^{1}and Dieter Braun

^{1,a)}

### Abstract

We show how fluid can be moved by a laser scanning microscope. Selected parts of a fluid film are pumped along the path of a moving warm spot which is generated by the repetitive motion of an infrared laser focus. With this technique, we remotely drive arbitrary two-dimensional fluid flow patterns with a resolution of . Pump speeds of are reached in water with a maximal temperature increase in the local spot of . Various experiments confirm that the fluid motion results from the dynamic thermal expansion in a gradient of viscosity. The viscosity in the spot is reduced by its enhanced temperature. This leads to a broken symmetry between thermal expansion and thermal contraction in the front and the wake of the spot. As result the fluid moves opposite to the spot direction due to both the asymmetric thermal expansion in the spot front and the asymmetric thermal contraction in its wake. We derive an analytical expression for the fluid speed from the Navier–Stokes equations. Its predictions are experimentally confirmed without fitting parameters under a number of different conditions. In water, this nonlinearity leads to a fluid step of for each passage of the spot. Since the spot movement can be repeated in the kilohertz regime, fluid speeds can exceed . Using this technique, we pump nanoparticles over millimeters through a gel. An all-optical creation of a dilution series of DNA and biomolecules by aliquotation and mixing is demonstrated for fluids sandwiched between untreated and unstructured, disposable microscope cover slips. The shown optical remote control of fluid flow expands the microfluidic paradigm into previously inaccessible regimes of tiny volumes, closed flow paths, fast switching between flow patterns, and remote fluid control under extreme fluid conditions.

We thank Joseph Egger for help in solving the fluid dynamics, Jonas A. Kraus and Thomas Franosch for discussions, Philipp Baaske and Ingmar Schön for reading the manuscript, Klaus Stierstadt for comments, Stefan Duhr and Ingmar Schön for assistance, and Hermann Gaub for hosting our Emmy-Noether Group. This work was funded by the Emmy Noether Program of the Deutsche Forschungsgemeinschaft (DFG) and supported by the Center for Nanoscience Munich (CENS) and the Nanosystems Initiative Munich (nim).

I. INTRODUCTION

II. EXPERIMENT

A. Optical setup and imaging

B. Chambers and light driven microfluidics

C. Finite element calculations

III. RESULTS AND DISCUSSION

A. Basic mechanism

B. Analytical solution

C. Numerical solution

D. Experimental tests

E. Pump path width

F. Pumping an Oseen tensor

G. Toward light driven nanofluidics

H. Light driven microfluidics in 2D fluid films

I. Light driven aliquotation of DNA

J. Enhanced speed using light absorption layer

K. Possibilities and limitations

IV. CONCLUSIONS

### Key Topics

- Viscosity
- 22.0
- Fluid flows
- 18.0
- Gels
- 18.0
- Thermal expansion
- 18.0
- Microfluidics
- 13.0

## Figures

Pumping water optically along arbitrary patterns. Fluid flow along the letters “LASER PUMP” is driven by dynamically heating a thin fluid film with a laser scanning microscope. As seen, complex flow patterns are easily accomplished. No channels restrict the fluid flow. Local pumping of the fluid film is the result of thermoviscous fluid movements for each passage of the laser focus. We visualize the water flow by fluorescent tracer particles. Movies of flow patterns can be found in the supplementary materials section.

Pumping water optically along arbitrary patterns. Fluid flow along the letters “LASER PUMP” is driven by dynamically heating a thin fluid film with a laser scanning microscope. As seen, complex flow patterns are easily accomplished. No channels restrict the fluid flow. Local pumping of the fluid film is the result of thermoviscous fluid movements for each passage of the laser focus. We visualize the water flow by fluorescent tracer particles. Movies of flow patterns can be found in the supplementary materials section.

Basic mechanism for pumping fluid with a moving laser focus. (a) A warm temperature spot moving to the left generates thermal expansion in its front and contraction in its wake, shown with dark gray arrows. These local fluid movements cancel out each other at constant viscosity. (b) The temperature dependency of the fluid’s viscosity breaks this symmetry and results in a net liquid flow to the right (black arrows). If the fluid expands upon heating and shows a decreased viscosity, this flow is directed against the movement of the spot as indicated by the black arrows.

Basic mechanism for pumping fluid with a moving laser focus. (a) A warm temperature spot moving to the left generates thermal expansion in its front and contraction in its wake, shown with dark gray arrows. These local fluid movements cancel out each other at constant viscosity. (b) The temperature dependency of the fluid’s viscosity breaks this symmetry and results in a net liquid flow to the right (black arrows). If the fluid expands upon heating and shows a decreased viscosity, this flow is directed against the movement of the spot as indicated by the black arrows.

