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Nonlinear Dynamics in Experimental Two-Phase Microfluidics Timeseries
Nonlinear Dynamics in Experimental Two-Phase Microfluidics Timeseries Francesca Sapuppo*, Florinda Schembri*, Maide Bucolo* *Dipartimento di Ingegneria Elettrica Elettronica, University of Catania, 95125 Catania, Italy (Tel: +39-095-7382603; e-mail: [email protected]) Abstract: In this work experimental studies on nonlinear dynamics in two phase microfluidics were carried out. Nonlinear analysis methods were applied to time series coming from experimentation in microfluidics. The time series were captured using high sensitive photodiodes on a magnified image of the microfluidic phenomena under investigation, and they represent the time domain dynamics of a two phase flow in an in vitro serpentine mixer, in particular of bubbles flow patterns. This study allows capturing significant chaotic features in the time domain for different experimental conditions, and it is useful for a first understanding of two phase flow patterns in complex microchannels. Keywords: Nonlinear dynamics; Two phase flow; Micromixer; Signal processing; Temporal chaos. 1. INTRODUCTION In microfluidics, mixing depends mainly on molecular diffusion because microfluidic systems are restricted to laminar flow, so conventional mixing process are not applicable. Several applications are based on microchannel mixers: from reaction, gas absorption to emulsification and foaming (Hessel et al., 2005). Moreover, formation and manipulation of microscopic droplets or bubbles is used for meter and mix small volume of fluids efficiently having the advantage in producing fast mixing in low cost device controlled by external flows. In this scenario chaotic trajectories in flow inside droplets and bubbles have been widely studied (Bringer et al., 2004), showing how they are determined both by the type of the input flow and the channel geometry (Stremler et al, 2004). In order to control the dynamics of bubbles and thus the rate of mixing inside the droplet the influence of the input flow rate to the droplet generation and motion must be understood. In this work the complex flow of bubbles in a typical serpentine microchannel has been qualitatively analyzed in experimental studies, varying the flow rate and frequency of the input fluids. The observation of the two phase flow (airwater) has been carried out using an experimental workbench (Sapuppo et al, 2007) allowing the magnification and extraction of significant time series representing the complex temporal dynamics of bubbles (Bucolo et al., 2008). The nonlinear analysis of such time series represents the basis for the study of bubble flow pattern control. 2. EXPERIMENTAL SYSTEM 2.1 The microfluidic device The serpentine snake mixer slide (SMS0104, Thinxxs), here considered, belongs to the class of passive snake mixer with two streams and has section of 640 µm and internal and
external radius of curvatures of 280 µm and 920 µm. In the case here presented, air and water flow through the mixer obtaining a two-phase flow. The control variables for the two-phase flow are the water and air flow rates. In the experimental setup the piezoelectric twin pumps (TPS1304, Thinxxs) are manually tuned through an electronic control (EDP0704, Thinxxs) acting on the frequencies (fair, fwater) of the piezo-driven diaphragm micropumps vibration and therefore generating sinusoidal input flow rates for both air and water (Fig. 1).
Fig. 1 Microfluidic Setup 2.2 The electro optical system The micro-scales of microfluidic devices requires an optical system for the extraction of parameters and variable involved in the microfluidic process. The optical system (Fig. 2.a,b) provides image magnification suitable for acquisition and processing. It has been set up with flexible discrete optomeccanical components mounted on a breadboard in order to provide an optical path accessible at any point where the information needs to be acquired (Sapuppo et al., 2007).
2.3 Photodiode sensing system
the embedding dimension and the time delay that characterize the dynamics of the time series could be applied.
Two photodiodes (SLD-70BG2A, Silonex) have been chosen to capture light variation due to the bubbles passage in the microchannel.
