Microfabricated integrated circuits revolutionized computation by vastly reducing the space, labor, and time required for calculations. Microfluidic systems hold similar promise for the large-scale automation of chemistry and biology, suggesting the possibility of numerous experiments performed rapidly and in parallel, while consuming little reagent. While it is too early to tell whether such a vision will be realized, significant progress has been achieved, and various applications of significant scientific and practical interest have been developed. Here a review of the physics of small volumes (nanoliters) of fluids is presented, as parametrized by a series of dimensionless numbers expressing the relative importance of various physical phenomena. Specifically, this review explores the Reynolds number Re, addressing inertial effects; the Peclet number Pe, which concerns convective and diffusive transport; the capillary number Ca expressing the importance of interfacial tension; the Deborah, Weissenberg, and elasticity numbers De, Wi, and El, describing elastic effects due to deformable microstructural elements like polymers; the Grashof and Rayleigh numbers Gr and Ra, describing density-driven flows; and the Knudsen number, describing the importance of noncontinuum molecular effects. Furthermore, the long-range nature of viscous flows and the small device dimensions inherent in microfluidics mean that the influence of boundaries is typically significant. A variety of strategies have been developed to manipulate fluids by exploiting boundary effects; among these are electrokinetic effects, acoustic streaming, and fluid-structure interactions. The goal is to describe the physics behind the rich variety of fluid phenomena occurring on the nanoliter scale using simple scaling arguments, with the hopes of developing an intuitive sense for this occasionally counterintuitive world.

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Microfluidics: Fluid physics at the nanoliter scale

Microfluidic devices for manipulating fluids are widespread and finding uses in many scientific and industrial contexts. Their design often requires unusual geometries and the interplay of multiple physical effects such as pressure gradients, electrokinetics, and capillarity. These circumstances lead to interesting variants of well-studied fluid dynamical problems and some new fluid responses. We provide an overview of flows in microdevices with focus on electrokinetics, mixing and dispersion, and multiphase flows. We highlight topics important for the description of the fluid dynamics: driving forces, geometry, and the chemical characteristics of surfaces.

Engineering flows in small devices

The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

Wetting phenomena are ubiquitous in nature and technology. A solid substrate exposed to the environment is almost invariably covered by a layer of fluid material. In this review, the surface forces that lead to wetting are considered, and the equilibrium surface coverage of a substrate in contact with a drop of liquid. Depending on the nature of the surface forces involved, different scenarios for wetting phase transitions are possible; recent progress allows us to relate the critical exponents directly to the nature of the surface forces which lead to the different wetting scenarios. Thermal fluctuation effects, which can be greatly enhanced for wetting of geometrically or chemically structured substrates, and are much stronger in colloidal suspensions, modify the adsorption singularities. Macroscopic descriptions and microscopic theories have been developed to understand and predict wetting behavior relevant to microfluidics and nanofluidics applications. Then the dynamics of wetting is examined. A drop, placed on a substrate which it wets, spreads out to form a film. Conversely, a nonwetted substrate previously covered by a film dewets upon an appropriate change of system parameters. The hydrodynamics of both wetting and dewetting is influenced by the presence of the three-phase contact line separating "wet" regions from those that are either dry or covered by a microscopic film only. Recent theoretical, experimental, and numerical progress in the description of moving contact line dynamics are reviewed, and its relation to the thermodynamics of wetting is explored. In addition, recent progress on rough surfaces is surveyed. The anchoring of contact lines and contact angle hysteresis are explored resulting from surface inhomogeneities. Further, new ways to mold wetting characteristics according to technological constraints are discussed, for example, the use of patterned surfaces, surfactants, or complex fluids.

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Wetting and Spreading

The dynamics and stability of thin liquid films have fascinated scientists over many decades: the observations of regular wave patterns in film flows down a windowpane or along guttering, the patterning of dewetting droplets, and the fingering of viscous flows down a slope are all examples that are familiar in daily life. Thin film flows occur over a wide range of length scales and are central to numerous areas of engineering, geophysics, and biophysics; these include nanofluidics and microfluidics, coating flows, intensive processing, lava flows, dynamics of continental ice sheets, tear-film rupture, and surfactant replacement therapy. These flows have attracted considerable attention in the literature, which have resulted in many significant developments in experimental, analytical, and numerical research in this area. These include advances in understanding dewetting, thermocapillary- and surfactant-driven films, falling films and films flowing over structured, compliant, and rapidly rotating substrates, and evaporating films as well as those manipulated via use of electric fields to produce nanoscale patterns. These developments are reviewed in this paper and open problems and exciting research avenues in this thriving area of fluid mechanics are also highlighted.

