3716
`
`Thomas Ward
`Magalie Faivre
`Manouk Abkarian
`Howard A. Stone
`
`Division of Engineering
`and Applied Sciences,
`Harvard University Cambridge,
`Cambridge, MA, USA
`
`Electrophoresis 2005, 26, 3716–3724
`Microfluidic flow focusing: Drop size and scaling in
`pressure versus flow-rate-driven pumping
`
`We experimentally study the production of micrometer-sized droplets using micro-
`fluidic technology and a flow-focusing geometry. Two distinct methods of flow control
`are compared: (i) control of the flow rates of the two phases and (ii) control of the inlet
`pressures of the two phases. In each type of experiment, the drop size l, velocity U and
`production frequency f are measured and compared as either functions of the flow-rate
`ratio or the inlet pressure ratio. The minimum drop size in each experiment is on the
`order of the flow focusing contraction width a. The variation in drop size as the flow
`control parameters are varied is significantly different between the flow-rate and inlet
`pressure controlled experiments.
`
`Keywords: Microfluidics; Flow focusing; Multiphase flow
`
`DOI 10.1002/elps.200500173
`
`1 Introduction
`
`Emulsions, foams, aerosols and other dispersions are
`examples of multiphase fluids. Traditionally these materi-
`als find applications and/or arise in processing of home
`and personal care products,
`foods, petroleum-based
`products, mineral flotation, etc. In many of these applica-
`tions the dispersions are polydisperse and poorly con-
`trolled, though variants of bulk emulsification methods
`can achieve control of the drop size and the size distribu-
`tion. Renewed interest in these fields that involve emul-
`sions has been brought about by the demonstration that
`microfluidic technology can provide accurate control of
`droplet size at the micrometer scale, allows precise con-
`trol of the chemical composition and the thermal environ-
`ment, and can be utilized so that individual drops act as
`isolated chemical containers. This control has been
`demonstrated to be useful in new applications and for
`developing new technologies, in particular for providing
`new routes for addressing questions, measurements and
`products of interest to the biological and chemical com-
`munities.
`
`One significant advantage of microfluidic devices is the
`ability to control the formation of drops at the very scale at
`which drops are desired. Thus, mm-sized drops can be
`readily formed using channels and other plumbing that
`are at the mm scale, and standard visualization proce-
`
`Correspondence: Dr. Howard A. Stone, Division of Engineering
`and Applied Sciences, Pierce Hall, Harvard University, 29 Oxford
`St, Cambridge, MA 02138 USA
`E-mail: has@deas.harvard.edu
`Fax: 11-617-495-9837
`
`Abbreviation: PDMS, poly(dimethylsiloxane)
`
` 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
`
`dures can be employed simultaneously. Several different
`geometries have been investigated for the precise control
`of drop sizes including liquid-liquid and liquid-gas sys-
`tems [1–6]. A general theme in all of these studies is that
`they show the results are reproducible for a wide range of
`parameters and that there exists the potential for precise
`control of individual droplets.
`
`It has been recognized that small droplets can act as iso-
`lated chemical containers, whose composition and other
`environmental variables can be carefully controlled and
`monitored. Thus, for example, biological species may be
`grown in a precision controlled environment to control
`population density and specimen yield [7–9]. Microfluidic
`formation of droplets also enables a variety of encapsu-
`lation methods such as producing double emulsions [10,
`11], and variants of these strategies have been used for
`the microfluidic production of vesicles [12] and colloidal
`crystals [13]. Alternatively, there are many opportunities in
`biology and chemistry to utilize a microfluidic platform
`capable of
`integrating and controlling thousands of
`channels, which has the potential for rapidly performing
`many experiments [14] using nanoliters or even picoliters
`of reagent (note that a spherical droplet with radius 10
`microns has a volume of 4 pL and a droplet with radius
`100 mm has a volume of 4 nL). Also, single experiments
`with high precision can be performed where the micro-
`fluidic device is used for sensing, detection and separa-
`tion. For example, it may be possible to rapidly determine
`if an airborne pathogen is present or detect and sort bio-
`logical species such as DNA or cells [15]. Recent work
`has also shown how a microfluidic device containing a
`viscoelastic liquid can behave much like an electronic
`transistor as a switch [16]. Further developments are likely
`to integrate several of the above ideas.