The nonlinear analysis on photodiode signals has been performed by means of the software TISEAN (Hegger et al., 1999). In particular the time delay τ has been calculated by the mutual information method (Fraser et al., 1986) while the embedding dimension m with the method of false nearest (Kennel, et al). Taking into account the divergence between trajectoies (dj), the dj asymptotic value (d∞) has been calculated as a significant parameter related to both stretching and folding dynamics (Bonasera et al. 2003). Finally the prediction error between very close trajectories was used to calculate the largest Lyapunov exponent (λmax.) (Aurell et al. 1997).
a 4. RESULTS AND DISCUSSION
b
c
Fig. 2. (a) Experimental setup CAD (b) Electro optical system (c) Photodiode board positioning . They are placed on the magnified image of the channel, on an axis parallel to the bubble flow (Fig. 2c). An opportune conditioning circuit has been implemented in order to convert the photodiode current into amplified voltage signals. Fig. 2c shows the photodiode board, also protected from out coming light. 3. SIGNAL PROCESSING 3.1 Pre-processing analysis and filtering A spectral analysis on photodiode signals has been necessary in order to reduce noise components due to environmental light and electrical interference signals. All the signal have been filtered using a low pass filter with cut off frequency of 60 Hz. Further filtering of single components of electrical noise was necessary in order to remove noise coming from the pumps vibration. 3.2 Nonlinear time series analysis methods The possibility to transfer nonlinear theory and methods applied to physical phase space of deterministic nonlinear dynamical systems to the characterization of experimental time series has been discovered in 1980 (Packard et al., 1980). It is possible to reconstruct from a scalar time series a state space that is equivalent in a diffeomorphism to the unknown original state space of the experimental nonlinear system (Parlits, 1998). From the reconstructed states, methods for the extraction of important parameters such as
An experimental campaign has been carried out consisting of a series of experiments obtained by varying the control frequencies of the pump control in a range between 5 Hz to 60 Hz for water and air, creating four couples of inputs fair and fwater. as described in Table I. Such range of frequencies corresponds to a range of input flow rates of 8-16ml/min for the air and of 0.2-1.2 ml/min for the water. During each experiment the pumps frequencies were kept constant. Moreover a reference experiment has been considered, controlling the water pump with a frequency of 10Hz and leaving the air pump off, therefore having only water passing through the channel. This was used as a reference signal for comparison to the other signal representing the two phase flow dynamics. Table 1. Experimental Campaign Water Frequency [Hz] 10 30 30 30 50
Air Frequency [Hz] OFF 5 25 60 25
Flow Rate Water [ml/min] 3 6.2 6.2 6.2 3.8
Flow Rate Air [ml/min] 0 2.5 6 12.5 6
4.1 Photodiode signals and bubbles patterns The image pattern related to the passage of the bubbles in the channel is shown in the frame sequence representing the two phase flow in the microchannel (Fig. 3). The comparison between the signals related to the reference experiment (green line) and a sample two-phase flow experiment (blu line, water 30 Hz, air 5Hz) is shown in Fig 4a. It shows how the bubble passage is sensed by the photodiodes and creates a temporal pattern in the time series. As a confirmation of such behaviour, such comparison is shown for all the case studies in Fig. 4b. Such result makes clear how the mean value of the reference signal is higher
than the ones related to the two-phase flow times series, since the passage of the bubbles creates a shadow on the images lowering the average light intensity. This fact is confirmed by the graph showing the mean intensity value of the voltage signal related to the control frequency of the air pump(Fig. 4c).
Table 2. Characteristic time of bubbles flow Water Frequency [Hz] 30 30 30 50
Air Frequency [Hz] 5 25 60 25
Characteristic time [s] 0.20 0.12 0.17 0.10
4.2 Nonlinear time series analysis results
Fig. 3. Frames of a video showing bubble flow in the microchannel.
