Spatiotemporal pattern formation occurs in a variety of nonequilibrium physical and chemical systems. Here we show that a microfluidic device designed to produce reverse micelles can generate complex, ordered patterns as it is continuously operated far from thermodynamic equilibrium. Flow in a microfluidic system is usually simple—viscous effects dominate and the low Reynolds number leads to laminar flow. Self-assembly of the vesicles into patterns depends on channel geometry and relative fluid pressures, enabling the production of motifs ranging from monodisperse droplets to helices and ribbons

Arnold, Frances H.

Quake, Stephen R.

Roberts, Richard W.

Thorsen, Todd

English

Caltech Authors - Main

VOLUME 86, NUMBER 18 P H Y S I C A L R E V I E W L E T T E R S 30 APRIL 2001Dynamic Pattern Formation in a Vesicle-Generating Microfluidic Device
Todd Thorsen,1 Richard W. Roberts,1 Frances H. Arnold,1 and Stephen R. Quake2
1Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125
2Department of Applied Physics, California Institute of Technology, Pasadena, California 91125
(Received 9 January 2001)
Spatiotemporal pattern formation occurs in a variety of nonequilibrium physical and chemical systems.
Here we show that a microfluidic device designed to produce reverse micelles can generate complex, or-
dered patterns as it is continuously operated far from thermodynamic equilibrium. Flow in a microfluidic
system is usually simple—viscous effects dominate and the low Reynolds number leads to laminar flow.
Self-assembly of the vesicles into patterns depends on channel geometry and relative fluid pressures,
enabling the production of motifs ranging from monodisperse droplets to helices and ribbons.
DOI: 10.1103/PhysRevLett.86.4163 PACS numbers: 82.40.Ck, 47.54.+r, 61.30.Pq, 82.70.UvComplex pattern formation is ubiquitous in nature. Ef-
forts to understand these effects have led to important in-
sights into nonlinear dynamical systems and fundamental
nonequilibrium physics [1]. Fluid systems have been fer-
tile ground for pattern formation, with classic examples
such as Rayleigh-Benard convection, Taylor-Couette flow
in rotary systems, nonlinear surface waves, liquid crystals,
and falling droplets [2]. Key ideas that have emerged from
the study of pattern formation are the central roles of insta-
bility and nonlinearity, as well as the influence of pertur-
bations and boundary conditions on the morphology of the
patterns. The elements of instability and nonlinearity are
generally not present in microfluidic devices because the
length scales are small enough that inertial effects in the
fluid can be neglected. As most microfluidic devices oper-
ate at low Reynolds number [3], the Navier-Stokes equa-
tion for fluid flow becomes linear and the flow is laminar.
This result has many practical consequences for efforts to
miniaturize biological assays and produce lab on a chip
system [4,5]. In this Letter, we show how the interaction
between two immiscible fluids can be used to introduce
nonlinearity and instability in a microfluidic device. The
resulting complex pattern formation is an unexpected and
fascinating example of self-organization in a dynamic sys-
tem far from equilibrium.
Emulsions are formed by shearing one liquid into a sec-
ond immiscible one, often in the presence of a surfactant,
to create small droplets. The droplets can be remarkably
stable, maintaining their shape and distribution for years
[6]. Significant advances have been made in the past few
years to produce emulsions that are monodisperse, with
standard deviations in droplet size less than 5% [7–10].
Unlike the standard crossflow techniques for generating
water-in-oil emulsions, in which the discontinuous phase
is forced through narrow pores [8,10] or capillaries [7,11]
into an open continuous phase, we accomplish droplet for-
mation at the junction of two microfluidic channels con-
taining water and an oil surfactant mixture, respectively.
The water partially obstructs flow at the junction, but is not
broken off at the channel interface as in traditional cross-
flow devices. Droplet formation is achieved by high shear0031-90070186(18)4163(4)$15.00forces generated at the leading edge of the water perpen-
dicular to the oil flow, generating picoliter-scale droplets.
Although the system remains at low Reynolds number, the
flow is nonlinear because of interactions on the boundary
between the two fluids. The two important effects are that
the boundary is not static and that the motion of one fluid
can entrain the other [12]. The resulting instability that
drives droplet formation is a well-known competition be-
tween surface tension and shear forces [13].
The emergence of static crystalline structure in emul-
sions has been documented previously [7,9]. In our experi-
ments, the droplets self-assemble into a variety of coherent,
moving patterns as they are formed. We examine the con-
trol parameters that lead to vesicle formation and organi-
zation in an emulsion in a microfluidic device, illustrating
the relationship between droplet pattern formation, pres-
sure, and the geometric boundary conditions of the system.
The droplet size and frequency can be precisely controlled
by modifying the relative pressure of water and oil, en-
abling the production of a wide range of vesicle shapes
and patterns. Under conditions where the water pressure
is lower than the oil pressure, monodisperse separated re-
verse micelles are formed. As the relative water pressure
is increased at fixed oil pressure, the droplets become or-
dered into a single continuous stream. At water pressures
that exceed the oil pressure, complex, organized patterns
begin to emerge in the stream, ranging from helical-like
structures to coherent ribbon motifs.
The microfluidic devices utilized in our experiments are