`
`1
`
`

`
`General
`
`Electrophoresis 2005, 26, 3716–3724
`
`Microfluidic flow focusing: Pressure versus flow-rate-driven pumping
`
`3717
`
`In this paper, we study drop formation in a flow-focusing
`microfluidic device and compare two methods of sup-
`plying the fluids: a syringe-pump methodology that dic-
`tates the volume flow rates and a method whereby the
`inlet fluid pressure is controlled. The flows are at low
`Reynolds numbers for which the pressure gradient and
`flow rate are linearly related in a single phase flow. Con-
`sequently it might be expected that the two distinct
`control methods, flow rate versus pressure, would lead
`to the same details of drop formation.
`Instead, we
`document distinct differences between these two
`approaches for the multiphase flows of interest. Although
`we have not yet succeeded in rationalizing all of these
`differences, it is clear that the significant influence of
`surface tension, which is known to introduce non-
`linearities into the theoretical description (due to the
`coupling of the shape and the fluid stresses, such as
`pressure, in the boundary conditions) is likely to be the
`major contributor to the significant differences we have
`identified between volume flow rate and pressure control
`of drop formation in multiphase flows.
`
`In Section 2 we describe the experimental setup for the
`formation of water drops in oil streams. We then report in
`Section 3 the differences between the two methods by
`varying the ratio of the dispersed to continuous phase
`flow parameter, either the ratio of flow rates Qw/Qo or the
`ratio of applied pressures Pw/Po. In order to characterize
`the drop formation process we report the droplet speed
`U, frequency of droplet production f, distance between
`two consecutive drops d, and the drop length l (which is a
`measure of the drop size). In the Section 4 we discuss
`surface tension effects as a possible difference between
`the two experiments.
`
`2 Materials and methods
`
`The microfluidic experiments use relief molds that are
`produced using common soft-lithography techniques
`[17]. The experiments are performed using a flow-focus-
`ing geometry [3, 5] with a single input channel for both
`fluids. An illustration of the setup is shown in Fig. 1. The
`channel height is 75 mm, as measured by a profilometer,
`the inlet channel width is 100 mm, and the outlet channel
`width is 100 mm with a 50 mm contraction. The channels
`are made of poly(dimethylsiloxane) (PDMS) and bonded
`to a thin sheet of PDMS by surface plasma treatment in
`air.
`
`Figure 1. Microfluidic setup for the two-phase flow in a
`flow-focusing geometry. The channel height is 75 mm. (a)
`Side view. (b) Top view.
`
`systems [18]. We assume that all of the fluid properties
`remain constant at their established values at room tem-
`perature.
`
`The fluids are driven two ways: by static pressure and by
`a syringe pump. These methods are distinct since the use
`of specified pressures sets up corresponding flow rates of
`the two phases while a syringe pump maintains a speci-
`fied volume flow rate by appropriate adjustment of the
`force applied to the syringe, which changes the pressure
`in the fluid. Although in a single-phase flow these two
`types of experiment would be equivalent, the same is not
`true in a multiphase flow since the shape of the fluid-fluid
`interface, which is determined by the flow itself, compli-
`cates the relationship between the pressure drop and the
`flow rate. In the static pressure pumping approach the
`fluids are placed in syringe tubes and the air pressure
`above the fluid is regulated. The pressure is controlled by
`precision regulators (Bellofram Type 10) with a sensitivity
`of 5 mPsi. The pressure is measured using a test gauge
`(Wika) with a resolution of 150 mPsi. All pressures are
`reported relative to atmosphere. The syringes are con-
`nected to their respective inputs of either the dispersed or
`the continuous phase channel. The syringe approach for
`fluid delivery requires the choice of a tube diameter and
`then uses standard motor-driven pumping to achieve a
`given flow rate.
