2. Two-phase flow in microchannels

In two-phase flow microfluidics, dispersed and continuous phase fluids generally were dispensed separately into the microfluidic device. The continuous and dispersed phase channels typically meet at a junction, depending on the specific geometry of a microfluidic device. Each shape of the junction helps to define the local flow fields that deform the interface between the two immiscible fluids [6, 7]. These configurations are also shown schematically in Table 1. When the instabilities of free surface between phases are sufficiently large, drops in microscale emerge and eventually pinch-off from the dispersed phase. There are many controlling parameters that will affect the microdroplet generation regime such as, interfacial tension [8, 9], surface Microdroplets Advancement in Newtonian and Non-Newtonian Microfluidic Multiphase System http://dx.doi.org/10.5772/intechopen.75358 143


Reproduced from Ref. [7, 15–17] with permission from the Physics of Fluids, Physical Review Letter and Lab on Chip.

Table 1. Main approaches for droplet breakup in microchannel.

velocity flow, the shear stress in a steady, laminar flow condition is having a linear relationship with its shear rate. In contrast, non-Newtonian fluids, examples of which in daily life including chocolate, toothpaste, lubricating oils experience a non-linear relationship for its shear stress versus shear rate curve due to the fact that the fluid viscosity is a variable at any given flow condition, i.e., temperature and pressure. Such fluid properties lead to much differences in term of the emulsion productions thus far in all engineering approaches. Conventionally, emulsion is prepared using high-pressure homogenizers and colloid mill. These devices apply high mechanical shear force to break up the large emulsion into smaller ones that are subsequently stabilized by the use of emulsifier [2]. However, such method contributed to large size distribution of emulsions formed [3], leading to material loss as well as emulsion function efficiency issue. This dispersity issued was then resolved by integration of microfluidic technology that was firstly developed in 1950s [4]. Microfluidic technology is defined as a branch of fluid mechanics that focuses on the understanding, designing, fabrications and operations of system that convey liquids inside channels with two of the three geometry length scales in the order of microns [5]. Furthermore, with the length scale associated within the microchannel, the flow regime formed in a microfluidic channel will not develop into turbulent flow, enables fluid to be manipulated that will form emulsion with high monodispersity. The wide range of technology options from decades of microfluidic multiphase system developments has

In this chapter, we aim to summarize the main technologies for emulsion formation non-Newtonian microfluidic multiphase system using Newtonian fluid as a comparison. The chapter will start with the review of fundamental two-phase flow in microfluidic followed by discussion of the fundamental flow physics of microdroplets translocation and breakup phenomena in microfluidics. The detailed differences between Newtonian and non-Newtonian flow system will be illustrated and compared. Emphasis will be placed on the advancement of emulsions formed, i.e. encapsulation and fission from single emulsion. Finally, we conclude with an outlook to the future of the field. This chapter is meant to familiarize readers who may be new to the field of microdroplets formation in Newtonian and non-Newtonian fluid systems, as well as those readers who are new to the field of microdroplets formation via encapsulation and fission approaches, and eventually bridge the knowledge gap between the two

In two-phase flow microfluidics, dispersed and continuous phase fluids generally were dispensed separately into the microfluidic device. The continuous and dispersed phase channels typically meet at a junction, depending on the specific geometry of a microfluidic device. Each shape of the junction helps to define the local flow fields that deform the interface between the two immiscible fluids [6, 7]. These configurations are also shown schematically in Table 1. When the instabilities of free surface between phases are sufficiently large, drops in microscale emerge and eventually pinch-off from the dispersed phase. There are many controlling parameters that will affect the microdroplet generation regime such as, interfacial tension [8, 9], surface

allowed emulsion to be generated and manipulated.

correlations, disciplinary fields.

142 Microfluidics and Nanofluidics

2. Two-phase flow in microchannels

wettability or wall adhesion [10], the volumetric flow rate [11, 12], viscosities of both immiscible fluids [9, 11, 13] and channel geometry [14].

