Passive and active droplet generation with microfluidics: a review

Abstract

Precise and effective control of droplet generation is critical for applications of droplet microfluidics ranging from materials synthesis to lab-on-a-chip systems. Methods for droplet generation can be either passive or active, where the former generates droplets without external actuation, and the latter makes use of additional energy input in promoting interfacial instabilities for droplet generation. A unified physical understanding of both passive and active droplet generation is beneficial for effectively developing new techniques meeting various demands arising from applications. Our review of passive approaches focuses on the characteristics and mechanisms of breakup modes of droplet generation occurring in microfluidic cross-flow, co-flow, flow-focusing, and step emulsification configurations. The review of active approaches covers the state-of-the-art techniques employing either external forces from electrical, magnetic and centrifugal fields or methods of modifying intrinsic properties of flows or fluids such as velocity, viscosity, interfacial tension, channel wettability, and fluid density, with a focus on their implementations and actuation mechanisms. Also included in this review is the contrast among different approaches of either passive or active nature.

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Pingan Zhu

Pingan Zhu received his bachelor's degree in Safety Science and Engineering from the University of Science and Technology of China in 2013. He is currently a PhD candidate in the department of Mechanical Engineering, the University of Hong Kong. His research interest is in microfluidic droplet generation and droplet-templated materials synthesis.

Liqiu Wang

Prof. L. Q. Wang received his PhD from University of Alberta in 1995 and is currently a professor in the Department of Mechanical Engineering, the University of Hong Kong. He is also a Qianren Scholar (Zhejiang) and serves as the director and the chief scientist for the Laboratory for Nanofluids and Thermal Engineering, Zhejiang Institute of Research and Innovation, the University of Hong Kong. Prof. Wang has over 20 years of university experience in thermal and power engineering, energy and environment, transport phenomena, nanotechnology, biotechnology and applied mathematics in Canada, China/Hong Kong, Singapore and the USA, and has led a team in developing a state-of-the-art thermal control system for the Alpha Magnetic Spectrometer (AMS) on the International Space Station. Prof. Wang's current research is mainly on microfluidic bubbles/droplets/particles, soft materials, flow bifurcation and stability, heat transfer with thermal waves and resonance, numerical simulation and nonlinear computation.

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1. Introduction

Droplet-based microfluidics has emerged as a versatile tool for widespread applications, attributed to the following advantages: a small volume of reagents consumed, massive production of monodisperse droplets, high surface-area-to-volume ratio that facilitates fast reaction, and independent control of each droplet.¹ In general, applications of microfluidic droplets arise from two distinct but complementary aspects.² One exploits droplets with well-defined components and structures as templates in materials science, for example, synthesis of microcapsules,^3–5 microparticles,^6–9 and microfibers¹⁰ applicable to pharmaceuticals, cosmetics, and foods; another involves lab-on-a-chip applications where droplets are used as microreactors to perform chemical and biochemical reactions. In most of these applications, highly uniform droplets are desired to ensure constant, controlled and predictable outcomes. In addition, a wide range of tunable droplet volumes, typically from femtolitres to nanolitres, is preferable. Some applications require even smaller droplets, such as several hundred nanometers in diameter, for the production of nanodroplets and nanoparticles.^11,12 Moreover, instead of utilizing uniform droplets, some applications favor well-controlled sequences of droplets with different volumes, for example, multi-volume droplet digital polymerase-chain-reaction (PCR) for precise and quantitative detection of genetic targets.¹³ Aside from commonly used passive methods, more sophisticated techniques have been adopted to actively control droplet generation. Reliable generation of droplets with accurate control over their size and size distribution is therefore of vital importance to meet the increasingly high demands in various applications. To this end, it is critical to have a deep and systematic understanding of microfluidic droplet formation, including both passive and active techniques.

Droplet generation originates from fluid instabilities. In passive microfluidic devices, introduction of one immiscible fluid (dispersed fluid) into another (continuous fluid) typically leads to the formation of droplets in one of the five modes (Fig. 1): squeezing,¹⁴ dripping,¹⁵ jetting,¹⁵ tip-streaming¹⁶ and tip-multi-breaking.¹⁷ The squeezing arises from a quite different mechanism from the capillary (Rayleigh–Plateau) instability that is responsible for the other four modes. Channel confinement plays a dominant role in the squeezing regime and inhibits capillary instability so that breakup exhibits quasi-static mechanisms until the last stage of thread pinch-off. The other four modes of breakup come from the capillary instability as interfacial tension forces seek to minimize the interfacial area according to the thermodynamic principle of minimum interfacial energy. In these cases, viscous and inertial forces that act to deform the liquid interface counteract interfacial tension forces that resist the deformation. It is the competition of these forces that determines the specific breakup mode of droplet generation for a given set of parameters.

Fig. 1 Schematic of droplet generation in passive and active methods. In the passive method, droplets can be produced in squeezing, dripping, jetting, tip-streaming and tip-multi-breaking modes, depending on the competition of capillary, viscous, and inertial forces. In active control, droplet generation can be manipulated by either applying additional forces from electrical, magnetic, and centrifugal controls, or modifying intrinsic forces via tuning fluid velocity and material properties including viscosity, interfacial tension, channel wettability, and fluid density.

In comparison with passive methods, active techniques add another level of controllability in modulating droplet formation with the aid of additional energy input by external elements. Apart from the advantage of fast response time, active controls are essential in some extreme situations and applications, such as breaking fluid threads in aqueous two-phase systems (ATPS) with ultra-low interfacial tension,¹⁸ producing droplets from highly viscous liquids,¹⁹ and on-demand droplet generation, for example, to encapsulate cells with deterministic cell numbers.²⁰ According to the nature of the external energy input, active techniques are categorized into electrical, magnetic, centrifugal, optical, thermal, and mechanical controls. The additional energy input modifies the force balance on the interface and thus manipulates interfacial instabilities. In principle, the interfacial force balance can be modified by two basic strategies in active controls: (i) introducing additional forces like electrical, magnetic, and centrifugal force; (ii) modifying viscous, inertial, and capillary force by varying intrinsic parameters like flow velocity and material properties (Fig. 1).

Previous reviews of droplet formation have concentrated mainly on passive approaches^1,2,21–24 with an exception of a very recent one exclusively on active droplet generation.²⁵ Rather than focusing on either passive or active approach in those reviews, we aim to present a unified review of both in a systematic manner from the point of view of physical mechanisms. Our focus is on droplet generation in liquid–liquid biphasic flows, though some examples of gas–liquid¹⁴ and liquid–gas²⁶ flows are cited for the sake of elucidating breakup mechanisms. We also limit our discussion to cases where continuous phase fluids preferentially wet the channel walls, so that dispersed droplets are completely surrounded by continuous fluids. In what follows, we outline the governing equations and fundamental dimensionless parameters for droplet generation in section 2. We summarize the commonly used devices in producing microfluidic droplets in section 3. In section 4, we discuss breakup modes of passive droplet generation, especially, the generation process, mechanism, condition of occurrence, characteristics, and nature of fluid instabilities for each mode. Finally, we highlight, in section 5, the state-of-the-art techniques for active droplet generation, organized according to their mechanisms of manipulation: techniques utilizing additional forces followed by those via modulating intrinsic parameters.

2. Dimensionless numbers

Fluid motions with various characteristics could occur in microflows, which are normally determined by competing physical effects, such as force balance. Dimensionless numbers characterize the relative predominance of different effects and are capable of contrasting flows in parameter space, thereby unifying flowing features between different systems. To explore the key dimensionless numbers characterizing the droplet generation (Table 1), we start from the governing equations that determine microfluidic two-phase flows.

Table 1 Dimensionless parameters in microfluidic droplet generation

Symbol	Name	Formula	Physical meaning
Re	Reynolds number		Inertial force/viscous force
Ca	Capillary number		Viscous force/interfacial tension
We	Weber number		Inertial force/interfacial tension
Bo	Bond number		Buoyancy/interfacial tension
λ	Viscosity ratio		Dispersed viscosity/continuous viscosity
φ	Flow rate ratio		Dispersed flow rate/continuous flow rate

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Liquid–liquid two-phase microflows are well described with the continuum hypothesis.²⁷ The incompressible continuity equation reads, for both dispersed and continuous phase fluids:

∇·u_s = 0, (1)

where ∇ is the del operator, u is the velocity vector of the fluid, and subscript “s” denotes either “d” or “c”, corresponding to dispersed or continuous phase fluid, respectively.

The momentum equation is the well-known Navier–Stokes equation of incompressible Newtonian fluids:

where t is the time, ρ the density of the fluid, p the pressure, η the dynamic shear viscosity, and f the body force vector per unit volume. The left side of eqn (2) describes inertial acceleration composed of time-dependent acceleration ρ_s∂u_s/∂t and convective acceleration ρ_su_s·∇u_s (spatial effect), while the right side represents force densities (forces per unit volume).²⁸ The diffusion term η_s∇²u_s is the divergence of viscous stresses σ_s = η_s[∇u_s + (∇u_s)^T]. The body force densities f_s can cover effects from those like gravity, electrical, magnetic, and centrifugal effects. If no other external fields are applied except gravity, f_s = ρ_sg with g being the gravitational acceleration.

Droplet formation in microfluidic devices involves the deformation and breaking of the liquid–liquid interface. As such, several interfacial boundary conditions should be specified. Among them, the first comes from the continuity of normal velocity at the immiscible interface:²⁹

u_d·n = u_c·n, (3)

with n being the unit normal vector outward to the interface. Secondly, the tangential viscous stress should be continuous (eqn (4)), and the normal stress difference between the dispersed and the continuous phases is balanced by capillary pressure (eqn (5)):²⁹

σ_d·t = σ_c·t, (4)

T_d·n − T_c·n = − γκ, (5)

where t is the unit tangential vector at the interface, T_s = −p_sI + σ_s the stress tensor that contains pressure p_s and viscous stress σ_s, I the identity matrix, γ the interfacial tension, and κ = R₁⁻¹ + R₂⁻¹ twice of the mean curvature of the interface, with R₁ and R₂ being the principal radii of the curvature.

The above-mentioned governing equations indicate four types of forces that govern the droplet generation: inertial force, viscous force, gravity (for the case without other external force fields) and capillary force. To compare the relative importance of various forces, we unify them in the form of stress (forces per unit area) based on which simple scaling arguments are derived. Considering a volume of fluid flowing at velocity u_s in a microfluidic device of characteristic length L, inertial stress scales as f_i ∼ ρ_su_s², viscous stress f_v ∼ η_su_s/L, gravity f_g ∼ ρ_sgL, and capillary pressure f_γ ∼ γ/L. Based on this scaling, we can develop several dimensionless numbers useful in studying droplet generation.

In principle, the ratio of any two stresses of the four defines a dimensionless number. The Reynolds number Re represents the relative importance of inertia to viscous force, and thus Re = f_i/f_v:

Re usually ranges between 10⁻⁶ and 10 in microfluidic flows. Therefore, the viscous stress dominates over fluid inertia, yielding a laminar flow of microflow typically.

For interfacial flows that are relevant to droplet generation, Reynolds number is rarely used. During the process of the droplet generation, three major steps are normally involved:²³ initially, the dispersed and continuous phase fluids meeting at the junction to form an immiscible interface, followed by the large deformation of the interface to an unstable state, and finally the unstable interface fragmenting spontaneously and decaying into disconnected droplets. Interfacial tension thereby plays a key role in droplet generation. Consequently, we focus on examining the following three dimensionless numbers that characterize the relative importance of interfacial tension to the other three forces. Among them, the most commonly used one is the capillary number Ca, which is the ratio of viscous stress to capillary pressure f_v/f_γ:

The inertial stress f_i does not depend on the device length-scale L directly. Shrinking of device length-scale L would, however, enhance both viscous stress f_v and capillary pressure f_γ and weaken the gravitational effect via reducing f_g. Therefore, viscous and capillary forces would become dominant over the other two if L is sufficiently small. This attributes Ca to be the most frequently used dimensionless number in characterizing microfluidic droplet generation. In microflows, Ca is usually in the range of 10⁻³ to 10.

Although fluid inertia is negligible for most microfluidic flows, it can be important for the cases such as jet¹⁵ and nonlinear bubble³⁰ formation at high flow velocity, or in the vicinity of droplet and bubble pinch-off.^31,32 The competition between inertia and capillary pressure, f_i/f_γ, yields the Weber number We in the form of,

For most microfluidic flows, We < 1.

The relative importance of gravity to capillary pressure, f_g/f_γ, yields the Bond number Bo,

where Δρ is the density difference between dispersed and continuous phases. Typically, Δρ and L are small for liquid–liquid microflows, thereby Bo ≪ 1 in microfluidic droplet generation.

Another two dimensionless numbers are also relevant: the viscosity ratio λ and the flow rate ratio φ of dispersed to continuous phase fluids defined by,

λ = η_d/η_c, (10)

and

φ = Q_d/Q_c. (11)

3. Device geometry

The microfluidic channel provides the boundary of microflow and thus its geometry would impact the droplet generation as well. Typical microfluidic geometries used in generating droplets are shown in Fig. 2. Among them, some use viscous shear forces to break off droplets with three types of most frequently used structures being cross-flow, co-flow and flow-focusing geometries. The others employ variations of channel confinement to facilitate or drive droplet generation, such as step emulsification, microchannel emulsification, and membrane emulsification. Herein, we discuss these droplet generators in terms of their features, developments, and variations and compare the fabrication technique, typical droplet generation frequency, droplet size and monodispersity for each droplet generator in Table 2.

Fig. 2 Schematic of various microfluidic device geometries (not to scale). (a) Cross-flow. (i) “T-junction” where the continuous and dispersed phase fluids meet perpendicularly (θ = 90°). (ii) “T-junction” in which the two fluids intersect at an angle θ (0° < θ < 90°). (iii) “Head-on” geometry (θ = 180°). (iv) Y-shaped junction with intersection angle of θ (0° < θ < 180°). (v) Double T-junction with droplet pairs generated at the same location. (vi) Double T-junction that produces droplet pairs at separated parallel T-junctions. (vii) “K-junction”. (viii) “V-junction”. (b) Co-flow. (i) Quasi-2D planner co-flow. (ii) 3D co-flow. (c) Flow-focusing. (i) Axisymmetric flow-focusing geometry. (ii) Planner flow-focusing geometry. (iii) Microcapillary flow-focusing device. (iv) Microcapillary device combining co-flow and flow-focusing geometries. (d) Step emulsification. (i) Horizontal step. (ii) Vertical step. The inlet channel has a high aspect ratio, and the reservoir is wider and deeper, with an abrupt topographic step in between the inlet channel and the reservoir. (e) Microchannel emulsification. (i) Grooved-type microchannel. (ii) Straight-through microchannel. (f) Membrane emulsification. (i) Direct membrane emulsification. (ii) Premix membrane emulsification. Q, w, h, and Δz denote the volumetric flow rate, channel width, channel height, and horizontal distance from the end of the dispersed microchannel to the orifice entrance, respectively. For planar devices, the channel height h is uniform. In the case of the geometry with a circular cross section, w represents the channel diameter. The subscripts “c”, “d”, “o”, and “or” stand for the continuous phase, dispersed phase, outlet channel, and orifice, respectively. (a,vii) is reprinted with permission from ref. 48. Copyright 2012, American Institute of Physics. (a,viii) is reprinted with permission from ref. 50. Copyright 2015, Royal Society of Chemistry.

Table 2 Comparisons of different droplet generators

Droplet generator	Fabrication technique	Droplet production rate	Minimum droplet size	Monodispersity
Cross-flow	For cross-flow, co-flow, and flow-focusing geometries, 2D devices can be fabricated with silicon and glass by etching and with polymers, such as poly(dimethylsiloxane) (PDMS), poly(methyl methacrylate) (PMMA), and poly(urethane), by soft lithography, hot embossing, injection molding, and laser ablation;^87 3D devices can be fabricated by assembly of glass capillaries, and with polymers by soft lithography and 3D printing. However, 3D devices are more complicated in fabrication: assembly of glass capillaries requires careful alignment; soft lithography of PDMS usually demands multi-layer fabrication;^323 3D printing is limited in the resolution.^324	Up to 7.4 kHz in air-bubble-triggered droplet formation^38	Limited by channel dimension, usually larger than 10 μm^33	CV <2% (ref. 34)
Co-flow		Several hundred Hz to tens of kHz (ref. 54)	Several hundred nanometers^169	CV <3% (ref. 54)
Flow-focusing		Up to tens of kHz (ref. 62)	Several hundred nanometers^166	CV <5% (ref. 61)
Step emulsification	Step emulsification devices are made with PMMA by micromachining,^67 and with SU-8 (ref. 75) and PDMS^70 by photolithography.	Up to 15 kHz (ref. 70)	Several hundred nanometers^70	CV <1% (ref. 70)
Microchannel emulsification	Microchannel array devices are fabricated with silicon by photolithography and wet etching.^86	Up to 44 Hz per microchannel^88	1 μm (ref. 86)	CV <5% (ref. 86)
Membrane emulsification	Shirasu porous glass (SPG) membrane is fabricated by phase separation; microsieve membranes are fabricated with nickel by LIGA process, aluminium and stainless steel by laser drilling, and silicon nitride by reactive ion etching.^86	Several tons of dispersed phase per hour (ref. 86)	∼0.1 μm (ref. 86)	CV ∼20% (ref. 114)

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3.1 Cross-flow

The cross-flow geometry refers to the one where dispersed and continuous phase fluids meet at an angle θ (0° < θ ≤ 180°; Fig. 2a).²⁴ In microfluidics, the cross-flow structure is frequently implemented as a T-junction, where the dispersed and continuous fluids are fed orthogonally (Fig. 2a,i). The T-junction device was first reported by Thorsen et al.³³ to produce monodisperse water droplets in oil surroundings for generating ordered dynamic patterns in pressure controlled laminar flows. This geometry is widely used because of its simplicity and ability to produce monodisperse droplets. The coefficient of variation (CV, defined as the ratio of standard deviation to the mean of the droplet radius) of droplets is usually less than 2% in T-junctions.³⁴ Contrary to the typical operation conditions shown in Fig. 2a,i where the continuous phase is introduced from the main channel and the dispersed phase is fed from the side channel, the opposite was also proposed in T-junctions by controlling the wetting properties of channel walls.^22,35 Besides functioning as a droplet generator, T-junctions have recently been applied in designing microvalves³⁶ and microactuators³⁷ as well.

Variations in T-junction have been studied. A slightly different T-junction with arbitrary angle θ (Fig. 2a,ii)³⁸ was found to influence the process of droplet breakup³⁹ and resultant droplets.⁴⁰ Another variation from the typical T-junction is the “head-on device”,⁴¹ where the dispersed and continuous fluids are fed into the two straight channels from opposite directions, as shown in Fig. 2a,iii. Droplet formation in head-on devices, rich as the flow behaviour was found to be,⁴² has similar features to that in typical T-junctions. Furthermore, if the two fluids are not injected oppositely but at a crossing angle θ, the head-on device reduces into a Y-shaped junction⁴³ (Fig. 2a,iv).

A single T-junction with the above-mentioned simple structure cannot meet special requirements in some pragmatic applications. In performing chemical reactions via merging two droplet microreactors that contain different reagents,⁴⁴ or indexing the targeted droplet by the addition of a droplet marker,⁴⁵ for example, it is desired to generate droplet pairs in microchannels. A double T-junction (Fig. 2a,v) was designed to create stable alternating droplet pairs by Zheng et al.,⁴⁵ where two dispersed streams pinch off at the same junction alternately. Droplets can also be generated separately at different locations in the main channel and then flushed into a wider channel to initiate fusion.⁴⁶ In addition, a parallel dual T-junction that produces droplet pairs in two parallelized T-junctions (Fig. 2a,vi) was proposed to tune the size ratio of droplet pairs.⁴⁷ In biological and chemical assays, on-demand droplet generation with prescribed size, composition and frequency is usually required. To achieve this goal, modified T-junctions such as K-junction^48,49 (Fig. 2a,vii) and V-junction^50,51 (Fig. 2a,viii) were introduced. For mass production of droplets, the proposed strategies include the integration of parallel T-junction droplet generators⁵² and splitting primary droplets subsequently into daughter droplets.^46,53

3.2 Co-flow

In co-flow geometry, the dispersed and continuous phase fluids meet in parallel streams²⁴ (Fig. 2b). In liquid–liquid systems, the co-flow microfluidic device was first introduced by Umbanhowar et al.⁵⁴ Co-flow configurations can either be quasi-two-dimensional (2D) planar²⁴ (Fig. 2b,i) or three-dimensional (3D) coaxial⁵⁵ (Fig. 2b,ii). The former can be fabricated by standard soft lithographic methods,⁵⁶ while the latter is often made by inserting a tapered cylindrical glass capillary^54,55,57 into a rectangular microchannel or a square glass capillary. The polydispersity of the resulting droplets in co-flow geometries depends on the flow conditions: droplets generated in dripping mode are highly monodisperse with CV <3%,⁵⁴ while those generated in jetting mode are polydisperse.¹⁵ Usually, droplet sizes are larger than the dimension of dispersed microchannel w_d (Fig. 2b) when droplets are produced in dripping or widening jetting mode.¹⁵ Exceptions include the continuous phase flows at very high speeds so that a narrowing jet¹⁵ and even tip-streaming regime^58,59 occur.

3.3 Flow-focusing

To produce smaller droplets, it is recommended to use flow-focusing devices, where the two immiscible fluids are hydrodynamically focused and thereafter in elongating flows when passing through a contraction (Fig. 2c). Similar to co-flow geometries, flow-focusing microfluidic devices can also be categorized into two types: 3D axisymmetric^26,60–62 (Fig. 2c,i) and quasi-2D planar^24,63 (Fig. 2c,ii). Gañán-Calvo was the first to produce monodisperse micron-sized droplets in a surrounding gas stream²⁶ and mono-sized microbubbles in a focused liquid stream⁶⁰ using the axisymmetric flow-focusing configuration as shown in Fig. 2c,i. Later in 2003, Anna et al.,⁶³ for the first time, translated the planar flow-focusing geometry (Fig. 2c,ii) into a liquid–liquid microfluidic system. Compared to planar flow-focusing devices, 3D axisymmetric flow-focusing devices avoid issues like wetting of channel walls by the dispersed phase,⁶¹ therefore producing monodisperse droplets (CV <5%) with higher throughputs.⁶² A branch of 3D flow-focusing microfluidic devices is the microcapillary device⁵⁵ (Fig. 2c,iii), where the continuous and dispersed fluids are supplied into the device in opposite directions and meet at the entrance of the narrow orifice, then focusing downstream through the orifice and generating uniform droplets. In combination with co-flow geometry, the microcapillary device (Fig. 2c,iv) is able to generate monodisperse multiple emulsions in a single step.^55,64

3.4 Step emulsification

In contrast to the aforementioned microfluidic devices that use shear forces to break off fluids in generating droplets, other geometries generate uniform droplets by variations of channel confinement through which a sharp change in capillary pressure occurs and droplet pinch-off is driven by interfacial tension. Among them, a generic structure is the step emulsification geometry where the stabilized co-flow in the high aspect ratio inlet channel gives its way into droplets upon reaching an abrupt geometric step towards the released channel confinement in the reservoir²³ (Fig. 2d). Two groups of step emulsification are identified based on the geometric feature of the step:⁶⁵ horizontal step (Fig. 2d,i) and vertical step (Fig. 2d,ii). To the best of our knowledge, Chan et al.⁶⁶ were the first to use a vertical step change in a microfluidic device to generate nanoliter droplets for the synthesis of CdSe nanocrystals. Then the geometry of step emulsification was carefully characterized by Priest et al.,⁶⁷ who produced highly monodisperse droplets (CV <1.5%) and gel emulsions with high volume fraction of the dispersed phase. More recently, femtolitre droplets produced by step emulsification were used to perform PCR⁶⁸ or solidified to design colloidal building blocks.⁶⁹ In vertical step emulsification (Fig. 2d,ii), the resultant smallest droplet has its diameter about two to three times the height of the inlet channel h_c, such that submicron droplets could be generated via scaling h_c down to several hundred nanometers.^70,71

Step emulsification has several advantages. Firstly, droplet generation in step emulsification is less sensitive to flow rate or pressure fluctuations, resulting in probably the highest monodispersity of droplets (CV <1.5% reported by Priest et al.,⁶⁷ and CV ∼1% reported by Malloggi et al.⁷⁰). Secondly, droplets with unaffected size can be generated in a wide range of flow rates by keeping flow rate ratio φ constant, while the production rate can be tuned over two orders of magnitude.⁶⁷ Thirdly, taking advantage of producing foam-like emulsions in situ, step emulsification has been employed to study the assembly of microdroplets into 3D clusters.^72,73 Finally, step emulsification is easier to be parallelized into multiple channels for high-throughput droplet production, because only the flow of the dispersed phase needs to be precisely controlled.⁷⁴ For example, pairs of droplets with different contents can be produced by duplicating the dispersed inlet channels,⁷⁵ and numbering the dispersed channels up to 16 (ref. 76) and 252 (ref. 77) was also achieved.