`
`The fluids are mineral oil (viscosity n = 34.5 cSt at 407C
`and density r = 880 kg/m3) and deionized water for the
`continuous and dispersed phases respectively. The inter-
`facial tension is estimated to be g = O(10) mN/m from
`published data for surfactant-free mineral oil and water
`
`The images are obtained using high-speed video at
`frames rates O(1 – 10 kHz) as needed depending on the
`experiment and drop speeds are determined by tracking
`the center of mass of individual drops using commercially
`available software.
`
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`

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`T. Ward et al.
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`3 Results
`
`3.1 Qualitative differences in flow-rate versus
`pressure-controlled experiments
`
`We first contrast the qualitative features of drop produc-
`tion using flow-rate controlled (syringe pumping) versus
`pressure-controlled flow. Typical results are shown side
`by side in Fig. 2. A direct qualitative comparison is made
`by visual inspection of the change in the drop size as
`either the flow rate of the dispersed water phase Qw is
`increased for fixed continuous phase flow rate Qo of the
`oil phase (Fig. 2a) or as the inlet pressure of the dispersed
`phase fluid Pw is increased for fixed inlet pressure of the
`continuous oil phase Po. Note: All pressures reported are
`gauge so that they are relative to atmosphere. The pic-
`tures are taken one frame after the liquid thread at the
`focusing nozzle breaks into a drop. The pictures of drops
`produced using flow-rate driven pumping show a striking
`difference to their pressure-driven counterparts. The
`most notable difference is that as the dispersed phase
`pressure is increased (by not quite a factor of two) the
`drop size increases significantly. This response is not true
`for the flow-rate driven experiments where the drop sizes
`only change slightly as the water flow rate is increased by
`a factor of 10.
`
`Electrophoresis 2005, 26, 3716–3724
`
`3.2 Quantitative measurements
`
`In a typical experiment drops of a certain size are formed
`at a given frequency f and move down the channel at a
`speed U. The frequency f of drop production is measured
`by direct counting from high-speed video, and the speed
`U is measured by tracking the drop center of mass for a
`short distance down the channel. The photos shown in
`Fig. 2 gave a qualitative indication of the drop size, and
`we will return to this topic below. Now we consider con-
`trasting the speeds of the drops in the flow-rate driven
`versus the pressure-driven approaches.
`
`The drop speeds versus the flow-rate or pressure ratio for
`different conditions in the continuous phase are shown in
`Fig. 3. In this figure and all of the other figures below, we
`use closed symbols to represent the case where the
`continuous phase flow parameter is held constant, while
`the dispersed phase flow parameter is increased until a
`liquid jet is produced and drops no longer form at the ori-
`fice. The open symbols are the opposite experiment,
`where a liquid jet is initially formed at higher values of the
`operating conditions and then the dispersed phase flow
`parameter is decreased until the dispersed phase fluid no
`longer forms drops. In all these experiments we expect a
`small difference in velocity between the speed of the
`drops and the average velocity of the continuous phase
`
`Figure 2. Drops produced
`using (a)
`flow-rate controlled
`flow-focusing setup where the
`oil flow-rate is constant and the
`dispersed phase flow rate varies
`as indicated; and (b) pressure-
`controlled setup where the oil
`inlet pressure is constant and
`the dispersed phase inlet pres-
`sure varies as indicated.
`
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`Electrophoresis 2005, 26, 3716–3724
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`Microfluidic flow focusing: Pressure versus flow-rate-driven pumping
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`
`speeds increase approximately linearly with Qw. In con-
`trast, for pressure-driven pumping there is a minimum
`pressure ratio, or pressure of the water phase, below
`which no drops are formed, while above this critical
`pressure the speeds are approximately independent of
`the pressure until a higher critical value at which the
`speeds increase rather rapidly.
