3 μm}/(ΔD_{1 μm}R_{1 μm})~12.75. The predicted frequency range for the smaller microspheres should be a factor of 12.75 larger than the measured frequency interval of the 3μm particles f_{1 μm}~12.75f_{3 μm}=(293, 612) Hz, which partially overlaps with the measured cycle frequencies for the 1μm particle f_{1 μm}=(357, 1025) Hz. The optical force is proportional to . In this case the assumption that it is constant and has the same value for both particle sizes yields a reasonable approximation, because the difference in volume is compensated with a stronger intensity gradient ∇I for the 1μm particles that are always closer to the focus.
The measured frequencies and maximum displacements ΔD per cycle show that there is a tradeoff between operating frequency and expansion ratio as described in^{14}, as increasing one reduces the other.
The engine is very stable for 3μm particles, which could oscillate for more than 20 min with no sign of deterioration, while with 1μm particles it is unstable and worked for tens of seconds.
Bubble dynamics
The dynamics of the cavitation bubble (radius R(t)) that result from explosive vapourization are extremely fast and are not resolved by the high speed video recorder. The bubble appears blurred in a single frame (Fig. 3a,b, centre frames) that is used to extract the maximum bubble radius R_{max}. During expansion, the bubble does work against the hydrostatic pressure (p_{0}=0.1 MPa) of the surrounding liquid, while the vapour pressure p_{v} and temperature inside decrease at a very fast rate. Once the bubble reaches the maximum size the kinetic energy has converted into potential energy and the vapour inside has reached the room temperature values. Then the bubble collapses under the static pressure of the surrounding liquid. The collapse time τ_{C} of a spherical bubble can be calculated from the Rayleigh relation derived from the inertial Rayleigh equation^{15,16}.
where p_{v} is the vapour pressure inside the fully expanded bubble (2,330 Pa, at T_{0}=20 °C) and ρ_{l} is the water density.
The time elapsed between the initial expansion of the bubble and the collapse is the lifetime or oscillation time τ_{osc}, which for spherical bubbles τ_{osc}=2τ_{C} (refs 17, 18). Hence, R_{max} is reached in τ_{C}.
The energy of the bubble is also related to the maximum bubble radius R_{max}
The measured R_{max} are R_{max−3 μm}=3.4±0.3 μm and R_{max−1 μm}=1.6±0.1 μm. These measurements are not correlated with P, suggesting that the initial pressure and volume of the superheated vapour are similar in the explored range of P for each particle size.
The collapse times are calculated using equation (1) and the measured R_{max}, τ_{C−1 μm}=148±12 ns and τ_{C−3 μm}=317±30 ns. These very brief timescales explain why the bubble only appears in a single frame.
The bubble energies are calculated from equation (2) and are between 10^{−12} and 10^{−11}J (Table 1).
Bubbleparticle interaction
The maximum displacement ΔD for a given cycle that results from the interaction with the cavitation bubble depends on the initial distance between the centre of the particle and the centre of the bubble δ′ and R_{max} (Fig. 3c). If those distances are non dimensionalized as δ=δ′/R_{max} and Δx=ΔD/(2R_{p}), then the interaction is described by a single master curve^{19}. For δ in the interval (0, 0.7) the particle is pushed away (rejection), δ=0.7 corresponds to a neutral region Δx=0, while for δ>0.7 there is attraction.
In these experiments the initial distance from the centre of the bubble to the centre of the particle is simply the radius of the particle (δ′=R_{p}), as the bubble originates at the surface of the particle (Fig. 3c). In this way δ is in the rejection region, δ_{1 μm}~0.33 and δ_{3 μm}~0.47, resulting in a positive displacement that can be calculated from the master curve of^{19} Δx_{1 μm}~2.5–3 and Δx_{3 μm}~1.7–2; these values correspond to ΔD_{1 μm}~2.5–3 μm and ΔD_{3 μm}~5.4–6.3 μm similar to the measurements (Table 1).