Numerical finite element calculation of thermoviscous pumping. (a) A temperature spot (color coded), is moving to the left and results in an expansion in its front (left) and a contraction in its wake (right). The resulting fluid flow to the right is calculated in three dimensions and indicated by white arrows. The resulting flow has a parabolic flow profile, allowing its description with a thin film approximation. (b) Pumping velocities calculated using the thin film approximation in two dimensions. The average velocity is viewed from top and calculated in the coordinate system of the moving spot. The fluid flow is shown as white arrows, color coded is the temperature . [(c) and (d)] Velocity and pressure profile in the spot along the -axis. The integral over the velocity is only nonzero for due to extended tails in the positive and negative -directions. The pressure profile is positive in front of the spot as result of the thermal expansion and negative in its wake. Friction reduces for . Calculations were made using FEMLAB. The simulation files can be found in the supplementary materials.

Numerical finite element calculation of thermoviscous pumping. (a) A temperature spot (color coded), is moving to the left and results in an expansion in its front (left) and a contraction in its wake (right). The resulting fluid flow to the right is calculated in three dimensions and indicated by white arrows. The resulting flow has a parabolic flow profile, allowing its description with a thin film approximation. (b) Pumping velocities calculated using the thin film approximation in two dimensions. The average velocity is viewed from top and calculated in the coordinate system of the moving spot. The fluid flow is shown as white arrows, color coded is the temperature . [(c) and (d)] Velocity and pressure profile in the spot along the -axis. The integral over the velocity is only nonzero for due to extended tails in the positive and negative -directions. The pressure profile is positive in front of the spot as result of the thermal expansion and negative in its wake. Friction reduces for . Calculations were made using FEMLAB. The simulation files can be found in the supplementary materials.

Thermoviscous pumping can be described by the analytical model of Eq. (8). (a) The pump velocity is a linear function of the repetition rate for when the spot geometry remains Gaussian (inset: temperature image). At faster rates, the warm spot becomes elongated due to the finite thermal equilibration time of cooling. Accordingly, the pump velocity is enhanced beyond the linear prediction as the spot width b increases from in the thin fluid film. The solid line predicts the pump velocities based on extrapolated temperature profiles for each repetition rate . (b) The pump velocity rises with the square of the spot temperature, confirming the linear dependence on both the thermal expansion and the temperature dependence of the viscosity. Pump velocities are predicted by Eq. (8) without fitting parameters at a spot width . (c) By changing the overall chamber temperature , we can probe the dependence on and . For , the water contracts upon heating. As expected from Eq. (8), pump velocity reverses its direction (solid line). In all plots, error bars show standard errors (s.e.m.) from particle tracking.

Thermoviscous pumping can be described by the analytical model of Eq. (8). (a) The pump velocity is a linear function of the repetition rate for when the spot geometry remains Gaussian (inset: temperature image). At faster rates, the warm spot becomes elongated due to the finite thermal equilibration time of cooling. Accordingly, the pump velocity is enhanced beyond the linear prediction as the spot width b increases from in the thin fluid film. The solid line predicts the pump velocities based on extrapolated temperature profiles for each repetition rate . (b) The pump velocity rises with the square of the spot temperature, confirming the linear dependence on both the thermal expansion and the temperature dependence of the viscosity. Pump velocities are predicted by Eq. (8) without fitting parameters at a spot width . (c) By changing the overall chamber temperature , we can probe the dependence on and . For , the water contracts upon heating. As expected from Eq. (8), pump velocity reverses its direction (solid line). In all plots, error bars show standard errors (s.e.m.) from particle tracking.

Controlling fluid flow at optical resolution. (a) The measured lateral temperature of the spot (dashed line) is compared to experimental pump velocities (dots). The width of the pump path follows the significantly sharper -profile (solid line). As result, the resolution of the fluid flow is enhanced by a factor of . (b) To demonstrate the fluid flow resolution, the laser was moved along the pattern “nim” with minimal path separations of . The resulting flow has a resolution of (standard deviation) at a chamber thickness of . As seen, the flow paths do not interfere. The corresponding movie can be found in the supplementary materials section.

Controlling fluid flow at optical resolution. (a) The measured lateral temperature of the spot (dashed line) is compared to experimental pump velocities (dots). The width of the pump path follows the significantly sharper -profile (solid line). As result, the resolution of the fluid flow is enhanced by a factor of . (b) To demonstrate the fluid flow resolution, the laser was moved along the pattern “nim” with minimal path separations of . The resulting flow has a resolution of (standard deviation) at a chamber thickness of . As seen, the flow paths do not interfere. The corresponding movie can be found in the supplementary materials section.