`
`The corresponding results for the frequency of drop for-
`mation for the two methods of flow control are shown in
`Fig. 4. The results are reported as the number of drops per
`millisecond. Again, the error bars are typically the symbol
`size or smaller. For the conditions used in our experiments
`there are from zero up to more than 1000 drops formed
`per second for the flow-rate driven experiments. For the
`pressure-driven experiments the frequencies are about
`one-third to one-half as large. In both sets of experiments,
`the frequencies increase approximately linearly as the in-
`dependent variable is increased. Also,
`for all of the
`experiments shown in Figs. 3 and 4 there is essentially no
`difference to the results upon either
`increasing or
`decreasing the independent variables. For the flow-rate
`controlled experiments, we have also plotted the fre-
`quencies normalized by the external flow rate, and as
`shown in the inset of Fig. 3 this effectively collapses all of
`the data.
`
`We next provide data on the relative spacing d between
`the drops as a function of the speed and frequency of
`drop formation. If the drop spacing is sufficiently large
`relative to the channel width we expect the train of drops
`to be stable, though there is evidence for instabilities in
`the motion of a train of drops when the drops are too
`closely spaced (S. Quake, private communication). We
`expect fd = U on geometric grounds and have independ-
`ently measured all three quantities, d, f, and U. In Fig. 5 we
`present a plot of fd versus U in dimensionless terms by
`essentially converting to an equivalent Reynolds number
`(e.g., Ua/n, where 2a is the width of the channel). As
`expected the plot is nearly linear, independent of the flow
`control parameter (inlet pressure or flow-rate); we believe
`the small difference of the slope from unity reflects
`experimental errors in the independently measured
`quantities. It is also now clear that the Reynolds numbers
`Ua/n are all less than one for all of our experiments so
`viscous effects should be more important than inertial
`effects in the dynamical processes.
`
`We next consider in more detail quantitative aspects of
`the variation of the drop size, l, which, because of the
`rectangular shape of the channel, is convenient to report
`as the end-to-end distance. The results are normalized by
`the width of the orifice a, which is half the width of the
`channel for all of the experiments reported here. In Fig. 6
`the drop sizes are reported as a function of
`the
`
`Figure 3. Measured velocity U versus (a) flow-rate where
`o, d Qo = 2000 mL/h, u, n Qo = 1000 mL/h and (e, r Qo =
`500 mL/h and (b) inlet pressure ratio where o, d Po =
`12.5 Psi, u, n Po = 10 Psi and e, r Po = 7.5 Psi. The open
`and closed symbols are for experiments where the dis-
`persed phase fluid flow parameter is either increased or
`decreased respectively.
`
`due to hydrodynamics of the two-phase flow; we have
`not, however, tried to measure this slip velocity system-
`atically. The error bars are typically the symbol size or
`smaller. In Fig. 3, for flow-rate driven pumping the drop
`
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`T. Ward et al.
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`Electrophoresis 2005, 26, 3716–3724
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`Figure 5. Frequency f of drop formation times the distance
`between drops d versus measured drop velocity U. Both
`parameters are made dimensionless by scaling with n/a.
`
`Figure 4. Frequency f of drop formation versus (a) flow
`rate where o, d Qo = 2000 mL/h, u, n Qo = 1000 mL/h and
`e, r Qo = 500 mL/h and (b) inlet pressure ratio. The open
`and closed symbols are for experiments where the dis-
`persed phase fluid flow parameter is either increased or
`decreased, respectively.
`
`flow-rate ratio Qw/Qo. The corresponding drop sizes as a
`function of the pressure ratio Pw/Po are reported in Fig. 8.
`Comparisons are further shown by plotting the best fit line
`through log-log plots of the scaled data. This is not meant
`to suggest a linear relationship but rather to show the dif-
`ference in trends between the two experiments.