fabricated by pouring acrylated urethane (Ebecryl 270,
UCB Chemicals) on a silicon wafer mold containing
positive-relief channels patterned in photoresist (SJR5740,
Shipley), which is then cured by exposure to UV light. The
channels are fully encapsulated by curing the patterned
urethane on a coverslip coated with a thin layer of urethane
and bonding the two layers together through an additional
UV light exposure. The measured channel dimensions
are approximately 60 mm wide 3 9 mm high, tapering
to 35 mm 3 6.5 mm in the region where the water and
oil/surfactant mixture meet at the crossflow intersection
(Fig. 1). Input and post-crossflow junction channel lengths© 2001 The American Physical Society 4163
VOLUME 86, NUMBER 18 P H Y S I C A L R E V I E W L E T T E R S 30 APRIL 2001FIG. 1. Microfabricated channel dimensions at the point of
crossflow and photomicrograph of the discontinuous water phase
introduced into the continuous oil phase. Dashed rectangle in-
dicates area in photomicrograph.
(60 mm wide 3 9 mm) are 1 and 4 cm, respectively.
The fluids are introduced into the urethane microfluidic
devices through pressurized reservoirs containing water
and oil. The reservoirs are connected to the device through
approximately 30 cm of 500 mm i.d. Tygon tubing. Pres-
sure was applied to the reservoirs with compressed air,
and the device output channel was allowed to vent to the
atmosphere. All reported pressures are relative to atmo-
spheric pressure (psig). Various oils were tested in the
device, including decane, tetradecane, and hexadecane,
combined with the surfactant Span 80 concentrations
(yy) of 0.5%, 1.0%, and 2%. The device is equilibrated
prior to crossflow by priming the outflow channel with
oil/surfactant to eliminate water interaction with the
hydrophilic urethane. The production of reverse micelles
is then initiated by modifying relative oil/surfactant and
water pressures such that the water enters the crossflow
junction perpendicular to the oil stream, shearing off into
discrete droplets (Fig. 1).
The shape of the channels influences the size distribution
and morphology of the droplet patterning and can be modi-
fied by heating the photoresist mold on the silicon wafer
(80–110 ±C) to round the normally rectangular channels.
The photoresist flows during the heating process, creating
localized maxima and minima at the perpendicular inter-
section in the mold where the water is sheared into the
oil/surfactant phase and the transitions from the restricted
to the wide channels. Channels that have not been rounded4164produce only monodisperse reverse micelles with regular
periodicity that associate with the walls of the wide chan-
nel as they flow through the device (Fig. 2). The relative
water/oil-surfactant pressures determine the size and spac-
ing between the reverse micelles. The patterns in a rounded
channel are more complex, ranging from periodic droplets
to “ribbons,” “pearl necklaces,” and helical intermediate
structures. The self-organization of the reverse micelles
depends on the differential pressure between the water and
oil-surfactant phases, with higher relative water pressures
driving the formation of increasingly complex droplet ar-
rays (Fig. 3).
The diverse pattern formation found in the rounded
channels can be classified as follows. When the oil pres-
sure greatly exceeds the water pressure, the water stream is
held in check by surface tension and only the oil flows. As
the water pressure is increased past a critical point, single
monodisperse separated droplets are formed at a frequency
of 20–80 Hz. Small adjustments in the water pressure in
this range change the radii of the formed droplets, with
higher water pressures generating larger droplets. When
the relative oil and water pressures are approximately
balanced (Pw  Po), droplets are formed in a pearl-
necklacelike configuration [Figs. 3(D) and 3(E)]. They
stack up against each other during the transition from the
30 mm channel to the wider 60 mm channel due in part to
the increased drag of the necklace (which is larger than the
separated monodisperse droplets). At water pressures that
slightly exceed the oil pressure (Pw . Po), the packing
density of the droplets in the 60 mm channel increases.
The first complex structure that emerges with increas-
ing oil pressure is a transition from the pearl-necklace
shape into a zigzag pattern of droplets [Fig. 3(G)]. At
moderately higher water pressures (10% higher than the
relative oil pressure), shear occurs at both the crossflow
junction and the transition from the narrow to wide
microchannel. Polydisperse and bidisperse motifs appear
as helices and patterned multilayer ribbon structures. The
patterns remain coherent as the arrayed droplets flow
down the entire length of the channel from the breakpoint
FIG. 2. Reverse micelles in square channels. Photomicro-
graphs show the transition from the 30 mm wide channel to the
60 mm wide channel. Respective pressures for the water and
oil/surfactant (hexadecane2% Span 80) are noted in the figure.
VOLUME 86, NUMBER 18 P H Y S I C A L R E V I E W L E T T E R S 30 APRIL 2001FIG. 3. Droplet patterns in rounded channels at different
water and oil/surfactant pressures (noted in the figure) and the
corresponding phase diagram depicting the relationship between
the oil and water pressure differences and droplet morphology.
Solid lines are used to define approximate boundaries between
the following droplet states (top to bottom): solid water stream,
ribbon layer, pearl necklace, single droplets, and solid oil stream.
Symbol definition: solid water stream (solid circle); elongated
droplets (open circle); triple droplet layer (solid triangle);
double droplet layer (open triangle); jointed droplets (solid
square); separated droplet (open square). Photomicrographs