Variations in step emulsification have been proposed. One is using a gradual transition of channel height⁷⁸ instead of a sudden change from the inlet channel to the reservoir. The generated monodisperse droplet (CV ∼0.1%) is forced away by the gradient of surface energy. The size of droplets mainly depends on the channel geometry and is weakly influenced by flow rates and fluid properties. On-demand chemical reactions can be achieved by guiding droplets with local variations in channel height. In addition, high-throughput generation of droplets was demonstrated by parallelizing 256 parallel nozzles. Another variation is the emulsification method of edge-based droplet generation (EDGE) proposed by van Dijke et al.^74,79,80 In this device, the dispersed phase is introduced into a shallow but rather wide quasi-2D plateau and then breaks into droplets at the edge of the plateau adjacent to which a deeper channel feeds the continuous cross flow to flush away the produced droplets. In a regular and organized manner, many sites of droplet generation can occur at the edge of the same plateau.⁷⁴ Both experiments and simulations showed that the distance between droplet generation sites depends on the height of the plateau and applied pressure.⁸⁰ A parallelization of up to 196 EDGE units on a single chip was presented to generate quasi-monodisperse droplets (CV <10%) at large production frequency (around 300 kHz).⁷⁹ In addition, Dutka et al.⁸¹ recently incorporated a constriction and bypasses into the dispersed inlet channel of step emulsification geometry. In this geometry, monodisperse droplets were produced by injecting only the dispersed phase fluid into the device. The size of droplets depended weakly on the injection rate, therefore increasing the stability of droplet generation against flow rate fluctuations. Furthermore, pump-free step emulsification configurations that incorporate centrifugal^82–84 and magnetic⁸⁵ controls have been exploited, which will be detailed in section 5 (Active droplet generation).

3.5 Microchannel emulsification

In microchannel emulsification, the dispersed phase is forced through the inlet channel and then fragments into droplets at the end of the channel. Microchannel emulsification can be divided into two groups:^86,87 grooved-type microchannel (Fig. 2e, i) and straight-through microchannel (Fig. 2e,ii). To achieve high throughput, microchannel arrays are usually implemented.⁸⁸ The grooved-type microchannel arrays were first fabricated by Kikuchi et al.⁸⁹ in 1989, in which the continuous fluid can be either static (dead-end)⁹⁰ or cross-flow.^91,92 For the sake of higher rate of droplet production, cross-flow modules are better than dead-end ones because the former enables more microchannels incorporated on one chip and the continuous flow sweeps away the generated droplets, leaving more space for latter droplets. Compared with grooved-type microchannel arrays, straight-through modules possess higher throughput because of the better utilization of chip surface.⁸⁶ In straight-through microchannel arrays, the rectangular microchannel (Fig. 2e,ii) is better than the circular shape and can be either symmetric⁹³ or asymmetric.^94–96 However, the monodispersity of droplets is seriously constrained by channel dimensions. To produce uniform droplets (CV <2%), an aspect ratio of channel height to width must be larger than three.⁹⁷

3.6 Membrane emulsification

To produce fine emulsions at industrial scales, membrane emulsification was proposed in the early 1990s.⁹⁸ With this technique, the dispersed phase is forced through a porous membrane after which droplets detach into the continuous phase (Fig. 2f). Apart from directly injecting a pure dispersed phase into the membrane, the so-called direct membrane emulsification⁹⁹ (Fig. 2f,i), pre-emulsions containing coarse droplets are homogenized by pressing the premix through a membrane¹⁰⁰ or a packed bed of uniform particles,¹⁰¹ named premix membrane emulsification (Fig. 2f,ii). There are many operation modes of applying shear stress on the membrane surface to facilitate uniform droplet formation with high throughputs, such as cross-flow,⁹⁸ stirring,^102,103 rotating^104,105 and vibrating.^106,107 The most commonly used membranes are Shirasu porous glass membrane⁹⁸ and micro-engineered membranes.¹⁰⁸ Others include ceramic,¹⁰⁹ metallic^110,111 and polymeric^112,113 membranes. Membrane emulsification enables quasi-monodisperse droplet formation (CV ∼20%).¹¹⁴ The droplet size is affected by characteristics of the membrane (pore morphology, pore size distribution, porosity, spatial arrangement of the pores, surface wettability and charge, to name a few), material properties (viscosity of the two fluids, surfactant type and concentration), and dynamic parameters (transmembrane flux and pressure, shear stress at the membrane surface).¹¹⁵ Both analytical^116,117 and computational^118–120 models have been developed to predict droplet size in membrane emulsification.

In addition to the above geometries, some other structures have been introduced. For example, Amstad et al.¹²¹ proposed a microfluidic post-array device to process the crude emulsion into a fine one, which is potentially beneficial for industrial applications due to its high throughput. Other devices enable on-site formation of droplets with predetermined volume by using micro-patterned channels,^122,123 where a long microchannel connects large numbers of pre-defined microchambers, either in series¹²² or in parallel,¹²³ as sites for droplet generation.

4. Passive droplet generation

Droplet generation is of prime importance for various microfluidic applications. In the passive method, microfluidic two-phase flow is controlled by either syringe pumps that supply constant flow rates Q_d and Q_c or pressure regulators^124,125 and gravity-based pressure units⁸¹ that set stable source pressures P_d and P_c without additional energy input. During droplet formation, energy introduced from the syringe pumps or pressure controllers is partially converted into interfacial energy and thus facilitates the destabilization of the liquid–liquid interface, whereby discrete droplet shedding from the dispersed phase occurs.²³ Despite the complexity of channel geometry, droplet generation can be categorized into several groups due to the similar physical processes and underlying mechanisms involved. In the following subsection 4.1, we identify and characterize five fundamental modes and one transitional mode of droplet generation in most commonly used microfluidic cross-flow, co-flow, and flow-focusing configurations as well as droplet generation in step emulsification structures. The influence of surfactants on droplet generation is also included. Then, in subsection 4.2, we discuss the fluid instabilities of two-phase microflows and relate each of the breakup modes to these instabilities.

4.1 Breakup modes

Five fundamental modes of droplet generation have been observed in shear-based droplet generators: squeezing,¹²⁶ dripping,⁵⁴ jetting,⁵⁷ tip-streaming¹²⁷ and tip-multi-breaking.¹⁷ The first three have been observed in cross-flow, co-flow, and flow-focusing geometries; the last two have not been reported in cross-flow yet (Fig. 3a–e). In principle, droplets generated with squeezing mode is larger than the channel dimension and highly monodisperse (Fig. 3a), with dripping is somehow smaller than channel dimension and monodisperse (Fig. 3b), with jetting is polydisperse (Fig. 3c), with tip-streaming can be as small as submicrometer-scale and monodisperse (Fig. 3d), and with tip-multi-breaking are in droplet sequences with polydisperse but geometric-progression size distribution (Fig. 3e). Transitions between different modes can be achieved by changing dispersed and/or continuous phase capillary numbers (Fig. 3f).¹⁰ For example, the squeezing–dripping transitional regime occurs at an intermediate continuous-phase capillary number between squeezing and dripping. Similarly, different modes of droplet generation are identified in step emulsification: shear-induced generation, step-regime and jet-regime. In this sub-section, our focus is mainly on the process of droplet generation, mechanism, the condition of occurrence and characteristics of each mode (Table 3) as well as effects of surfactants.

Fig. 3 Images of droplet generation with different modes in cross-flow, co-flow and microcapillary flow-focusing geometries. (a) Squeezing mode. (b) Dripping mode. (c) Jetting mode. The upper image in co-flow is a narrowing jet while the lower one is a widening jet. (d) Tip-streaming mode. (e) Tip-multi-breaking mode. Neither tip-streaming nor tip-multi-breaking modes has been reported in cross-flow geometry yet. (f) Phase diagram in (Ca_c, Ca_d) plane for various modes observed in microcapillary flow-focusing devices.¹⁷ Cross-flows of (a)–(c) are reprinted with permission from ref. 131. Copyright 2010, American Chemical Society. Co-flow of (a) is reprinted with permission from ref. 133. Copyright 2007, American Physical Society. Co-flows of (b) and (c) are reprinted with permission from ref. 15. Copyright 2007, American Physical Society. Co-flows of (d) and (e) are reprinted with permission from ref. 58. Copyright 2013, Cambridge University Press.

Table 3 Characteristics of passive breakup modes

Breakup mode	Condition for occurrence	Characteristics	Specification	Ref.
Squeezing	Cac <O(10^−2)	L p/wc = ε + δφ	Eqn (12). Plug length is only determined by flow rate ratio and channel dimension. ε and δ are of order unity.	129
		V/(hwc^2) = Vfill/(hwc^2) + ζφ	Eqn (13). Expression for predicting droplet volume in T-junctions by taking leakage into account.	146
		V/(hwc^2) = αf + ζfφ	Eqn (14). Prediction of droplet volume in flow-focusing geometries.	147
Squeezing–dripping transition	Cac ≈ O(10^−2)		Eqn (20) and (21). Prediction of droplet volume in transitional regime.	152
		V/(hwc^2) = αlag + αfill + ζneckφ	Eqn (23). Prediction of droplet volume based on lagging, filling, and necking stages of droplet formation.	159
		fre^*drop = φ/(αlag + αfill + ζneckφ)	Eqn (24). Prediction of droplet generation frequency based on lagging, filling, and necking stages of droplet formation.	159
Dripping	O(10^−2) < Cac < O(1), and Wed < O(1)		Eqn (15). In co-flow geometry, droplet size is the solution of the third-order polynomial. Only shear forces and interfacial tension are considered.	54
			Solution of eqn (15) when φ ≪ 1. a and b are fitting parameters. Eqn (16) is recovered when a = b = 1.	54
			Eqn (17). In flow-focusing geometry, droplet size is the solution of the fourth-order polynomial.	153
			Eqn (18). Solution of eqn (17) when φ ≪ 1.	153
			Eqn (19). In cross-flow geometry, droplet size is the solution of the fourth-order polynomial.	155
Jetting	Cac + Wed≥ O(1)		Eqn (25), jet diameter in narrowing jetting regime.	161
		d jet/wo = (φ/2)^1/2	Eqn (26), jet diameter in narrowing jetting regime when φ ≪ 1.	161
			Eqn (27), jet length in narrowing jetting regime.	162
		D/djet = [3π/(2k*)]^1/3	Ratio of droplet diameter and jet diameter in narrowing jetting regime.	59
			Eqn (28), expression for dimensionless droplet diameter in narrowing jetting regime.	59
			Eqn (29), jet diameter in widening jetting regime.	59
		D ∼ (Qd/uc)^1/2	Eqn (30), scaling of droplet diameter in widening jetting regime.	15
			Eqn (31), unified expression for dimensionless droplet diameter in both narrowing and widening jetting regimes.	59
Tip-streaming	Rec ≪ 1, Red ≪ 1, Cac > Cacri, and φ ≪ 1		Eqn (28), expression for dimensionless droplet size. Tip-streaming can be regarded as narrowing jetting with φ ≪ 1.	59
Tip-multi-breaking	φ < φmin ≪ 1, and Cac < Cacri	R i = R1χ^i−1, i = 1, 2,…, n	Eqn (32), geometrical progression of droplet size distribution in a decreasing manner.	17
Step-regime	Cadwc/hc ≲ 0.38		Eqn (33). Prediction of droplet size in step emulsification geometry.	197

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4.1.1 Squeezing. Squeezing mode occurs at low capillary number, such as Ca_c <O(10⁻²) in T-junctions,^128–130 where viscous stress gives way to the confinement of channel walls. In this case, the dispersed-fluid protrusion obstructs the junction region in cross-flow,^129,131,132 co-flow,^16,133 and flow-focusing^14,126 structures as it grows, thereby restricting the continuous flow around the developing protrusion (Fig. 3a). Consequently, a pressure gradient in continuous fluid across the enlarging droplet is built up as the gap between the liquid interface and the opposing channel wall decreases. Once the pressure gradient is sufficiently large to overcome the pressure inside the dispersed droplet, the interface is “squeezed” to deform and necks into a droplet. The generated droplet is confined by channel walls, adopting a plug-like rather than a spherical shape. Constrained by the channel geometry, squeezing is also referred to as “geometry-controlled” mode in the literature.^{17,41,127,134}

The pressure rise in the continuous fluid is responsible for the dispersed fluid breakup in squeezing mode. During the breakup, the pressure profile in the dispersed phase fluctuates rather than remains constant. This mechanism was first unveiled by Sivasamy et al.¹³⁵ in a numerical study and later confirmed experimentally by using Laplace sensors to measure pressures in T-junction¹³⁶ and flow-focusing¹³⁷ microfluidic devices. As the emerging droplet blocks the junction region and then necks, the continuous-phase pressure fluctuates in the way of first increasing and then decreasing, in anti-phase with the dispersed-fluid pressure variation.^136,137 The amplitude of pressure fluctuations diminishes with the increasing of capillary number Ca_c.

In a plug-like shape, the droplet generated with squeezing mode has its size characterized by the ratio of the plug length L_p to the continuous phase channel width w_c (such as in T-junctions, Fig. 2a,i), L_p/w_c. To preliminarily determine the dimensionless length L_p/w_c, the process of droplet generation is usually divided into two stages in T-junctions.^129,138,139 First, the dispersed-liquid tip fills and blocks the junction region with its length estimated as L₁ = εw_c; then the dispersed droplet is squeezed by the continuous fluid and has the neck thinning rate proportional to the continuous fluid velocity u_c = Q_c/(hw_c), with h being the channel height. During the second “squeezing” stage, the dispersed plug elongates due to the inflation of dispersed fluid at the rate: u_d = Q_d/(hw_d) (w_d is the dispersed channel width, Fig. 2a,i). Accordingly, the length contribution from the second stage is about L₂ = u_d(d/u_c) = dw_cQ_d/(w_dQ_c) = dφw_c/w_d, where d is the initial neck diameter. Finally, the droplet length L_p = L₁ + L₂= εw_c + dφw_c/w_d. Therefore, L_p/w_c is approximated as,

L_p/w_c = ε + δφ, (12)

where ε and δ(δ = d/w_d) are fitting parameters of order unity. Depending on the channel geometry, various groups of ε and δ have been reported from both experimental^129,138,140 and numerical^141–143 works. Eqn (12) suggests that, in squeezing mode, the droplet length is determined by device geometry and flow rate ratio independent of the physical properties of fluids.

The continuous phase is not utterly blocked during droplet formation. This has been confirmed by the microscopic particle image velocimetry (μ-PIV) measurements of 3D flow fields around the emerging interface,^144,145 especially in rectangular microchannels, where the leakage of continuous fluid from the gutters between liquid interface and channel corners is significant. By considering the leakage, van Steijn et al.¹⁴⁶ proposed a closed-form expression to predict droplet and bubble volume V without fitting parameters in T-junctions,

V/(hw_c²) = V_fill/(hw_c²) + ζ_Tφ, (13)

where V_fill is the droplet/bubble volume at the end of the first “filling” stage depending only on channel geometry, and ζ_T depends on both geometry and the leakage. The analytical expressions for V_fill and ζ_T are available in ref. 146. Eqn (13) is consistent with eqn (12) in that the droplet volume V can be roughly estimated to be V = L_pw_ch.

Similar to eqn (13), Chen et al.¹⁴⁷ theoretically predicted the volume of droplet in flow-focusing geometries by accurately reconstructing the 3D shape of the droplet and determining the pressure drop along the droplet via force balance analysis:

V/(hw_c²) = α_f + ζ_fφ, (14)

where α_f denotes the dispersed flow into the droplet during the filling stage, and ζ_f accounts for the continuous flow into the junction during the necking stage. The model of eqn (14) includes the influence of device geometry, viscosity ratio, flow rate ratio, and continuous-phase capillary number. Based on the prediction of droplet volume, the droplet generation frequency and droplet spacing were also predicted, with a relative error of ±10%.

4.1.2 Dripping. As capillary number Ca_c increases, the mode transforms from squeezing to dripping,¹³⁰ where viscous forces that drag the interface to rupture dominate over interfacial tension effects that stabilize the emerging droplet against breakup. In dripping mode, breakup occurs right at the dispersed nozzle in cross-flow and co-flow geometries, or at the focusing orifice in flow-focusing structures (Fig. 3b). As a consequence of large viscous shear force, the dispersed fluid breaks up before the growing droplet obstructs microchannels, so that the droplet retains a spherical shape with its size smaller than the channel dimension (Fig. 3b). Given a constant viscous stress, highly monodisperse droplets can be generated.

The droplet diameter D can be scaled, by simply balancing the viscous stress of magnitude η_cu_c/L and capillary pressure γ/D,^15,33 to be D/L ∼Ca_c⁻¹, where L is the hydraulic length of the nozzle with L = 2hw_d/(h + w_d) in 2D geometries and L = w_d in 3D co-flow and flow-focusing structures. In the squeezing–dripping transitional regime, both shear stress and squeezing pressure play a role in determining the droplet size;^{130,143,148–150} the scaling power of Ca_c deviates from −1 to other values such as −0.25 numerically observed by van der Graaf et al.,¹³⁹ ranging from −0.3 to −0.2 by Liu and Zhang,^143,151 −1/3 by Christopher et al.,¹⁵² −0.34 by Lan et al.,¹⁵⁰ and −0.4 by De Menech et al.¹³⁰ Here, we focus our attention only on the purely shear-dominant dripping regime.

To predict the absolute droplet size, both the viscous shear and the interfacial tension forces should be analytically determined. In a 3D co-flow geometry, viscous force can be approximated by a modified Stokes formula after considering the shielding effect of the dispersed nozzle: F_v = 3πη_c(D − w_d)(u_c − u_d), where u_d = Q_d/(πD²). Interfacial tension force can be expressed by F_γ = πw_dγ. By equating F_v and F_γ, Umbanhowar et al.⁵⁴ found the dimensionless droplet diameter D̄ = D/w_d as a solution to the third-order polynomial,

with Λ = Q_d/(πw_d²u_c). Commonly, Q_d is negligible compared with Q_c in dripping regime. Therefore, dropping the last two terms with Λ at the left-hand side of eqn (15) arrives atAdditional fitting parameters are introduced to collapse the experimental data onto the analytical prediction of eqn (16). Depending on different values of nozzle diameter w_d, D̄ = a + b/(3Ca_c) takes different fitting values of a and b.

In microcapillary flow-focusing devices (Fig. 3b), a similar deduction was carried out to predict the droplet size. Erb et al.¹⁵³ defined the same forms of viscous and interfacial tension forces as used by Umbanhowar et al.⁵⁴ and approximated u_d = 4Q_d/(πD²) and u_c = 4Q_c/[π(w_or² − D)²], where w_or is the diameter of the focusing orifice. By assuming the condition for droplet rupture to be F_v/F_γ = Ca_cri, Erb et al.¹⁵³ pointed out that droplet diameter D̄ can be predicted by solving the following polynomial:

where Q̄_c = Q_c/Q₀, Q̄_d = Q_d/Q₀ and w̄_or = w_or/w_d with Q₀ = πw_d²γ/(12η_c). Similarly, when Q_d ≪ Q_c, all terms in eqn (17) involving Q̄_d can be neglected. An analytical solution can, therefore, be found by solving the quadratic equation Ca_criD̄² + Q̄_cD̄ − Ca_criw̄_or² − Q̄_c = 0 to determine the physically meaningful root:Note that Q̄_c = Q_c/Q₀ = 12η_cQ_c/(πw_d²γ) can be understood as the continuous-phase capillary number Ca_c at the dispersed nozzle. A single universal value of the fitting parameter Ca_cri = 0.1 is applicable to a wide range of flow rates, channel dimensions and viscosity ratios.¹⁵³

Eqn (18) can be generalized to predict droplet diameter in flow-focusing geometries with an arbitrary shape of the collection microchannel of the cross-sectional area Ar. In this situation, Q̄_c and w̄_or are redefined to be Q̄_c = 3η_cQ_c/(w_d²γ) and . Liu et al.¹⁵⁴ applied the generalized eqn (18) to successfully predict the absolute droplet size in a flow-focusing geometry with a square collection microchannel. Considering the shielding effect in the expression of F_v, the droplet diameter D is assumed to be larger than the nozzle diameter w_d in eqn (15)–(18), e.g. D̄ > 1. This assumption is however not always valid in flow-focusing configurations when Q_c is sufficiently large (see Fig. 3b, flow-focusing). Moreover, the influence of viscosity ratio on droplet diameter is not considered in eqn (15)–(18). Nevertheless, eqn (15)–(18) provide a fast and efficient preliminary determination of the drop size once a two-phase microflow is known.

In cross-flow geometries, viscous and interfacial tension force balance is still applicable to the prediction of droplet size. Husny and Cooper-White¹⁵⁵ estimated F_γ = πw_d²γ/D, and, by taking both dispersed and continuous phase viscosities into account, F_v = C_λπη_cD{2u_c[1 − (D_c − D)²/D_c²] − u_d} in a cross-flow geometry, where C_λ = (3 + 2λ)/(1 + λ), u_c = Q_c/(w_ch) and u_d = βu_c, with λ being the viscosity ratio, D_c = 2w_ch/(w_c + h) the hydraulic diameter of the continuous-phase microchannel, and β the ratio of the droplet velocity u_d to the average velocity of continuous flow u_c (0 < β < 2). By equating F_γ and F_v, a fourth-order polynomial that determines the dimensionless droplet diameter D̄ = D/D_c reads,²¹

where D̄_d = w_d/D_c, and Ca_c = η_cQ_c/(w_chγ). Husny and Cooper-White¹⁵⁵ used a fitting value of β = 0.6 to collapse all the experimental data onto the predicted curve, which, within the accuracy of measurements, represented good agreement for all investigated capillary number Ca_c and viscosity ratio λ. The proposed model (eqn (19)) suggests a decreasing of droplet diameter D as viscosity ratio λ increases at a fixed capillary number Ca_c, which was however not observed.¹⁵⁵ Instead, the measured droplet diameter appears to depend weakly on the viscosity ratio. In all cases, droplet generation frequency fre_drop can then be easily obtained based on mass conservation, fre_drop = 6Q_d/(πD³), provided that droplet diameter D is predicted.

4.1.3 Squeezing–dripping transition. A transitional regime occurs between squeezing and dripping, where the emerging droplet blocks the continuous-phase fluid partially and temporarily during its formation. Due to the partial blocking, the breakup dynamics is dominated by both viscous shear and squeezing pressure. The pressure drop across the droplet is studied by direct measurements of the pressure difference across the droplet^{130,136,137,149} and the distribution of local flow field.¹⁴⁸ The condition for the existence of transitional regime was found to be Ca_c ≈ O(10⁻²). For examples, De Menech et al.¹³⁰ reported a value of Ca_c = 0.015 for squeezing–dripping transition in their numerical simulations; Liu and Zhang¹⁴³ identified a transition criterion of Ca_c = 0.018; Xu et al.¹³⁸ experimentally observed the transitional regime with 0.002 < Ca_c < 0.01.