`
` 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
`
`Figure 6. Dimensionless drop length l/a versus flow rate
`ratio Qw/Qo. The inset shows same data using log-log
`coordinates, where the solid line has slope 0.25.
`
`In Fig. 6, the drop size as a function of the flow-rate ratio
`Qw/Qo shows very little variation even when the oil flow
`rate is varied. At the lowest flow-rate ratios the drop sizes
`are nearly on the order of the flow-focusing orifice size, l/a
`< 1, which indicates the geometric control of drop size
`common to many microfluidic experiments. The inset
`shows the same data on a log-log plot with more
`
`5
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`Electrophoresis 2005, 26, 3716–3724
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`Microfluidic flow focusing: Pressure versus flow-rate-driven pumping
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`
`by the orifice size a, versus the flow rate ratio Qw/Qo. At
`low flow-rate ratios the distance between drops is large
`and this distance decreases rapidly to a nearly constant
`value of about three times the width of the orifice; with this
`spacing the train of drops is stable. We then combine the
`above results by showing in Fig. 7b the relationship be-
`tween the drop size l scaled by the distance between
`drops d as a function of the flow-rate ratio. When plotted
`with log-log coordinates, as shown in the inset, the data
`again appears to be reasonably fit with a power-law rela-
`tion. Unlike the data scaled by the orifice width, the
`l/d < 1
`asymptotic
`value
`of
`is
`reached
`at
`(Qw/Qo)max < 1.5.
`
`The pressure-driven flow rate experiments have some
`similarities to the flow-rate driven results, though as we
`shall now see there are both quantitative and qualitative
`differences. In Fig. 8 we begin with the data for the drop
`size l/a as a function of the pressure ratio, Pw/Po. As
`already indicated above, there is a minimum pressure
`ratio (Pw/Po)min < 0.3 below which the dispersed phase
`fluid does not penetrate the continuous phase. The mini-
`mum drop size for each curve is about the orifice size
`(< 50 mm), which indicates the strong geometry depend-
`ence in these systems. When plotted in log-log coordi-
`nates, as shown in the inset though now there is only a
`small variation in Pw/Po, again there appears to be a rea-
`sonable fit with a power-law relation, l/a ! (Pw/Po)2.
`
`Figure 7.
`(a) Distance d between the center of mass of
`two consecutive drops versus the flow-rate ratio.
`(b)
`Dimensionless drop length l/a versus flow-rate ratio. The
`inset shows the same data plotted in log-log coordinates,
`where the solid line has slope 0.4, which appears to fit the
`data over two decades of the flow rate ratio Qw/Qo.
`
`than a factor of ten variation in Qw/Qo, which indicates an
`approximate power-law relationship, l/a ! (Qw/Qo)0.25,
`obtained by a best fit of the data.
`
`We continue with the data corresponding to the flow-rate
`driven experiments in Fig. 7a where we plot the distance,
`d, between the center of mass of two drops, normalized
`
` 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
`
`Figure 8. Dimensionless drop length l/a versus inlet pres-
`sure ratio Pw/Po. The inset shows the same data using
`log-log coordinates, where the solid line has
`slope 2.
`
`6
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`Electrophoresis 2005, 26, 3716–3724
`
`three times the orifice width, as observed in the flow-rate
`control experiments. As with the flow-rate driven experi-
`ment we combine the results by showing in Fig. 9b the
`relationship between the drop size l scaled by the dis-
`tance between drops d as a function of the inlet pressure
`ratio. When plotted in log-log coordinates, as shown in
`the inset with only a slight variation in Pw/Po, there again
`appears to be a reasonable fit with a power-law relation,
`l/d ! (Pw/Po)3. Unlike the data scaled by the orifice
`width, the asymptotic value of l/d < 1 is reached at
`(Pw/Po)max < 0.55.