In applying the Navier-Stokes equation, the analytical solutions of the Poiseuille flow for

1 k

where l, w and h are the length, width and height of the channel, respectively, ΔP is the driven pressure drop, η is the viscosity of liquid. Most real fluids exhibit non-Newtonian behavior, which means that the flow curve presents a nonlinear relationship between shear stress and shear rate or does not pass through the origin. The laminar velocity profile of a power-law fluid flowing through a rectangular duct to the governing equation is shown as below [24]:

> <sup>2</sup> <sup>þ</sup> <sup>1</sup> � �sin <sup>β</sup><sup>i</sup>

where constant α<sup>i</sup> and β<sup>i</sup> and Ai is the constants selected to minimize an integral. Six constants which are to be computed by the Ritz-Galerkin method are identical to the corresponding Fourier coefficient and tabulated as a function of aspect ratio and fluid-behavior index (Figure 1).

An emulsion contains a mixture of two immiscible liquids as one phase being dispersed throughout the other phase in small droplets. Most common emulsions include oil-in-water, or direct emulsions, and water-in-oil, or inverted emulsions [19]. The characteristics of emulsion products, foremost the droplet size distribution is the most important parameters that affect the stability, rheology, chemical reactivity, and physiological efficiency of any emulsion

Figure 1. Velocity profile for laminar (a) Newtonian (n = 1) and (b) non-Newtonian shear-thinning (n < 1) and (c) shearthickening flow (n > 1): Two-dimensional (2D) plot with velocity height expression and one-dimensional (1D) plot with parabolic velocity profile in rectangular microchannel at different power-law index n. w denotes the channel width and d

Aisin <sup>α</sup>i<sup>π</sup> <sup>z</sup> ð Þ <sup>=</sup><sup>h</sup>

<sup>3</sup> <sup>1</sup> � cosh <sup>k</sup><sup>π</sup> <sup>y</sup>

cosh kπ <sup>w</sup> 2h � � " #sin k<sup>π</sup>

Microdroplets Advancement in Newtonian and Non-Newtonian Microfluidic Multiphase System

h � �

> <sup>y</sup>ð Þ =<sup>w</sup> þ 1 2

z h

http://dx.doi.org/10.5772/intechopen.75358

� � (3)

� � (2)

145

X∞ <sup>k</sup>¼<sup>1</sup>, <sup>3</sup>, <sup>5</sup>…

rectangular cross-section are shown as follows:

uxð Þ¼ y; z

u yð Þ ; z <sup>U</sup> <sup>¼</sup> <sup>X</sup> 6

denotes the depth of the microchannel.

i¼1

2.1. Microdroplets translocation and breakup phenomena in microfluidics

4h<sup>2</sup> ΔP ηπ<sup>3</sup>l

For microdroplet formation to be made possible in microfluidics, a defined flow conditions must be fulfilled in order to achieve the dripping regime [16, 18–20]. In the case of Newtonian fluids, the dripping regime occurs when the Weber number (We) of the dispersed phase and the Capillary number (Ca) of the continuous phase are less than one. Under these conditions, droplets formed are highly uniform and their uniformity is unaffected over a wide range of flow rates. However, non-Newtonian fluids rarely fulfill these conditions and pose a challenge in achieving monodispersed microdroplets due to their complex rheological characteristics. As general thumb rule, non-Newtonian fluids can be further classified intro three groups, i.e. purely viscous fluids, time-dependent fluids, and viscoelastic fluids. Each fluid group possess distinct characteristic respectively; however, there is no single constitutive equation that has been established to describe the rheogram for these fluids. For instance, the extensional viscosity of the fluid can resist the pinching at the tip of the capillary that is required for a microdroplet to form. This results in a long cylinder of fluid forming from the capillary tip to a distance downstream from the tip before breaking into non-uniform sized or polydispersed microdroplets formation due to the Rayleigh-Plateau instability [21]. Non-Newtonian fluids also exhibit a variety of behaviors that are unique to their chemical compositions, mixture combinations and many physical conditions that include flow rate, temperature, etc. Such complexities in the characteristics of non-Newtonian fluids prevent a thorough understanding of the dispersion stability and break-up of individual emulsions in microfluidics. Moreover, it is also known that an emulsifying non-Newtonian solutions in microfluidics in a controllable manner is a persistent problem [21] that prevents these techniques from being suitable for industrial applications. Understanding the dynamical mechanisms of microdroplets formation of non-Newtonian fluids in microfluidic channels is essential to ensure microdroplets can be created based on the droplet size, patterns, and productivity.

