PRL 96, 134505 (2006)
PHYSICAL REVIEW LETTERS
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Selection of Two-Phase Flow Patterns at a Simple Junction in Microfluidic Devices
W. Engl, K. Ohata, P. Guillot, A. Colin, and P. Panizza*
Laboratoire du Futur, Rhodia/CNRS FRE 2177, 178 Avenue A. Schweitzer, 33608 Pessac, France
(Received 18 January 2006; published 7 April 2006)
We study the behavior of a confined stream made of two immiscible fluids when it reaches a T junction.
Two flow patterns are witnessed: the stream is either directed in only one sidearm, yielding a preferential
flow pathway for the dispersed phase, or splits between both. We show that the selection of these patterns
is not triggered by the shape of the junction nor by capillary effects, but results from confinement. It can be
anticipated in terms of the hydrodynamic properties of the flow. A simple model yielding universal
behavior in terms of the relevant adimensional parameters of the problem is presented and discussed.
DOI: 10.1103/PhysRevLett.96.134505
PACS numbers: 47.56.+r, 45.70.Qj, 47.61.Jd
An important class of ‘‘out of equilibrium’’ patterns
occurs by forcing immiscible fluids in interconnected
thin gap networks, a phenomenon frequently encountered
in nature as well as in many man-made systems [1,2].
Illustrative examples include the flows of hydrocarbons
and water through porous rocks during oil recovery, parallel flows used for various microfluidic applications [3], as
well as flow of air in the lungs. These fluid-fluid displacements generate preferential flow pathways along one of the
fluid flows, forming patterns ranging from compact to
ramified and fractal [4]. The formation and selection of
these patterns are challenging problems in the field of
nonequilibrium physics [5] and central for many industrial
applications. For instance ‘‘tongues of water in oil’’ is a
limiting factor in secondary oil recovery.
One of the major difficulties in achieving a good understanding of preferential pathways in interconnected networks is the large number of parameters potentially
involved. These include the viscosity of the two fluids,
their respective flow rates, the surface tension between
them, as well as the topology of the network [6].
The aim of this Letter is to address an essential element
of immiscible flows in porous media or branched networks,
which is the effect of a bifurcation where the flow is separated into two streams. We focus on the simplest configuration, namely, a T junction. As sketched in Fig. 1, our goal
is to understand how a laminar stream and its carrier phase
divide between these two branches. We start by reporting
experiments which show the existence of two possible
hydrodynamic regimes: the stream is either directed in
only one of the arms, yielding a preferential flow path for
the dispersed phase, or splits between both. Surprisingly,
the mechanism at work in this phenomenon is not triggered
by the shape of the junction nor by capillary effects but
results from the 3D structure of the confined stream. A
simple hydrodynamic model yielding universal behavior in
terms of the different adimensional parameters of the
problem is presented and confronted with our experimental
data.
Experiments are performed in microchannels, fabricated
into poly(dimethylsiloxane) (PDMS) transparent elasto0031-9007=06=96(13)=134505(4)$23.00
mers, using classical soft lithography techniques. The microfluidic devices are prepared by placing a PDMS slab
with the following channel features, height h 20 m
and width d 100 m, onto either a glass cover slip or
a flat PDMS slab. For each device, laminar streams are
formed at a symmetrical cross flow junction (Fig. 1). The
two Newtonian immiscible fluids used in our experiments
are millipore water and silicone oil (Fluka), whose respective viscosities at T 20 C are w 1 mPa s and o 20 mPa s. Using separate syringe pumps, the flow rates of
these two fluids are independently controlled and adjusted
in order to form a water in oil (W-O) stream. For both
fluids, the Reynolds numbers can be estimated using Re Qf f =hf , where Qf , f , and f are, respectively, the
fluid flow rate and its mass volume and viscosity. The
values obtained for our experiments are smaller than 0.25
indicating that flows are laminar. To study the behavior of
the laminar stream at a junction, we work with the microfluidic analogs of electric current splitting devices [7]
(Fig. 1). Briefly, the stream is directed towards a T junction, referred from now on as the inlet T junction. In order
to achieve the same pressure drop in both side arms, the
two arms merge at another T junction, referred as to the
outlet junction (see Fig. 1). As shown later on, by adjusting
the relative lengths L1 and L2 of the two sidearm channels
it is possible to control the behavior of the jet in the device.