`
`4 Discussion
`
`In this section we present some ideas for rationalizing the
`experimental results reported above. To the best of our
`knowledge there are no general quantitative theories for
`two-phase flow in a geometry such as the flow-focusing
`configuration. In the special case that a very narrow
`thread is formed at and downstream of the orifice, then
`Ganan-Calvo et al. [2] have given quantitative models that
`are in good agreement with the measurements. All the
`results presented here are not in this regime. Furthermore,
`in two-phase flows, the viscosity ratio l = mw/mo, where
`mw,mo are the dispersed and continuous phase fluid dy-
`namic viscosity, between the drops is a parameter and
`our experiments have only considered one viscosity ratio,
`while varying other parameters and considering two
`methods of flow control.
`
`4.1 The role of flow control: flow-rate versus
`pressure
`
`Now we discuss a possible mechanism for the difference
`in flow-rate and inlet pressure driven droplet production
`seen in our experiments. To illustrate we qualitatively
`relate the global transport of momentum for the dispersed
`phase with the dimensionless expression Pw ! g(a, Po,g,
`mw,mo)Q where the function g(a, Po,g, mw,mo) is the sum of
`the external resistances that depend on geometry a,
`external phase fluid pressure Po, surface tension g and
`absolute viscosity of the two fluids mw and mo.
`
`for the pressure-controlled experi-
`Most significantly,
`ments there is a minimum water pressure below which no
`drop formation is observed. Just below this pressure the
`continuous phase fluid pressure is too high for the dis-
`persed phase fluid to penetrate into the main channel
`after the contraction. But the inlet pressure for the dis-
`persed phase fluid is not zero and there must be another
`force to create the static or zero flow rate dispersed
`phase flow. At pressures just below this critical pressure
`(Pw/Po)min, the external flow exerts shear and normal
`
`Figure 9. (a) Distance between the center of mass of two
`consecutive drops versus inlet pressure ratio. (b) Dimen-
`sionless drop length l/a versus flow-rate ratio. The inset
`shows the same data plotted in log-log coordinates,
`where the solid line has slope 3, which appears the fit the
`data over two decades of the flow rate ratio Pw/Po.
`
`In Fig. 9 we show data corresponding to the inlet pres-
`sure-controlled experiments. In Fig. 9a we plot the dis-
`tance, d, between the center of mass of two drops, nor-
`malized by the orifice size a, versus the inlet pressure ratio
`Pw/Po. At low inlet pressure ratios Pw/Po ? (Pw/Po)min the
`distance between drops is large and this distance
`decreases rapidly to a nearly constant value of a little over
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`Microfluidic flow focusing: Pressure versus flow-rate-driven pumping
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`stresses on the liquid-liquid interface that are greater than
`the forces exerted by the internal fluid, and a static flow
`situation is observed at the orifice for the internal fluid
`where a static spherical cap of fluid is formed, which
`suggests the significant role of surface tension.
`
`The pressure distribution can have a substantial influence
`on penetration into the main channel of the spherical cap
`formed at the orifice. To illustrate this point we use a
`microscopic force balance to rationalize what happens at
`the liquid-liquid interface. The boundary conditions for the
`interface in dimensionless form are written as
`Dp þ t ¼ Ca1k
`
`(1)
`
`Here, Dp is the pressure difference between the dis-
`persed and continuous phase, t = mn?ru?n is the viscous
`stress components where n is a unit vector pointing nor-
`mal to the interface, u is the velocity and r is the gradient
`operator. k is the mean curvature along the interface
`where Ca = mU/g is the capillary number.
`
`We estimate the capillary number for our pressure-driven
`system to be Ca < 0.1 – 1, based on a drop velocity of U
`, O(1 – 10 cm/s), indicating the mean curvature term in
`Eq. (1) is at least of the same order as the fluid pressure
`and viscous stresses. As a droplet forms near the orifice
`the volume increases which indicates a decrease in the
`mean curvature until the outer fluid pressure and viscous
`forces are high enough to break the drop.
