# Thermocapillarity in Microfluidics—A Review

^{*}

## Abstract

**:**

## 1. Introduction

**Figure 1.**Surface Marangoni flow induced due to surface temperature gradient from hot to cold region. The flow propagates inside due to drag and reverse flow forms to hold mass conservation (

**a**). Droplet on a solid surface with temperature gradient leans toward the cold region due to higher surface tension at the inception of movement and Marangoni circulation forms inside it (

**b**). In the case of channel flows, the carrier liquid obeys a bulk flow from hot to cold region due to surface tension forces and the confined droplet moves to the opposite direction (toward the hot region) to conserve the mass (

**c**).

## 2. Theory

**Figure 2.**Components of the stress tensor, surface tension gradient, and normal and tangential unit vectors on a fluid surface element.

- The normal component of the hydrodynamic stresses must balance with the portion of surface tension due to curvature (Equation (1)). This equation is called Laplace equation in the absence of viscosity.
- The tangential component of the hydrodynamic stresses must balance the tangential portion of the surface tension gradient (Equation (2)).

Number | Symbol | Definition |
---|---|---|

Capillary Number | Ca | $\frac{\mathrm{Viscosity}}{\mathrm{Capillarity}}$ |

Marangoni Number | Ma | $\frac{\mathrm{Thermocapillarity}}{\mathrm{Viscosity}}$ |

Reynold Number | Re | $\frac{\mathrm{Inertia}}{\mathrm{Viscosity}}$ |

Weber Number | We | $\frac{\mathrm{Inertia}}{\mathrm{Capillarity}}$ |

Bond Number | Bo | $\frac{\mathrm{Gravity}}{\mathrm{Capillarity}}$ |

Ohnesorge Number | Oh | $\frac{\sqrt{\mathrm{We}}}{\mathrm{Re}}\equiv \frac{\mathrm{Viscosity}}{\sqrt{\mathrm{Inertia}\cdot \mathrm{Capillarity}}}$ |

Froude Number | Fr | $\frac{\mathrm{Inertia}}{\mathrm{Gravity}}$ |

Biot Number | Bi | $\frac{\mathrm{Convective\; Heat\; Transfer}(\mathrm{liquid})}{\mathrm{Conductive\; Heat\; Transfer}(\mathrm{Solid})}$ |

Nusselt Number * | Nu | $\frac{\mathrm{Convective\; Heat\; Transfer}}{\mathrm{Conductive\; Heat\; Transfer}}$ |

Prandtl Number | Pr | $\frac{\mathrm{Viscous\; Diffusion\; Rate}}{\mathrm{Thermal\; Diffusion\; Rate}}$ |

Peclet Number * | Pe | $\frac{\mathrm{Convective\; Heat\; Transfer}}{\mathrm{Diffusive\; Heat\; Transfer}}$ |

_{c}is the critical temperature of the liquid, i.e., the end point of its phase equilibrium curve. This relationship turned out to be exact for the temperatures below the critical point of the liquid. As is shown in Figure 3 for water in contact with air, surface tension of most of the liquids is a linear function of temperature which decreases by increasing the temperature.

## 3. Thermocapillary Instability of Thin Films

## 4. Thermocapillary Effects in Evaporating Droplets

## 5. Thermocapillary-Induced Droplet/Bubble Actuation

#### 5.1. Droplet Spreading

**Figure 5.**The flow field (left figures) and isotherms (right figures) inside the droplet with radius of R = 4 mm at t = 3.5 s (

**a**), R = 5 mm at t = 3.5 s (

**b**), and R = 5.5 mm at t = 2.5 s (

**c**). Reprinted with permission from [86]. Copyright 2010, AIP Publishing LLC.

**Figure 6.**Thermocapillary actuation of a binary heptanol-water drop. Stage (

**a**): heptanol spread against the temperature gradient; Stage (

**b**): heptanol and water drop mixed together; Stage (

**c**): the binary drop was driven to the cold side. Reprinted with permission from [94]. Copyright 2011, AIP Publishing LLC.