Arising from the coexistence of shear stress and squeezing pressure, the droplet size depends on both capillary number Ca_c and flow rate ratio φ in transitional regime,^138,151,156 different from squeezing where droplet size is purely determined by flow rate ratio, and dripping where droplet size solely depends on the capillary number. Semi-empirical models were developed to predict droplet size by fitting experimental and numerical data respectively with the scaling law of L_p/w_c = ε + δφⁿCa_c^m (ref. 138) or L_p/w_c = (ε + δφ)Ca_c^m (ref. 151) to determine the empirical parameters ε, δ, n, and m. Applying the balance of capillarity force (F_γ≈ −γh), viscous force (F_v≈ η_cQ_cb/(w_c − b)² with b being the depth of droplet penetration into the continuous channel) and squeezing pressure (F_p≈ η_cQ_cb²/(w_c − b)³), Christopher et al.¹⁵² derived an analytical formula for the prediction of droplet volume in transitional regime,

where Γ = w_d/w_c is the channel width ratio, and b̄ = b/w_c is expressed as

Eqn (20) includes the influence of channel geometry Γ, in addition to flow rate ratio and capillary number, on droplet volume. This approximate model agrees reasonably well with numerical^141,157 and experimental¹⁵⁸ results, but underpredicts the droplet volume and does not consider the influence of viscosity ratio.¹⁵²

A more robust model was later developed by Glawdel et al.,¹⁵⁹ who experimentally identified three sequential stages of droplet formation in the transitional regime: lagging, filling, and necking.¹⁶⁰ The last two stages are similar to those observed in squeezing regime,¹²⁹ while the initial lagging stage is featured by a receding back of the interface into the injection channel of dispersed fluid after the previous droplet pinch-off. The retraction distance of the interface L_lag is evaluated by a balance between the viscocapillary velocity in a two-fluid system³¹ (u_v-c ∼ γλ^1/2/η_d) and the injection velocity of dispersed phase (u_d ∼ Q_d/(hw_d)), L_lag/w_c ∼ u_v-c/u_d:

L_lag/w_c ∼ λ^1/2/Ca_d. (22)

The lagging affects not only the downstream droplet spacing but also the droplet volume.¹⁶⁰ Taking the three stages into consideration, Glawdel et al.¹⁵⁹ developed a predictive model in T-junctions

V/(hw_c²) = α_lag + α_fill + ζ_neckφ, (23)

fre^*_drop = φ/(α_lag + α_fill + ζ_neckφ), (24)

by a detailed analysis of the 3D droplet shape and a force balance similar to that used by Garstecki et al.¹²⁹ and Christopher et al.¹⁵² In eqn (23) and (24), α_lag, α_fill, and ζ_neck account for contributions from the lagging, filling and necking, respectively, with their mathematical expressions available in ref. 159; fre*_drop = fre_drophw_c²/Q_c is the dimensionless droplet generation frequency. This model has considered the influences of channel width ratio, channel aspect ratio (h/w_c), flow rate ratio, viscosity ratio, and capillary number on droplet volume and droplet generation frequency and is with a ±10% error compared with experimental data.¹⁵⁹

4.1.4 Jetting. By increasing either the continuous-fluid flow rate Q_c or dispersed-fluid flow rate Q_d, dripping to jetting transition can occur,¹⁵ where an extended liquid jet emits from the dispersed channel and ultimately breaks up into droplets at the end of the jet due to Rayleigh–Plateau instability (Fig. 3c). Subjected to capillary perturbations, jetting results in more polydisperse droplets compared with squeezing and dripping. For co-flowing liquid streams, jetting occurs when the sum of viscous forces exerted by continuous-fluid and dispersed-fluid inertia overcomes interfacial tension forces, summarized by Ca_c + We_d≥ O(1).¹⁵ According to this force balance argument, jetting in co-flow geometries can be divided into two groups: narrowing jets that occur when Ca_c≥ O(1) (upper image in Fig. 3c, co-flow) and widening jets that arise from We_d≥ O(1) (lower image in Fig. 3c, co-flow). In narrowing jetting regime, the viscous-drag force overwhelms the capillary force, thus the jet stretches downstream with a thinner diameter. The widening jet is, however, the opposite, where the dispersed-fluid velocity u_d is larger than the continuous-fluid velocity u_c; consequently, the jet is decelerated as moving downstream due to the velocity difference induced viscous shear at the interface, which renders the shape of jets widening.

The characteristics of jetting mode include the jet diameter d_jet, intact jet length L_jet (the length from the dispersed nozzle to the end of the jet), and droplet diameter D. In narrowing jetting regime, the jet diameter d_jet can be approximated as the diameter of an unperturbed jet with constant shape governed by Stokes equations for both continuous and dispersed fluids at low Reynolds number: η_s∇²u_s = ∇p_s. By solving Stokes equations, the jet diameter is found to be,¹⁶¹

where w_o is the diameter of the outlet channel with a circular cross section. In the case of flow rate ratio φ ≪ 1, eqn (25) reduces into an asymptotic scaling law,

d_jet/w_o = (φ/2)^1/2, (26)

that has been validated both experimentally^15,162 and numerically.¹⁶¹

The intact jet length L_jet in narrowing jetting is simply estimated as the product of the jet velocity u_d = 4Q_d/(πd_jet²) = 8Q_c/(πw_o²) (by applying eqn (26)) and the viscous-capillary time t_v-c = d_jetη_d/γ.¹⁶² Cubaud and Mason,¹⁶² therefore, derived the formula for the jet length L_jet to be

where C_jet is a fitting parameter. The jet length L_jet thus depends on flow rates, device geometry, and fluid properties. By experimentally determining C_jet, the prediction shows satisfactory agreement with the measurements.¹⁶² Cubaud and Mason¹⁶² proposed that viscosity ratio barely affects the jet length within their experimental operation.

The droplet size in narrowing jetting regime can be determined by mass balance. Subjected to Rayleigh–Plateau instability, the jet breaks up due to the growth of capillary perturbations with the maximum growth rate and wavelength λ_w. To estimate droplet diameter D, the volume of the droplet is assumed to be equal to the volume of the jet with wavelength λ_w: πD³/6 = πd_jet²λ_w/4. Since λ_w = πd_jet/k*, with k* being the dimensionless wave number (k* = kd_jet/2, with k being the wave number), the droplet diameter D is scaled with d_jet as D/d_jet = [3π/(2k*)]^1/3. If d_jet is replaced by w_o in eqn (26), then

D/w_o = g(k*)φ^1/2, (28)

where the prefactor g(k*) = (9π²/32)^1/6(k*)^−1/3. The dimensionless wavenumber k* depends on the viscosity ratio and can be obtained by solving the dispersion relation deduced by Tomotika.¹⁶³ For instance, the calculation gives λ_w = 5.48d_jet for viscosity ratio λ = 0.1, which results in D/d_jet ≈2 and g(k*) ≈1.43, consistent with experimental results.¹⁵ Cubaud and Mason¹⁶² obtained from experiments that D/d_jet ≈ 2.9 ± 0.3 for viscosity ratio in the range of 22.4 to 439, while the corresponding calculation of k* gives 2.4 < D/d_jet < 3.1, in good agreement with experiments.

The characteristics of widening jetting differ from those of narrowing jetting as a result of different dominant effects. With a widening shape, the jet velocity is not fully developed prior to droplet pinch-off. Therefore, the steady-state Stokes equations fail to predict the jet diameter d_jet. Castro-Hernández et al.⁵⁹ proposed a formula to find d_jet by using linear stability analysis:

where ω_i,max is the maximum growth rate of the capillary perturbations and can be determined from Tomotika's dispersion relation.¹⁶³Eqn (29) should be solved iteratively since ω_i,max also depends on jet diameter d_jet.⁵⁹ The droplet volume V is approximated as V = τQ_d, with τ being the droplet formation time and roughly equal to the advection time of the growing droplet τ ∼ D/u_c.¹⁵ Consequently, the droplet diameter in widening jetting regime scales as

D ∼ (Q_d/u_c)^1/2. (30)

This simple scaling argument has been validated by experiments for viscosity ratio λ = 0.1. However, for λ ≫1, its accuracy might lose.¹⁵

Despite the distinct dominant effects, droplet sizes in both narrowing and widening jetting regimes can be unified,⁵⁹

where u_jet is the advection velocity of the jet at its most downstream position. The difference between narrowing and widening jetting regimes lies in the different expressions for the jet diameter d_jet and advection velocity of the jet u_jet. For narrowing jets, d_jet is given by eqn (26), or more simply estimated as d_jet = [4Q_d/(πu_c)]^1/2; u_jet = u_c. In this case, eqn (31) recovers the same scaling of eqn (28): D ∼ φ^1/2. In widening jetting regime, d_jet is found by solving eqn (29), and u_jet is determined experimentally. The dimensionless wavenumber k*, corresponding to the fastest growth of the perturbations, can be determined by solving Tomotika's dispersion relation¹⁶³ for both narrowing and widening jets. Eqn (31) also includes the influence of viscosity ratio λ on the droplet diameter, as k* depends on λ. Eqn (31) was validated for both narrowing and widening regimes under various Q_c, Q_d and λ with the relative errors being ±30%.⁵⁹

4.1.5 Tip-streaming. In the pioneering work by Taylor,¹⁶⁴ an isolated droplet was stretched to have pointed tips when subjected to an extensional or shear flow. Fine jets or tiny daughter droplets were seen to issue from the pointed tips, a phenomenon now commonly known as tip-streaming.¹⁶⁵ Similarly, in microfluidics, droplets are generated in tip-streaming mode (Fig. 3d) when the dispersed liquid tip is deformed into a conical shape from whose apex a fine jet emanates and subsequently breaks into exceptionally small droplets.^{11,12,16,58,166–170} The thin jet is two to three orders of magnitude smaller than the injection nozzle in radius. In such a way, micrometer and even submicron-sized droplets can be readily produced.^{11,12,166,168–170}

Two distinct mechanisms account for the occurrence of tip-streaming: surfactant-mediated and surfactant-free tip-streaming. The former was proposed by De Bruijn,¹⁷¹ who ascribed tip-streaming phenomenon to the presence of surfactants on the liquid–liquid interface and identified an optimal range of surfactant concentration for tip-streaming to occur. These findings were consistent with later experiments in both simple shear¹⁷² and plane hyperbolic¹⁷³ flows. Subsequent numerical simulations further unveiled the mechanism of surfactant-mediated tip-streaming from free drops¹⁷⁴ and bubbles¹⁷⁵ in extensional flows. More recently, Anna et al. reported the occurrence of tip-streaming, which was called thread formation by them too, in planar microfluidic flow-focusing devices.^{24,127,176,177} They argued that the coupling between surfactant transport and bulk flow plays a critical role²⁴ and developed a semi-analytical model to predict the conditions for the occurrence of tip-streaming.¹⁷⁷ Tip-streaming phenomenon was also observed in the presence of an interfacial chemical reaction that produces a surfactant.¹⁷⁸ On the other hand, no evidence suggested the possibility of surfactant-free tip-streaming until Zhang¹⁷⁹ predicted the elegant steady-state solution for the vanishing thin spout ejecting from the perfectly conical tip when the capillary number of external straining fluid is above a critical value. Then more conclusive proof was provided by simulations¹⁶ and experiments¹⁶⁶ which demonstrated surfactant-free tip-streaming in a double flow-focusing arrangement. In what follows, we focus only on the type of surfactant-free tip-streaming, which is referred to as tip-streaming for short hereafter.

Tip-streaming is featured by the steady cone–jet structure (Fig. 3d) of the dispersed fluid, which very much resembles the Taylor cone in charged liquids during electrospraying.^180,181 In the cone region, the continuous viscous stress induces the recirculating flow pattern within the dispersed phase.¹⁶ Then a speeding up of the dispersed flow is observed upon approaching the apex of the cone, which is the cone–jet transition region.¹⁶ Sufficiently far downstream from the apex, the dispersed flow is nearly plug flow and the jet takes a cylindrical shape with an almost constant diameter of d_jet.¹⁶ The cone–jet transition is affected by both the flow process and the fluid properties. For larger continuous-phase capillary number Ca_c and smaller viscosity ratio λ, the transition region becomes less slender,^168,182 and the local jet slope decreases. Experimental observations reveal that the cone–jet transition is stable for liquid–liquid systems, but not for gas–liquid.^168,169,182 This can be qualitatively understood as that the very small viscosity ratio λ for gas–liquid systems requires a quite low jet slope,¹⁸² which is, however, difficult to achieve in experiments. The cone–jet geometrical profile has been deduced by Zhang¹⁷⁹ in the extreme limit Q_d → 0, and by Castro-Hernández et al.¹⁶⁸ as a function of Ca_c, φ, λ and device geometry in co-flow configurations. The resultant droplet size can also be precisely predicted by eqn (28),^59,169 because tip-streaming can be regarded as narrowing jetting with extremely thin jet diameter.^169,182 However, an accurate prediction for the intact jet length has yet to be developed.

Several conditions should be met for the occurrence of tip-streaming. Note that device geometry has a significant effect. No tip-streaming has been observed yet in cross-flow geometry. According to Tseng and Prosperetti,¹⁸³ tip-streaming occurs when the local streamlines converge near the interface. This is easy to achieve in co-flow and flow-focusing, but difficult in cross-flow configuration (Fig. 3d). To generate tip-streaming, no flow separation should exist between the continuous flow and the drop interface, which requires creeping flow condition, Re_c ≪ 1. Then, to stretch the dispersed fluid into a conical shape, the continuous-fluid viscous stress must overcome the capillary pressure at the injection nozzle, therefore requiring Ca_c > Ca_cri (Ca_cri is a critical constant of the order of unity). For the extremely thin jet to issue from the cone, φ ≪ 1 should also be maintained. The final condition Re_d ≪ 1 ensures that the average dispersed velocity equals continuous velocity in the thin jet region due to large momentum diffusion, and the fine jet is thus almost a cylinder far downstream from the cone. These four conditions are sufficient to generate the tip-streaming.¹⁶⁹

Some effort has been devoted to exploring the value of Ca_cri. Suryo and Basaran¹⁶ found that the critical capillary number Ca_cri is a function of viscosity ratio λ and flow rate ratio φ. Gañán-Calvo et al.^166,182 then theoretically determined Ca_cri as a function of λ by using a spatiotemporal stability analysis and concluded that once the jet surface velocity u_js is above a critical value, u_js > γ(η_dη_c)^−1/2Ca_cri, the jet can be made arbitrarily thin (down to the continuum limit) without transition to dripping as the dispersed flow rate Q_d vanishes. More recently, Gordillo et al.⁵⁸ found that Ca_cri is less sensitive to flow rate ratio φ than to viscosity ratio λ, who also applied a global stability analysis to obtain the λ-dependent Ca_cri, below which the cone–jet structure is unsteady and the resultant droplet size is not uniform.⁵⁸

4.1.6 Tip-multi-breaking. Unlike the aforementioned four breakup modes (Fig. 2a–d), in which continuous droplet streams are generated, tip-multi-breaking^17,184 generates droplets intermittently in periodic sequences (Fig. 3e). Similar to tip-streaming, a conical dispersed tip is observed in tip-multi-breaking mode (Fig. 2d and e). However, the dispersed cone structure is unsteady in tip-multi-breaking, in sharp contrast with the steady cone–jet in tip-streaming. Discrete droplets are ejected from the apex of the conical tip as a result of Rayleigh–Plateau instability. Oscillation of the conical meniscus then cuts off the continuous generation of droplet stream and leads to periodic trains of droplets with non-uniform size distribution (Fig. 3e). Note that the periodic generation of droplet-train is synchronized with the oscillation of the tip.

The unsteady feature of dispersed tip has been widely observed in gas–liquid, liquid–gas and liquid–liquid systems. Garstecki et al.^30,126 found that the gas–liquid system destabilized and transitioned from period-1 (dripping, one bubble generation in one period) state to higher-order periodic or chaotic behaviour (as exampled by period-2, period-3 and period-4 behaviour in Fig. 4a,i) when Q_c is above some critical value. The transition was ascribed to the inertial-dominated dynamics of the gas–liquid interface.³⁰ Similar phenomena were identified in a multi-orifice system¹⁸⁵ and later in a multi-section flow-focusing junction.¹⁸⁶ More recently, Evangelio et al.¹⁸⁷ observed the unsteady gas meniscus when the continuous liquid velocity u_c is lower than the threshold u* (Fig. 4a,ii). In parallel, for a liquid jet focused by a gas, the liquid meniscus is unsteady when the dispersed flow Q_d is below the minimum value Q_min (ref. 181, 188–191) (Fig. 4b). These observations suggest that dynamics of both dispersed and continuous fluids plays a role in rendering the meniscus unsteady. In a liquid–liquid co-flow system, Gordillo et al.⁵⁸ determined the critical continuous-phase capillary number Ca_cri as a function of viscosity ratio λ, which sets an upper limit for the occurrence of the unsteady conical tip. Zhu et al.¹⁷ reported that the number of the droplet in one droplet-train increases with increasing continuous-phase capillary number Ca_c in a microcapillary flow-focusing device (Fig. 4c). Attributed to the feature that the unsteady dispersed fluid tip breaks off repeatedly into multiple droplets in one cycle, Zhu et al.¹⁷ named this mode of droplet breakup as tip-multi-breaking.

Fig. 4 Tip-multi-breaking mode in microfluidic devices. (a) Unsteady gas meniscus during bubble generation in a gas–liquid microsystem. (i) Period-1, period-2, period-4 and period-3 gas thread behaviour with increasing continuous flow rate, respectively (from left to right, top to bottom). (ii) Irregular bubble generation when the continuous-fluid velocity u_c is lower than the critical threshold. (b) Unsteady liquid meniscus when a liquid jet is focused by a gas stream. (c) Tip-multi-breaking mode in liquid–liquid microfluidic flows by which droplets of decreasing size are generated in periodic sequences. The number of droplets in one sequence increases as the continuous-phase capillary number increases (from left to right, top to bottom). (d) Recirculation cell in the dispersed meniscus. When Q_d is smaller than the minimum flow rate Q_min, the recirculation cell is large and penetrates into the dispersed nozzle, in which case tip-multi-breaking takes place. (e) Droplet size distribution in the form of “constant–decreasing”. (f) Droplet size distribution in the form of “increasing–constant–decreasing”. (g) Temporal evolution of the tip diameter at the focusing orifice in three situations. (h) Schematic of the decreasing droplet size distribution in a geometrical progression. (i) Tip-multi-breaking mode in three-phase microfluidics. Snapshots from left to right are one four-droplet train, one five-droplet train and two droplet trains encapsulated in one middle phase droplet. (a,i) is adapted with permission from ref. 30. Copyright 2005, American Physical Society. (a,ii) is reprinted with permission from ref. 187. Copyright 2015, Cambridge University Press. (b) is reprinted with permission from ref. 189. Copyright 2011, American Physical Society. (d) is adapted with permission from ref. 127. Copyright 2006, American Institute of Physics.

Since the unsteady meniscus gives rise to the generation of a droplet-train, the condition of destabilizing the dispersed meniscus determines the occurrence of tip-multi-breaking mode. While higher-order periodic and chaotic behavior of the gas thread was ascribed to the dominant inertial effects,³⁰ the mechanism responsible for the unsteady liquid meniscus seems to be different. From the perspective of energy balance, once the kinetic energy of the liquid meniscus is overwhelmed by the viscous dissipation and/or free surface energy, the steady emission of continuous droplets is halted.¹⁸¹ Corresponding to the above three energy terms, the dispersed-phase inertial, viscous and capillary forces scale respectively as ∼ρ_du_d², ∼η_du_d/d_tip and ∼γ/d_tip, where d_tip is the liquid tip diameter at the focusing orifice. With the assumption of Q_d ∼ u_dd_tip² and Q_d/Q_c ≈ (d_tip/w_or)² at the focusing orifice (w_or is the diameter of the focusing orifice), inertia dominated by viscous forces (ρ_du_d² < η_du_d/d_tip) gives (Q_dQ_c)^1/2 < η_d/(ρ_dw_or), and inertia overwhelmed by capillary forces (ρ_du_d² < γ/d_tip) leads to Q_d^1/2Q_c^3/2 < γ/(ρ_dw_or³) for the dispersed meniscus to be unsteady. Both of the criteria suggest a minimum dispersed flow rate Q_min below which a steady liquid tip cannot be formed when Q_c is fixed.

In liquid–liquid systems, the Q_min yields the condition that φ < φ_min ≪ 1 for the occurrence of tip-multi-breaking.¹⁷ Actually, for the dispersed fluid with low viscosity and sufficiently small flow rate, the flow displays a recirculation cell whose size increases as Q_d decreases (Fig. 4d). When Q_d ≤ Q_min, the recirculation cell is large and penetrates into the injection capillary, consuming a large amount of momentum due to viscous friction between the liquid and the solid wall.^181,188,190 Simultaneously, as Q_d decreases, the decreasing d_tip finally makes the capillary pressure too large to be overcome in generating a new free surface for droplet production, thus destabilizing the dispersed tip.¹⁸¹

Similarly, the continuous fluid also reserves a critical flow rate Q_cri lower than which the shear force exerted by the continuous phase is not sufficient to hold a steady liquid meniscus. To have unsteady meniscus at the focusing orifice, the shear stress ∼η_cu_c/w_or should be dominated by the capillary pressure ∼γ/w_or. This criterion then leads to the existence of a critical capillary number Ca_cri which sets an upper limit for the liquid tip to be unsteady, Ca_c < Ca_cri. In liquid–liquid systems, Ca_cri is a function of viscosity ratio and weakly depends on flow rate ratio φ.⁵⁸ Apparently, Ca_c cannot be infinitely small, otherwise, the system would transfer into other modes, such as dripping and squeezing.¹⁷ It is worth noting that the two conditions, φ < φ_min ≪ 1 and Ca_c < Ca_cri, should be met simultaneously for the occurrence of tip-multi-breaking mode in liquid–liquid systems. Experiments revealed that φ_min, to some extent, depends on and decreases with Ca_c.¹⁷

The droplet size distribution in tip-multi-breaking mode is highly synchronized with the dynamics of the tip evolution at the focusing orifice. The normalized droplet diameter D/d_tip scales as D/d_tip ∼ Ca_c⁻¹ in tip-multi-breaking mode.¹⁸⁴ Depending on device geometry, the droplet size distribution can be either monotonically decreasing (Fig. 4c), or first keeping constant and then decreasing (“constant–decreasing” in Fig. 4e), or first increasing followed by keeping constant and finally decreasing (“increasing–constant–decreasing” in Fig. 4f), as the spacing L (Fig. 4f) between the left injection and right focusing orifices increases.¹⁸⁴Fig. 4g shows the temporal evolution of d_tip/w_or for the three cases of droplet size distribution. The very apparent oscillation of the liquid tip is observed with period T. During one cycle of oscillation, d_tip can undergo any one of the three evolutional dynamics (insets in Fig. 4g), and simultaneously droplets are ejected from the tip in sequences. This verifies that the non-uniform droplet and unique droplet size distribution is a consequence of the d_tip oscillation. It is worth clarifying that an initial very-fast-growing stage of d_tip in all three cases (non-shadow areas in insets of Fig. 4g) is due to the penetration of the liquid tip into the focusing orifice, and no droplet is generated at this stage.

The size distribution in one droplet-train is predictable. Since the thinning rate of d_tip is nearly linear for all the cases (yellow region in insets of Fig. 4g), the decreasing size of droplet, very interestingly, is in the distribution of geometric progression,¹⁷

R_i = R₁χⁱ⁻¹ ,i = 1, 2,  …, n, (32)

where R_i is the radius of the ith droplet (Fig. 4h), and χ < 1 is the common ratio. The common ratio χ can be determined either geometrically (Fig. 4h) as χ = (1 − sin θ)/(1 + sin θ) or dynamically once the linear thinning rate of d_tip and material properties are known.¹⁷ As a first-order approximation, χ has a correlation with the number of droplets n in the form of 1 – χⁿ⁻¹ ≈ χ, for the droplet train with decreasing size in Fig. 4c.¹⁷Eqn (32) is deduced for, but is not limited to, the decreasing size distribution. It can also be applied to the increasing size distribution in Fig. 4f because the increasing rate of d_tip is also almost constant (inset in Fig. 4g for “increasing–constant–decreasing”). More effort should be devoted to further characterize the tip-multi-breaking mode, for instance, the condition that distinguishes the three types of droplet size distribution in Fig. 3c, e and f, respectively.