`
`This suggests that for a system where the pressure is
`specified the dominant contributions to the resistance will
`come from surface tension, not viscous forces. This may
`or may not be the case for the flow-rate driven experi-
`ments. In either situation, to fully understand this part of
`the problem we would need to know more about the
`relationship between viscous stress and surface forces in
`confined geometries.
`
`Looking further downstream, it is well known in the vis-
`cous flow literature that small drops moving in circular or
`rectangular channels move at nearly the same speed as
`the external phase fluid. Whether they lag or lead the flow
`depends on geometry in a very complicated way that has
`not been completely characterized, but in any event the
`velocity difference between the two phases is propor-
`tional to a small power of the external phase capillary
`number, i.e., to Ca1/3 = (mU/g)1/3, when the capillary num-
`ber is small [19]. In our case mU/g < 0.1 – 1, depending on
`how the flow is driven, so existing theories may yet be
`useful for providing quantitative insight. Nevertheless the
`results presented show strong evidence that varying the
`type of boundary conditions, i.e., fixing fluid flow rate or
`pressure, when performing experiments and analysis for
`two-phase fluid systems can produce varying results.
`
` 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
`
`4 Conclusions
`
`We have presented results of microfluidic experiments
`conducted to study micrometer-sized droplet production
`using a flow-focusing geometry. We study droplet pro-
`duction using either the flow-rate ratio or the inlet pres-
`sure ratio as the flow-control parameter. In each experi-
`ment the drop size l, velocity U and production frequency
`f are measured and compared for the different flow-con-
`trol parameters. Perhaps surprisingly, there are significant
`differences between these two methods of flow control.
`The minimum drop size in each experiment is on the order
`of the flow-focusing contraction width, and illustrates one
`aspect of the geometric control possible if the results are
`to be scaled down (or up). The transition in drop size as
`we vary the flow-control parameter contrasts sharply in
`the two distinct two-phase flow experiments. It is this
`distinct difference between the two different manners of
`flow control used here, which has not been previously
`noted, that is one of the major conclusions of the present
`paper.
`
`In each set of experiments the data range for the drop size
`is qualitatively similar when plotted versus the ratio of the
`flow parameter, i.e., Qw/Qo for the flow-rate driven case
`and Pw/Po for the inlet pressure controlled case where w
`and o denote the dispersed (water) and continuous
`(mineral oil) phase fluid, respectively. Similarly the max-
`imum value for the onset of a jet which is easily deter-
`mined using the dynamic length scale d = U/f is the same
`in both sets of experiments, i.e., the normalized distance
`d/l between the center of mass of twoo consecutive drops
`which can never be less than 1. But a power-law fit
`through the data for the drop size scaled by either a or d
`versus the flow control parameter ratio shows quantitative
`differences, in spite of the viscous (low-Reynolds number)
`flow conditions where pressure gradient and average ve-
`locity are expected to be proportional. These results
`suggest a fundamental difference in drop break-up be-
`tween the two types of experiments, and the influence of
`surface tension on this free-surface flow is likely to be the
`origin of the differences.
`
`Droplets rapidly made and manipulated in microfluidic
`devices have been used in creative ways for chemical and
`biological applications, such as the measurement of
`reaction rates and for identifying conditions for protein
`crystallization [20]. Almost certainly, additional uses of
`droplets as isolated containers, small reactors, chemical
`delivery agents, templates for interfacial assembly, etc.,
`will be identified and demonstrated. In such cases, the
`method for the control of the drop formation process must
`be chosen and the results reported here contrasting vol-
`ume flow rate and pressure control should then be of
`interest.
`
`8
`
`

`
`3724
`
`T. Ward et al.
`
`Electrophoresis 2005, 26, 3716–3724
`
`We thank Unilever Research and the Harvard MRSEC
`(DMR-0213805) for support of this research. We also
`thank F. Jousse and colleagues at Unilever and Harvard
`for helpful conversations. We thank D. Link for helpful
`feedback on a draft of the paper.
`
`Received February 28, 2005
`Revised June 8, 2005
`Accepted July 6, 2005
`
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` 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
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