Dynamics and stability of thin liquid films

We demonstrate the active manipulation of nanoliter liquid samples on the surface of a glass or silicon substrate by combining chemical surface patterning with electronically addressable microheater arrays. Hydrophilic lanes designate the possible routes for liquid migration while activation of specific heater elements determine the trajectories. The induced temperature fields spatially modulate the liquid surface tension thereby providing electronic control over the direction, timing, and flow rate of continuous streams or discrete drops. Temperature maps can be programed to move, split, trap, and mix ultrasmall volumes without mechanically moving parts and with low operating voltages of 2–3 V. This method of fluidic actuation allows direct accessibility to liquid samples for handling and diagnostic purposes and provides an attractive platform for palm-sized and battery-powered analysis and synthesis.

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Microfluidic actuation by modulation of surface stresses

We characterize the selective deposition of liquid microstructures on chemically heterogeneous surfaces by means of dip coating processes. The maximum deposited film thickness depends critically on the speed of withdrawal as well as the pattern size, geometry, and angular orientation. For vertically oriented hydrophilic strips, we derive a hydrodynamic scaling relation for the deposited film thickness which agrees very well with interferometric measurements of dip-coated liquid lines. Due to the lateral confinement of the liquid, our scaling relation differs considerably from the classic Landau–Levich formula for chemically homogeneous surfaces. Dip coating is a simple method for creating large area arrays of liquid microstructures for applications involving chemical analysis and synthesis, biochemical assays, or wet printing of liquid polymer or ink patterns.

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Selective dip-coating of chemically micropatterned surfaces

The recent introduction of actuation mechanisms for microfluidic transport based on free surface flows raises a number of interesting questions involving efficient mixing configurations, especially in systems with small aspect ratios. This work investigates the characteristics of convective and diffusive mixing in continuous-mode streaming of thermocapillary microflows on chemically micropatterned surfaces. Mixing times and mixing lengths relevant to chemical microreactors or gas sensors are investigated for various geometries and parameter ranges. Scaling arguments and full numerical solutions are presented to extract optimal operating conditions. Confocal fluorescence microscopy measurements of the interfacial diffusive broadening in adjacent flowing streams confirm numerical predictions. Three important mixing regimes, based on analogues of purely diffusive dynamics, Rhines-Young shear-augmented diffusion and Taylor-Aris dispersion are identified and investigated for use in free surface flows with large surface-to-volume ratios.

A study of mixing in thermocapillary flows on micropatterned surfaces

Recent investigations of microfluidic flows have focused on manipulating the motion of very thin liquid films by modulating the surface tension through an applied streamwise temperature gradient. The extent to which the choice of contact line model affects the flow and stability of such thermocapillary-driven films is not completely understood. Regardless of the contact line model used, the linearized disturbance operator corresponding to the evolution of the film height is non-normal, and a generalized non-modal stability analysis is required. Surprisingly, early predictions of frontal instability that stemmed from conventional modal analysis of thermocapillary flow on a flat, infinite precursor film showed excellent agreement with experiment. Within the more rigorous framework provided by a generalized stability analysis, this work investigates the transient dynamics and amplification of optimal disturbances subject to a finite precursor film generated by attractive van der Waals forces. Convergence of the disturbance growth rates and perturbed shapes to the asymptotic solutions obtained by conventional linear stability analysis occurs early in the spreading process. In addition, the level of transient disturbance amplification is minimal. The equations governing thermocapillary-driven spreading exhibit a small degree of non-normality, which explains the source of agreement between modal theory and experiment. The more rigorous generalized stability analysis presented here, however, affords critical insight into the types of disturbances leading to maximum unstable growth and the exact influence of the contact line model used.

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Influence of attractive van der Waals interactions on the optimal excitations in thermocapillary-driven spreading.

Thin liquid films driven to spread on homogeneous surfaces by thermocapillarity can undergo frontal breakup and parallel rivulet formation with well-defined wavelength. Previous modal analyses have relieved the well-known divergence in stress that occurs at a moving contact line by matching the front region to a precursor film. Because the linearized disturbance operator is non-normal, a generalized, nonmodal analysis is required to probe film stability at all times. The effect of the contact line model on nonmodal stability has not been previously investigated. This work examines the influence of boundary slip on thermocapillary driven spreading using a transient stability analysis, which recovers the conventional modal results in the long-time limit. In combination with earlier work on thermocapillary driven spreading, this study verifies that the dynamics and stability of this system are rather insensitive to the choice of contact line model and that the leading eigenvalue is physically determinant, thereby assuring results that agree with the eigenspectrum. Modal results for the flat precursor film model are reproduced with appropriate choice of slip coefficient and contact line slope.

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Jeffrey M. Davis

Papers

Microfluidic actuation by modulation of surface stresses

Selective dip-coating of chemically micropatterned surfaces

A study of mixing in thermocapillary flows on micropatterned surfaces

Influence of attractive van der Waals interactions on the optimal excitations in thermocapillary-driven spreading.

Influence of boundary slip on the optimal excitations in thermocapillary driven spreading