Download The hydrostatic pressure in the dispersed and continuous phases

Force balance for Emerging Droplet Interface
The hydrostatic pressure in the dispersed and continuous phases, Ph,d and Ph,c, are governed by
the driving pressure applied to the fluids at the inlet device, as well as the microfluidic channel
network, which provides fluidic resistance to the flow and induces a pressure drop along its
distance according to the Hagen-Poiseuille equation:
∆𝑃 =
𝑄
(3a)
𝑅
where Q = the volumetric flow rate of the fluid and R is the fluidic resistance in square
microfluidic channels, given by:
𝑅=
12𝜇𝐿
(3b)
𝑤ℎ3
In a single phase system where fluid is driven from the dispersed phase inlet, fluid flow may
proceed to either the main channel or the waste channel, denoted as “m” and “w” in the figure
below. The resistance difference between the channels can be examined in detail by taking into
account actual geometries of the channels as well as the viscosities of the fluids. Using equation
3b and substituting in actual values of the microfluidic channels, we arrive at resistance values as
shown below –
Continuous phase resistance (Rm)12(29.2 mPa ∗ s)(4.55 ∗ 10−3 m)
mPa ∗ 𝑠
= 6.377 ∗ 1016
−6
−6
3
(200 ∗ 10 m)(50 ∗ 10 m)
m3
Dispersed phase resistance (Rw)-
12(1.002 mPa ∗ s)(4.991 ∗ 10−3 m)
mPa ∗ 𝑠
= 6.401 ∗ 1015
(75 ∗ 10−6 m)(50 ∗ 10−6 m)3
m3
It can be observed that the dispersed phase resistance is an order of magnitude smaller than that
of the continuous phase and this results in the flow of the dispersed phase towards the waste
line.
In addition to these hydrostatic pressures, the presence of two phases at the entrance to the main
channel introduces a pressure drop due to surface tension. The introduction of this force, in
conjunction with the viscosity effect noted above, allows the dispersed phase to be directed
toward the waste channel instead of entering the main channel.
In the two-phase system, the force balance on the interface between the two phases, which
occurs at the intersection of the inlet and main fluidic channels, is given by:
𝐹ℎ,𝑑 = 𝐹ℎ,𝑐 + 𝐹𝛾
(1)
where Fh,d is the hydrostatic force applied to the emerging droplet interface from the dispersed
phase, Fh,c is the hydrostatic force applied to the emerging droplet from the continuous phase,
and Fγ is the surface tension force across the droplet interface.(Figure 1)
The above equation may be expanded using the equation F = PA, where A is the area over which
the pressure is applied at the front of the emerging droplet, to:
(𝑃ℎ,𝑑 ) �
2
𝜋𝑟𝑖𝑛𝑡
2
2
2
2
𝜋𝑟𝑖𝑛𝑡
� = 𝛾 �ℎ + 𝑟 � �
𝑖𝑛𝑡
2
2
𝜋𝑟𝑖𝑛𝑡
� + �𝑃ℎ,𝑐 � �
2
� 𝑐𝑜𝑠𝜃
(2)
In equation 1, the force on the left acts to drive the dispersed phase into either the main channel
or the waste channel. The surface tension pressure across the interface from the dispersed to the
continuous phase, and the hydrostatic pressure exerted by the continuous phase act to balance the
hydrostatic pressure exerted by the dispersed phase.
Prior to actuation of the membrane to produce the droplet, fluid from the inlet channel flows
toward the junction and out of the device through the waste channel. In this situation, the surface
tension force and hydrostatic force from the continuous phase balance the hydrostatic force from
the dispersed phase, as shown in Figure 3a.
Upon actuation of the membrane, a volume of dispersed phase fluid is displaced from the
chamber. This translates to a sudden increase in flow rate – Q. As demonstrated in equation 3a,
an increase in Q with fixed R cause P to rise. In this specific case, Ph,d increases rapidly. Similar
to the examination of the relative resistance above, the change in flow rate Q can be estimated by
taking into account the geometries involved in the actuation of the membrane. The volumes of
the actuation chamber and an equivalent segment of channel is as shown below, and given by V
= lwh, where w = width, h = height, and l = a given length of channel –
Volume of chamber -
Volume of channel -
150µm ∗ 200µm ∗ 50µm = 1.5 ∗ 106 µm3
150µm ∗ 25µm ∗ 50µm = 1.875 ∗ 105 µm3
The presence of the expansion volume in the chamber provides for storage of nearly 10 times as
much fluid as an equivalent section of channel without the expansion volume. When the
membrane is actuated, this expansion volume provides for an increase in the volumetric flow rate
of nearly 10 times, which provides for an increase in the hydrostatic pressure of the dispersed
phase and brings the hydrostatic pressures of the continuous and dispersed phases to the same
magnitude. This order-of-magnitude analysis explains the ability of the chamber to deform the
dispersed fluid into the main fluidic channel, to create a droplet.
Since Ph,c is constant with time, the increase in Ph,d concomitant with actuation of the membrane
allows the dispersed phase fluid to move into the main channel. The balance of the two pressures
therefore serves as an effective design and operation guideline for the system where the pressures
are balanced when Qc*Rc = Qd*Rd. When the resultant pressure on the dispersed fluid side
exceeds that of the continuous phase side, a protrusion is formed in the main fluidic channel,
allowing a droplet to be formed. As this occurs, the radius of the droplet increases rapidly.
From equation 2 above, Ph,d ~ rint2 and Ph,c ~ rint2, while Pγ ~ rint. Thus as rint increases, the
hydrostatic pressure terms dominate over the surface tension force term and the sudden increase
in the hydrostatic pressure of the dispersed phase allows the droplet to grow.
As the droplet continues to grow, however, it is subjected to an increasing hydrostatic force from
the continuous phase. Although the hydrostatic pressure of the continuous phase remains
constant, the growing droplet increases in surface area, which subjects the droplet to increased
force and begins to shear the droplet. At a critical droplet size, governed by the balance between
the three forces indicated in the force balance equation, the droplet begins to pinch off from the
inlet channel and form a “neck”. As the droplet experiences this force; the neck narrows with
time. During this process, some of the fluid that initially emerged into the main channel upon
membrane actuation is driven back toward the waste channel as this neck forms. This
phenomenon occurs due to the development of two quasi- radii of curvature. As the droplet
begins to neck, the radius of curvature on the back of the droplet begins to decrease, and a radius
of curvature begins to develop at the interface between the waste and main microfluidic
channels. This decrease in the quasi- radii of curvature produces an increase in the surface
tension force, according to the Fγ term in the force balance equation. As the neck reaches a
critical thickness, a droplet is sheared from the dispersed phase flow into the main channel as a
combination of the shear force provided by the continuous phase flow and the subsequent
increase in surface tension force produced as the radii of curvature decrease.