#### 5.2. Bubble/Droplet Migration

**Figure 7.**The droplet shape and streamlines of the flow in the y = 0 plane. The value of the surface tension ratio γ

_{12}= 0 corresponds to a spherical droplet (

**a**) and γ

_{12}= 1 results in a slender droplet (

**b**). All other parameters are fixed at unity. Reprinted with permission from [99]. Copyright 2009, AIP Publishing LLC.

#### 5.2.1. Mechanical Heating

**Figure 8.**Periodic back and forth movement of the oil plug due to changing the direction of thermocapillary force by switching the left and right heaters.

**Figure 9.**Encapsulation of a water drop inside an oil plug in a microtube under thermocapillary effect observed by Jiao et al. Reproduced with permission from [132], Z. Jiao, N.-T. Nguyen, and X. Huang, “Thermocapillary actuation of a water droplet encapsulated in an oil plug”, Journal of Micromechanics and Microengineering, vol. 17, p. 1843, 2007. Copyright 2007, IOP Publishing, all rights reserved.

#### 5.2.2. Optical Heating

**Figure 10.**(

**a**) Microscopic image of the rapid flow around the micro bubble induced by the photothermal conversion. The cross mark shows the laser position, the small black dots are the polystyrene (PS) spheres, and the big black circle is the micro bubble with the diameter of around 65 µm. Scale bar: 50 µm. (

**b**) Sketch of the flow around the bubble. Reprinted with permission from [135]. Copyright 2015, AIP Publishing LLC.

**Figure 11.**(

**a**) Experimental, and (

**b**) numerical results of the fluid flow that forms around a laser spot on the interface of the droplet stopping it from further movement due to thermocapillary phenomenon. Reprinted figures with permission from [136], C.N. Baroud, J.-P. Delville, F. Gallaire, and R. Wunenburger, Physical Review E, vol. 75, p. 046302, 2007. Copyright 2007, American Physical Society.

**Figure 12.**(

**a**) Localized fusion and merging of the two adjacent drops; (

**b**) Blocking the first drop by a laser valve until the second drop arrives and merging the two by another laser beam at the interface; (

**c**) A droplet sampler that decides precisely what portion of the drop should go to each specific outlet. Republished with permission of Royal Society of Chemistry, from [140], “An optical toolbox for total control of droplet microfluidics”, C.N. Baroud, M.R. de Saint Vincent, and J.-P. Delville, Lab on a Chip, vol. 7, pp. 1029–1033, 2007; Permission conveyed through Copyright Clearance Center, Inc.

**Figure 13.**Studying the flow pattern inside a droplet by emitting a point laser beam on its interface. (

**a**) The experimental apparatus, (

**b**) the actual flow pattern inside a confined drop due to two point laser beams located on its interface symmetrically, (

**c**,

**d**) simulated flow pattern under the same conditions as in (

**b**). Reprinted with permission from [141]. Copyright 2009, American Chemical Society.

#### 5.3. Thermocapillary Mixing

**Figure 14.**Streamlines of the Taylor (

**A**), dipole (

**B**), and quadrupole (

**C**) flow at the midplane of the droplet. Reprinted with permission from [150]. Copyright 2005, AIP Publishing LLC.

**Figure 15.**Advection of dye by the dipole flow shown in the midplane of the droplet. The initial state (

**a**) and stretching in steady dipole flow at t = 6 (

**b**) and t = 24 (

**c**). Stretching and folding in a time-periodic flow obtained by rotating its direction by 90° in the horizontal plane every six time units. At t = 12 (

**d**), t = 18 (

**e**), and t = 24 (

**f**). Reprinted with permission from [150]. Copyright 2005, AIP Publishing LLC.

**Figure 16.**Snapshots of the position of passive tracer showing the enhancement of mixing of the lower and upper halves of the droplet using two stationary (not altering) laser beams. Reproduced with permission from [155], M.L. Cordero, H.O. Rolfsnes, D.R. Burnham, P.A. Campbell, D. McGloin, and C.N. Baroud, “Mixing via thermocapillary generation of flow patterns inside a microfluidic drop”, New Journal of Physics, vol. 11, p. 075033, 2009. Copyright 2009, IOP Publishing & Deutsche Physikalische Gesellschaft. CC BY-NC-SA.

