Micromixers based on chaotic advection - Micromixers Fundamentals Design and Fabrication - page 219

and play a significant role in mixing in microscale. Segmented flows have the advantage of fast mixing
because of chaotic advection and reduced axial dispersion due to the confined fluid segments [39] .

6.4.1.1 Passive droplet formation
A common system for the formation of droplet consists of a solid channel wall, an aqueous phase, and
an immiscible oil phase. The relation between the contact angle and the interfacial tensions in a liquid/
liquid/solid system can be described by the Young equation
s sw s so ¼ s wo cos q (6.10)

where s sw , s so , s wo , and cos q are the interfacial tensions of solid/water, solid/oil, and the contact angle
at the triphasic line. Adding a surfactant such as Span80 to the oil decreases s so and s wo . As a result,
the term cos q should increase to keep the above equation in balance. If the concentration of the
surfactant is high enough, cos q reaches its maximum value of 1 and q ¼ 0. In this case, the oil totally
wets the channel wall causing the water droplet to detach from the solid surface. Adding surfactant to
the oil can help to control the passive formation of aqueous droplets in an oil flow.
The forces involved in the passive formation of droplets, bubbles, plugs, and slugs are gravitational,
interfacial, inertial, and viscous forces. Besides the Reynolds number often used for single-phase
flows, the relation between the above forces can be represented in a multiphase flow through a number
of dimensionless numbers, such as the Bond number, the capillary number, and the Weber number.
The Bond number represents the ratio between the gravitational buoyancy force and the interfacial
force:

Interfacial force ¼ Dr gD h

Gravitational force

Bo ¼ (6.11)

s

where Dr ¼ r 2 r 1 is the density difference between the two immiscible fluids 1 and 2, and D h is the

hydraulic diameter of the microchannel. The capillary number represents the ratio between the viscous
force and the interfacial force:

Viscous force ¼ m u

Inertial force

Ca ¼

(6.12)

s

where m is the dynamic viscosity of the carrier fluid and u is the average velocity. The Weber number
represents the ratio between inertial force and interfacial force:

Interfacial force ¼ ru 2 D h

Inertial force

We ¼ : (6.13)

s

Another important parameter of the formation process is the sample fraction:

Q d

r dc ¼

Q d þ Q c ;

(6.14)

and Q d and Q c are the flow rates of the droplet liquid and carrier fluid, respectively.
The physics of the formation process is determined by a critical capillary number Ca cr z 10 2 . For

Ca Ca cr , shear stress

becomes insignificant and the formation process works in a squeezing regime. The formation of
bubbles is typically in the squeezing regime, while the formation of droplets can be in both squeezing
and shearing regimes.

> Ca cr , the shear force plays an important role in the formation process. For Ca <