FIG. 1. Schematic representations of the experimental setup.
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© 2006 The American Physical Society
PRL 96, 134505 (2006)
PHYSICAL REVIEW LETTERS
Images of the flows are captured and recorded using an
inverted microscope equipped with a CCD camera.
Experiments are performed by fixing the value of the water
flow rate (Qw 600 l=h), while varying that of the oil
(Qo ). Figure 2 shows the evolution of the stream at the inlet
T junction as a function of q Qo =Qw , for LL21 1:75
and for different shapes of the junction.
For small values of q, the stream splits at the junction,
forming two secondary coflows in each sidearm. At the
outlet T junction, these two coflows merge to reform the
initial stream. As shown on Fig. 2, the asymmetry between
the two secondary coflows becomes more and more important as q increases. When q is slightly increased above a
critical value, qc 0:13, an unexpected and surprising
phenomenon occurs: the initial stream no longer splits at
the inlet T junction, but instead deviates towards the
shorter sidearm. In this flow regime, the junction therefore
acts as a perfect filter, since water is no longer flowing in
the longest arm. In the shortest arm, a coflow configuration
is witnessed since the deviated stream is in contact with
one of the two channel vertical edges. At a critical value of
q, it suddenly loses contact with this edge. The position of
the stream in the shortest outlet then becomes more and
more central upon increasing the value of q. The (S-F)
transition between splitting and filtering is independent of
the angle between the three arms of the junction (Fig. 2). In
addition, it persists if the flat PDMS cover slab is replaced
by a glass cover slip or if, , the surface tension between
oil and water is varied (as later shown on Fig. 5), indicating
that the physical mechanism at play in this phenomenon is
not likely triggered by wetting properties.
To elucidate the origin of this phenomenon, we first
characterize the three-dimensional structure of the flow
using confocal microscopy experiments. A minute amount
of rhodamine (Fluka) is added to the aqueous phase.
Figure 2 displays a 3D image of the flow, obtained by
reconstruction from several juxtaposed horizontal sections.
For channel features (h d), we observe a quite unexpected result since the water stream is not wrapped by an
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oil film but instead wets both the top and bottom surfaces
of the channel, independently of their chemical nature
(glass or PDMS). At first sight, this result is very striking
considering that the wettability affinity for PDMS is much
stronger for silicone oil than for water and that the capillary
numbers at play in our experiments are very low (typically
smaller than 102 ). This counterintuitive phenomenon results from confinement [8]. Briefly, a laminar stream flowing along a constant cross section channel must present the
same constant curvature shape in any perpendicular cross
section of the channel. For a given set Qo ; Qw and a given
channel shape, only one value of the stream curvature, c,
exists. This value depends on o =w , q, and . Under
certain experimental conditions [8], c1 may eventually be
larger than h, leading to truncated cylindrical streams
being then in contact with the two horizontal channel
surfaces. This situation is favored when the channel features are very anisotropic (h d) and q and are small.
Two parameters are then necessary to describe the stream:
its width w and c, the curvature of the W-O interface.
When w=h > 1, w depends only on and q at first order
(the wetting properties of the two fluids and their affinity
for the substrate imposes c, but have second order effects
on w). This coincides with our experimental observations.