#### 5.4. Thermocapillary Coalescence and Nonwetting

**Figure 17.**Head-on collision of binary droplets with increased temperature of the incoming droplet at releasing height of 13 mm (T

_{stationary}= T

_{room}= 25 °C, ΔT = T

_{incoming}− T

_{stationary}). Reprinted by permission from Macmillan Publishers Ltd: Scientific Reports [162], Copyright 2014.

**Figure 18.**Using a heater to study the coalescence of two droplets under thermocapillary effect. Reprinted with permission from [163]. Copyright 2012, AIP Publishing LLC.

**Figure 19.**Streamline pattern of a compound drop moving with temperature gradient. Reproduced with permission of Cambridge University Press, from [168], L. Rosenfeld, O. Lavrenteva, and A. Nir, “On the thermocapillary motion of partially engulfed compound drops”, Journal of Fluid Mechanics, vol. 626, pp. 263–289, 2009; Permission conveyed through Copyright Clearance Center, Inc.

**Figure 20.**Streamlines of spontaneous thermocapillary flow in and around two spherical drops in contact (

**a**), and at a separation distance (

**b**). Reprinted with permission from [171]. Copyright 2002, AIP Publishing LLC.

**Figure 21.**A display of a dynamic deformation and migration of two interacting drops. (

**a**) R = 1 Ca = 1; (

**b**) R = 0.6 Ca = 0.67 (with R being the ratio of the radius of two drops and Ca the capillary number). Reprinted with permission from [171]. Copyright 2002, AIP Publishing LLC.

#### 5.5. Drop Manipulation on Free Surface

**Figure 22.**Liquid surface deformation in existence of thermocapillarity and levitation of a spherical droplet with an air film separating it from the liquid film. Reproduced with permission of Cambridge University Press, from [175], R. Savino, D. Paterna, and M. Lappa, “Marangoni flotation of liquid droplets”, Journal of Fluid Mechanics, vol. 479, pp. 307–326, 2003; Permission conveyed through Copyright Clearance Center, Inc.

**Figure 23.**(

**a**) Two photographs showing a drop of radius 2 mm as it bounces on the liquid surface. The arrows show the direction of bath motion. (

**b**) A large floating drop as seen from the side (top). The interference fringes of the air film as observed when the drop is lit from the top with white light. The black dot at the center is the reflection of the camera’s lens. Reprinted figures with permission from [176], Y. Couder, E. Fort, C.-H. Gautier, and A. Boudaoud, Physical Review Letters, vol. 94, p. 177801, 2005. Copyright 2005, American Physical Society.

**Figure 24.**(

**a**) Flow pattern in and out of a droplet in presence of thermal gradient, (

**b**) flow pattern in a liquid film driven by surface temperature gradient. Reproduced with permission from [178], A.S. Basu and Y.B. Gianchandani, “Virtual microfluidic traps, filters, channels and pumps using Marangoni flows”, Journal of Micromechanics and Microengineering, vol. 18, p. 115031, 2008. Copyright 2008, IOP Publishing, all rights reserved.

**Figure 25.**Aqueous drops resting on the free surface of fluorocarbon liquid (FC-43) in a spherical mode (

**a**) and lens mode (

**b**). Reprinted with permission from [181]. Copyright 2010, AIP Publishing LLC.

**Figure 26.**Deformation and flow generated at the free surface of thin immiscible liquid layer drives levitated-spherical droplets toward the hotter end of a thermal gradient. Copyright 2011, IEEE. Reprinted and modified, with permission, from [182], E. Yakhshi-Tafti, R. Kumar, and H. Cho, “Thermally-actuated high speed droplet manipulation platform”, in Proceesings of 2011 16th International Solid-State Sensors, Actuators and Microsystems Conference (TRANSDUCERS), 2011, pp. 1484–1487.

