Biomedical Engineering Reference
In-Depth Information
In particular the model explains why the upper limit of the droplet domain is
the same for any couple of liquids. The boundary corresponds to the limit between
droplet regime and flow reversal inside the central (alginate) channel (i.e.,
Q
i
= 0).
By setting
q
*
= 0 in (4.107), we find
p
*
= 1 -
B
;
B
being purely a geometrical param-
eter—under the condition that
f
stays close to unity—the upper limit of the droplet
regime in the pressure diagram is a fixed straight line, independent of the viscosity
h
i
of the discontinuous phase. This property has been checked experimentally using
deionized water (DIW), as well as 1 to 1.75% Keltone alginate solutions.
Scaling Rules
Are there scaling rules between FFDs? In this section, we focus on the geometrical
scaling-up between a small FFD (50
m
m) and a larger one (100
m
m to 1 mm). We
determine the driving pressure conditions to obtain the droplet regime at any size.
It is shown that a similar behavior of the device is obtained by a scaling-up of the
dimensions by a ratio
k
and a scaling-down of the driving pressures by the ratio
1/
k
—or a scaling-up of the flow rates by
k
2
[72].
There are two conditions to fulfill to produce monodispersed droplets. First, the
flow rates or driving pressures must be adapted to correspond to the droplet regime.
Second, the flow rates or driving pressures must be chosen to obtain a dripping re-
gime, avoiding the jetting regime. It is recalled that the jetting regime is obtained at
larger velocities and is characterized by a continuous thread of alginate that extends
far into the outlet channel, breaking randomly and producing rather polydisperse
droplets (Figure 4.73). Hence, the dripping regime is chosen in order to be certain
to obtain monodispersed droplets, which is a requirement for cell encapsulation.
In the following we shall see that geometrical scaling rules are determined by
the condition of a droplet regime, and the flow conditions, e.g. driving pressures
(or flow rates) scaling rules by the condition of a dripping regime. Let us begin by
the first condition.
Geometrical Scaling
We have shown that there are two limits for the droplet regime. One is flow rever-
sal in the central channel (e.g., when the flow rate in the central channel changes
direction). This limit is given by
Q
i
= 0 or
q
*
= 0 , or, in a pressure actuated system
P
i
/P
e
=
p
*
=
cste
=
p
*
1,2
. The second is the droplet-plug regime limit. This limit only
Figure 4.73
(a) Dripping regime: monodisperse droplets are produced at the nozzle; (b) jetting
regime: polydispersed droplets are produced at the end of an alginate jet (photo courtesy of S. Le
Vot, CEA-LETI).