Work done by the engine
The particle dynamics, speed v(t) and the magnitude of the acceleration a(t) are extracted from the distance data D(t) processed with standard left finite difference with fixed time step Δt=3.3 μs (300,000 fps videos). The maximum accelerations during the explosion (at t_{0}) for the cycles analysed in Table 1 are a_{max−3 μm}=2.02±0.59 × 10^{5}ms^{−2} and a_{max−1 μm}=1.15±0.66 × 10^{5}ms^{−2}. The values of a_{max} are proportional to the largest forces and the largest contributions to the total work per cycle. However, the temporal resolution Δt is not sufficient to resolve the dynamics during the explosion, as the particle is displaced a distance R_{max} in a time τ_{C}≪Δt. This results in an underestimation of speeds and accelerations only during the explosive expansion (one frame), since the expected accelerations should have an order of R_{max}/τ_{C}^{2}, which yield values of 7.3 and 3.4 × 10^{7}ms^{−2} for the 1 and 3μm particles, respectively. In this way, the calculations that follow (with fixed Δt) for the maximum force and mean work per cycle can be considered lower bounds that are underestimated by at least two orders of magnitude.
The magnitude of the resultant force F(t) acting on the particle is calculated from the particle dynamics. During the explosion (a(t_{0})=a_{max}, one frame), the impulsive force (maximum force in a cycle) is simply F_{max}=V_{p}ρ_{p}a_{max}, where V_{p} is the particle volume and ρ_{p} the particle density. Notice that there is no added mass, as it is assumed that the particle moves with the velocity of the liquid that is displaced by the bubble during the explosion^{20}. After the bubble collapses, the flow induced by the bubble ceases and the force is calculated considering the added mass F(t)=V_{p}(C_{A}ρ_{l}+ρ_{p})a(t), with C_{A}=(1/2) the added mass coefficient for a sphere and ρ_{l} is the density of water. The impulsive forces during the explosion are F_{max−3 μm}=4.07±1.18 × 10^{−9}N and F_{max−1 μm}=9.24±5.36 × 10^{−11}N.
The work done by the particle is w(t)=∫F(t)dD(t) where the numerical integration is done with the trapezoidal method. Plots of the work done by a 1μm particle and by a 3μm particle in the liquid as a function of time are shown in Fig. 4a,b. As expected, the work increases in sharp steps due to the large acceleration a_{max} during the explosive expansion of the bubble. The mean work per cycle is and . A fraction of the mechanical energy of the explosion is transferred to the microparticle (piston) and it is estimated by the ratio , which has the order of 5 × 10^{−5}–6 × 10^{−4}.
The average power is , which lies in the intervals (3.3, 9.4) × 10^{−14}W and (2.37, 4.94) × 10^{−13}W for the 1 and 3μm particles, respectively. If the engine volume is defined using the maximum bubble radius R_{max} then the power density is , which yields about 3 kWm^{−3} for both particle sizes. The results of this subsection are summarized in Table 2.
Finally, the work done by the engine (3μm) in the surrounding liquid results in streaming, which can be observed by seeding the liquid with neutrally buoyant 1μm polystyrene microparticles, which are transparent to the laser wavelength and thus not heated by the beam (Fig. 4c,d). The flow induced by the engine is extracted from standard PIV (Thielicke, W., Stamhuis, E. J. PIVlab http://dx.doi.org/10.6084/m9.figshare.1092508). The vectors in Fig. 4c,d show the average velocities (during 1 s) of the induced flow. Close to the engine the average speeds are ~80 μms^{−1}.
The effect of periodic cavitation (without the microbead piston) is a symmetric attractive flow around the bubble for particles located at a distance δ′>0.7R_{max}~2.4 μm from the centre of the bubble as shown in ref. 19. The movement of the microparticle (piston) displaces liquid creating an asymmetry in the flow that depends in the oscillation direction, which is random (Fig. 4c,d, see Supplementary Movie 3). This effect is similar to streaming produced by oscillating liquidgas boundaries driven by transducers or pulsed lasers ^{19,21,22,23}.
Discussion
In this work, for the first time, a steam engine is implemented with a length scale of one micrometer. This study also shows that it is possible to exert impulsive forces with optical tweezers. Considering the underestimation of two orders of magnitude, the maximum forces during the explosive displacement of the microbead are in the range of 10^{−9}−10^{−7}N, three to five orders of magnitude larger than in regular optical tweezers (pN).
The mean work per cycle extracted from the microscopic steam engine is six to eight orders of magnitude (eight to ten considering underestimation) larger than in the stochastic heat engine^{3}, while the cycle frequency is two to three orders of magnitude larger than in that study. The power density of ~3 kWm^{−3} (or 300 kWm^{−3} if the factor of 100 is considered) is much smaller than that of other microscopic heat engines. For example, the piezoresistive heat engine^{1} that operates with the compression and expansion of a solid has a power density of 36 GWm^{−3}, which is larger than that of car engines (tens of MWm^{−3}). However, some advantages of this implementation are that microfabrication is not required and that the engine operates in a regime dominated by fluctuations.