Now that the structure of the flow is well established, the
S-F transition can be understood only using hydrodynamic
considerations. Let us start with Stokes equations and
incompressible flows for both fluids with no slip boundary
conditions. In view of experimental results, reasonable
assumptions are to neglect the geometry details of the inlet
T junction, namely, the influence of the angle between the
three arms of the inlet T junction, to consider that the water
stream wets both top and bottom surfaces and that the
interface between both phases can be well approximated
by a plane [Fig. 3(b)]. Within this framework, let us first
analyze the hydrodynamic configuration where the stream
splits at the inlet junction (Fig. 3).
First, look at a steady state coflow in a straight channel
of constant cross section S h d and length L. For a
given position of the interface between the water and the
FIG. 2. Behavior of the stream at the
inlet junction for different values of q
(given in the images) and o 20 mPa s.
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L2
L1
x2
features. Since the water stream wets both the top and
the bottom channel surfaces, no exchange of oil between
the two sides of the incoming stream is possible at the inlet
junction. It therefore follows that for a symmetric oil
injection (Fig. 1) the oil flow rates must be equal in both
sidearms of the junction (Fig. 3). For i 1; 2, such a
condition writes
Pfh; d
1
Qo
(5)
G 1 Xi ; :
Li o
2
x1
h
o w
o
d
x2
Q0/2
x1
Q0/2
h
d
FIG. 3 (color online). (a) 3D reconstructed confocal image for
a splitting flow pattern and (b) its schematic representation.
oil domains, X dx , the velocity profiles of both fluids and
therefore their respective flow rates can be numerically
computed using a finite discretization of the Stokes equations. The computations are performed with the scientific
software SCILAB, developed by INRIA (France). A rapid
analysis shows that the expressions of the two flow rates
follow
Pfh; d
GX; Lw
(1)
Pfh; d
1
G 1 X; ;
Lo
(2)
Qw and
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PHYSICAL REVIEW LETTERS
PRL 96, 134505 (2006)
By dividing Eqs. (3) and (4) by Eq. (5), a system of two
nonlinear equations between only X1 and X2 and the three
experimental parameters , , and q is obtained, yielding
the existence of universal behaviors .
To test this, we first compute the variations of X1 and X2
from this set of equations as a function of q for 1:75
and 20 and compare them to our experimental data.
As shown on Fig. 4, all experimental data obtained for
different water and oil flow rates fall on the master curve
predicted by the model. Above a critical value qc 0:14,
all the solutions admit X2 0, indicating that for q > qc
the stream can no longer split at the inlet T junction and
therefore all the dispersed phase must flow through the
shortest channel. This is the filter regime.
Another test is to compare numerical results to experimental results when q and are fixed but when is
allowed to vary (see Fig. 5). To do so, experiments are
now performed at 1:75 using water and silicone oils of
different viscosities. As before, we set Qw and change Qo .
In agreement with our numerical results, we observe that
the S-F transition strongly depends on when q and are
fixed (Fig. 5). As expected by the model, a universal
behavior is observed by plotting X2 as a function of q
(inset of Fig. 5). The value of the critical flow rate ratio
X1 /d
1
1
Qw Qo Pfh; d
GX1 ; GX2 ; ;
L2 w
Pfh;d
1
1
:
G 1 X1 ; G 1 X2 ;
L2 o
(3)
(4)
The third equation results from the unexpected shape of
the incoming stream (Fig. 3) observed for our channel
0.8
0.6
0.8
0.4
X2/d
where fh; d and GX; are two functions having, respectively, m4 and no units, and P the pressure drop
between the two extremities of the channel. For the
branched network of (Fig. 1), three variables are necessary
to describe a flow situation where the stream splits at the
junction, namely, the positions of the water-oil interfaces in
each side arms X1 and X2 and P, the pressure drop
between the inlet and outlet T junctions. On this basis,
three relations are necessary to solve this problem. Two
predictable equations are obtained from the conservations
of the water and the oil flow rates at the junction:
0.6
0.2
0.4
0
-3
10
10
-2
10
-1
10
0q
0.2
0
10
-3
10
-2
10
-1
10
0
10
1
q
FIG. 4. Positions of the W-O interface in the longest side arm
(and in the shortest arm for the inset) as a function of q. The two
fluids are water and silicone oil (o 20 mPa s). The water
flow rate is fixed, and the oil flow rate varied. The (䊐), (4), (),
and (5) symbols correspond to experiments performed, respectively, at Qw 200, Qw 300, Qw 400, and Qw 600 l=h with a junction angle of 90. The solid symbols
correspond to experiments performed at Qw 600 l=h but
with junction angle 45 (䉬) and 135 (䊉). The
continuous and dashed lines correspond to the numerical solutions of the model.