**Figure 27.**(

**a**) Release height as a function of drop diameter, and (

**b**) We as a function of Oh. (

**I**) region in which drops collapse into submerged form; (

**II**) region in which drops assume a spherical ball-shape above the surface; and (

**III**) Region in which the drops bridge the gap between the dispensing tip and the liquid interface where free falling pendant drops cannot be formed. Reprinted from Journal of Colloid and Interface Science, vol. 350, E. Yakhshi-Tafti, H.J. Cho, and R. Kumar, “Impact of drops on the surface of immiscible liquids”, pp. 373–376 [183]. Copyright 2010, with permission from Elsevier.

**Figure 28.**Stable configurations of aqueous droplets at oil-air interface. (

**a**) Non-coalescent droplet resting on a stretched and deformed free surface. (

**b**) Cap-bead droplet with triple contact line. (

**c**,

**d**) The effect of droplet size on the deformation of free surfaces. Reprinted with permission from [184]. Copyright 2013, AIP Publishing LLC.

## 6. Device Applications

#### 6.1. Thermocapillary Pumps, Mixers, and Actuators

**Figure 29.**Schematic of an integrated DNA analysis device proposed by Burns et al. which had five different sections: (A) Injection entry ports; (B) thermocapillary pumping channels; (C) thermally controlled reaction chamber; (D) electrophoresis channel; and (E) DNA band migration detector. Reprinted with permission from [187], M.A. Burns, et al., “Microfabricated structures for integrated DNA analysis“, Proceedings of the National Academy of Sciences, vol. 93, pp. 5556–5561. Copyright 1996, National Academy of Sciences, USA.

**Figure 30.**Droplet actuation and mixing in the Y-channel by thermocapillary effect. Reprinted with permission from [187], M.A. Burns, et al., “Microfabricated structures for integrated DNA analysis”, Proceedings of the National Academy of Sciences, vol. 93, pp. 5556–5561. Copyright 1996, National Academy of Sciences, USA.

**Figure 31.**The thermocapillary pumping device proposed by Sammarco and Burns (

**a**) actuating a confined droplet in a hydrophilic (

**b**) and hydrophobic (

**c**) channel. Reprinted with permission from [188], T.S. Sammarco and M.A. Burns, “Thermocapillary pumping of discrete drops in microfabricated analysis devices”, AIChE Journal, vol. 45, pp. 350–366. Copyright 1999, John Wiley and Sons.

**Figure 32.**(

**a**) Cross sectional view of a portion of the device containing two microheaters and an overlying droplet; (

**b**) Top view of a liquid droplet moving along a liquophilic microstripe; (

**c**) Top view of the resistor and contact layout; (

**d**) A droplet moving along the heater array. The images sequence represent times (1) t = 0 s, (2) 44 s, (3) 88 s, and (4) 132 s. Copyright 2003, IEEE. Reprinted with permission from [190], A.A. Darhuber, J.P. Valentino, S.M. Troian, and S. Wagner, “Thermocapillary actuation of droplets on chemically patterned surfaces by programmable microheater arrays”, Journal of Microelectromechanical Systems, vol. 12, pp. 873–879, 2003.

**Figure 33.**Top (

**a**) and cross sectional (

**b**) views of microfluidic device fabricated on glass substrate, (

**c**) optical micrograph of a 1 mm wide liquid filament separated electronically into (

**d**) one, (

**e**) two, and (

**f**) three droplets. Republished with permission of Royal Society of Chemistry, from [195], “Planar digital nanoliter dispensing system based on thermocapillary actuation”, A.A. Darhuber, J.P. Valentino, and S.M. Troian, Lab on a Chip, vol. 10, pp. 1061–1071, 2010; Permission conveyed through Copyright Clearance Center, Inc.