To date, studies on the unsteady dispersed meniscus have mainly focused on its hydrodynamic interest but underestimate its potential in applications. This is because droplets generated in tip-multi-breaking mode are non-uniform, while monodisperse droplets are usually in favor for most applications. Nevertheless, some applications prefer non-uniform droplets. For example, in multi-volume droplet digital PCR (MV-dPCR),¹³ droplets with various volumes provide simultaneous measurements of a sample at different copies per droplet. Compared with single-volume digital PCR, MV-dPCR retains higher detection reproducibility, wider dynamic range, and better resolution while reducing the total number of droplets/wells required for the measurements.^13,192,193 Tip-multi-breaking mode can, therefore, fit well in the application of MV-dPCR due to its nature in producing identical droplet trains with a tunable number of droplets and predictable size distribution. Besides, the droplet trains can be successfully encapsulated into middle-phase droplets (Fig. 4i), which would be beneficial for chemical and biomedical applications, for example, in the design of chemically communicating droplet networks.¹⁹⁴

4.1.7 Droplet generation in step emulsification. In droplet generators with the step emulsification, the inlet channel with strong confinement suppresses the interfacial instability, thus stabilizing the interface between the two immiscible liquids.^195,196 On reaching the reservoir, the relaxed confinement destabilizes the interface and triggers droplet pinch-off as a result of minimizing interfacial energy. Three regimes of droplet generation have been identified with the increasing flow rate ratio φ:⁶⁷ breakup at the junction where the two fluids meet, step-regime where fluid filaments break up upon reaching the step, and jet-regime where droplets are generated downstream from the step. The first is the shear-induced breakup,^66,67 while the latter two are unique to the step emulsification and thus attract recent studies.^{65,70,197–199}

Dangla et al.¹⁹⁸ investigated the step-regime in a vertical step. They ascribed the breakup to the quasi-static balance between the curvature of the dispersed thread in the inlet channel and that of the droplet protrusion in the reservoir, based on which a lower bound of droplet size is determined. Recently, the closed-form expression for predicting droplet size in step-regime is theoretically derived by Li et al.,¹⁹⁷

where C is the fitting parameter of a value about 2, and w_c and h_c are the width and height of the inlet channel, respectively. Eqn (33) suggests a constant droplet size by keeping the flow rate ratio φ constant when the device geometry and material properties are fixed. As such, droplet generation frequency fre_drop (fre_drop = 6Q_d/(πD³)) is proportional to the dispersed-phase capillary number Ca_d, fre_drop ∼ Ca_d, when φ is constant.⁶⁷

With the increase in dispersed phase flow rate, the step-regime gives its way to the jet-regime at a critical dispersed-phase capillary number.^67,199 The condition for such a transition is experimentally determined to be¹⁹⁷

In jet-regime, the dispersed thread shrinks in its width and results in a tongue-like tip upstream near the step, which was ascribed to the capillary focusing effect.⁷⁰ The quasi-static shape of the tongue has been determined by numerical simulation¹⁹⁹ and simplified theoretical modeling,¹⁹⁷ both of which were confirmed by experiments. Aside from single step-regime or jet-regime, the coexistence of different regimes on the same dispersed-phase filament was observed recently by Hein et al.,⁶⁵ which implies a symmetry breaking in filament breakup even under symmetric flow conditions.

4.1.8 Effects of surfactants. Surfactants are routinely used in droplet microfluidics to reduce interfacial tension and prevent droplet coalescence.²⁰⁰ In the presence of surfactants, the interfacial tension is determined by the competition between interfacial deformation and surfactant convection, diffusion and adsorption–desorption kinetics during droplet generation.²⁴ A faster interfacial deformation and slower mass transfer process render the surface coverage of surfactants smaller, thereby interfacial tension larger. In microfluidic flows, convection enhances the mass transportation of surfactants.^201–203 Owing to internal convection, the mass transport inside the droplet is faster than that outside.²⁰³ Increasing surfactant concentration would increase the rate of mass transport.²⁰⁴ Droplet generation is subjected to a constant interfacial tension provided that surfactant concentration is high enough,^202,205 or the droplet generation time t_drop is either much longer²⁰⁶ or much shorter²⁰⁷ than surfactant mass transfer time t_m, where surfactants are at saturated adsorption in the first case, at equilibrium adsorption in the second case, and nearly absent on the interface in the last case. Contrary to being constant, the interfacial tension is varying with time, known as dynamic interfacial tension, when the surfactant is at intermediate concentrations and the two timescales (t_drop and t_m) are comparable.

Variations of interfacial tension dramatically alter the dynamics of liquid thread pinch-off. During pinch-off, the presence of surfactants lowers the interfacial tension and may induce the depletion of surfactants from the pinch region.²⁰⁸ The reduction in interfacial tension gives rise to a decrease in capillary pressure (∼γ/L) that drives surfactants out of the pinching neck; surfactant depletion induces an interfacial tension gradient and a Marangoni stress counteracting the outflow. Considering the pinch-off of a surfactant-covered fluid thread at a small surfactant Peclet number, Pe (the parameter characterizing the relative strength of the surfactant convection over the diffusion), the interfacial tension starts at the equilibrium value and then increases due to surfactant depletion as pinch-off approaches.^209–211 Provided a strong Marangoni stress, a third stage of surface tension reduction can occur,²¹¹ which was also observed for a two-fluid thread pinch-off in microchannels.²¹⁰ The temporal variation of dynamic interfacial tension, therefore, changes the thinning rate of the pinching thread.^209,210 Besides, the thread shape near the pinch-off is also altered by surfactants,^210–212 which induces variations in pinch-off location^211,212 and influences the size of satellite droplets.²¹³

Several models have been developed to predict the dynamic interfacial tension during droplet formation. The experimental measurements of dynamic interfacial tension are carried out by first determining the relationship between interfacial tension and variables such as the droplet generation frequency,²¹⁴ droplet size,^{201–206,215} droplet deformation,²¹⁶ and the pressure drop of the continuous phase^217,218 and then measuring these variables. Semi-empirical models are developed by fitting the model with experimental data to determine the empirical parameters, such as those in ref. 201–206, 215, 217 and 218. However, semi-empirical models are highly sensitive to the fluidic system, and the results are normally not transferable. On the other hand, Glawdel and Ren²¹⁶ developed a theoretical model to predict the dynamic interfacial tension during droplet generation in squeezing–dripping transitional regime in T-junctions, based on which droplet volume is well predicted with surfactants. Nevertheless, no experimental method is available for capturing the spatial distribution of surfactants on the interface, which is usually uneven during droplet formation.²¹⁹

4.2 Fluid instabilities in microfluidic biphasic flows

The fragmentation of fluid threads into droplets is initiated by fluid instabilities. The currently identified five fundamental modes of droplet formation are related to different types of instabilities, of which droplet pinch-off in squeezing mode is imposed by pressure buildup in the continuous flow, while dripping, jetting, tip-streaming and tip-multi-breaking are associated with capillary instability. Such a difference originates from the distinct degree of channel confinement between squeezing and the other four modes. In this subsection, we briefly summarize the characteristics of hydrodynamic instabilities for each of the five breakup modes in microfluidic biphasic flows (Table 4).

Table 4 Relationship between breakup mode and hydrodynamic instabilities

Droplet breakup mode	Interfacial instability	Capillary instability
		Absolute instability	Convective instability	Global instability
a Fine balance of forces is required to generate absolute instability.^225 b Dripping, jetting, and tip-streaming modes are globally stable.
Squeezing	√
Dripping		√
Widening jet^a		√
Narrowing jet			√
Tip-streaming			√
Tip-multi-breaking^b				√

---

4.2.1 Interfacial instability of squeezing. Geometrical confinement dramatically suppresses capillary instability, thereby stabilizing the fluid–fluid interface during thread pinch-off in squeezing mode. Basically, the collapse of a gas thread surrounded by an outer fluid and confined in a microchannel undergoes two stages:¹⁴ the neck, with the minimal width of w_m, shrinks at a constant rate, then followed by a nonlinear rapid collapse (Fig. 5a). In the first linear collapse stage, the shrinkage speed dw_m/dt is proportional to the continuous flow rate Q_c, but independent of the gas pressure P_d, liquid viscosity η_c, and interfacial tension γ. Besides, dw_m/dt is one to three orders of magnitude smaller than the speed of capillary waves. These observations signify that the collapse is not driven by surface tension. Instead, Garstecki et al.¹⁴ argued that the confined collapse proceeds through a series of equilibria of minimal surface energy (Fig. 5b), in contrast to the non-equilibrium capillary instability. A similar linear collapse process was identified by other groups in flow-focusing and T-junction, as well.^32,144

Fig. 5 Fluid instability of squeezing mode. (a) Upper diagram: evolution of the minimal width w_m, and the axial curvature of the gas–liquid interface in a typical breakup event. Bottom images: optical micrographs of the gas–liquid interface along the breakup trajectory. (b) Comparison of the minimal width w_m of the simulated (equilibrium) interface (V₀ − V_thread, solid dots connected by solid lines) and the experimentally measured minimal widths w_m ((t − t₀)q, unconnected symbols). Insets of “b–d” represent the equilibrium shape of the gas–liquid interface along the trajectory leading to final breakup. The interface shown in inset “d” (rapid collapse stage) is also in equilibrium. (c) The gas thread of minimal width w_m is first squeezed inwards from the sides by the surrounding liquid (2D collapse, left two graphs). The gas thread experiences finally a fast pinch-off, where it is squeezed radially (3D collapse, right two graphs). (d) In-plane velocity at 0.1h from the top wall. Left: before the rapid collapse, flow in the gutters runs towards the tip end. Right: the thread starting to collapse, liquid now runs from the tip to the neck through all four gutters. (e) Top diagram: evolution of the radii of curvature r and R. Bottom diagram: evolution of the pressure drop over the gutters, indicating that the collapse coincides with reversal of pressure drop in the gutters. (a) and (b) are reprinted with permission from ref. 14. Copyright 2005, American Physical Society. (c) is reprinted with permission from ref. 32. Copyright 2008, American Physical Society. (d) and (e) are reprinted with permission from ref. 144. Copyright 2009, American Physical Society.

The mechanism responsible for the second nonlinear rapid collapsing stage is controversial. Dollet et al.³² argued that the onset of rapid collapse is initiated from capillary instability, coinciding with the detachment of the shrinking thread from the wall (Fig. 5c). Then the shrinkage of the gas thread is driven solely by liquid and gas inertia, with the minimal neck width scaling as w_m ∼ (t_c − t)^1/3, where t_c is the critical time for thread breakup. In contrast to the above explanation, comparisons of experimental and simulated static interfaces reveal a quasi-static pinch-off mechanism until the last stage of breakup (Fig. 5b).¹⁴ van Steijn et al.¹⁴⁴ later demonstrated that the reversal of liquid flow in corner gutters, from the end tip of the gas bubble to the shrinking neck (Fig. 5d), triggers the rapid collapse of gas thread. The flow reversal was attributed to the onset of an adverse liquid pressure drop over the gutters, which determines a critical thread radius r_cri for the onset to occur, r_cri = hw_c/[2(h + w_c)], in straight microchannels (Fig. 5e), incompatible with r_cri = min[w_c/2, h/2] as predicted by capillary instability. Combining all the evidence together, one concludes that the interfacial instability of a squeezing fluid thread does not originate from capillary instability. Consequently, highly monodisperse droplets can be produced in squeezing regime due to the elimination of satellite droplets that are almost inevitable for breakups governed by capillarity instability.

4.2.2 Capillary instability. Different from squeezing mechanism, dripping, jetting, tip-streaming and tip-multi-breaking are initiated by capillary instability. These four modes differ from each other in their distinct instability natures, of which jetting–dripping transition is related to the transition from local convective to absolute instability, and tip-streaming to tip-multi-breaking transition corresponds to the transition from global stability to global instability. Here, terms “local” and “global” describe the stability/instability of the local velocity profile and of the entire flow field, respectively.²²⁰ If growing disturbances are advected downstream from the source, the velocity profile is recognized as convective instability. If by contrast disturbances propagate both upstream and downstream and contaminate the entire domain when amplifying, the velocity profile is said to be absolute instability.²²⁰ In this subsection, we briefly summarize the basic principles and concepts in hydrodynamic instability and their applications to biphasic flows in microfluidic devices. The reader is referred to Huerre and Monkewitz²²⁰ and references therein for a detailed review on the absolute/convective and local/global instability concepts.

A linear stability analysis²²¹ is usually performed to assess the local instability. In doing this, initial infinitesimal perturbations ψ(z, y, t) that develop in space and time are applied to a given parallel basic flow, where z, y, and t represent streamwise, cross-stream and time coordinates, respectively. Perturbations are then decomposed into elementary solutions ψ(z, y, t) = ξ(y; k)exp[i(kz − ωt)] after linearization around the base state, with complex wavenumber k = k_r + ik_i, complex frequency ω = ω_r + iω_i and eigenfunction ξ(y; k) that describes the cross-stream distribution of perturbations. In the eigenvalue problem, non-trivial solutions for ξ(y; k) exist if and only if k and ω satisfy a dispersion relation in a manner:

D(k, ω; S) = 0, (35)

where S is the control parameter, for example, the dimensionless parameters discussed in section 2. The following analysis is denoted as temporal stability analysis if k is real and ω is complex, as spatial stability analysis if k is complex and ω is real, and as spatiotemporal stability analysis if both k and ω are complex.

Typically, a temporal stability analysis is adopted to determine whether a system is stable or not by the following criterion:

where ω_i,max is the maximum temporal growth rate determined by ω_i,max = ω_i(k_max) when ∂ω_i/∂k(k_max) = 0. For example, if a co-flowing jet is stable in microfluidic devices, pinch-off will not occur, thus no generation of droplets. Conversely, if a jet is unstable, it ultimately breaks up into discrete droplets. As such, flows with dripping, jetting, tip-streaming and tip-multi-breaking are temporally unstable. For temporally unstable capillary jets, an additional spatiotemporal stability analysis is applied to distinguish convective from absolute instability. To that end, one seeks complex pair (ω₀, k₀) satisfying eqn (35), D(k₀, ω₀; S) = 0, and zero group velocity ∂ω/∂k(k₀) = 0 at a fixed spatial location. The corresponding ω₀ = ω(k₀) is called the absolute frequency, and the absolute growth rate gives ω_0,i = ω_i(k₀). The following criterion determines the absolute/convective nature of the instability:

In general, ω₀ are branch-point singularities of k(ω) with two spatial branches k⁺(ω) and k⁻(ω). As an additional requirement, the physically relevant complex pair (ω₀, k₀) must satisfy the Briggs–Bers criterion,²²⁰ in which branches k⁺(ω) and k⁻(ω) originate respectively from the upper and lower half of the complex k-plane when ω_i is decreased from positive values to zero. Convective instability, such as narrowing jetting, displays extrinsic dynamics as a spatial amplifier, where external noises are amplified when advected downstream, while absolute instability, such as dripping, exhibits intrinsic dynamics that induces breakup at a fixed spatial location and at a self-sustained frequency intrinsic to the system. This explains experimental observations that droplets produced by dripping are highly monodisperse, but by jetting are more polydisperse.

Guillot et al.¹³³ performed a linear spatiotemporal stability analysis to parametrically map jetting and dripping regions for confined capillary jets in cylindrical co-flowing microchannels. Consistent with criterion eqn (37), a temporally unstable flow is convectively unstable (jetting) if the front velocity of the trailing edge of the wave packet is positive (perturbations convected downstream), and is otherwise absolutely unstable (dripping) if (upstream propagation of perturbations). Accordingly, a marginal stability criterion was obtained for the jetting–dripping transition:¹³³

with

E(x, λ) = −4x + (8 − 4λ⁻¹)x³ + 4(λ⁻¹ − 1)x⁵, (40)

where viscosity ratio λ = η_d/η_c, the degree of confinement of the unperturbed jet with , and capillary number Ca_c = 8η_cQ_c[1 + λ(α − 1)]²/(πR_c²γ) (R_c, the inner radius of the cylindrical channel wall). Criterion eqn (38) is plotted in the (x, Ca_c) plane as operational conditions for the jetting–dripping transition with fixed λ (Fig. 6a, where Ka denotes Ca_c). Guillot et al.²²² then extended the analysis to square and rectangular microchannels in which dripping is found to be promoted in comparison to cylindrical channels. Adopting lubrication approximation at low Reynolds numbers (neglecting inertia), eqn (38) is simple and robust in predicting jetting–dripping transition for highly confined jets (Fig. 6a) but fails for the low degree of confinement. Herrada et al.²²³ proposed axisymmetric models based on the analysis by Guillot et al.¹³³ to improve the accuracy of prediction in low confinement cases. In addition, jetting–dripping transition in an open flow-focusing configuration (Fig. 2c,i) was studied by considering a viscous jet flowing in an unbounded co-flowing viscous liquid domain.²²⁴ The transition is mapped in the parametrical space of (Re, We) and shows good agreement with experiments (Fig. 6b).

Fig. 6 Capillary instability in microfluidic biphasic flows. (a) Flow behaviour in the (x, Ka) plane for a given value of the viscosity ratio λ = 0.23. Symbols of light colour correspond to droplets (absolute instability), and those of dark colour represent jet (convective instability), both of which are experimental data. The line is the theoretical prediction of absolute-to-convective instability transition. (b) Theoretical jetting–dripping transition, for various (α, β, Re, We). α and β stand respectively for density ratio and viscosity ratio of continuous to dispersed phase liquids.²²⁴ Re and We are Reynolds number and Weber number of the dispersed liquid, respectively. (c) Top images: jet length increases with increasing continuous capillary number C_out (C_out defined in ref. 225). The oscillations on the jet gradually die out as the length increases at the critical value, C*_out. Bottom diagram: plot of the jet length as a function of C_out. The arrow marks C*_out obtained from the linear stability analysis. Inset: Linear stability analysis (W_in represents the dispersed phase Weber number). Below C*_out the jet breaks due to an absolute instability, while above, it breaks due to a convective instability. For these experiments λ = 0.01. (d) The critical capillary number, Ca* (defined in ref. 58) as a function of the viscosity ratio λ, for the transition between globally stable to globally unstable flows. (a) is reprinted with permission from ref. 133. Copyright 2007, American Physical Society. (b) is reprinted with permission from ref. 224. Copyright 2006, Cambridge University Press. (c) is reprinted with permission from ref. 225. Copyright 2008, American Physical Society. (d) is reprinted with permission from ref. 58. Copyright 2013, Cambridge University Press.

Absolute instability is also observed in widening jet breakup (Fig. 6c).²²⁵ This is supported by several experimental identifications: (i) the neck diameter oscillates with exponentially growing amplitude until pinch-off occurs, implying a positive temporal growth rate ω_i; (ii) the temporal oscillations of the widening jet remain nearly stationary in space, suggesting the zero group velocity at a fixed spatial location; (iii) despite that We_d > O(1) of a widening jet at injection, droplet pinch-off at the end of the jet occurs only when the neck diameter is sufficiently widened such that We_d ≲ O(1), analogous to droplet formation in dripping mode due to absolute instability. Utada et al.²²⁵ predicted the critical capillary number Ca_cri = 0.69 (with λ = 0.01) of the continuous phase by performing a spatial stability analysis. Above Ca_cri, the absolute instability of widening jets transitions to convective instability together with a significant increase in the jet length (Fig. 6c). The theoretical prediction is in good agreement with the experimental result of Ca_cri ≈ 0.65 and independent of dispersed Weber number We_d (Fig. 6c).

Spatially developing flows, for example, capillary jets emanating from a nozzle, may exhibit self-sustained global modes when flows have a sufficient region of absolute instability.^220,226 In flow-focusing microfluidic devices, the tapering liquid meniscus of the dispersed phase oscillates periodically with time and spatially between the injection and the focusing nozzle, emitting droplets in sequence by tip-multi-braking mode,^17,184 which displays global instability.^189,190 In comparison, flows in dripping, jetting and tip-streaming modes are globally stable because of their stable menisci (Fig. 3a–d, flow-focusing). Similar to flows in flow-focusing, highly stretched jets in co-flowing streams set off global instability with an unsteady conical meniscus when the continuous-phase capillary number is below the critical value Ca_cri.⁵⁸ To determine Ca_cri that separates tip-streaming from tip-multi-breaking mode, Gordillo et al.⁵⁸ performed a global stability analysis by taking the real shape of the stretched jets into consideration with no simplification of parallel flows. It was found that Ca_cri is a decreasing function of viscosity ratio λ but insensitive to flow rate ratio φ provided that λ ≲ 0.1 and φ ≪ 1. The theoretical predictions by global stability analysis are in fair agreement with experiments and more accurate than predictions by local stability analysis of absolute/convective transition (Fig. 6d).

5. Active droplet generation

Modulating droplet generation with the aid of additional energy input by active controls has several advantages over passive droplet generation, for example, (i) active droplet generation offers additional handles and higher flexibility in controlling droplet size and production rate, and in some cases enables on-demand droplet generation (e.g., droplet with prescribed volume produced at a required frequency), therefore largely facilitating practical applications of microfluidic droplets; it is however almost impossible, in passive droplet generation, to independently control droplet size and generation frequency, as they correlate with each other by mass conservation; (ii) system response time, the time needed for stabilized droplet production, is much shorter in active control than in passive method. For example, the response time can be down to several milliseconds in an active approach compared with several seconds or even minutes in a passive method.

The energy imbalance modulates the nature of force balance on the interface for droplet generation. According to the nature of force balance variation, the active droplet generation can be categorized into two groups: introducing additional forces and modifying intrinsic forces. As discussed in section 2, microfluidic droplet formation is controlled by external forces and intrinsic ones of inertial, viscous, and capillary effects scaling as f_i ∼ ρ_su_s², f_v ∼ η_su_s/L, and f_γ ∼ γ/L, respectively. An active control is effective in modulating droplet generation if it affects at least one type of the aforementioned four forces. Normally, additional forces are exploited by applying external electric, magnetic and centrifugal fields, and modifying the intrinsic inertial, viscous and capillary forces is realized by manipulating the dynamic velocity u and material properties, including viscosity, interfacial tension, channel wettability, and fluid density. A recent review²⁵ classified active methods according to the energy type: electrical, magnetic, thermal and mechanical. We note that the mechanism of active control could be different even for the same energy input (Table 5). Taking electrical control for example, dielectrophoresis, electrowetting (EW), electrochemical, electrocapillary and electrorheological effects can be utilized. In dielectrophoresis an external dielectric force is applied on the liquid–liquid interface; in EW the wettability of the microchannel substrate is changed; in electrochemical reaction and electrocapillarity, the interfacial tension is modified; by using electrorheological fluids the viscosity is actively controlled (Table 5). As such, we organize this section based on the force-balance argument of active droplet generation.

Table 5 Comparison of different active droplet generation techniques

Type of control	Mechanism	Implementation	Device geometry	Type of droplet generation^a	Ref.
a Tunable – droplet generation with tunable size and production rate; on-demand – on-demand droplet generation.
Electrical	Electric force	Constant DC	Flow-focusing	Tunable	19, 227, 231 and 232
		DC pulse	Step emulsification	On-demand	229 and 233
		Low-frequency AC	Flow-focusing	Tunable	230
		High-frequency AC	Flow-focusing	Tunable	228 and 234–236
	Electrorheology	DC pulse	T-junction	On-demand	295
			Flow-focusing	On-demand	295 and 299
	EW effect	High-frequency AC	Flow-focusing	Tunable	316–319
			Flow-focusing	On-demand	320
	Electrochemical and electrocapillary effects	Constant DC and low-frequency AC	Flow-focusing	Tunable	313
Magnetic	Magnetic force	Uniform field, in-plane	Flow-focusing	Tunable	238–241 and 243
			T-junction	Tunable	240
		Uniform field, out-of-plane	T-junction	Tunable	242
		Non-uniform field	T-junction	Tunable	237
			Step emulsification	On-demand	85
Centrifugal	Centrifugal/Coriolis/Euler forces	Rotation	Dispenser nozzle	Tunable	247–249
			Flow-focusing	Tunable	250 and 251
			Cross-flow	Tunable	252 and 253
			Step emulsification	Tunable	82–84
Optical	Modifying fluid pressure	Laser pulse	T-junction	On-demand	254
	Marangoni effect	Laser heating	T-junction	Tunable	308
			Flow-focusing	Tunable	307, 309 and 310
			Co-flow	Tunable	311
	Photosensitive surfactant	UV/blue light	Flow-focusing	Tunable	315
Thermal	Temperature dependence of viscosity and interfacial tension	Heating	T-junction	Tunable	302 and 303
			Flow-focusing	Tunable	297, 300, 301 and 304
	Marangoni effect	Heating	T-junction	Tunable	305 and 306
	Phase change	Heating and cooling	Flow-focusing	Tunable	321
Mechanical	Modifying fluid pressure	Mechanical vibration	Co-flow	Tunable	255–260
		Off-chip valve	T-junction	On-demand	261–265
			Flow-focusing	On-demand	18
			Step emulsification	On-demand	266
		SAW	T-junction	Tunable	276
				On-demand	273 and 274
			Flow-focusing	Tunable	275
	Channel deformation/blocking	Piezoelectric actuator	Flow-focusing	Tunable	269–271
			T-junction	On-demand	267, 268 and 272
			Flow-focusing	On-demand	268
		On-chip microvalve	T-junction	Tunable	13, 279, 280, 282, 287 and 291
				On-demand	283–286, 289 and 290
			Flow-focusing	Tunable	277, 278, 280, 281, 288 and 292–294
				On-demand	284

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5.1 Additional forces in electrical control

Similar to the cases of electrospray and electrohydrodynamic atomization,¹⁸⁰ electric fields can be employed to manipulate droplet generation in microfluidics. To apply an electric field, a microfluidic device is embedded with electrodes, one²²⁷ or pairs of²²⁸ which are connected to the high voltage and the others are grounded. Taking water–oil systems, for example, high voltage U_e is applied to the water phase, and the grounded oil phase behaves as an insulator (Fig. 7a). Application of electric fields induces the migration and accumulation of charges on the water–oil interface. With appropriate distribution of the electric field, the interaction between the surface charges and the electric field offers an additional handle to control droplet generation. Electrical control is categorized into two groups based on the type of current: direct current (DC)²²⁷ and alternating current (AC) control.²²⁸ In DC control the voltage is only in one direction, whereas the polarity of voltage reverses periodically in AC control. DC control can be applied at constant²²⁷ or pulsatile voltage,²²⁹ while AC control is subcategorized into low-frequency (on the order of Hz)²³⁰ and high-frequency (on the order of kHz)²²⁸ control, depending on whether the frequency of applied voltage is lower (low-frequency) or higher (high-frequency) than that of droplet generation.