In microfluidic systems, the length scales demands that all flow to be laminar. Based on the Newtonian and non-Newtonian flow through the microchannel with rectangular crosssection, the axial velocity in the fully developed region is a function of two independent variables and the study of the hydrodynamic behavior in a rectangular microchannel requires a two-dimensional or three-dimensional analyses. The non-linear partial differential momentum Navier-Stokes equation with the associated boundary condition of zero velocity at the wall for a pressure-driven, steady-state, incompressible, constant viscosity, known as Newtonian fluid and the flow without body forces in microchannel is being presented as follows:

$$0 = -\frac{\partial P}{\partial x} + \eta \left(\frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}\right) \text{ for } -\frac{w}{2} < y < \frac{w}{2} \text{ and } 0 < z < h \tag{1}$$

where u is average velocity characteristic of the flow, dynamic viscosity is η, and P is the pressure, x is the coordinate axes along the channel length, y is the coordinate axes along the channel width, z is the coordinate axes along the channel height, w is the channel width, and h is the channel height, respectively. With no-slip boundary condition, the geometry of the crosssection of the rectangular is fixed and an analytical solution solutions is possible [22, 23]. In applying the Navier-Stokes equation, the analytical solutions of the Poiseuille flow for rectangular cross-section are shown as follows:

wettability or wall adhesion [10], the volumetric flow rate [11, 12], viscosities of both immiscible

For microdroplet formation to be made possible in microfluidics, a defined flow conditions must be fulfilled in order to achieve the dripping regime [16, 18–20]. In the case of Newtonian fluids, the dripping regime occurs when the Weber number (We) of the dispersed phase and the Capillary number (Ca) of the continuous phase are less than one. Under these conditions, droplets formed are highly uniform and their uniformity is unaffected over a wide range of flow rates. However, non-Newtonian fluids rarely fulfill these conditions and pose a challenge in achieving monodispersed microdroplets due to their complex rheological characteristics. As general thumb rule, non-Newtonian fluids can be further classified intro three groups, i.e. purely viscous fluids, time-dependent fluids, and viscoelastic fluids. Each fluid group possess distinct characteristic respectively; however, there is no single constitutive equation that has been established to describe the rheogram for these fluids. For instance, the extensional viscosity of the fluid can resist the pinching at the tip of the capillary that is required for a microdroplet to form. This results in a long cylinder of fluid forming from the capillary tip to a distance downstream from the tip before breaking into non-uniform sized or polydispersed microdroplets formation due to the Rayleigh-Plateau instability [21]. Non-Newtonian fluids also exhibit a variety of behaviors that are unique to their chemical compositions, mixture combinations and many physical conditions that include flow rate, temperature, etc. Such complexities in the characteristics of non-Newtonian fluids prevent a thorough understanding of the dispersion stability and break-up of individual emulsions in microfluidics. Moreover, it is also known that an emulsifying non-Newtonian solutions in microfluidics in a controllable manner is a persistent problem [21] that prevents these techniques from being suitable for industrial applications. Understanding the dynamical mechanisms of microdroplets formation of non-Newtonian fluids in microfluidic channels is essential to ensure microdroplets can be