**Figure 34.**(

**a**) The device proposed by Jiao et al. to produce a temperature field and actuate a confined droplet by changing the voltage supplied to the four heaters on four sides of the device. (

**b**) Droplet migration for a specific heating code. Reproduced with permission from [198], Z. Jiao, X. Huang, and N.-T. Nguyen, “Manipulation of a droplet in a planar channel by periodic thermocapillary actuation”, Journal of Micromechanics and Microengineering, vol. 18, p. 045027, 2008. Copyright 2008, IOP Publishing, all rights reserved.

**Figure 35.**(

**a**) Concept of a contactless microdroplet manipulator based upon Marangoni flows. The flow is driven by heat sources of various geometries suspended above the oil layer. The projected heat fluxes (shown in red) generate flows which emulate droplet channels, reservoirs, and mixers. (

**b**) A microfluidic trap generated by a point heat source. Schematic is showing the suspended heat source, the projected Gaussian heat flux profile, and the toroidal flow region in the oil layer. (

**c**) Virtual droplet channels generated by parallel linear heat sources. The channel boundaries are defined by the heat flux projected by two heated wires with separation s held parallel to the liquid surface. Target sized droplets are pulled into the channel by the subsurface flows. Orders are excluded. Marangoni flows are shown in green and with arrows. (

**d**) Single droplet trapping with an annular heat flux. Schematic is showing the annular heat flux projected on the surface, the trapped droplet, and the exclusion of larger and smaller droplets. Reproduced with permission from [178], A.S. Basu and Y.B. Gianchandani, “Virtual microfluidic traps, filters, channels and pumps using Marangoni flows”, Journal of Micromechanics and Microengineering, vol. 18, p. 115031, 2008. Copyright 2008, IOP Publishing, all rights reserved.

**Figure 36.**Fabricated device (

**a**) for moving a levitated droplet of water on a film of FC-43 oil toward the hot spot where the underlying heater is switched on (

**b**,

**c**). Pictures (

**c**) is taken by a FLIR SC5000 infrared thermal camera and shows the temperature gradient. Copyright 2011, IEEE. Reprinted and modified, with permission, from [182], E. Yakhshi-Tafti, R. Kumar, and H. Cho, “Thermally-actuated high speed droplet manipulation platform”, in Proceedings of 2011 16th International Solid-State Sensors, Actuators and Microsystems Conference (TRANSDUCERS), 2011, pp. 1484–1487.

#### 6.2. Thermocapillary Valves, Switches, and Traps

**Figure 37.**(

**a**) Overview of the experimental device by Selva et al. (

**b**) Top view of their experimental setup. The liquid fills the cavity positioned above the chromium resistors which have been placed in series and are connected by gold wires. Reproduced with permission from [200], B. Selva, J. Marchalot, and M.-C. Jullien, “An optimized resistor pattern for temperature gradient control in microfluidics”, Journal of Micromechanics and Microengineering, vol. 19, p. 065002, 2009. Copyright 2009, IOP Publishing, all rights reserved.

**Figure 38.**(

**a**) Switching and (

**b**) trapping of the droplets are shown using the device proposed by Selva et al. Republished with permission of Royal Society of Chemistry, from [199], “Thermocapillary actuation by optimized resistor pattern: bubbles and droplets displacing, switching and trapping”, B. Selva, V. Miralles, I. Cantat, and M.-C. Jullien, Lab on a Chip, vol. 10, pp. 1835–1840, 2010; Permission conveyed through Copyright Clearance Center, Inc.

**Figure 39.**Image sequence showing how the drop order can be changed: The initial drop is sent down and held stationary, then successive droplets are sent up. The dashed lines overlay the position of laser patterns. Reprinted with permission from [204]. Copyright 2008, AIP Publishing LLC.