The microscopic steam engine could be used in applications that previously required pulsed lasers or transducers. For example, optical pumps in microfluidic applications^{24}, cell poration^{21,22} and cell lysis^{25}.
Methods
Experimental setup
The trapping laser is a single mode fibre coupled diode with a wavelength of 975 nm focused by a microscope objective ( × 100, oil, 1.25 NA). The mean diameter of the microspheres (Promag, Bangs) are 1.05 and 3.16 μm (Promag 1 and 3, surfactant free, Bangs) with a density of 1,330 and 1,225 kgm^{−3}, respectively. These particles contain magnetite (Fe_{3}O_{4} and Fe_{2}O_{3}) and a unspecified polymer. The optical properties of these microparticles are not provided. A study on silica gels containing magnetite^{26} shows that the optical transmittance vanishes at wavelengths below 400 nm and increases to about 60% at 800 nm (largest wavelength in that work). The absorption coefficients decrease monotonically with increasing wavelength with the minimum also at 800 nm. If the optical properties of the magnetic microparticles are similar to the silica gels with magnetite, the high transmittance and low absorption in the near infrared would be consistent with the ability to trap these particles at low optical powers and induce explosive vapourization for higher powers closer to the beam waist.
The aqueous solution with the microparticles is placed between two microscope coverslides (Number 1, 0.13–0.16 mm thick) separated with spacers with a height ~100 μm and sealed with nail polish. The video recorder is a Photron SA 1.1.
Image processing
The 3d dynamics are extracted from the high speed videos. First, all frames are divided by the background, then the images are processed with a 3 pixel averaging filter to remove some of the noise and the centre (in XY) is extracted with standard video microscopy techniques, with a resolution of 50 nm (±25 nm) for the 3μm particles and 100 nm for the 1μm particles (fewer points).
The axial position is extracted from the images by first doing a calibration in Z like with a quadrant photodiode (not used due to small linear range ~1 μm), videos are recorded under the same conditions of the experiment at each height. The sample is raised in steps of 125 nm with a piezo stage in closed loop for a total of 160 steps, starting at a position of Z=−16 μm to Z=4 μm (the geometrical focus is at Z=0). At each step Z_{i}, the normalized intensity centred at the particle and averaged over the azimuthal angle is extracted for every value of in a range from (0, 7 μm). Hence, for each axial position Z_{i} an f(Z_{i},ρ) is extracted for several frames. The signal is averaged for at least 80 frames. Sample signals are shown in Fig. 5a. Interpolation is done between the 160 samples f(Z_{i},ρ) to get a continuous function f(Z,ρ). Calibrations were done with several particles to take into account small variations in the size. Figure 5b shows sample fits from videos recorded at 300,000 fps. The axial resolution is 100 nm for 3μm particles and 400 nm in the case of 1μm particles.
Additional information
How to cite this article: QuintoSu, P.A. A microscopic steam engine implemented in an optical tweezer. Nat. Commun. 5:5889 doi: 10.1038/ncomms6889 (2014).
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Acknowledgements
I thank P. QuintoDiez for discussions about heat engines. This work is partially funded by the following grants: CONACYT (153821) and DGAPA, UNAM (PAPIIT TB100312).
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P.A.Q.S. conceived and designed the research, performed the experiments, analysed the data and wrote the manuscript.
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Supplementary information
Supplementary Movie 1
3 μm particle engine. The duration of the movie is 300 ms and the frame height is 30.0 μm. (AVI 4411 kb)
Supplementary Movie 2
1 μm particle engine. The duration of the movie is 150 ms and the frame height is 11.4 μm. (AVI 1242 kb)
Supplementary Movie 3
3 μm particle engine with 1 μm polystyrene tracer particles. The duration of the movie is 10 s (real time) and the frame height is 60.6 μm. (AVI 2904 kb)
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QuintoSu, P. A microscopic steam engine implemented in an optical tweezer. Nat Commun 5, 5889 (2014). https://doi.org/10.1038/ncomms6889
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DOI: https://doi.org/10.1038/ncomms6889
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