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1
1
2
10
X2/d
0.8
0.8
X2/d
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PHYSICAL REVIEW LETTERS
PRL 96, 134505 (2006)
0.6
0.4
0.6
0.2
0.4
0 -2
10
-1
10
0
10
qη
η
1
10
1
10
0.2
0
10
-3
10
-2
10
-1
10
0
10
1
q
0
10
FIG. 5. Positions of the W-O interface in the longest sidearm
as a function of q for different values of . (), (䉱), and (5)
symbols correspond to experiments performed with different
silicone oils ( 0:043 N m1 ) having respective viscosity
numbers: 20, 50, and 100, for Qw 600 l=h.
(䊉) correspond to an experiment performed with hexadecane:
0:052 N m1 and 3, for Qw 200 l=h. The continuous lines are the numerical solutions of the model.
above which the filter regime exists is then given by qc 2:9.
To test the influence of the last adimensional parameter,
on the S-F transition, we set a value for the product q
and compute X2 as a function of . A systematic numerical
study reveals that, for any values of q, solutions of the set
of equations present a X2 0 domain when becomes
larger than a critical value c , satisfying c 1 2:15=q (Fig. 6). To check this prediction, we have fabricated several microfluidic devices having the same channel cross but different values of . For each device,
experiments are performed with water and silicone oil
(o 20 mPa s) by changing the values of q (we set
Qw and change Qo ). Under these conditions, we observe
that the value of qc , characterizing the S-F transition
decreases when increases. Consequently, as predicted
by the model, if the value of the product q is fixed, a S-F
transition can therefore be witnessed by merely increasing
the value of (inset of Fig. 6).
In summary, the features of the S-F transition are well
captured by our simple hydrodynamic model. The origin of
this phenomenon lies in the absence of oil exchange between the two sides of the incoming stream due to its
strong confinement. This effect can therefore be witnessed
whether the injection of an oil stream is symmetric (see
Fig. 1) or not. For a symmetric injection of oil, the universality of this phenomenon can be summarized by a diagram
which maps the nature of the selected flow pattern as a
function of the two relevant adimensional numbers of the
problem: q and (Fig. 6). Such a general effect opens up
a way to digital and integrated microfluidic devices. Thus,
the (S-F) transition occurring at the inlet T junction operates either as a flow rate or as a viscosity comparator [9]
10
filter
regime
splitting
regime
-1
0
10
1
λ -1
10
FIG. 6. Universal diagram mapping the nature of selected flow
pattern: filter or splitting as a function of the two pertinent
adimensional parameters of the problem. The continuous line,
q 2:15 11 , is the transition line between the two flow
regime predicted by the model, whereas () symbols corresponds to experimental data. Inset: flow patterns observed at the
junction for (a) 1:25, (b) 2, and (c) 5 when 20 and q 0:05.
depending on which adimensional variable, or q, is
known. Since the initial stream reforms at the outlet T
junction, such comparator elements whose threshold values are fixed by can be integrated into cascade to design
inexpensive digital microfluidic devices. In its simplest
version, a succession of comparators with different values
of simply obtains an estimation of a fluid viscosity.
Along this line, work is in progress to develop a continuous
measurement device.
*Electronic
address:
pascal.panizza-exterieur@eu.
rhodia.com
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