show 60 mm channel regions downstream of the point of
crossflow.
to the outlet (4 cm). At excessive water pressure, water
coflows with the oil as separate streams, as one would
expect for laminar flow of two conjoined streams.
The simplest model for droplet formation is based on the
shear forces generated between the water and oil surfactant
at the crossflow junction. The predicted size of a droplet
under external shear force is approximated by equating the
Laplace pressure with the shear force [13]:
r 
s
h ´
, (1)
where r is the final droplet radius, s is the interfacial
tension between the water/oil-surfactant, h is the viscosity
of the continuous phase, and ´ is the shear rate.
In the microfluidic device, a shear gradient is estab-
lished as water tries to expand into the pressurized con-
tinuous phase. The water stream never completely blocks
the flow of the continuous phase, and the oil surfactantflows through the restriction at velocities up to 6.4 cms.
Equation (1) gives a good approximation of the droplet
sizes generated in the microfluidic device when the shear
rate is estimated as ´  2yy0, where y0 is the channel ra-
dius at the center of flow estimated by triangular approxi-
mation, and y is the velocity of the fluid through the gap.
Predicted droplet sizes are within a factor of 2 of actual
droplet size measured by video microscopy for monodis-
perse droplets generated at water and oil pressures ranging
from 8.0–22.4 psi (Fig. 4).
Pattern formation in the microchannels appears to be
driven by the drag force of the droplets and contain fric-
tion with the floor and ceiling of the device. As the droplets
transition from the narrow crossflow junction to the 60 mm
channel, they slow down significantly relative to the oil
phase. At higher droplet frequencies, they begin to col-
lide, stacking up into organized patterns at the transition
between the 30 60 mm channel. Complex structures form
in rounded channels at high relative water pressures as
colliding droplets are pushed from the center of the flow
stream. The pattern formation results as a trade off be-
tween the interfacial tension of the droplets and the shape
of the channels —droplets prefer to stay in the middle of
the rounded channels in order not to pay an energy penalty
for deformation in the crevices at the edge of the channels.
Secondary shearing at the slowing junction also affects pat-
tern formation — if the initial droplet is not commensurate
with the size selected by the junction, then size dispersity
is introduced to the stream and asymmetric motifs appear
[Figs. 3(J) and 3(L)].
We have mapped a crude phase diagram that shows the
pattern morphology is predominantly dependent on only
the dimensionless differential input pressure. However,
some of the most interesting patterns are found only at cer-
tain absolute values of the input pressure (Fig. 3). Some-
what surprisingly, these structures maintain a high degree
FIG. 4. Predicted vs actual drop size at different water and
oil/surfactant pressures. The predicted sizes were calculated
using Eq. (1). Open symbols, predicted size; solid symbols,
experimental.4165
VOLUME 86, NUMBER 18 P H Y S I C A L R E V I E W L E T T E R S 30 APRIL 2001of coherence, despite the fact that they are formed dynami-
cally when the system is far from thermodynamic equilib-
rium. Furthermore, this system shows an unusual richness
in the variety of phases it can display, especially consid-
ering that the boundary conditions (which also function as
order parameters) are simply constant pressure applied to
fluid inputs.
Pattern formation also occurs in granular materials,
which have some striking similarities with our system.
In both cases, the fundamental particles are so large
that thermal fluctuations are negligible. Also, granular
systems can display a “jamming” phenomena, in which
the particles get trapped in metastable configurations that
are difficult to escape from. The pearl necklaces and
zigzag patterns in our system shown an ability to get into
jammed states of high stress. The joined droplets behave
similar to a spring, continually trying to relieve the added
strain within the system by trying to orient themselves
in the center of the stream. This behavior is shown by
multiparticle defects that propagate as waves through
the pearl necklaces with a speed greater than the droplet
stream [Fig. 3(A)].
In conclusion, we have shown how instability can de-
velop as a competition between shear forces and surface
tension in a microfluidic device. The system is technically
at low Reynolds number, but the equations of motion are
nonlinear because the boundary between the two fluids is
not static. Although we have outlined some of the basic
physics leading to the vesicle forming instability and sub-
sequent pattern formation, it is clear that more work needs
to be done to achieve a complete understanding of the sys-
tem. Since geometric effects play a significant role in the
pattern formation, one should be able to take advantage
of the powerful microfabrication technology both to ex-
plore the consequences of this observation and to provide
stringent tests of theoretical models. This system may also4166find application as a component in a microfluidic screen-
ing chip, since it has been shown that subnanoliter vesicles
have significant potential as tools for screening of biologi-
cal and synthetic compounds [14–16].
We thank R. Goldstein for helpful discussions. This
work was partially supported by Research Corporation and
the NSF.
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Dynamic Pattern Formation in a Vesicle-Generating Microfluidic Device

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Dynamic Pattern Formation in a Vesicle-Generating Microfluidic Device

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