Fig. 7 Electrical control of droplet generation. (a) A high DC voltage U_e is applied to the dispersed phase, and the continuous phase is grounded. Two methods of implementation have been demonstrated: inserting electrodes into the liquids,¹⁹⁴ or patterning indium tin oxide (ITO) electrodes on the substrate.¹⁹⁰ (b) Droplet generation in a DC electric field. (i) Stable jet. Flow rate ratio φ = 80/350. Voltage U_e = 1200 V. (ii) Electrospray. Flow rate ratio φ = 5/350. Voltage U_e = 2000 V. (c) Top: contour plot of the electric field strength at one time-step during the droplet formation. The black lines indicate the electric field lines. High voltage is applied to the left side inlet. Bottom: contour plot of the volume fraction of the dispersed phase in the regions enclosed by the red dashed curves. The white arrows indicate the vectors of the electric body force induced on the fluid interface. (d) A sequence of images showing the electrogeneration of water droplets with a DC voltage pulse. (i–iii) Images of two droplet formation in one pulse. (i) Advancing of the water jet. (ii) Primary droplet formation. (iii) Secondary droplet formation. (iv) Formation of one droplet per pulse. (e) Droplet series generated using an AC electric field of the triangular waveform. A–I: the first half period. J–R: the second half period. (f) Schematic of a flow-focusing device under an AC electric field. The electrodes are in orange and the dispersed phase is in blue. (b) is adapted with permission from ref. 231. Copyright 2007, American Institute of Physics. (c) is reproduced with permission from ref. 19. Copyright 2015, Royal Society of Chemistry. (d) is adapted with permission from ref. 233. Copyright 2006, American Chemical Society. (e) is adapted with permission from ref. 230. Copyright 2010, American Institute of Physics.

In DC control where constant high voltage is applied to the dispersed phase (Fig. 7a), the electric stress on the water–oil interface scales as²²⁷

f_e ∼ ε_cE², (42)

where E ∼ U_e/L is the electric field strength, ε_c the permittivity of continuous liquid (usually oil). The electric stress f_e increases with U_e and competes with surface tension ∼γ/L. When U_e is sufficiently large, the electric stress decreases the droplet size, consistent with experimental observations.^227,231,232 Flow rate ratio and surfactant concentration play roles in electrical control. At high flow rate ratio φ and high voltage U_e, jetting appears and bridges the dispersed conducting liquid and the grounded electrode, inducing no breakup (Fig. 7b,i).²³¹ At low φ, increasing the voltage above a threshold otherwise gives rise to the formation of a Taylor cone, where droplets with a diameter less than 1 μm are generated in electrospray regime (Fig. 7b,ii).²³¹ The threshold voltage for electrospray decreases with surfactant concentration.²³² At constant surfactant concentration, the semiangle of the Taylor cone in microfluidic chips was found to decrease as the flow rate ratio decreases and always smaller than that in air.²³²

Variation in DC implementation affects the process of droplet formation. In a flow-focusing geometry where two continuous inlets are connected to high DC power supplier and ground (Fig. 7c), the electric force points from the continuous phase to the dispersed phase due to the smaller ε_d (ε_d, the permittivity of dispersed phase) than ε_c.¹⁹ Numerical simulation¹⁹ demonstrates three distinct stages of droplet size variation with the increase in voltage: decreasing, increasing and decreasing again. This trend is explained by the counterbalance between the x and y components of electric forces (Fig. 7c). The x component, perpendicular to the flow direction, accelerates the squeezing rate of the viscous droplet, whereas the y component, opposite to the flow direction of dispersed phase, exerts a retardation effect on the fluid interface.

DC pulse is capable of producing droplet on demand.^229,233 Initially, the water–oil interface is stabilized by capillary pressure at the orifice. Upon pulse initiation, the interface is forced by electric effects into the narrow orifice channel (Fig. 7d,i) at a nearly constant speed before reaching its equilibrium displacement.²²⁹ The maximum equilibrium displacement increases with the pulse amplitude. At high amplitude a droplet is emitted at the end of the protruded jet (Fig. 7d,ii), then the jet either re-protrudes and issues a secondary droplet (Fig. 7d,iii) or directly retracts (Fig. 7d,iv) after the pulse.²³³ The secondary droplet is only observed in a narrow channel with a high axial aspect ratio (ratio of channel length L to equivalent radius wh/(w + h)). The secondary droplet formation was ascribed to Rayleigh instability, different from the primary one of electrohydrodynamic origin.²³³

In low-frequency AC control, droplet generation is asynchronous with applied voltage and displays hysteresis.²³⁰ The droplet size is independent of the polarity of the electric field, but displays time dependence as electric fields vary with time (Fig. 7e). The irregular size distribution gives rises to uneven charges in individual droplets.²³⁰ In addition, the dispersed flow rate Q_d issued from the Taylor cone exhibits relaxation oscillation that might be responsible for the hysteresis of droplet size variation.²³⁰

In high-frequency AC control, Tan et al.²²⁸ designed a microfluidic setup where electrodes are not in contact with fluids (Fig. 7f). This method enables reliable droplet generation and manipulation with response time in the order of milliseconds, based on which a musical interpretation of droplet microfluidics was demonstrated.²³⁴ In this design, dripping to jetting transition is observed when the voltage increases.²²⁸ At a fixed high voltage, systematic variation of the voltage frequency fre_e and electrical conductivity of the dispersed liquid k_e leads to three regimes: dripping, unstable production of droplets, and jetting.²²⁸ The dripping and unstable production regimes are separated by the boundary of fre_e/k_e ∼ 5 × 10⁵ mF⁻¹, while the transition from jetting to unstable production occurs when fre_e is below a certain value fre_mess.^228,235 In dripping regime, the droplet size is a function of electric field, and can be quantitatively related to an effective capillary number Ca_eff of the form Ca_eff = Ca_c/(1 – B_e), where B_e is the electric bond number.²²⁸ The unstable production of droplets is likely to be induced by nonaxisymmetric instabilities when the electric field at the jet surface is of a critical value.²³⁵ In jetting regime,^235,236 the jet length L_jet increases with applied voltage and scales as L_jet ∼ (fre_e/k_e)^−0.5 at a constant voltage. The electrical potential at the tip of jets was found to be about 550 V, which might confirm that jet breakup occurs for a given electric field distribution around the tip.²³⁵

5.2 Additional forces in magnetic control

A ferrofluid is a type of nanofluids containing nanometer-sized magnetic particles (≤10 nm), which is magnetized by applying a magnetic field, and demagnetized when the magnetic field is withdrawn. Consequently, ferrofluid is a good candidate for magnetic control of droplet generation in microfluidics. In the presence of a magnetic field, magnetic forces per unit volume f_m are expressed as,

f_m = μ₀M∇H, (43)

where μ₀ is the magnetic permeability of free space, M represents the magnetization, and ∇H stands for the gradient of the magnetic field strength. Accordingly, droplet generation can be harnessed by varying either M or ∇H or both in magnetic control. In this subsection, we focus only on droplet generation actively controlled by the extrinsic magnetic forces where ferrofluid is used as the dispersed phase. We do not consider magnetorheology because the strong Brownian motion of magnetic nanoparticles renders this effect very weak in a ferrofluid.

Implementation of magnetic fields can be adjusted in various aspects: the type of magnet, the location, uniformity, direction, and polarity of the magnetic field. Both permanent magnets^85,237,238 and electromagnets^239–241 have been used. The magnetic field can either cover the entire microfluidic device^{85,237–240,242} or be localized,²⁴¹ and be either uniform^238–242 or non-uniform^85,237 (Fig. 8a). For the uniform magnetic field, it can be either in-plane^238–241 or out-of-plane²⁴² of the microfluidic chip, and either parallel to^238–240 or perpendicular to²³⁸ the main flow direction (Fig. 8a). Finally, the polarity of the uniform magnetic field can be inverted as well.

Fig. 8 Magnetic control of droplet generation. (a) Schematic of microfluidic devices under magnetic fields, as exampled by flow-focusing (left) and cross-flow (right) geometries. In-plane magnetic field: ①, ②, ④, ⑤. The direction of magnetic field perpendicular (①, ④) to and parallel (②, ⑤) to the main flow direction. Out-of-plane magnetic field with two polarities: ③. Localized magnetic field: ④. The entire device exposed to a magnetic field: ⑤. Uniform magnetic field: ④, ⑤. Non-uniform magnetic field, exampled by a downstream magnet: ⑥. Note that all types of magnetic fields (①–⑥) can be applied to any type of fluidic structures. In addition, ④ and ⑥ can be placed at any locations of microfluidic devices. (b) Formation of ferrofluid droplets by dripping in a flow-focusing device. (i) Magnetic field parallel to the main flow direction. During pinch-off, the droplet is stretched in the main flow direction. (ii) Magnetic field perpendicular to the main flow direction. During pinch-off, the droplet is expanded to block the outlet channel. (c) Alignments of the clustered magnetic nanoparticles in the direction of the magnetic field. (i) Alignment parallel to the dispersed flow direction in a flow-focusing device. (ii) Alignment perpendicular to the dispersed flow direction in a T-junction. (d) Maximum contraction position C_max (after droplet formation) and droplet size decreasing with magnetic flux density (from left to right). (b) is adapted with permission from ref. 238. Copyright 2013, Royal Society of Chemistry. (c) is adapted with permission from ref. 240. Copyright 2011, American Physical Society. (d) is adapted with permission from ref. 241. Copyright 2015, Springer.

Droplet generation depends on the specific implementation of the magnetic field. First of all, the direction of the magnetic field matters. In T-junctions, the droplet size increases when a permanent magnet is placed upstream of the junction, whereas the size decreases when the magnet is positioned downstream.²³⁷ Changing the position of the permanent magnet varies the direction of the non-uniform magnetic field as well as that of the magnetic forces f_m. In the upstream case, f_m pulls the droplet back and delays the breakup, resulting in larger droplets. In contrast, the downstream case has the opposite effect. In addition, by employing a uniform magnetic field in T-junction, the in-plane application causes a decrease in droplet size,²⁴⁰ while the out-of-plane implementation induces the droplet size increase.²⁴² Moreover, in flow-focusing configuration, magnetic fields parallel to and perpendicular to the main flow mainly affect the breakup (Fig. 8b,i) and expanding (Fig. 8b,ii) process of the dispersed thread, respectively.²³⁸ Interestingly, the polarity of the magnetic field has no influence on droplet size.²⁴⁰

Secondly, the device geometry plays a significant role in modulating droplet size. In the case of a uniform magnetic field parallel to the main flow, droplet size increases in flow-focusing but decreases in T-junction geometries as magnetic flux density increases.²⁴⁰ This difference results from the orientation of the magnetic field relative to the dispersed flow direction. When subjected to an external magnetic field, the magnetic nanoparticles align with the magnetic field. In flow-focusing configuration, the alignment is in the same direction as the dispersed flow, elongating the fluid tip and delaying breakup (Fig. 8c,i).^240,243 However, in T-junction, the alignment is perpendicular to the dispersed flow direction (Fig. 8c,ii), thereby accelerating breakup.²⁴⁰ In step emulsification, droplet size is not affected by the magnetic field but depends on the channel dimension.⁸⁵ Kahkeshani and Di Carlo⁸⁵ recently showed that magnetic body forces, counteracted by interfacial and viscous forces, drive the dispersed fluid into the continuous reservoir. In a device with fixed channel dimension, increasing magnetic forces, or decreasing interfacial and viscous forces enables a larger rate of droplet generation, while droplet size keeps constant.⁸⁵

Finally, the location of the magnetic field impacts droplet size variation. In flow-focusing geometry with a uniform magnetic field perpendicular to the main flow, the droplet size increases when magnetic field covers the entire device²³⁸ but decreases when the field locates at continuous inlet microchannel.²⁴¹ In the former case, the magnetic field accelerates the expanding velocity of the dispersed tip and results in larger expanding tips as well as larger droplets (Fig. 8b,ii).²³⁸ In comparison, the latter observation is interpreted as a result of the magnetic drag.²⁴¹ The drag drives the dispersed tip to move downstream and block the outlet channel (Fig. 8d), leading to an increased continuous-fluid pressure buildup that accelerates droplet breakup.²⁴¹

In general, the effectiveness of magnetic control is attenuated by the increase in flow rate^237,238,240 because magnetic forces (eqn (43)) are counteracted by the elevated viscous and/or inertial forces. To the best of our knowledge, a closed-form expression or scaling law for predicting the droplet size under external magnetic fields has yet to be developed.

5.3 Additional forces in centrifugal control

Centrifugal microfluidics has been developed to fulfill many unit fluidic functions, such as mixing, valving, separation, detection, metering and droplet generation, and various applications ranging from nucleic acid analysis, immunoassays, cell lysis, blood plasma separation to water, food and soil analyses.^244,245 We herein focus exclusively on the droplet generation in centrifugal microfluidics. To this end, several additional forces due to the presence of rotation are introduced first. In a rotating frame of reference, three pseudo forces may act on a fluid volume: centrifugal force f_Ω, Coriolis force f_Co and Euler force f_E, respectively (Fig. 9a). The centrifugal force is caused by the rotary motion of the microchannel network, always pointing radially outward. The Coriolis force is generated by the relative motion of the object, therefore perpendicular to the direction of the motion. The Euler force is proportional to the rotational acceleration. Considering a fluid plug of density ρ at a radial position r rotating at an angular velocity Ω, the three forces scale as (in the form of forces per unit area):²⁴⁶

f_Ω ∼ ρΩ²rL, (44)

f_Co ∼ ρ²Ω³rL³/η, (45)

f_E ∼ ρ[capital Omega, Greek, dot above]rL, (46)

where L, η and [capital Omega, Greek, dot above] refer to the characteristic length scale of the fluid plug, dynamic viscosity of the fluid, and angular acceleration, respectively. During droplet generation, these forces counteract interfacial tension effects and facilitate the fluid breakup. Accordingly, two Bond numbers Bo_Ω and Bo_Co can be defined as the ratios of centrifugal and Coriolis forces to interfacial tension force, respectively:

Bo_Ω ∼ ρΩ²rL²/γ, (47)

Bo_Co ∼ ρ²Ω³rL⁴/ηγ. (48)

Fig. 9 Centrifugal control of droplet generation. (a) Three pseudo forces in centrifugal droplet generation. Centrifugal force f_Ω is proportional to Ω², pointing radially outward. Coriolis force f_Co ∝ Ω³, perpendicular to both Ω and fluid velocity u. Euler force is proportional to angular acceleration, f_E ∝ [capital Omega, Greek, dot above], perpendicular to both [capital Omega, Greek, dot above] and radial coordinate r. (b) Schematic of the experimental setup consisting of a centrifugal platform with a micronozzle and Eppendorf tube in a swinging bucket. (c) Centrifugal flow-focusing configuration. The centrifugal force F_ω and viscous drag F_d of the sheath flow ϕ_o support droplet formation. The emerging droplet is squeezed by the inertia (F_i) induced by the transversal component of the sheath flow. The surface tension counterforce F_σ prevents the droplet break-off until a critical mass is reached.²⁵⁰ (d) Droplet formation in a centrifugal cross-flow configuration. While surface tension F_σ indulges in maintaining the symmetric shape of the droplet, centrifugal force F_ω and Coriolis force F_C continuously pull the liquid finger until pinch-off.²⁵² (e) Centrifugal step emulsification device. The microfluidic device is located on a spinning disk (inset A) and consists of an inlet chamber (red), a channel (red) which connects the inlet chamber to a step emulsification nozzle (inset C), and a droplet collection chamber (PCR chamber, inset B). (b) is reproduced with permission from ref. 247. Copyright 2009, Elsevier. (c) is reproduced with permission from ref. 250. Copyright 2006, Springer. (d) is reproduced with permission from ref. 252. Copyright 2015, American Institute of Physics. (e) is reproduced from ref. 84. Copyright 2016, Royal Society of Chemistry.

To date, four different structures have been employed in centrifugal droplet generation: dispenser nozzle,^247–249 flow-focusing,^250,251 cross-flow^252,253 and step emulsification.^82–84 One scheme to generate droplets is to rotate a dispenser nozzle, by which individual droplets are issued into the intermediate air spacer before they are collected by the receiving liquid (Fig. 9b).^247–249 To generate a droplet, the pressure difference (Δp = ρ_dΩ²r²/2) of the dispersed liquid in the nozzle should exceed the capillary pressure (f_γ = 4γ/w_d). This gives a minimum rotation velocity:²⁴⁷

with w_d being the dispenser nozzle diameter. When the droplet detaches from the nozzle, the total surface tension force (F_γ = πw_dγ) is balanced by the total centrifugal force (F_Ω = πρ_dD³Ω²r/6), thereby the droplet diameter D is expressed as,^247,248

Since flow rate Q_d obeys Q_d ∝ Δp ∝ Ω²,²⁴⁵ the droplet generation frequency fre_drop for Newtonian fluids could be fre_drop ∝ Ω⁴ based on mass conservation. Rotating the dispenser nozzle has been employed to produce alginate/chitosan microparticles after solidification of droplets by cross-linking solutions.^247–249 The abilities to tailor the particles into non-spherical shape and with 3D multi-compartment structure have also been demonstrated by varying the distance between the nozzle and the surface of the cross-linking solution and by using a multi-barrelled capillary nozzle, respectively.²⁴⁸

In flow-focusing configuration, the dispersed liquid tip is pulled downstream by centrifugal and viscous drag forces. Once the tip is long enough, thread breakup is stimulated by the junction constriction and inertia coming from the transversal component of continuous flows (Fig. 9c).²⁵⁰ Depending on the channel geometry and Bond number Bo_Ω, squeezing, dripping and jetting modes of droplet generation are observed. In squeezing and dripping, highly monodisperse droplets (e.g., CV = 0.84% by dripping) are generated with the droplet size decreasing but generation frequency increasing as Ω increases.²⁵⁰ In contrast to this trend, both the bubble length L_p and bubble generation frequency fre_bubble are proportional to Bo_Ω (equivalently, L_p ∝ Ω² and fre_bubble ∝ Ω²) when Bo_Ω ≲ 1 in centrifugal bubble generation.²⁵¹ By programming the rotational speed, it is feasible to tailor the inter-bubble spacing as well as bubble size.²⁵¹

In cross-flow structure, the force balance depends on the rotation speed Ω. According to eqn (44) and (45), f_Ω ∝ Ω², whereas f_Co ∝ Ω³, growing faster with Ω than f_Ω. As such, at high Ω, f_Co would probably dominate f_Ω. In a Y-shaped microchannel with the rotation speed ranging from 104.72 rad s⁻¹ to 167.55 rad s⁻¹, the two Bond numbers are in the range of Bo_Ω = 0.22–0.2 and Bo_Co = 12.62–20.90.²⁵² The dominance of f_Co over f_Ω and f_γ indicates that the droplet generation is governed by the effective pulling of Coriolis force until the pinch-off (Fig. 9d). In this circumstance,²⁵² the droplet volume V is ruled by V ∝ Ω⁻¹, and droplet generation frequency fre_drop is thus proportional to Ω³. In addition, by executing the rotation speed in alternate clockwise and counterclockwise directions, Kar et al.²⁵² demonstrated the production of long serpentine threads, as a result of the repetitive alterations in the direction of f_Co, as well as the impact of Euler force at the momentary time stages of speed turnover.

Centrifugal step emulsification (Fig. 9e) takes the advantage of step emulsification that droplet size is less sensitive to pressure and flow rate fluctuations but only depends on the nozzle geometry and interfacial tension.⁸² Subsequently, uniform droplet generation (CV of 2–4%) could be achieved, which facilitates the afterward application of digital droplet recombinase polymerase amplification (RPA)⁸² and digital droplet PCR.⁸⁴ At high rotation speed (Ω ≈ 125.6 rad s⁻¹),^82,83 the setup enables in situ production of gel emulsions with very high internal volume fractions of >97.2%. Parallelization of 72 nozzles improves the droplet production rate up to 3700 Hz.⁸³

5.4 Modifying fluid velocity

Intrinsic forces such as inertial and viscous forces could be modified dynamically by manipulating fluid velocity u. In general, modulating the hydraulic pressure and flow resistance (e.g., changing channel dimension) leads to variations in velocity u, as implemented by off-chip and on-chip actuation. The former implementation includes optical control, mechanical vibration and an off-chip valve, while the latter involves piezoelectric control and an on-chip microvalve. In contrast to on-chip actuation, off-chip modules are simple in assembly and compatible with microfluidic devices.

5.4.1 Optical control. An intense laser pulse (Fig. 10a) is employed to actuate fluid pressure by inducing rapidly expanding cavitation bubbles, therefore enabling on-demand droplet generation.²⁵⁴ To achieve this, initially, the immiscible interface between dispersed water and continuous oil phase is set to be stable at the junction by adjusting flow rates. Then an intense laser pulse is focused on the laser-absorption water phase to break down water molecules and generate hot plasma.²⁵⁴ Arising from the dissipation of heat, a laser-induced cavitation bubble expands rapidly and destabilizes the oil–water interface. The expanding cavitation acts as a micropump to push water into the oil phase and produces aqueous droplets. Due to controllable laser pulsing intervals, on-demand droplet generation of production frequency as high as 10 kHz can be achieved, meanwhile maintaining high monodispersity (droplet volume variations <1%).²⁵⁴

Fig. 10 Active control by modulating the dynamic velocity of fluid flows. (a) Schematic of the pulse laser-driven droplet generation device that consists of two microfluidic channels connected by a nozzle-like opening. A highly focused intense laser pulse induces a rapidly expanding cavitation bubble to push the nearby water into the oil channel for droplet formation. (b) Schematic of dispersed fluid pressure perturbed by a mechanical vibrator by shocking the flexible microtubing. (c) Schematic of an off-chip valve regulating the dispersed fluid pressure. In addition, off-chip valves can also be applied to continuous phase fluid. (d) Schematic of piezoelectric actuation. (i) A piezoelectric actuator is incorporated into the dispersed fluid inlet channel. (ii) Droplet generation actuated by SAW. Implementation in both T-junction and flow-focusing devices has been demonstrated. (e) Schematic of on-chip valve actuation. (i) One actuation channel placed upstream of the junction. (ii) Two actuation channels located aside the junction. (iii) Chopping of droplet generation by several actuation channels placed atop the downstream fluidic channel. The actuation channels can also be placed aside the fluidic channel. (iv) Deformable droplet generator with an underneath actuation channel covering the entire fluidic channel with a T-junction. This implementation is also applicable to flow-focusing devices. (a) is reproduced with permission from ref. 254. Copyright 2011, Royal Society of Chemistry.