Using the nonlinear time series analysis different dynamics of the two phase flow have been observed. The extracted parameters are reported in Table 3, where the first line is related to the reference signal time series. For each case study, graphs with the time series, the frequency spectrum of the pre-processed series, the reconstructed 3D attractor and a 2D projection are shown. Moreover in Fig. 10 the divergence curves (dj) between trajectories are shown. These results indicate the variation on the nonlinear features of the time series related to the temporal bubbles dynamics as the input parameters (flow rate and frequency) change.
a Photodiode Signals 0.06 water 10Hz water 30Hz water 30Hz water 30Hz water 50Hz
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Table 3. Nonlinear Parameters Water Frequency [Hz] 10 30 30 30 50
Air Frequency [Hz] OFF 5 25 60 25
Delay τ 6 9 7 29 10
Embedding Dimension m 5 4 5 4 4
Largest Lyapunov Exponent
log(d∞)
2.58 2.80 2.14 1.45
-1.83 -1.64 -1.59 -1.95
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c Fig. 4. Photodiode signal related to different experiments.(a) reference and sample experiment (water 30Hx, air 5Hz) (b) All time series. (c) Photodiode time series mean values vs air frequency. A characteristic time related to the bubble passage in the channel is furthermore calculated for every experiments and it is shown in Table 2.
A mathematical nonlinear analysis of experimental time series was performed. The time series were obtained as optical information through standard photodiodes acquisition of a microfluidics two phase flow process. This study was performed on various time series of the process differenced by the input frequencies of the micro pumps and so the flow rates of the input fluids. Thus in the case studies the dynamics of the micromixing process changed according to the flow rate of the input fluids. Qualitative study has been performed on time series through the embedding space analysis. For this purpose, the proper embedding delay and embedding dimension has been calculated from the mutual information method and false nearest neighbour method. This study allows a first insight in the complex bubbles flow dynamics in microchannel. Through different results presented in this work we have shown that varying the inputs flow rate, the dynamics of bubbles changes significantly. These results are important for successive studies in bubbles flow dynamic control.
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Fig. 5. (a) Reference signal (no bubbles): experiments with only water pumped at 10 Hz (b) Signal spectrum (c) Embedding space reconstruction in 3D view. (d) yz section of the embedding space.
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Fig. 6. Experiments Water 30 Hz – Air 5 Hz. (a) Photodiode signal (b) Signal spectrum (c) Embedding space reconstruction in 3D view. (d) yz section of embedding space.
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Fig. 7. Experiments Water 50 Hz – Air 25 Hz. (a) Photodiode signal (b) Signal spectrum (c) Embedding space reconstruction in 3D view. (c) yz section of embedding space.
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Fig. 8. Experiments Water 30 Hz – Air 60 Hz. (a) Photodiode signal (b) Signal spectrum (c) Embedding space reconstruction in 3D view. (c) yz section of embedding space.
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Fig. 9. Experiments Water 30 Hz – Air 25 Hz. (a) Photodiode signal (b) Signal spectrum (c) Embedding space reconstruction in 3D view. (c) yz section of embedding space.