**Figure 40.**Both figures are superposition of successive snapshots to show the effect of laser heating on (

**a**) droplet switching, and (

**b**) droplet sorting. Reprinted with permission from [205]. Copyright 2008, AIP Publishing LLC.

#### 6.3. Thermocapillary Sensors

**Figure 41.**(

**a**) Schematic diagram of a complete sensing circuit consisting of two inverters, one resistor R

_{c}, and one capacitor C

_{c}, which are attached in parallel to the system shown in (

**b**) whose capacitance is given by C

_{sys}. Changes in C

_{sys}detectable by the oscillation frequency shift are measured with an external multimeter. (

**b**) Schematic diagram of the experimental setup containing the sensing electrode arrays. The liquid film thickness, d

_{liq}, is adjusted by vertical displacement of a (non-wetting polycarbonate sheet. The native electric field capacitance is denoted C

_{12}; the coupling capacitances with the brass block are labelled C

_{10}and C

_{20}. (

**c**) Top view of the actual device with and without droplet. Republished with permission of Royal Society of Chemistry, from [206], “Capacitive sensing of droplets for microfluidic devices based on thermocapillary actuation”, J.Z. Chen, A.A. Darhuber, S.M. Troian, and S. Wagner, Lab on a Chip, vol. 4, pp. 473–480, 2004; Permission conveyed through Copyright Clearance Center, Inc.

**Figure 42.**Schematic of the microdroplet detection and analysis proposed by Valentino et al. Reprinted with permission from [207]. Copyright 2005, AIP Publishing LLC.

**Figure 43.**(

**a**) The schematic of the device for micromirror actuation consisted of four quarter ring shaped heaters; (

**b**) the device loaded with a droplet with a microplate on top. Switching the heaters leads to actuation of the droplet and hence tilting the microplate. Copyright 2009, IEEE. Reprinted, with permission, from [209], R. Dhull, I. Puchades, L. Fuller, and Y. Lu, “Optical Micromirror Actuation using Thermocapillary Effect in Microdroplets”, in Processings of IEEE 22nd International Conference on Micro Electro Mechanical Systems (MEMS 2009), 2009, pp. 995–998.

## 7. Summary

Configuration | Heat Source | Application | Refs. |
---|---|---|---|

Drops and Bubbles in Microchannels | Embedded Al | Droplet Actuation and Mixing | [187] |

Embedded Poly-Si | Thermocapillary Pumping (TCP) | [188] | |

Embedded Cr/Ni | Bubble Micro-oscillator | [189] | |

Embedded Cr | Droplet Switching, Sorting and Trapping | [199,200,201,202] | |

Laser Beam | Droplet Manipulation | [204] | |

Laser Beam | Droplet Switching and Sorting | [205] | |

Drops on Solid Surfaces | Embedded Ti/Au | Droplet Actuator | [190,191] |

Embedded Ti | Droplet Actuator | [207] | |

Embedded Ti/Pt | Droplet Mixing/Reaction Measurement | [197,198] | |

Embedded Poly-Si | Micromirror Actuator | [209] | |

Embedded Au/Ti | Droplet Manipulation | [196] | |

Embedded Ti/Au | Nano-Dispenser | [195] | |

Embedded Ti/Au | Droplet Position, Size, Composition Measurement | [206] | |

Drops on Liquid Films | Suspended Heaters | Droplet Actuation, Mixing, Trapping, and Pumping | [178] |

Embedded Ti | Droplet Actuation, Mixing, Trapping, and Sorting | [182] | |

Miscellaneous | Laser Beam | Microbead Manipulation | [210] |

Suspended Heaters PCB | Mechanical Power Production | [211] | |

Cold Metal | Chemical Computation | [212] |

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Karbalaei, A.; Kumar, R.; Cho, H.J.
Thermocapillarity in Microfluidics—A Review. *Micromachines* **2016**, *7*, 13.
https://doi.org/10.3390/mi7010013

**AMA Style**

Karbalaei A, Kumar R, Cho HJ.
Thermocapillarity in Microfluidics—A Review. *Micromachines*. 2016; 7(1):13.
https://doi.org/10.3390/mi7010013

**Chicago/Turabian Style**

Karbalaei, Alireza, Ranganathan Kumar, and Hyoung Jin Cho.
2016. "Thermocapillarity in Microfluidics—A Review" *Micromachines* 7, no. 1: 13.
https://doi.org/10.3390/mi7010013