5.4.2 Mechanical vibration. Off-chip mechanical vibration can be achieved by perturbing the flexible dispersed-phase microtubing using a mechanical vibrator (Fig. 10b).^255–260 This technique demonstrates the capability of inducing corrugations on water–water interfaces with ultra-low interfacial tension in ATPS,²⁵⁵ breaking jets into droplets,^{255–257,259} and producing droplets with core–shell and multi-compartment structures.^257,258 Upon perturbing the microtubing, the dispersed-phase pressure pulsates and induces an oscillatory flow at the nozzle.²⁵⁹ Experimental findings suggested that vibration-enhanced dispersed-fluid inertia counteracts the viscous stress, inducing faster pinch-off of dispersed fluid threads and producing smaller droplets.²⁶⁰ In a certain range of frequency, the droplet generation was synchronized with the vibration, by which a droplet with predefined size is achievable regardless of the continuous flow rate, channel dimension, and fluid properties, provided that the frequency of vibration and dispersed flow rate are fixed.²⁵⁹ The lower limit of synchronization frequency is approximately the frequency of passive droplet generation, while the upper limit is determined by²⁵⁹where V_c is the critical volume for a drop protrusion to break up, Y is the dimensionless amplitude of vibration of the form Y = πR_tubing⁴ρ_dgY₀/(8Q_dη_dl_d), with R_tubing being the inner wall diameter of the microtubing, g the gravitational acceleration, Y₀ the amplitude of vibration in height, and l_d the distance between the vibration location and injection nozzle.

5.4.3 Off-chip valve. Mechanical valves have been externally connected to microfluidic devices for on-demand droplet generation (Fig. 10c).^18,261–265 In doing this, the dispersed-phase pressure is pulsated by valves in a square-wave where open and close statuses are denoted by “on” (duration t_on) and “off” (duration t_off), respectively. The continuous phase is driven at either constant volumetric flow rate,¹⁸ or constant pressure,²⁶¹ or pulsatile pressure.²⁶⁵ The droplet volume is proportional to t_on in the form of V ∼ t_onQ_d. Due to laminar flows in microchannels, the dispersed-phase flow rate Q_d is well described by the Hagen–Poiseuille law Q_d = ΔP_d/Rre_d, where ΔP_d is the pressure drop along the dispersed microchannel of length l_d, and Rre_d is the flow resistance, scaling as Rre_d ∼ η_dl_d/w_dh³ for rectangular channels. Consequently, prediction of droplet volume yields

V ∼ ΔP_dt_on/Rre_d. (52)

Eqn (52) denotes that droplet volume is proportional to the amplitude of pressure pulse ΔP_d and valve “on” duration t_on, in consistency with experiments.^18,261,265 In addition, experiments revealed that in ATPS with a constant continuous-phase flow rate Q_c, droplet volume is inversely proportional to the product of Q_c and t_off.¹⁸ As such, V scales as:

V ∼ ΔP_dt_on/Rre_dQ_ct_off. (53)

By programming the on–off cycle, sequences of droplets with various volumes are produced, and independent control over droplet volume and generation frequency is attainable. Moreover, Jung et al.²⁶⁶ demonstrated on-demand control of droplet numbers per pulse by varying ΔP_d and t_on in a step emulsification device equipped with an outside pressure regulator. Compared with systems actuated by an electromagnetic valve, those by a piezoelectric valve have a higher droplet generation frequency. The frequency is lower than 100 Hz for the former^18,261–264 and up to 400 Hz for the latter.²⁶⁵

5.4.4 Piezoelectric actuation. On-chip piezoelectric control is categorized into two groups: utilizing the pulse of piezoelectric actuator to deform microchannels/chambers (Fig. 10d,i), exploiting surface acoustic wave (SAW) (Fig. 10d,ii). For the former type, the piezoelectric actuator is placed atop the dispersed phase chamber (Fig. 10d,i). The devices are normally fabricated by polydimethylsiloxane (PDMS), and a flexible PDMS membrane separates the chamber and the actuator.^267–272 Piezoelectric pulsation can either have on-demand droplet generation^267,268 or speed up the droplet production.^270,271 In the former case, no droplet is generated in the absence of actuation. Upon actuation, the droplet is dispensed from the junction. The droplet size and generation frequency can be controlled independently. Experiments showed that droplet volume first increases linearly with the pulse amplitude and pulse duration before reaching a plateau.²⁶⁸ In addition, the shape of the pulse signal affects the droplet size and its uniformity.²⁶⁷ As for the speed-up of droplet generation, the droplet size and generation frequency is controlled by both the actuation and the hydrodynamic forces. In squeezing regime, piezoelectric actuation accelerates the droplet pinch-off²⁶⁹ and reduces the droplet size.²⁷⁰ A similar enhancement was observed in ATPS for jet breakup, where monodisperse hydrogel microspheres were produced under piezoelectric actuation.²⁷¹

Different from channel deformation by the pulse of piezoelectric actuator, SAW utilizes acoustic radiation force (ARF) to deform the liquid–liquid interface by advancing one liquid into another.²⁷³ In SAW, ARF arises from the gradient in the density of a traveling acoustic wave such as at an interface between two fluids with different acoustic impedances. In the implementation of SAW, a microfluidic device is placed on top of an interdigitated transducer (IDT) that comprises a series of comb-like metal electrodes arrayed on a piezoelectric substrate (Fig. 10d,ii).²⁷⁴ Both on-demand drop generation^273,274 and drop size modulation^275,276 are achievable by steering SAW. In water–oil systems where the dispersed water phase is pressurized by focused SAW, on-demand emulsification was realized by manipulating the applied power and pulse duration.^273,274 Single or a batch of droplets is produced during SAW actuation. The number of droplets is determined by pulse duration, while droplet size is affected by the applied power, pulse duration, continuous-phase flow rate and channel geometry.^273,274 As the continuous-phase flow rate increases, squeezing, dripping and jetting regimes are successively observed.²⁷³ Meanwhile, by using unfocused SAW, Schmid and Franke demonstrated the decrease in aqueous droplet size in both flow-focusing²⁷⁵ and T-junction.²⁷⁶ The size decrease is explained by a SAW-induced pressure increase in the inlet channel of the continuous phase.^275,276

5.4.5 On-chip microvalve. On-chip valves are usually integrated into microfluidic devices and activated pneumatically by compressed air or hydraulically by the pressurized liquid in actuation channels. Under valve actuation, the fluidic channel is deformed or even blocked, thereby modulating droplet generation. The actuation channel, usually separated by a membrane from the fluidic channel, can be placed upstream of, at, and downstream of the junction, or covering the entire fluidic channel (Fig. 10e). In implementation, the actuation channel has been designed to locate aside,^277,278 atop,²⁷⁹ underneath,²⁸⁰ and around²⁸¹ the fluidic channel, in which fabrication of aside actuation channels is easy, while the other three arrangements require multi-layer chip fabrication.

When located upstream of the junction, actuation valves are normally used to perturb the dispersed-fluid velocity. Both blocking and deformation of the dispersed channel have been demonstrated by actuation channels.^13,281–286 In the blocking case,^283–286 the resultant droplet size can be well predicted by eqn (52), in that underlying mechanisms are the same for both on-chip and off-chip valve-blocked systems. In the case of fluidic channel deformed by an actuation channel (Fig. 10e,i), monodisperse droplets are favored in synchronized regime, and polydisperse droplets are produced in quasiperiodic regime.²⁸⁷ The synchronized regime occurs when the frequency of valve pulse fre_pulse is close to multiple times the frequency of valve-free droplet generation fre₀, such as fre_drop/fre₀ ≈ 1, 3/2, 2, 3, 4, 5. The under-forcing droplet generation frequency fre_drop is synchronized with fre_pulse in the manner fre_drop/fre_pulse = 1, 2/3, 1/2, 1/4, 1/5.²⁸⁷ Apart from the above normally open valves, a normally closed micro-valve is used where the dispersed channel is initially closed by a PDMS membrane.²⁸⁸ Upon valve actuation, the PDMS membrane is deflected to open the dispersed channel. The degree of membrane deflection maneuvers the dispersed flow velocity so that droplet size is tunable.

Valves located at the junction modulate drop generation by directly affecting the liquid–liquid interface through fluidic channel deformation or blocking. In the deformation case, variations in junction dimension (Fig. 10e,ii) make the continuous-phase shear stress unsteady and the droplet size tunable.^277–279 In the case of channel blocking, the volume of the droplet V was found to increase, either linearly²⁸⁹ or nonlinearly,²⁹⁰ with the valve “on” time t_on:

where m₁ and m₂ (m₂ ≥ 1) are constants mainly depending on fluid driving pressures. The nonlinearity (m₂ > 1) might come from the transient dispersed-phase flow rate during valve actuation.²⁹⁰ Increasing the dispersed driving pressure leads m₂ close to 1, probably because the transient effect is depressed.²⁹⁰

The downstream microvalve (Fig. 10e,iii) can break discrete droplets secondarily into daughter droplets²⁹¹ and chop the prefocusing stable stream into individual droplets.^292–294 In both situations, the resultant droplet sizes decrease with increased pressure that activates the valve. The stable stream is fragmented through a chopping method^292–294 where two or more choppers deflect the fluidic channel. The resultant droplets are monodisperse (CV <3%),²⁹² and the size depends on the flow rate ratio and pulse frequency of the chopper. However, the working frequency was relatively low (<17.4 Hz).^293,294

Raj et al.²⁸⁰ recently proposed a deformable droplet generator (Fig. 10e,iv) where the actuation channel lay underneath the entire fluidic channel. By characterizing the deformation of fluidic channel in response to the actuation pressure, Raj et al.²⁸⁰ determined the relationship between droplet size and channel deformation, flow rate ratio, and viscosity ratio. The results show that monodisperse droplets are generated in dripping regime for a wider range of capillary number Ca_c in comparison to the passive method. It is also possible to produce droplets of size slightly smaller than the junction size in this deformable device.

5.5 Modifying material properties

5.5.1 Viscosity. Three methods have been adopted to actively manipulate the viscosity of liquids: electrorheological (ER) fluid by electrical control,²⁹⁵ magnetorheological (MR) fluid by magnetic control,²⁹⁶ and heating the liquids by thermal control.²⁹⁷ In electrical control, ER fluid consists of non-conducting but electrically active particles dispersed in a carrier insulating liquid (usually oil).²⁹⁸ When subjected to an electric field, the suspended particles are electrically polarized and aggregate into chains/columns along the field direction (Fig. 11a). These chains/columns restrict the fluid flow perpendicular to the applied field direction by which the ER fluid exhibits increased apparent viscosity and even solid-like behavior.²⁹⁸ The rheological variation of ER fluid is reversible and displays fast response to an electric field (response time down to milliseconds).²⁹⁵ Zhang et al.²⁹⁵ utilized a type of ER fluid with the giant ER effect (GER)²⁹⁸ as the continuous phase to actively control droplet generation. Upon the turn-on of a sufficiently strong electric field, the continuous ER fluid stops flowing, and only the dispersed water advances into the junction. When the electric field is decreased to a critical value, the ER fluid rushes into the junction and breaks off the water thread. By programming the electrical control signal, tunable droplet size and droplet generation frequency are attainable. The GER fluid was also used as the dispersed phase by the same group.²⁹⁹

Fig. 11 Active droplet generation via modifying material properties. (a) Schematic of chain and column formation with increasing electric field in ER fluid. (b) Images of water drops generated in mineral oil at different temperatures. (c) Laser control of droplet generation. (i) Overlaid images showing the motion of seeding particles near the hot spot. Note that the motion along the interface is directed towards the hot spot. (ii) Optocapillary deformation of the liquid interface by focused laser beam. Top two images: a thick thread (A <1) pinches while a thinner one (A >1) bulges (indicated by red arrows). Bottom two images: for large enough power, the thread eventually breaks in either pinching (A = 0.4) or bulging (A = 6.3) case (indicated by black arrows). (d) Plot of size (diamonds) and production rate (squares) versus DC voltage for both electrochemical and electrocapillary cases. (e) Effect of UV illumination for different flow rates of continuous fluid (Q_oil) and AzoTAB solution (Q_aq) in a flow-focusing device. (f) EW actuation of droplet generation. Top: schematic of EW implementation. Upon actuation, the contact angle of water on Teflon substrate decreases (dashed curve) and induces a smaller mean curvature of the water meniscus. Bottom: images of dripping, tip-streaming and conical spray regimes (from left to right, top to bottom) under EW actuation. (b) is reproduced with permission from ref. 300. Copyright 2009, American Chemical Society. (c,i) is reproduced with permission from ref. 307. Copyright 2007, American Physical Society. (c,ii) is adapted with permission from ref. 310. Copyright 2015, American Physical Society. (d) is reproduced with permission from ref. 313. Copyright 2015, Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim. (e) is reproduced with permission from ref. 315. Copyright 2011, Royal Society of Chemistry. Images in (f) are adapted with permission from ref. 319. Copyright 2008, American Institute of Physics.

MR fluid exhibits tunable rheological characteristics due to the formation of chains/columns under a magnetic field. To display the MR effect, the magnetic particles should be large enough (on the order of 100 nm to 10 μm) to dominate over the effect of Brownian motion, different from those in ferrofluid with size smaller than 10 nm.²⁹⁶ Despite the unique characteristics of MR fluid, few applications of MR fluid to droplet generation were reported in the literature. Although it is weak, the MR effect exists in ferrofluid under certain circumstances. In droplet generation with the dispersed phase being water-based ferrofluid containing Fe₃O₄ nanoparticles (average size: 100 nm),²⁴¹ the nanoparticles align to form chain-like structures along the direction of the magnetic field (Fig. 8c). Ferrofluid droplets with magnetically increased viscosity are larger than water droplets with lower viscosity.²⁴¹ However, the MR effect is limited by the small particle size and counteracted by the magnetic drag that tends to decrease the droplet size.

Thermal control exploits the dependence of fluid viscosity and interfacial tension on temperature. The microfluidic device is heated by microheaters²⁹⁷ or heat exchangers.³⁰⁰ The resultant droplet size D(T) is a function of the combined effect of temperature-dependent viscosity and interfacial tension:^297,300

where T is the temperature. For most liquids, viscosity and interfacial tension decrease as temperature increases, but the decrease in viscosity is faster than that of interfacial tension. Consequently, increasing temperature will enlarge the droplet size (Fig. 11b).^297,300 By selectively choosing continuous fluids that have a much larger variation of viscosity than that of interfacial tension, Stan et al.³⁰⁰ demonstrated the increase in water droplet volume by almost two orders of magnitude when the temperature is increased from 0 to 80 °C. As droplet size increases with temperature, droplet generation mode could transition from dripping to squeezing.³⁰¹ In squeezing mode, the droplet size is mainly determined by the flow rate ratio (eqn (12)), thus depending weakly on temperature. Channel height also influences the correlation between droplet size and temperature. A decreased channel height shows an enhanced temperature dependence of droplet size due to the larger temperature gradient.³⁰²

Besides water, water-based nanofluid droplet generation by thermal control has been investigated in both flow-focusing³⁰¹ and T-junction^302,303 configurations. In flow-focusing geometry, water and nanofluid droplet formation exhibit similar characteristics at different temperatures.³⁰¹ In T-junctions, however, the size of the water droplet can be either larger or smaller than that of the nanofluid droplet depending on the channel dimension and flow rate ratio.^302,303 Moreover, the shape of the nanoparticle impacts the dependence of droplet size on temperature. The droplet size increases with increasing temperature for nanofluids with spherical-shaped nanoparticles, but decreases for nanofluids with cylindrical-shaped nanoparticles.³⁰² The underlying mechanism is unclear yet.

Different from the above studies with both continuous and dispersed fluids heated, heating the dispersed phase exclusively can also be able to tune the droplet size. Yeh et al.³⁰⁴ reported the generation of uniform gelatin emulsions (CV <5%) by heating the gelatin phase. When the temperature is over 25 °C by heating, the gelatin transforms from a gel to an aqueous solution with decreased viscosity. The viscosity decrease is responsible for the hastened release of the gelatin droplet, which reduces the continuous phase supply during the formation of each droplet and renders the droplet larger.³⁰⁴

5.5.2 Interfacial tension. Active control of the interfacial tension can be achieved by thermal control,^{297,300,305,306} optical control^307–312 and electrical control.³¹³ Thermal control involves either temperature-dependent interfacial tension^297,300 or Marangoni effect.^305,306 Similarly, optical control exploits the Marangoni effect by local laser heating.^307–311 In parallel, electrical control modulates the interfacial tension by electrochemical and electrocapillary effects.³¹³ Note that laser heating is occasionally referred to as thermal control.²⁵ Here we group laser heating into optical control due to its use of a laser beam. In this subsection, we discuss exclusively the manipulation of interfacial tension at the liquid–liquid interface, distinct from that at the liquid–solid interface which is classified into tunable channel wettability in the next subsection.

Thermal control can be categorized into uniform and non-uniform heating. In uniform heating, the heating region covers the entire liquid–liquid interface by which the interfacial tension is tuned homogeneously by temperature in the absence of interfacial tension gradient. In this situation, droplet size is controlled by eqn (55), as we have discussed. In contrast, non-uniform heating involves locally heating part of the liquid–liquid interface and producing a thermal gradient. The thermal gradient gives rise to a gradient of interfacial tension that generates Marangoni stress (tangential stress along the liquid–liquid interface. The Marangoni stress, in turn, induces an interfacial flow that affects the droplet formation. In the simulation by Suryo and Basaran,³⁰⁵ heating the liquid–nozzle contact line gave rise to droplet generation owing to the competition between Marangoni stress and capillary pressure. Moreover, in a T-junction configuration, Miralles et al.³⁰⁶ demonstrated the control over droplet size and generation frequency by heating the dispersed entrance of the junction.

In optical control involving laser heating, the laser beam is focused through a microscope objective, locally heating the centerline of the flow downstream of the junction. A dye (a fluorescein for example) is added to the dispersed aqueous phase to mediate laser absorption.^307–309 In the presence of Marangoni stress, dispersed bubbles and/or droplets would be forced to migrate with velocity u_m. Recent experiments suggested that the migration velocity u_m is proportional to the product of the channel width w and the mean temperature gradient 〈∇T〉 for aqueous droplets in confined rectangular microchannels:³⁰⁹

u_m ∝ w〈∇T〉, (56)

with 〈∇T〉 proportional to the laser power. In the presence of surfactants (where interfacial tension was found to rise with increasing temperature^307,314), locally heating of the advancing interface of a growing droplet leads to a migration velocity against the basic flow in the microchannel, thus slowing down or even blocking the droplet (Fig. 11c,i).^307–309 As such, active tuning of the droplet generation is enabled by local laser heating.

Laser-heating-mediated droplet generation has been demonstrated in both dripping^307–309 and jetting^310,311 flows, and in both flow-focusing³⁰⁷ and T-junction³⁰⁸ microchannels. In dripping regime, the advancing water–oil interface is blocked at the laser heating spot once the heating power exceeds the threshold, in which case the migration velocity u_m is larger than the velocity of the droplet imposed by the basic flow.³⁰⁹ The net force exerted on the droplet due to the Marangoni effect is theoretically³⁰⁷ and experimentally³¹² estimated to be of the order ∼0.1 μN, a sufficiently large force accounting for the blocking. For constant-pressure-driven basic flow, the length of the droplet with laser blocking (L_block) is shorter than that of the unblocked droplet (L₀), while for the constant-flow-rate-driven case, L_block is larger than L₀ and proportional to blocking time τ_b:³⁰⁹

The difference in droplet size variation arises from the different dynamics of neck width evolution. During blocking, the neck width thins exponentially but increases linearly with time at the constant-pressure and constant-flow-rate case, respectively.³⁰⁹

In jetting regime, the interface of the jet is deformed by the Marangoni stress.³¹⁰ The orientation of the deformation is predicted by the following dimensionless ratio A:³¹⁰

with H_d and H_c representing the height of the dispersed jet and that of the continuous fluid, respectively (H_c and H_d are related by H_c + H_d = 0.5h, with h being the channel height). Provided that ρ_d > ρ_c and the laser heating spot has the largest interfacial tension, A >1 (thin jet) indicates the deformation pointing from the dispersed to the continuous phase, and the jet bulges. Conversely, A< 1 (thick jet) stands for the pinching of the jet (Fig. 11c,ii).³¹⁰ When laser power is above a critical value, the Marangoni stress is sufficiently large to break the jet into droplets. In the bulging case, the breakup occurs upstream of the heating spot. On the contrary, fragmentation of the jet occurs downstream of the hot spot where the pinched neck becomes locally unconfined (Fig. 11c,ii).³¹⁰ With continuous heating, the droplet production exhibits a transition from period-1 with high monodispersity to period-2 with two distinct droplets per sequence, and back to period-1 regime as laser power increases.³¹⁰ In addition, Cordero et al. observed the breakup of an aqueous jet under sinusoidal laser heating.³¹¹ The jet's oscillation can synchronize to laser frequency and produce droplets with decreased polydispersity.

In electrical control employing electrochemical and electrocapillary effects,³¹³ the liquid metal EGaIn (75% gallium, 25% indium) and glycerol–NaOH solution are used as the dispersed and continuous phase, respectively. The voltage is applied by connecting the anode and the cathode to the liquid metal and glycerol–NaOH solution inlet tubes, respectively, or vice versa. Either case can lower the interfacial tension between the two fluids. In the application of a positive (oxidative) bias to the liquid metal phase, electrochemically induced formation of a surface oxide acts as a surfactant and lowers the interfacial tension. In the opposite case where the liquid metal phase is negatively charged, the interfacial tension is decreased due to electrocapillary effect. As such, the droplet size decreases with applied voltage in both cases (Fig. 11d) owing to the increase in capillary number Ca_c. The size decrease from electrochemical effect is in a larger range than that from electrocapillary effect (Fig. 11d). Different from DC voltage that produces uniform droplets, low-frequency AC control demonstrates its ability to generate sequences of droplets with various sizes. The size variation originates from the varying magnitude of the voltage.

5.5.3 Channel wettability. Both optical and electrical controls have been used to vary the wettability of the channel wall/substrate, where the former utilizes a photosensitive surfactant, and the latter makes use of the EW effect. In optical control, the surfactant azobenzene trimethylammonium bromide (AzoTAB), with a photosensitive apolar tail, is introduced into the dispersed aqueous phase.³¹⁵ AzoTAB is available in two configurations: trans (less polar) and cis (more polar) forms. The initial trans (cis) form isomerizes to the cis (trans) form upon UV (blue light) illumination, and the transition is reversible when the light is off. Compared to trans-AzoTAB, cis-AzoTAB has a lower substrate wettability. Consequently, the UV-induced dewetting transition, associated with imposed flow rates, results in three behaviors: (i) preserving the initially dripping regime but producing larger droplets; (ii) breaking the initially stable jet into successive droplets; (iii) stabilizing the jet but expanding in its radius (Fig. 11e). In contrast, a blue-light-induced cis-to-trans transition converts the dripping into the stable jet regime.

In electrical control using the EW effect, high-frequency AC is applied³¹⁶ to change the wetting state of the microchannel substrate (Fig. 11f). Under applied voltage U_e, no droplet is generated until the dispersed pressure P_d exceeds an onset value ^316,317

where P_c is the pressure of the continuous phase, P_L(U_e) is the Laplace pressure, and C is a fitting parameter depending on the fluidic resistance of the channel. Laplace pressure P_L(U_e) is expressed as P_L(U_e) = γ(R_w⁻¹ + R_h⁻¹), with R_w and R_h being the principal radii of curvature in the horizontal and vertical planes, respectively. Upon applying the voltage, the contact angle of the dispersed liquid on the channel substrate decreases (Fig. 11f) and R_h, therefore, increases (R_h is initially positive). As γ and R_w are constant, P_L(U_e) and thus decrease with the increase in voltage U_e.^316,317 With appropriate design of EW control, it is even possible to have negative values of R_h and P_L(U_e).³¹⁸ Consequently, spontaneous imbibition of the dispersed fluid into the microchannel can be achieved.³¹⁸

Three regimes of droplet generation in EW have been identified: dripping, tip-streaming and conical spray (Fig. 11f).³¹⁹ Dripping occurs when the dispersed pressure P_d is relatively high, while tip-streaming and conical spray occur at intermediate P_d (at low P_d, no droplet generation). Compared with the tip-streaming, the conical spray occurs at a higher voltage where droplets are charged and repulse each other. While the size of generated droplets is comparable to the dimension of the orifice in dripping regime, they can be smaller than 10 μm in diameter in tip-streaming and conical spray regimes. Generally, the droplet size increases monotonically with increasing P_d.³¹⁹ Nevertheless, the influence of applied voltage is complicated. When the continuous-phase viscous force (∼η_cu_c/L) is weak (e.g., low P_c) compared with the electrostatic force (∼ε_cE²), the droplet size increases with increasing voltage U_e;³¹⁷ when the viscous force is comparable to electrostatic force, the droplet size is nearly constant as U_e increases,^317,319 but the frequency of droplet generation can be varied by one order of magnitude.³¹⁷ The latter offers possible on-demand generation of droplets.³²⁰

5.5.4 Fluid density. Modulating droplet size via varying fluid density is enabled by a phase change process. Wang et al.³²¹ proposed a temperature-controlled vapor bubble condensation process by which the dispersed liquid (n-pentane) is first vaporized in the feeding pipe and then breaks up into vapor bubbles in the co-flow sub-millimeter fluidic channels, followed by a cooling process where vapor bubbles condense into microdroplets. The droplet diameter D is found to be proportional to the vapor bubble diameter D_bubble: D = 0.24D_bubble. This method demonstrates the capability of producing uniform (CV of 2.2–4.0%) microdroplets (43–200 μm in diameter) at a moderate generation frequency (1.3–2000 Hz) in sub-millimeter fluidic devices.