Aurell, E., Boffetta, G. Crisanti, A. Paladin, G. and Vulpiani, A. (1997). Predictability in the large: an extension of the concept of Lyapunov exponent, J. Phys. A 30, 1 Bonasera, A., Bucolo M., Fortuna L., Frasca M., Rizzo A. (2003).A New Characterization of Chaotic Dynamics: the d-infinite Parameter, Non linear Phenomena in Complex Systems, 6, 3, 779-786, Bringer, M. R., Gerdts, C. J., Song, H., Tice, J. D. and Ismagilov, R. F. (2004). Microfluidic systems for chemical kinetics that rely on chaotic mixing in droplets, Phylos. Trans. Royal Society (London), 362, 1087-1104. Bucolo, M., Fortuna, L., Sapuppo, F. and Schembri, F. (2008). Chaotic dynamics in microfluidics experiments, MTNS 2008, Blacksburg, Virginia, USA. Fraser A. M. and Swinney H. L. (1986). Independent coordinates for strange attractors from mutual information, Phys.Rev. A 33, 1134. Hegger R., Kantz H., and Schreiber T. (1999). Practical implementation of nonlinear time series methods: The TISEAN package, CHAOS 9, 413. Hessel, V., Holger, L. and Schonfeld, F. (2005). Micromixera review on passive and active mixing principles, Elsevier, Chemical Engineering Science, 60, 2479-2501. Kennel M. B., Brown R., and Abarbanel H. D. I. (1992). Determining embedding dimension for phase-space reconstruction using a geometrical construction”, Phys. Rev. A, 45,3403. Packard, N.H., J.P. Crutchfield, J.D. Farmer, R.S. Shaw. (1980). Geometry from a time series, Phys. Rev. Lett, 45, 712-716. Parlits, U., (1998). Nonlinear time series analysis, .A.K. Suykens, J.Vanderwalle, Kluwer Academic Publisher, Boston, 209-239. Sapuppo, F., Cantelli, G., Fortuna, L., Arena, P. and Bucolo M. (2007). Real-Time monitoring system for microfluidics, SPIE Europe Microtechnologies for the New Millennium, Maspalomas, Spain. Stremler, M., A., Haselton, F., R. and Aref, H. (2004). Design for chaos: application of chaotic advection at the microscale, Phylos. Trans. Royal Society (London), 362, 1019-1036.
external radius of curvatures of 280 µm and 920 µm. In the case here presented, air and water flow through the mixer obtaining a two-phase flow. The control variables for the two-phase flow are the water and air flow rates. In the experimental setup the piezoelectric twin pumps (TPS1304, Thinxxs) are manually tuned through an electronic control (EDP0704, Thinxxs) acting on the frequencies (fair, fwater) of the piezo-driven diaphragm micropumps vibration and therefore generating sinusoidal input flow rates for both air and water (Fig. 1).
Fig. 1 Microfluidic Setup 2.2 The electro optical system The micro-scales of microfluidic devices requires an optical system for the extraction of parameters and variable involved in the microfluidic process. The optical system (Fig. 2.a,b) provides image magnification suitable for acquisition and processing. It has been set up with flexible discrete optomeccanical components mounted on a breadboard in order to provide an optical path accessible at any point where the information needs to be acquired (Sapuppo et al., 2007).
2.3 Photodiode sensing system
the embedding dimension and the time delay that characterize the dynamics of the time series could be applied.
Two photodiodes (SLD-70BG2A, Silonex) have been chosen to capture light variation due to the bubbles passage in the microchannel.
The nonlinear analysis on photodiode signals has been performed by means of the software TISEAN (Hegger et al., 1999). In particular the time delay τ has been calculated by the mutual information method (Fraser et al., 1986) while the embedding dimension m with the method of false nearest (Kennel, et al). Taking into account the divergence between trajectoies (dj), the dj asymptotic value (d∞) has been calculated as a significant parameter related to both stretching and folding dynamics (Bonasera et al. 2003). Finally the prediction error between very close trajectories was used to calculate the largest Lyapunov exponent (λmax.) (Aurell et al. 1997).