Pumping a finite-sized Oseen tensor. (a) The focus is moved along a short path by a short deflection and a subsequent flyback with the laser switched off, both implemented by the acousto-optical deflector. (b) The generated flow field is inferred from particle tracking of beads shown for a recording interval of (black). Their movement is well described by a finite-sized Oseen tensor (red and gray, see supplementary material). Any solution of the laminar, two-dimensional Navier–Stokes equation can be generated by the superposition of Oseen tensors. Individual pump patterns can be superposed by the laser scanning. As result we can generate arbitrary solutions of the laminar two-dimensional Navier–Stokes equation on the tens of micrometer scale.

Pumping a finite-sized Oseen tensor. (a) The focus is moved along a short path by a short deflection and a subsequent flyback with the laser switched off, both implemented by the acousto-optical deflector. (b) The generated flow field is inferred from particle tracking of beads shown for a recording interval of (black). Their movement is well described by a finite-sized Oseen tensor (red and gray, see supplementary material). Any solution of the laminar, two-dimensional Navier–Stokes equation can be generated by the superposition of Oseen tensors. Individual pump patterns can be superposed by the laser scanning. As result we can generate arbitrary solutions of the laminar two-dimensional Navier–Stokes equation on the tens of micrometer scale.

Methods to enhance the pump speed. (a) Viscous fluids are pumped equally well, shown for viscous water-glycerol mixtures. The increased pump speed is the result of an increased at a rising glycerol concentration. The parallel increase in viscosity does not quench pump velocities. (b) Pump speeds also increase for thinner chambers since the repetition rate can be enhanced quadratically with decreasing chamber thickness.

Methods to enhance the pump speed. (a) Viscous fluids are pumped equally well, shown for viscous water-glycerol mixtures. The increased pump speed is the result of an increased at a rising glycerol concentration. The parallel increase in viscosity does not quench pump velocities. (b) Pump speeds also increase for thinner chambers since the repetition rate can be enhanced quadratically with decreasing chamber thickness.

Millimeter-scale unidirectional flow of beads with diameter along the letters “LASER.” The laser focus guides the fluid with the beads through a low melting agarose gel from an aqueous reservoir on the top left through a gel and back into the aqueous reservoir on the right side of the image.

Millimeter-scale unidirectional flow of beads with diameter along the letters “LASER.” The laser focus guides the fluid with the beads through a low melting agarose gel from an aqueous reservoir on the top left through a gel and back into the aqueous reservoir on the right side of the image.

Light driven mixing of DNA hairpins in dynamically created gel pockets. (a) Two liquids are mixed with different ratios simultaneously. Fluorescein-Dextran (MW 40.000) is pumped from a liquid boundary (bottom) into three chambers of different size. It is mixed with the dye-free liquid (top) in equally sized areas at volume mixing ratios of 4:1, 1:1, and 1:4. The mixing is performed all optically without any microfluidic walls and without any contact to the liquid other than the unstructured cover slips that border the fluid film. At the used repetition frequency the mixing sequence took . (b) A 40-base DNA Hairpin (bottom, bright) is mixed with target DNA (top, dark) at mixing ratios 1:8, 1:4, 1:3, 1:2, and 1:1. The mixing is provided from a single channel into pockets of variable size. The overall mixing time is at . The corresponding movies are supplied as supplementary material.

Light driven mixing of DNA hairpins in dynamically created gel pockets. (a) Two liquids are mixed with different ratios simultaneously. Fluorescein-Dextran (MW 40.000) is pumped from a liquid boundary (bottom) into three chambers of different size. It is mixed with the dye-free liquid (top) in equally sized areas at volume mixing ratios of 4:1, 1:1, and 1:4. The mixing is performed all optically without any microfluidic walls and without any contact to the liquid other than the unstructured cover slips that border the fluid film. At the used repetition frequency the mixing sequence took . (b) A 40-base DNA Hairpin (bottom, bright) is mixed with target DNA (top, dark) at mixing ratios 1:8, 1:4, 1:3, 1:2, and 1:1. The mixing is provided from a single channel into pockets of variable size. The overall mixing time is at . The corresponding movies are supplied as supplementary material.

Fast light driven creation of a dilution series. Biomolecules are aliquoted and mixed from an interface of two neighboring gels. First, three volumes of 65, 40, and are created. In a second step the fluid is mixed by repeatedly pumping rectangular ring flows with perpendicular orientation. The result is a dilution series with volume ratios of 4:1, 1:1, and 1:4 in equal volumes. Due to the light absorbing chamber wall composite material, protocol time drops to . The corresponding movie can be found in the supplementary materials section.

Fast light driven creation of a dilution series. Biomolecules are aliquoted and mixed from an interface of two neighboring gels. First, three volumes of 65, 40, and are created. In a second step the fluid is mixed by repeatedly pumping rectangular ring flows with perpendicular orientation. The result is a dilution series with volume ratios of 4:1, 1:1, and 1:4 in equal volumes. Due to the light absorbing chamber wall composite material, protocol time drops to . The corresponding movie can be found in the supplementary materials section.

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