6. Concluding remarks

This review is on fundamentals of passive and active generation of microfluidic droplets. Five fundamental modes of droplet generation, squeezing, dripping, jetting, tip-streaming, and tip-multi-breaking, are identified in passive generation. Each of them has its unique characteristics owing to distinct flow instabilities. Suppression of capillary instability by confinement primarily leads to the superior control and monodispersity of droplet size in the squeezing regime. Dripping, from an absolute flow instability that exhibits intrinsic oscillations, prevents external noises, thus producing monodisperse droplets as well. Convectively unstable flows that amplify perturbations of various frequencies render, on the other hand, droplets non-uniform in the jetting regime. Tip-streaming mode enables droplet microfluidics to produce droplets of size (down to submicrometer-scale) unrestricted by the characteristic length scale of devices. The periodic oscillation of the globally unstable tip offers precise control over droplet size distribution in tip-multi-breaking mode. A deep understanding regarding the effects of viscous, inertial and capillary forces and governing dimensionless parameters (e.g., capillary and Weber numbers) is critical for effective production of droplets with the desired outcome.

Active droplet generation is achieved by incorporating additional components into microfluidic systems. The state-of-the-art techniques mainly apply electrical, magnetic, centrifugal, optical, thermal and mechanical methods, by which electrical, magnetic and centrifugal forces are introduced, and the flow, viscosity, interfacial tension, channel wettability and fluid density are varied. In principle, any approach that could affect interfacial force balance by applying external forces and/or modifying intrinsic viscous, inertial and capillary forces is capable of actively controlling droplet generation. Based on this speculation, new techniques for active control can be developed, for example using magnetorheological effect²⁹⁶ to tune viscosity, thermo-sensitive surfactant (such as diethylhexyl sodium sulphosuccinate³²²) to tune interfacial tension, and magnetowetting effect²⁹⁶ to tune channel wettability. Active control, therefore, offers additional flexibility and a tool in manipulating droplet formation.

Several outstanding issues remain to be resolved in order to produce droplets precisely and effectively. In passive generation, it is essential to design a system that enables a stable tip-streaming over a long period of time. As it occurs at quite low flow rate ratio (φ ≪ 1), tip-streaming is stable typically in less than several minutes before being destabilized by the variation in flow rate from syringe pumps. Using constant-pressure-driven flows in 3D devices is beneficial for the stabilization.¹² Further studies of tip-multi-breaking dynamics are required to produce digital droplets with precise inter-correlation and thus develop novel applications of droplet microfluidics. For active droplet generation, one of the current challenges is to miniaturize and parallelize the system. In addition, most microfluidic chips with active components suffer from complicity in fabrication. Compared with the on-chip integration (e.g., electrodes, microheaters, piezoelectric actuators), the off-chip-actuated devices are easier to fabricate. Moreover, in the on-demand droplet generation, limited systems are able to produce high-throughput (>kHz) monodisperse (CV <5%) droplets in a wide range of tunable volumes (e.g., from femtolitre-scale to nanolitre-scale). This requires the smart design of microfluidic junctions and development of fast-responding actuation. The pulse laser-driven droplet generation²⁵⁴ and piezoelectric dispenser²⁶⁷ appear promising. The delivery of the promising potential of droplet microfluidics relies on our fundamental understanding of the mechanisms responsible for and the development of effective techniques for droplet generation.

Acknowledgements

The financial support from the Research Grants Council of Hong Kong (GRF 17237316, 17211115, 17207914 and 717613E) and the University of Hong Kong (URC 201511159108, 201411159074 and 201311159187) is gratefully acknowledged. The work is also supported in part by the Zhejiang Provincial, Hangzhou Municipal, and Lin'an County Governments.