a 4. RESULTS AND DISCUSSION
b
c
Fig. 2. (a) Experimental setup CAD (b) Electro optical system (c) Photodiode board positioning . They are placed on the magnified image of the channel, on an axis parallel to the bubble flow (Fig. 2c). An opportune conditioning circuit has been implemented in order to convert the photodiode current into amplified voltage signals. Fig. 2c shows the photodiode board, also protected from out coming light. 3. SIGNAL PROCESSING 3.1 Pre-processing analysis and filtering A spectral analysis on photodiode signals has been necessary in order to reduce noise components due to environmental light and electrical interference signals. All the signal have been filtered using a low pass filter with cut off frequency of 60 Hz. Further filtering of single components of electrical noise was necessary in order to remove noise coming from the pumps vibration. 3.2 Nonlinear time series analysis methods The possibility to transfer nonlinear theory and methods applied to physical phase space of deterministic nonlinear dynamical systems to the characterization of experimental time series has been discovered in 1980 (Packard et al., 1980). It is possible to reconstruct from a scalar time series a state space that is equivalent in a diffeomorphism to the unknown original state space of the experimental nonlinear system (Parlits, 1998). From the reconstructed states, methods for the extraction of important parameters such as
An experimental campaign has been carried out consisting of a series of experiments obtained by varying the control frequencies of the pump control in a range between 5 Hz to 60 Hz for water and air, creating four couples of inputs fair and fwater. as described in Table I. Such range of frequencies corresponds to a range of input flow rates of 8-16ml/min for the air and of 0.2-1.2 ml/min for the water. During each experiment the pumps frequencies were kept constant. Moreover a reference experiment has been considered, controlling the water pump with a frequency of 10Hz and leaving the air pump off, therefore having only water passing through the channel. This was used as a reference signal for comparison to the other signal representing the two phase flow dynamics. Table 1. Experimental Campaign Water Frequency [Hz] 10 30 30 30 50
Air Frequency [Hz] OFF 5 25 60 25
Flow Rate Water [ml/min] 3 6.2 6.2 6.2 3.8
Flow Rate Air [ml/min] 0 2.5 6 12.5 6
4.1 Photodiode signals and bubbles patterns The image pattern related to the passage of the bubbles in the channel is shown in the frame sequence representing the two phase flow in the microchannel (Fig. 3). The comparison between the signals related to the reference experiment (green line) and a sample two-phase flow experiment (blu line, water 30 Hz, air 5Hz) is shown in Fig 4a. It shows how the bubble passage is sensed by the photodiodes and creates a temporal pattern in the time series. As a confirmation of such behaviour, such comparison is shown for all the case studies in Fig. 4b. Such result makes clear how the mean value of the reference signal is higher
than the ones related to the two-phase flow times series, since the passage of the bubbles creates a shadow on the images lowering the average light intensity. This fact is confirmed by the graph showing the mean intensity value of the voltage signal related to the control frequency of the air pump(Fig. 4c).
Table 2. Characteristic time of bubbles flow Water Frequency [Hz] 30 30 30 50
Air Frequency [Hz] 5 25 60 25
Characteristic time [s] 0.20 0.12 0.17 0.10
4.2 Nonlinear time series analysis results
Fig. 3. Frames of a video showing bubble flow in the microchannel.
Using the nonlinear time series analysis different dynamics of the two phase flow have been observed. The extracted parameters are reported in Table 3, where the first line is related to the reference signal time series. For each case study, graphs with the time series, the frequency spectrum of the pre-processed series, the reconstructed 3D attractor and a 2D projection are shown. Moreover in Fig. 10 the divergence curves (dj) between trajectories are shown. These results indicate the variation on the nonlinear features of the time series related to the temporal bubbles dynamics as the input parameters (flow rate and frequency) change.
a Photodiode Signals 0.06 water 10Hz water 30Hz water 30Hz water 30Hz water 50Hz
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Table 3. Nonlinear Parameters Water Frequency [Hz] 10 30 30 30 50
Air Frequency [Hz] OFF 5 25 60 25
Delay τ 6 9 7 29 10
Embedding Dimension m 5 4 5 4 4
Largest Lyapunov Exponent
log(d∞)
2.58 2.80 2.14 1.45
-1.83 -1.64 -1.59 -1.95
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c Fig. 4. Photodiode signal related to different experiments.(a) reference and sample experiment (water 30Hx, air 5Hz) (b) All time series. (c) Photodiode time series mean values vs air frequency. A characteristic time related to the bubble passage in the channel is furthermore calculated for every experiments and it is shown in Table 2.