In microfluidic systems, the length scales demands that all flow to be laminar. Based on the Newtonian and non-Newtonian flow through the microchannel with rectangular crosssection, the axial velocity in the fully developed region is a function of two independent variables and the study of the hydrodynamic behavior in a rectangular microchannel requires a two-dimensional or three-dimensional analyses. The non-linear partial differential momentum Navier-Stokes equation with the associated boundary condition of zero velocity at the wall for a pressure-driven, steady-state, incompressible, constant viscosity, known as Newtonian fluid and the flow without body forces in microchannel is being presented as follows:

for � <sup>w</sup>

where u is average velocity characteristic of the flow, dynamic viscosity is η, and P is the pressure, x is the coordinate axes along the channel length, y is the coordinate axes along the channel width, z is the coordinate axes along the channel height, w is the channel width, and h is the channel height, respectively. With no-slip boundary condition, the geometry of the crosssection of the rectangular is fixed and an analytical solution solutions is possible [22, 23].

<sup>2</sup> <sup>&</sup>lt; <sup>y</sup> <sup>&</sup>lt;

w

<sup>2</sup> and 0 <sup>&</sup>lt; <sup>z</sup> <sup>&</sup>lt; <sup>h</sup> (1)

fluids [9, 11, 13] and channel geometry [14].

144 Microfluidics and Nanofluidics

created based on the droplet size, patterns, and productivity.

∂2 u ∂y<sup>2</sup> þ

∂2 u ∂z<sup>2</sup> 

<sup>0</sup> ¼ � <sup>∂</sup><sup>P</sup> ∂x þ η

$$\mathbf{u}\_x(y, z) = \frac{4h^2 \Delta P}{\eta \pi^3 l} \sum\_{k=1,3,5\dots}^{\infty} \frac{1}{k^3} \left[ 1 - \frac{\cosh\left(k \pi \frac{y}{h}\right)}{\cosh\left(k \pi \frac{w}{2h}\right)} \right] \sin\left(k \pi \frac{z}{h}\right) \tag{2}$$

where l, w and h are the length, width and height of the channel, respectively, ΔP is the driven pressure drop, η is the viscosity of liquid. Most real fluids exhibit non-Newtonian behavior, which means that the flow curve presents a nonlinear relationship between shear stress and shear rate or does not pass through the origin. The laminar velocity profile of a power-law fluid flowing through a rectangular duct to the governing equation is shown as below [24]:

$$\frac{u(y,z)}{U} = \sum\_{i=1}^{6} A\_i \sin\left[\alpha\_i \pi \frac{(z/\_h)}{2} + 1\right] \sin\left[\beta\_i \frac{(y/\_w + 1)}{2}\right] \tag{3}$$

where constant α<sup>i</sup> and β<sup>i</sup> and Ai is the constants selected to minimize an integral. Six constants which are to be computed by the Ritz-Galerkin method are identical to the corresponding Fourier coefficient and tabulated as a function of aspect ratio and fluid-behavior index (Figure 1).

#### 2.1. Microdroplets translocation and breakup phenomena in microfluidics

An emulsion contains a mixture of two immiscible liquids as one phase being dispersed throughout the other phase in small droplets. Most common emulsions include oil-in-water, or direct emulsions, and water-in-oil, or inverted emulsions [19]. The characteristics of emulsion products, foremost the droplet size distribution is the most important parameters that affect the stability, rheology, chemical reactivity, and physiological efficiency of any emulsion

Figure 1. Velocity profile for laminar (a) Newtonian (n = 1) and (b) non-Newtonian shear-thinning (n < 1) and (c) shearthickening flow (n > 1): Two-dimensional (2D) plot with velocity height expression and one-dimensional (1D) plot with parabolic velocity profile in rectangular microchannel at different power-law index n. w denotes the channel width and d denotes the depth of the microchannel.

[25–27]. For most microfluidic applications, the Reynolds number is much smaller than 1, indicating the effects of volume-based inertia and gravity are not as significant as that in macroscale. The surface-based interfacial tension, flow rates, surface chemistry, and the viscosity become more significant and play in controlling flow behavior of multiphase flow in microscale. Moreover, the degree of confinement, channel aspect ratio, and geometrical structure also significantly induce the impact on capillary pressure, which promote the destabilization of the interface and the drop formations.