References

1. S. Y. Teh, R. Lin, L. H. Hung and A. P. Lee, Lab Chip, 2008, 8, 198–220.
2. C. N. Baroud, F. Gallaire and R. Dangla, Lab Chip, 2010, 10, 2032–2045.
3. S. S. Datta, A. Abbaspourrad, E. Amstad, J. Fan, S.-H. Kim, M. Romanowsky, H. C. Shum, B. Sun, A. S. Utada, M. Windbergs, S. Zhou and D. A. Weitz, Adv. Mater., 2014, 26, 2205–2218.
4. T. Y. Lee, T. M. Choi, T. S. Shim, R. A. M. Frijns and S.-H. Kim, Lab Chip, 2016, 16, 3415–3440.
5. T. T. Kong, J. Wu, K. W. K. Yeung, M. K. T. To, H. C. Shum and L. Q. Wang, Biomicrofluidics, 2013, 7, 044128.
6. T. T. Kong, J. Wu, M. To, K. W. K. Yeung, H. C. Shum and L. Q. Wang, Biomicrofluidics, 2012, 6, 034104.
7. T. T. Kong, Z. Liu, Y. Song, L. Q. Wang and H. C. Shum, Soft Matter, 2013, 9, 9780–9784.
8. J. Wu, T. T. Kong, K. W. K. Yeung, H. C. Shum, K. M. C. Cheung, L. Q. Wang and M. K. T. To, Acta Biomater., 2013, 9, 7410–7419.
9. Z. X. Kang, T. T. Kong, L. Y. Lei, P. A. Zhu, X. W. Tian and L. Q. Wang, J. Micromech. Microeng., 2016, 26, 075011.
10. J. Nunes, S. Tsai, J. Wan and H. Stone, J. Phys. D: Appl. Phys., 2013, 46, 114002.
11. T. D. Martz, D. Bardin, P. S. Sheeran, A. P. Lee and P. A. Dayton, Small, 2012, 8, 1876–1879.
12. W.-C. Jeong, J.-M. Lim, J.-H. Choi, J.-H. Kim, Y.-J. Lee, S.-H. Kim, G. Lee, J.-D. Kim, G.-R. Yi and S.-M. Yang, Lab Chip, 2012, 12, 1446–1453.
13. Y. Zeng, M. Shin and T. Wang, Lab Chip, 2013, 13, 267–273.
14. P. Garstecki, H. A. Stone and G. M. Whitesides, Phys. Rev. Lett., 2005, 94, 164501.
15. A. S. Utada, A. Fernandez-Nieves, H. A. Stone and D. A. Weitz, Phys. Rev. Lett., 2007, 99, 094502.
16. R. Suryo and O. A. Basaran, Phys. Fluids, 2006, 18, 082102.
17. P. A. Zhu, T. T. Kong, Z. X. Kang, X. W. Tian and L. Q. Wang, Sci. Rep., 2015, 5, 11102.
18. B.-U. Moon, S. G. Jones, D. K. Hwang and S. S. H. Tsai, Lab Chip, 2015, 15, 2437–2444.
19. Y. Li, M. Jain, Y. Ma and K. Nandakumar, Soft Matter, 2015, 11, 3884–3899.
20. D. J. Collins, A. Neild, A. deMello, A.-Q. Liu and Y. Ai, Lab Chip, 2015, 15, 3439–3459.
21. G. Christopher and S. Anna, J. Phys. D: Appl. Phys., 2007, 40, R319–R336.
22. C.-X. Zhao and A. P. J. Middelberg, Chem. Eng. Sci., 2011, 66, 1394–1411.
23. R. Seemann, M. Brinkmann, T. Pfohl and S. Herminghaus, Rep. Prog. Phys., 2012, 75, 016601.
24. S. L. Anna, Annu. Rev. Fluid Mech., 2016, 48, 285–309.
25. Z. Z. Chong, S. H. Tan, A. M. Gañán-Calvo, S. B. Tor, N. H. Loh and N.-T. Nguyen, Lab Chip, 2016, 16, 35–58.
26. A. M. Gañán-Calvo, Phys. Rev. Lett., 1998, 80, 285.
27. H. A. Stone, A. D. Stroock and A. Ajdari, Annu. Rev. Fluid Mech., 2004, 36, 381–411.
28. T. M. Squires and S. R. Quake, Rev. Mod. Phys., 2005, 77, 977.
29. G. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, 1967.
30. P. Garstecki, M. J. Fuerstman and G. M. Whitesides, Phys. Rev. Lett., 2005, 94, 234502.
31. J. R. Lister and H. A. Stone, Phys. Fluids, 1998, 10, 2758–2764.
32. B. Dollet, W. van Hoeve, J.-P. Raven, P. Marmottant and M. Versluis, Phys. Rev. Lett., 2008, 100, 034504.
33. T. Thorsen, R. W. Roberts, F. H. Arnold and S. R. Quake, Phys. Rev. Lett., 2001, 86, 4163.
34. J. Xu, S. Li, J. Tan, Y. Wang and G. Luo, AIChE J., 2006, 52, 3005–3010.
35. J. H. Xu, G. S. Luo, S. W. Li and G. G. Chen, Lab Chip, 2006, 6, 131–136.
36. L. Q. Wang, Y. X. Zhang and L. Cheng, Chaos, Solitons Fractals, 2009, 39, 1530–1537.
37. K. Khoshmanesh, A. Almansouri, H. Albloushi, P. Yi, R. Soffe and K. Kalantar-zadeh, Sci. Rep., 2015, 5, 9942.
38. A. R. Abate and D. A. Weitz, Lab Chip, 2011, 11, 1713–1716.
39. L. Ménétrier-Deremble and P. Tabeling, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2006, 74, 035303.
40. A. Abate, A. Poitzsch, Y. Hwang, J. Lee, J. Czerwinska and D. Weitz, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2009, 80, 026310.
41. L. Shui, F. Mugele, A. van den Berg and J. C. T. Eijkel, Appl. Phys. Lett., 2008, 93, 153113.
42. L. Shui, A. van den Berg and J. C. T. Eijkel, J. Appl. Phys., 2009, 106, 124305.
43. M. L. Steegmans, K. G. Schroën and R. M. Boom, Langmuir, 2009, 25, 3396–3401.
44. H. Song, D. L. Chen and R. F. Ismagilov, Angew. Chem., Int. Ed., 2006, 45, 7336–7356.
45. B. Zheng, J. D. Tice and R. F. Ismagilov, Anal. Chem., 2004, 76, 4977–4982.
46. Y. X. Zhang and L. Q. Wang, in Advances in Transport Phenomena 2010, Springer-Verlag, Berlin Heidelberg, 2011, pp. 171–294.
47. L. Frenz, J. Blouwolff, A. D. Griffiths and J.-C. Baret, Langmuir, 2008, 24, 12073–12076.
48. R. Lin, J. S. Fisher, M. G. Simon and A. P. Lee, Biomicrofluidics, 2012, 6, 024103.
49. M. Rhee, P. Liu, R. J. Meagher, Y. K. Light and A. K. Singh, Biomicrofluidics, 2014, 8, 034112.
50. Y. Ding, X. C. i Solvas and A. deMello, Analyst, 2015, 140, 414–421.
51. U. Tangen, A. Sharma, P. Wagler and J. S. McCaskill, Biomicrofluidics, 2015, 9, 014119.
52. T. Nisisako and T. Torii, Lab Chip, 2008, 8, 287–293.
53. D. Link, S. L. Anna, D. Weitz and H. Stone, Phys. Rev. Lett., 2004, 92, 054503.
54. P. B. Umbanhowar, V. Prasad and D. A. Weitz, Langmuir, 2000, 16, 347–351.
55. A. Utada, L.-Y. Chu, A. Fernandez-Nieves, D. Link, C. Holtze and D. Weitz, MRS Bull., 2007, 32, 702–708.
56. J. R. Anderson, D. T. Chiu, H. Wu, O. Schueller and G. M. Whitesides, Electrophoresis, 2000, 21, 27–40.
57. C. Cramer, P. Fischer and E. J. Windhab, Chem. Eng. Sci., 2004, 59, 3045–3058.
58. J. M. Gordillo, A. Sevilla and F. Campo-Cortés, J. Fluid Mech., 2014, 738, 335–357.
59. E. Castro-Hernández, V. Gundabala, A. Fernández-Nieves and J. M. Gordillo, New J. Phys., 2009, 11, 075021.
60. A. M. Gañán-Calvo and J. M. Gordillo, Phys. Rev. Lett., 2001, 87, 274501.
61. S. Takeuchi, P. Garstecki, D. B. Weibel and G. M. Whitesides, Adv. Mater., 2005, 17, 1067–1072.
62. L. Yobas, S. Martens, W.-L. Ong and N. Ranganathan, Lab Chip, 2006, 6, 1073–1079.
63. S. L. Anna, N. Bontoux and H. A. Stone, Appl. Phys. Lett., 2003, 82, 364.
64. A. Utada, E. Lorenceau, D. Link, P. Kaplan, H. Stone and D. Weitz, Science, 2005, 308, 537–541.
65. M. Hein, J.-B. Fleury and R. Seemann, Soft Matter, 2015, 11, 5246–5252.
66. E. M. Chan, A. P. Alivisatos and R. A. Mathies, J. Am. Chem. Soc., 2005, 127, 13854–13861.
67. C. Priest, S. Herminghaus and R. Seemann, Appl. Phys. Lett., 2006, 88, 024106.
68. M. Leman, F. Abouakil, A. D. Griffiths and P. Tabeling, Lab Chip, 2015, 15, 753–765.
69. B. Shen, J. Ricouvier, F. Malloggi and P. Tabeling, Adv. Sci., 2016, 3, 1600012.
70. F. Malloggi, N. Pannacci, R. Attia, F. Monti, P. Mary, H. Willaime, P. Tabeling, B. Cabane and P. Poncet, Langmuir, 2010, 26, 2369–2373.
71. L. Shui, A. van den Berg and J. C. Eijkel, Microfluid. Nanofluid., 2011, 11, 87–92.
72. L. Shui, E. Stefan Kooij, D. Wijnperle, A. van den Berg and J. C. T. Eijkel, Soft Matter, 2009, 5, 2708–2712.
73. J. Guzowski and P. Garstecki, Phys. Rev. Lett., 2015, 114, 188302.
74. K. C. van Dijke, G. Veldhuis, K. Schroën and R. M. Boom, AIChE J., 2010, 56, 833–836.
75. V. Chokkalingam, S. Herminghaus and R. Seemann, Appl. Phys. Lett., 2008, 93, 254101.
76. C. Cohen, R. Giles, V. Sergeyeva, N. Mittal, P. Tabeling, D. Zerrouki, J. Baudry, J. Bibette and N. Bremond, Microfluid. Nanofluid., 2014, 17, 959–966.
77. N. Mittal, C. Cohen, J. Bibette and N. Bremond, Phys. Fluids, 2014, 26, 082109.
78. R. Dangla, S. C. Kayi and C. N. Baroud, Proc. Natl. Acad. Sci. U. S. A., 2013, 110, 853–858.
79. K. van Dijke, G. Veldhuis, K. Schroen and R. Boom, Lab Chip, 2009, 9, 2824–2830.
80. K. van Dijke, R. de Ruiter, K. Schroen and R. Boom, Soft Matter, 2010, 6, 321–330.
81. F. Dutka, A. S. Opalski and P. Garstecki, Lab Chip, 2016, 16, 2044–2049.
82. F. Schuler, F. Schwemmer, M. Trotter, S. Wadle, R. Zengerle, F. von Stetten and N. Paust, Lab Chip, 2015, 15, 2759–2766.
83. F. Schuler, N. Paust, R. Zengerle and F. von Stetten, Micromachines, 2015, 6, 1180–1188.
84. F. Schuler, M. Trotter, M. Geltman, F. Schwemmer, S. Wadle, E. Dominguez-Garrido, M. Lopez, C. Cervera-Acedo, P. Santibanez, F. von Stetten, R. Zengerle and N. Paust, Lab Chip, 2016, 16, 208–216.
85. S. Kahkeshani and D. Di Carlo, Lab Chip, 2016, 16, 2474–2480.
86. G. T. Vladisavljević, I. Kobayashi and M. Nakajima, Microfluid. Nanofluid., 2012, 13, 151–178.
87. G. T. Vladisavljević, N. Khalid, M. A. Neves, T. Kuroiwa, M. Nakajima, K. Uemura, S. Ichikawa and I. Kobayashi, Adv. Drug Delivery Rev., 2013, 65, 1626–1663.
88. I. Kobayashi, Y. Wada, K. Uemura and M. Nakajima, Microfluid. Nanofluid., 2010, 8, 255–262.
89. Y. Kikuchi, H. Ohki, T. Kaneko and K. Sato, Biorheology, 1989, 26, 1055.
90. T. Kawakatsu, Y. Kikuchi and M. Nakajima, J. Am. Oil Chem. Soc., 1997, 74, 317–321.
91. T. Kawakatsu, H. Komori, M. Nakajima, Y. Kikuchi and T. Yonemoto, J. Chem. Eng. Jpn., 1999, 32, 241–244.
92. T. Kawakatsu, G. Trägårdh, Y. Kikuchi, M. Nakajima, H. Komori and T. Yonemoto, J. Surfactants Deterg., 2000, 3, 295–302.
93. I. Kobayashi, M. Nakajima, K. Chun, Y. Kikuchi and H. Fujita, AIChE J., 2002, 48, 1639–1644.
94. I. Kobayashi, S. Mukataka and M. Nakajima, Langmuir, 2005, 21, 7629–7632.
95. S. Sugiura, T. Oda, Y. Izumida, Y. Aoyagi, M. Satake, A. Ochiai, N. Ohkohchi and M. Nakajima, Biomaterials, 2005, 26, 3327–3331.
96. I. Kobayashi, Y. Murayama, T. Kuroiwa, K. Uemura and M. Nakajima, Microfluid. Nanofluid., 2009, 7, 107–119.
97. I. Kobayashi, S. Mukataka and M. Nakajima, J. Colloid Interface Sci., 2004, 279, 277–280.
98. T. Nakashima, M. Shimizu and M. Kukizaki, Key Eng. Mater., 1991, 61–62, 513–516.
99. T. Nakashima, M. Shimizu and M. Kukizaki, Adv. Drug Delivery Rev., 2000, 45, 47–56.
100. K. Suzuki, I. Shuto and Y. Hagura, Food Sci. Technol. Int., Tokyo, 1996, 2, 43–47.
101. M. Yasuda, T. Goda, H. Ogino, W. R. Glomm and H. Takayanagi, J. Colloid Interface Sci., 2010, 349, 392–401.
102. S. R. Kosvintsev, G. Gasparini, R. G. Holdich, I. W. Cumming and M. T. Stillwell, Ind. Eng. Chem. Res., 2005, 44, 9323–9330.
103. M. M. Dragosavac, M. N. Sovilj, S. R. Kosvintsev, R. G. Holdich and G. T. Vladisavljević, J. Membr. Sci., 2008, 322, 178–188.
104. G. T. Vladisavljević and R. A. Williams, J. Colloid Interface Sci., 2006, 299, 396–402.
105. M. S. Manga, O. J. Cayre, R. A. Williams, S. Biggs and D. W. York, Soft Matter, 2012, 8, 1532–1538.
106. J. Zhu and D. Barrow, J. Membr. Sci., 2005, 261, 136–144.
107. R. G. Holdich, M. M. Dragosavac, G. T. Vladisavljević and S. R. Kosvintsev, Ind. Eng. Chem. Res., 2010, 49, 3810–3817.
108. N. A. Wagdare, A. T. Marcelis, O. B. Ho, R. M. Boom and C. J. van Rijn, J. Membr. Sci., 2010, 347, 1–7.
109. G. T. Vladisavljević and H. Schubert, J. Dispersion Sci. Technol., 2003, 24, 811–819.
110. P. J. Dowding, J. W. Goodwin and B. Vincent, Colloids Surf., A, 2001, 180, 301–309.
111. X. Liu, D. Bao, W. Xue, Y. Xiong, W. Yu, X. Yu, X. Ma and Q. Yuan, J. Appl. Polym. Sci., 2003, 87, 848–852.
112. I. Kobayashi, M. Yasuno, S. Iwamoto, A. Shono, K. Satoh and M. Nakajima, Colloids Surf., A, 2002, 207, 185–196.
113. L. Giorno, N. Li and E. Drioli, J. Membr. Sci., 2003, 217, 173–180.
114. M. M. Dragosavac, G. T. Vladisavljević, R. G. Holdich and M. T. Stillwell, Langmuir, 2011, 28, 134–143.
115. G. T. Vladisavljević, Adv. Colloid Interface Sci., 2015, 225, 53–87.
116. G. De Luca, F. P. Di Maio, A. Di Renzo and E. Drioli, Chem. Eng. Process., 2008, 47, 1150–1158.
117. N. C. Christov, K. D. Danov, D. K. Danova and P. A. Kralchevsky, Langmuir, 2008, 24, 1397–1410.
118. A. Timgren, G. Trägårdh and C. Trägårdh, Chem. Eng. Res. Des., 2010, 88, 229–238.
119. A. Abrahamse, A. Van der Padt, R. Boom and W. De Heij, AIChE J., 2001, 47, 1285–1291.
120. M. Rayner, G. Trägårdh, C. Trägårdh and P. Dejmek, J. Colloid Interface Sci., 2004, 279, 175–185.
121. E. Amstad, S. Datta and D. Weitz, Lab Chip, 2014, 14, 705–709.
122. L. Wu, G.-P. Li, W. Xu and M. Bachman, Appl. Phys. Lett., 2006, 89, 144106.
123. X. Huang, W. Hui, C. Hao, W. Yue, M. Yang, Y. Cui and Z. Wang, Small, 2014, 10, 758–765.
124. Y. X. Zhang and L. Q. Wang, Nanoscale Microscale Thermophys. Eng., 2009, 13, 228–242.
125. T. Ward, M. Faivre, M. Abkarian and H. A. Stone, Electrophoresis, 2005, 26, 3716–3724.
126. P. Garstecki, I. Gitlin, W. DiLuzio, G. M. Whitesides, E. Kumacheva and H. A. Stone, Appl. Phys. Lett., 2004, 85, 2649–2651.
127. S. L. Anna and H. C. Mayer, Phys. Fluids, 2006, 18, 121512.
128. J. D. Tice, A. D. Lyon and R. F. Ismagilov, Anal. Chim. Acta, 2004, 507, 73–77.
129. P. Garstecki, M. J. Fuerstman, H. A. Stone and G. M. Whitesides, Lab Chip, 2006, 6, 437–446.
130. M. De Menech, P. Garstecki, F. Jousse and H. Stone, J. Fluid Mech., 2008, 595, 141–161.
131. M. Zagnoni, J. Anderson and J. M. Cooper, Langmuir, 2010, 26, 9416–9422.
132. J. Fan, Y. X. Zhang and L. Q. Wang, Nano, 2010, 5, 175–184.
133. P. Guillot, A. Colin, A. S. Utada and A. Ajdari, Phys. Rev. Lett., 2007, 99, 104502.
134. B. R. Benson, H. A. Stone and R. K. Prud'homme, Lab Chip, 2013, 13, 4507–4511.
135. J. Sivasamy, T.-N. Wong, N.-T. Nguyen and L. T.-H. Kao, Microfluid. Nanofluid., 2011, 11, 1–10.
136. A. R. Abate, P. Mary, V. van Steijn and D. A. Weitz, Lab Chip, 2012, 12, 1516–1521.
137. P. A. Romero and A. R. Abate, Lab Chip, 2012, 12, 5130–5132.
138. J. Xu, S. Li, J. Tan and G. Luo, Microfluid. Nanofluid., 2008, 5, 711–717.
139. S. van der Graaf, T. Nisisako, C. Schroen, R. van der Sman and R. Boom, Langmuir, 2006, 22, 4144–4152.
140. J. D. Tice, H. Song, A. D. Lyon and R. F. Ismagilov, Langmuir, 2003, 19, 9127–9133.
141. A. Gupta and R. Kumar, Microfluid. Nanofluid., 2010, 8, 799–812.
142. Y. Ba, H. Liu, J. Sun and R. Zheng, Int. J. Heat Mass Transfer, 2015, 90, 931–947.
143. H. Liu and Y. Zhang, J. Appl. Phys., 2009, 106, 034906.
144. V. van Steijn, C. R. Kleijn and M. T. Kreutzer, Phys. Rev. Lett., 2009, 103, 214501.
145. V. van Steijn, M. T. Kreutzer and C. R. Kleijn, Chem. Eng. Sci., 2007, 62, 7505–7514.
146. V. van Steijn, C. R. Kleijn and M. T. Kreutzer, Lab Chip, 2010, 10, 2513–2518.
147. X. Chen, T. Glawdel, N. Cui and C. L. Ren, Microfluid. Nanofluid., 2015, 18, 1341–1353.
148. D. Funfschilling, H. Debas, H.-Z. Li and T. Mason, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2009, 80, 015301.
149. K. Xu, C. P. Tostado, J.-H. Xu, Y.-C. Lu and G.-S. Luo, Lab Chip, 2014, 14, 1357–1366.
150. W. Lan, S. Li and G. Luo, Chem. Eng. Sci., 2015, 134, 76–85.
151. H. Liu and Y. Zhang, Phys. Fluids, 2011, 23, 082101.
152. G. F. Christopher, N. N. Noharuddin, J. A. Taylor and S. L. Anna, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2008, 78, 036317.
153. R. M. Erb, D. Obrist, P. W. Chen, J. Studer and A. R. Studart, Soft Matter, 2011, 7, 8757–8761.
154. Y. Liu and L. Yobas, Appl. Phys. Lett., 2015, 106, 174101.
155. J. Husny and J. J. Cooper-White, J. Non-Newtonian Fluid Mech., 2006, 137, 121–136.
156. Y. X. Zhang, J. Fan and L. Q. Wang, Curr. Nanosci., 2009, 5, 519–526.
157. A. Gupta, S. M. S. Murshed and R. Kumar, Appl. Phys. Lett., 2009, 94, 164107.
158. T. Fu, Y. Ma, D. Funfschilling, C. Zhu and H. Z. Li, Chem. Eng. Sci., 2010, 65, 3739–3748.
159. T. Glawdel, C. Elbuken and C. L. Ren, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2012, 85, 016323.
160. T. Glawdel, C. Elbuken and C. L. Ren, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2012, 85, 016322.
161. A. Sauret and H. C. Shum, Int. J. Nonlinear Sci. Numer. Simul., 2012, 13, 351–362.
162. T. Cubaud and T. G. Mason, Phys. Fluids, 2008, 20, 053302.
163. S. Tomotika, Proc. R. Soc. London, Ser. A, 1935, 150, 322–337.
164. G. Taylor, Proc. R. Soc. London, Ser. A, 1934, 146, 501–523.
165. H. A. Stone, Annu. Rev. Fluid Mech., 1994, 26, 65–102.
166. A. M. Gañán-Calvo, R. González-Prieto, P. Riesco-Chueca, M. A. Herrada and M. Flores-Mosquera, Nat. Phys., 2007, 3, 737–742.
167. O. A. Basaran and R. Suryo, Nat. Phys., 2007, 3, 679–680.
168. E. Castro-Hernández, F. Campo-Cortés and J. M. Gordillo, J. Fluid Mech., 2012, 698, 423–445.
169. A. G. Marín, F. Campo-Cortés and J. M. Gordillo, Colloids Surf., A, 2009, 344, 2–7.
170. J.-U. Shim, R. T. Ranasinghe, C. A. Smith, S. M. Ibrahim, F. Hollfelder, W. T. Huck, D. Klenerman and C. Abell, ACS Nano, 2013, 7, 5955–5964.
171. R. A. De Bruijn, Chem. Eng. Sci., 1993, 48, 277–284.
172. J. Janssen, A. Boon and W. Agterof, AIChE J., 1994, 40, 1929–1939.
173. J. Janssen, A. Boon and W. Agterof, AIChE J., 1997, 43, 1436–1447.
174. C. D. Eggleton, T.-M. Tsai and K. J. Stebe, Phys. Rev. Lett., 2001, 87, 048302.
175. M. Booty and M. Siegel, J. Fluid Mech., 2005, 544, 243–275.
176. W. Lee, L. M. Walker and S. L. Anna, Phys. Fluids, 2009, 21, 032103.
177. T. M. Moyle, L. M. Walker and S. L. Anna, Phys. Fluids, 2012, 24, 082110.
178. T. Ward, M. Faivre and H. A. Stone, Langmuir, 2010, 26, 9233–9239.
179. W. W. Zhang, Phys. Rev. Lett., 2004, 93, 184502.
180. J. Fernández de la Mora, Annu. Rev. Fluid Mech., 2007, 39, 217–243.
181. A. M. Gañán-Calvo and J. M. Montanero, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2009, 79, 066305.
182. A. M. Gañán-Calvo, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2008, 78, 026304.
183. Y.-H. Tseng and A. Prosperetti, J. Fluid Mech., 2015, 776, 5–36.
184. P. A. Zhu, T. T. Kong, L. Y. Lei, X. W. Tian, Z. X. Kang and L. Q. Wang, Sci. Rep., 2016, 6, 21527.
185. P. Garstecki, M. J. Fuerstman and G. M. Whitesides, Nat. Phys., 2005, 1, 168–171.
186. M. Hashimoto and G. M. Whitesides, Small, 2010, 6, 1051–1059.
187. A. Evangelio, F. Campo-Cortes and J. M. Gordillo, J. Fluid Mech., 2015, 778, 653–668.
188. M. A. Herrada, A. M. Gañán-Calvo, A. Ojeda-Monge, B. Bluth and P. Riesco-Chueca, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2008, 78, 036323.
189. J. M. Montanero, N. Rebollo-Muñoz, M. A. Herrada and A. M. Gañán-Calvo, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2011, 83, 036309.
190. E. J. Vega, J. M. Montanero, M. A. Herrada and A. M. Gañán-Calvo, Phys. Fluids, 2010, 22, 064105.
191. T. Si, F. Li, X.-Y. Yin and X.-Z. Yin, J. Fluid Mech., 2009, 629, 1–23.
192. J. E. Kreutz, T. Munson, T. Huynh, F. Shen, W. Du and R. F. Ismagilov, Anal. Chem., 2011, 83, 8158–8168.
193. F. Shen, B. Sun, J. E. Kreutz, E. K. Davydova, W. Du, P. L. Reddy, L. J. Joseph and R. F. Ismagilov, J. Am. Chem. Soc., 2011, 133, 17705–17712.
194. J. Guzowski, K. Gizynski, J. Gorecki and P. Garstecki, Lab Chip, 2016, 16, 764–772.
195. K. J. Humphry, A. Ajdari, A. Fernández-Nieves, H. A. Stone and D. A. Weitz, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2009, 79, 056310.
196. D. Saeki, S. Sugiura, T. Kanamori, S. Sato, S. Mukataka and S. Ichikawa, Langmuir, 2008, 24, 13809–13813.
197. Z. Li, A. M. Leshansky, L. M. Pismen and P. Tabeling, Lab Chip, 2015, 15, 1023–1031.
198. R. Dangla, E. Fradet, Y. Lopez and C. N. Baroud, J. Phys. D: Appl. Phys., 2013, 46, 114003.
199. M. Hein, S. Afkhami, R. Seemann and L. Kondic, Microfluid. Nanofluid., 2014, 18, 911–917.
200. J.-C. Baret, Lab Chip, 2012, 12, 422–433.
201. M. L. Steegmans, A. Warmerdam, K. G. Schroen and R. M. Boom, Langmuir, 2009, 25, 9751–9758.
202. J. H. Xu, P. F. Dong, H. Zhao, C. P. Tostado and G. S. Luo, Langmuir, 2012, 28, 9250–9258.
203. Y. Chen, G.-T. Liu, J.-H. Xu and G.-S. Luo, Chem. Eng. Sci., 2015, 132, 1–8.
204. Y. Chen, J.-H. Xu and G.-S. Luo, Chem. Eng. Sci., 2015, 138, 655–662.
205. K. Wang, Y. Lu, J. Xu and G. Luo, Langmuir, 2009, 25, 2153–2158.
206. K. Wang, L. Zhang, W. Zhang and G. Luo, Langmuir, 2016, 32, 3174–3185.
207. J.-C. Baret, F. Kleinschmidt, A. El Harrak and A. D. Griffiths, Langmuir, 2009, 25, 6088–6093.
208. Y.-C. Liao, E. I. Franses and O. A. Basaran, Phys. Fluids, 2006, 18, 022101.
209. M. R. de Saint Vincent, J. Petit, M. Aytouna, J.-P. Delville, D. Bonn and H. Kellay, J. Fluid Mech., 2012, 692, 499–510.
210. M. Roché, M. Aytouna, D. Bonn and H. Kellay, Phys. Rev. Lett., 2009, 103, 264501.
211. Q. Xu, Y.-C. Liao and O. A. Basaran, Phys. Rev. Lett., 2007, 98, 054503.
212. F. Jin, N. R. Gupta and K. J. Stebe, Phys. Fluids, 2006, 18, 022103.
213. N. M. Kovalchuk, E. Nowak and M. J. H. Simmons, Langmuir, 2016, 32, 5069–5077.
214. N.-T. Nguyen, S. Lassemono, F. A. Chollet and C. Yang, IEEE Sens. J., 2007, 7, 692–697.
215. K. Muijlwijk, E. Hinderink, D. Ershov, C. Berton-Carabin and K. Schroën, J. Colloid Interface Sci., 2016, 470, 71–79.
216. T. Glawdel and C. L. Ren, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2012, 86, 026308.
217. X. Wang, A. Riaud, K. Wang and G. Luo, Microfluid. Nanofluid., 2015, 18, 503–512.
218. W. Lan, C. Wang, X. Guo, S. Li and G. Luo, AIChE J., 2016, 62, 2542–2549.
219. J. M. Park, M. Hulsen and P. Anderson, Eur. Phys. J.: Spec. Top., 2013, 222, 199–210.
220. P. Huerre and P. A. Monkewitz, Annu. Rev. Fluid Mech., 1990, 22, 473–537.
221. J. Eggers and E. Villermaux, Rep. Prog. Phys., 2008, 71, 036601.
222. P. Guillot, A. Colin and A. Ajdari, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2008, 78, 016307.
223. M. A. Herrada, A. M. Gañán-Calvo and P. Guillot, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2008, 78, 046312.
224. A. M. Gañán-Calvo and P. Riesco-Chueca, J. Fluid Mech., 2006, 553, 75–84.
225. A. S. Utada, A. Fernandez-Nieves, J. M. Gordillo and D. A. Weitz, Phys. Rev. Lett., 2008, 100, 014502.
226. J. Chomaz, P. Huerre and L. Redekopp, Phys. Rev. Lett., 1988, 60, 25.
227. D. R. Link, E. Grasland-Mongrain, A. Duri, F. Sarrazin, Z. Cheng, G. Cristobal, M. Marquez and D. A. Weitz, Angew. Chem., Int. Ed., 2006, 45, 2556–2560.
228. S. H. Tan, B. Semin and J.-C. Baret, Lab Chip, 2014, 14, 1099–1106.
229. M. He, J. S. Kuo and D. T. Chiu, Appl. Phys. Lett., 2005, 87, 031916.
230. P. He, H. Kim, D. Luo, M. Marquez and Z. Cheng, Appl. Phys. Lett., 2010, 96, 174103.
231. H. Kim, D. Luo, D. Link, D. A. Weitz, M. Marquez and Z. Cheng, Appl. Phys. Lett., 2007, 91, 133106.
232. C.-H. Yeh, M.-H. Lee and Y.-C. Lin, Microfluid. Nanofluid., 2012, 12, 475–484.
233. M. He, J. S. Kuo and D. T. Chiu, Langmuir, 2006, 22, 6408–6413.
234. S. H. Tan, F. Maes, B. Semin, J. Vrignon and J.-C. Baret, Sci. Rep., 2014, 4, 4787.
235. E. Castro-Hernández, P. García-Sánchez, J. Alzaga-Gimeno, S. H. Tan, J.-C. Baret and A. Ramos, Biomicrofluidics, 2016, 10, 043504.
236. E. Castro-Hernández, P. García-Sánchez, S. H. Tan, A. M. Gañán-Calvo, J.-C. Baret and A. Ramos, Microfluid. Nanofluid., 2015, 19, 787–794.
237. S.-H. Tan, N.-T. Nguyen, L. Yobas and T. G. Kang, J. Micromech. Microeng., 2010, 20, 045004.
238. Y. Wu, T. Fu, Y. Ma and H. Z. Li, Soft Matter, 2013, 9, 9792–9798.
239. J. Liu, S.-H. Tan, Y. F. Yap, M. Y. Ng and N.-T. Nguyen, Microfluid. Nanofluid., 2011, 11, 177–187.
240. S. H. Tan and N.-T. Nguyen, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2011, 84, 036317.
241. Q. Yan, S. Xuan, X. Ruan, J. Wu and X. Gong, Microfluid. Nanofluid., 2015, 19, 1377–1384.
242. C.-P. Lee, T.-S. Lan and M.-F. Lai, J. Appl. Phys., 2014, 115, 17B527.
243. J. Liu, Y. F. Yap and N.-T. Nguyen, Phys. Fluids, 2011, 23, 072008.
244. O. Strohmeier, M. Keller, F. Schwemmer, S. Zehnle, D. Mark, F. von Stetten, R. Zengerle and N. Paust, Chem. Soc. Rev., 2015, 44, 6187–6229.
245. M. Madou, J. Zoval, G. Jia, H. Kido, J. Kim and N. Kim, Annu. Rev. Biomed. Eng., 2006, 8, 601–628.
246. J. Ducrée, S. Haeberle, S. Lutz, S. Pausch, F. Von Stetten and R. Zengerle, J. Micromech. Microeng., 2007, 17, S103–S115.
247. D. Mark, S. Haeberle, R. Zengerle, J. Ducree and G. T. Vladisavljević, J. Colloid Interface Sci., 2009, 336, 634–641.
248. K. Maeda, H. Onoe, M. Takinoue and S. Takeuchi, Adv. Mater., 2012, 24, 1340–1346.
249. S. Haeberle, L. Naegele, R. Burger, F. v. Stetten, R. Zengerle and J. Ducrée, J. Microencapsulation, 2008, 25, 267–274.
250. S. Haeberle, R. Zengerle and J. Ducrée, Microfluid. Nanofluid., 2007, 3, 65–75.
251. D. Chakraborty and S. Chakraborty, Appl. Phys. Lett., 2010, 97, 234103.
252. S. Kar, S. Joshi, K. Chaudhary, T. K. Maiti and S. Chakraborty, Appl. Phys. Lett., 2015, 107, 244101.
253. Y. Ren and W. W. F. Leung, Micromachines, 2016, 7, 17.
254. S.-Y. Park, T.-H. Wu, Y. Chen, M. A. Teitell and P.-Y. Chiou, Lab Chip, 2011, 11, 1010–1012.
255. A. Sauret, C. Spandagos and H. C. Shum, Lab Chip, 2012, 12, 3380–3386.
256. H. C. Shum, J. Varnell and D. A. Weitz, Biomicrofluidics, 2012, 6, 012808.
257. A. Sauret and H. C. Shum, Appl. Phys. Lett., 2012, 100, 154106.
258. J. Li, N. Mittal, S. Y. Mak, Y. Song and H. C. Shum, J. Micromech. Microeng., 2015, 25, 084009.
259. P. A. Zhu, X. Tang and L. Q. Wang, Microfluid. Nanofluid., 2016, 20, 47.
260. P. A. Zhu, X. Tang, Y. Tian and L. Q. Wang, Sci. Rep., 2016, 6, 31436.
261. H. Zhou and S. Yao, Microfluid. Nanofluid., 2014, 16, 667–675.
262. K. Churski, P. Korczyk and P. Garstecki, Lab Chip, 2010, 10, 816–818.
263. M. E. Dolega, S. Jakiela, M. Razew, A. Rakszewska, O. Cybulski and P. Garstecki, Lab Chip, 2012, 12, 4022–4025.
264. M. Yu, Y. Hou, H. Zhou and S. Yao, Lab Chip, 2015, 15, 1524–1532.
265. S. Jakiela, P. R. Debski, B. Dabrowski and P. Garstecki, Micromachines, 2014, 5, 1002–1011.
266. S.-Y. Jung, S. T. Retterer and C. P. Collier, Lab Chip, 2010, 10, 2688–2694.
267. J. Xu and D. Attinger, J. Micromech. Microeng., 2008, 18, 065020.
268. A. Bransky, N. Korin, M. Khoury and S. Levenberg, Lab Chip, 2009, 9, 516–520.
269. Y. N. Cheung and H. Qiu, J. Micromech. Microeng., 2012, 22, 125003.
270. Y. N. Cheung and H. Qiu, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2011, 84, 066310.
271. I. Ziemecka, V. van Steijn, G. J. M. Koper, M. Rosso, A. M. Brizard, J. H. van Esch and M. T. Kreutzer, Lab Chip, 2011, 11, 620–624.
272. J. Shemesh, A. Nir, A. Bransky and S. Levenberg, Lab Chip, 2011, 11, 3225–3230.
273. J. C. Brenker, D. J. Collins, H. Van Phan, T. Alan and A. Neild, Lab Chip, 2016, 16, 1675–1683.
274. D. J. Collins, T. Alan, K. Helmerson and A. Neild, Lab Chip, 2013, 13, 3225–3231.
275. L. Schmid and T. Franke, Lab Chip, 2013, 13, 1691–1694.
276. L. Schmid and T. Franke, Appl. Phys. Lett., 2014, 104, 133501.
277. A. R. Abate, M. B. Romanowsky, J. J. Agresti and D. A. Weitz, Appl. Phys. Lett., 2009, 94, 023503.
278. C.-Y. Lee, Y.-H. Lin and G.-B. Lee, Microfluid. Nanofluid., 2009, 6, 599–610.
279. H.-W. Wu, Y.-C. Huang, C.-L. Wu and G.-B. Lee, Microfluid. Nanofluid., 2009, 7, 45–56.
280. A. Raj, R. Halder, P. Sajeesh and A. K. Sen, Microfluid. Nanofluid., 2016, 20, 102.
281. B. Cai, R. He, X. Yu, L. Rao, Z. He, Q. Huang, W. Liu, S. Guo and X.-Z. Zhao, Microfluid. Nanofluid., 2016, 20, 56.
282. J.-W. Choi, S. Lee, D.-H. Lee, J. Kim and S.-I. Chang, RSC Adv., 2014, 4, 20341–20345.
283. S. Zeng, B. Li, X. o. Su, J. Qin and B. Lin, Lab Chip, 2009, 9, 1340–1343.
284. K. Churski, J. Michalski and P. Garstecki, Lab Chip, 2010, 10, 512–518.
285. F. Guo, K. Liu, X.-H. Ji, H.-J. Ding, M. Zhang, Q. Zeng, W. Liu, S.-S. Guo and X.-Z. Zhao, Appl. Phys. Lett., 2010, 97, 233701.
286. J.-C. Galas, D. Bartolo and V. Studer, New J. Phys., 2009, 11, 075027.
287. H. Willaime, V. Barbier, L. Kloul, S. Maine and P. Tabeling, Phys. Rev. Lett., 2006, 96, 054501.
288. J.-H. Wang and G.-B. Lee, Micromachines, 2013, 4, 306–320.
289. W. S. Lee, S. Jambovane, D. Kim and J. W. Hong, Microfluid. Nanofluid., 2009, 7, 431–438.
290. B.-C. Lin and Y.-C. Su, J. Micromech. Microeng., 2008, 18, 115005.
291. J.-H. Choi, S.-K. Lee, J.-M. Lim, S.-M. Yang and G.-R. Yi, Lab Chip, 2010, 10, 456–461.
292. C.-T. Chen and G.-B. Lee, J. Microelectromech. Syst., 2006, 15, 1492–1498.
293. S.-K. Hsiung, C.-T. Chen and G.-B. Lee, J. Micromech. Microeng., 2006, 16, 2403–2410.
294. C.-H. Lee, S.-K. Hsiung and G.-B. Lee, J. Micromech. Microeng., 2007, 17, 1121–1129.
295. M. Zhang, J. Wu, X. Niu, W. Wen and P. Sheng, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2008, 78, 066305.
296. N.-T. Nguyen, Microfluid. Nanofluid., 2012, 12, 1–16.
297. N.-T. Nguyen, T.-H. Ting, Y.-F. Yap, T.-N. Wong, J. C.-K. Chai, W.-L. Ong, J. Zhou, S.-H. Tan and L. Yobas, Appl. Phys. Lett., 2007, 91, 084102.
298. P. Sheng and W. Wen, Annu. Rev. Fluid Mech., 2012, 44, 143–174.
299. X. Niu, M. Zhang, J. Wu, W. Wen and P. Sheng, Soft Matter, 2009, 5, 576–581.
300. C. A. Stan, S. K. Tang and G. M. Whitesides, Anal. Chem., 2009, 81, 2399–2402.
301. S.-H. Tan, S. S. Murshed, N.-T. Nguyen, T. N. Wong and L. Yobas, J. Phys. D: Appl. Phys., 2008, 41, 165501.
302. S. S. Murshed, S. H. Tan, N. T. Nguyen, T. N. Wong and L. Yobas, Microfluid. Nanofluid., 2009, 6, 253–259.
303. S. S. Murshed, S.-H. Tan and N.-T. Nguyen, J. Phys. D: Appl. Phys., 2008, 41, 085502.
304. C.-H. Yeh, K.-R. Chen and Y.-C. Lin, Microfluid. Nanofluid., 2013, 15, 775–784.
305. R. Suryo and O. A. Basaran, Phys. Rev. Lett., 2006, 96, 034504.
306. V. Miralles, A. Huerre, H. Williams, B. Fournie and M.-C. Jullien, Lab Chip, 2015, 15, 2133–2139.
307. C. N. Baroud, J.-P. Delville, F. Gallaire and R. Wunenburger, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2007, 75, 046302.
308. C. N. Baroud, M. R. de Saint Vincent and J.-P. Delville, Lab Chip, 2007, 7, 1029–1033.
309. M. R. de Saint Vincent and J.-P. Delville, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2012, 85, 026310.
310. M. R. de Saint Vincent, H. Chraïbi and J.-P. Delville, Phys. Rev. Appl., 2015, 4, 044005.
311. M. L. Cordero, F. Gallaire and C. N. Baroud, Phys. Fluids, 2011, 23, 094111.
312. E. Verneuil, M. a. Cordero, F. Gallaire and C. N. Baroud, Langmuir, 2009, 25, 5127–5134.
313. S.-Y. Tang, I. D. Joshipura, Y. Lin, K. Kalantar-Zadeh, A. Mitchell, K. Khoshmanesh and M. D. Dickey, Adv. Mater., 2016, 28, 604–609.
314. B. Berge, O. Konovalov, J. Lajzerowicz, A. Renault, J. Rieu, M. Vallade, J. Als-Nielsen, G. Grübel and J. Legrand, Phys. Rev. Lett., 1994, 73, 1652.
315. A. Diguet, H. Li, N. Queyriaux, Y. Chen and D. Baigl, Lab Chip, 2011, 11, 2666–2669.
316. F. Malloggi, S. A. Vanapalli, H. Gu, D. van den Ende and F. Mugele, J. Phys.: Condens. Matter, 2007, 19, 462101.
317. F. Malloggi, H. Gu, A. Banpurkar, S. Vanapalli and F. Mugele, Eur. Phys. J. E: Soft Matter Biol. Phys., 2008, 26, 91–96.
318. H. Gu, M. H. Duits and F. Mugele, Lab Chip, 2010, 10, 1550–1556.
319. H. Gu, F. Malloggi, S. A. Vanapalli and F. Mugele, Appl. Phys. Lett., 2008, 93, 183507.
320. H. Gu, C. U. Murade, M. H. Duits and F. Mugele, Biomicrofluidics, 2011, 5, 011101.
321. K. Wang, L. Xie, Y. Lu and G. Luo, Lab Chip, 2013, 13, 73–76.
322. G. Bolognesi, A. Hargreaves, A. D. Ward, A. K. Kirby, C. D. Bain and O. Ces, RSC Adv., 2015, 5, 8114–8121.
323. M. Zhang, J. Wu, L. Wang, K. Xiao and W. Wen, Lab Chip, 2010, 10, 1199–1203.
324. S. Waheed, J. M. Cabot, N. P. Macdonald, T. Lewis, R. M. Guijt, B. Paull and M. C. Breadmore, Lab Chip, 2016, 16, 1993–2013.

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