A mathematical nonlinear analysis of experimental time series was performed. The time series were obtained as optical information through standard photodiodes acquisition of a microfluidics two phase flow process. This study was performed on various time series of the process differenced by the input frequencies of the micro pumps and so the flow rates of the input fluids. Thus in the case studies the dynamics of the micromixing process changed according to the flow rate of the input fluids. Qualitative study has been performed on time series through the embedding space analysis. For this purpose, the proper embedding delay and embedding dimension has been calculated from the mutual information method and false nearest neighbour method. This study allows a first insight in the complex bubbles flow dynamics in microchannel. Through different results presented in this work we have shown that varying the inputs flow rate, the dynamics of bubbles changes significantly. These results are important for successive studies in bubbles flow dynamic control.
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Fig. 5. (a) Reference signal (no bubbles): experiments with only water pumped at 10 Hz (b) Signal spectrum (c) Embedding space reconstruction in 3D view. (d) yz section of the embedding space.
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Fig. 6. Experiments Water 30 Hz – Air 5 Hz. (a) Photodiode signal (b) Signal spectrum (c) Embedding space reconstruction in 3D view. (d) yz section of embedding space.
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20 25 30 Frequency [Hz]
35
40
45
b
Recostructed Attractor
0.06
0.052 0.05
0.055
0.048 0.046 x3
x3
0.05 0.045
0.044 0.042
0.04
0.04 0.038 0.06
0.06 0.06
0.05 0.04
0.045
0.055
0.06
0.05 0.04
0.04
0.03
x2
0.05
x1
c
0.045
Recostructed Attractor
0.055
0.04
0.03
x2
0.05
x1
c
Recostructed Attractor
0.056
0.052
0.054 0.05
0.052 0.048
0.05
0.046 x3
x3
0.048 0.046
0.044
0.044 0.042
0.042 0.04
0.04 0.038 0.038
0.04
0.042
0.044
0.046
0.048 x2
0.05
0.052
0.054
0.038 0.038
0.056
d
Fig. 7. Experiments Water 50 Hz – Air 25 Hz. (a) Photodiode signal (b) Signal spectrum (c) Embedding space reconstruction in 3D view. (c) yz section of embedding space.
0.04
0.042
0.044
0.046 x2
0.048
0.05
0.052
d
Fig. 8. Experiments Water 30 Hz – Air 60 Hz. (a) Photodiode signal (b) Signal spectrum (c) Embedding space reconstruction in 3D view. (c) yz section of embedding space.
dj
Photodiode Signal 0.06
-1.4
0.058
-1.6
0.056
-1.8 -2
0.052
-2.2
0.05
log(dj)
Voltage [V]
0.054
0.048 0.046
-2.4 -2.6
0.044
Water 30Hz Air 5Hz Water 30Hz Air 25Hz Water 30Hz Air 60Hz Water 50Hz Air 25Hz
-2.8
0.042
-3 0.04
5
5.2
5.4
5.6
5.8
6 Time [s]
6.2
6.4
6.6
6.8
7
a
-3.2 -3.4
0
0.05
0.1
0.15
0.2
0.25 Time [s]
0.3
0.35
0.4
0.45
0.5
FFT 6
Fig. 10. Logarithmic divergence curves (dj) for the experimental time series.
5
Magnitude
4
3
REFERENCES
2
1
0
0
5
10
15
20 25 30 Frequency [Hz]
35
40
45
b
Recostructed Attractor
0.056 0.054 0.052
x3
0.05 0.048 0.046 0.044 0.042 0.055 0.055
0.05 0.05 0.045
0.045 0.04
x2
0.04
x1
c
Recostructed Attractor 0.056
0.054
0.052
x3
0.05
0.048
0.046
0.044
0.042 0.042
0.044
0.046
0.048
0.05 x2
0.052
0.054
0.056
d
Fig. 9. Experiments Water 30 Hz – Air 25 Hz. (a) Photodiode signal (b) Signal spectrum (c) Embedding space reconstruction in 3D view. (c) yz section of embedding space.
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