We introduce a novel technique that enables pressure measurements to be made in microfluidic chips using optical trapping. Pressure differentials across droplets in a microfluidic channel are determined by monitoring the displacements of a bead in an optical trap. We provide physical interpretation of the results. Our experiments reveal that our device has high sensitivity and can be operated over a wide range of pressures from several Pascals to several thousand Pascals.
© 2012 OSA
Microfluidic techniques have been in rapid development in recent years and have found numerous applications in physics, chemistry, biology and interdisciplinary studies [1,2]. The transport and manipulation of emulsions, especially microfluidic droplets, are at the heart of many devices [3–5], and the extra pressure caused by droplets is often a key factor in their design and functionality [6,7]. For example, some “digital” microfluidic devices use droplets or bubbles as logic signals and rely heavily on the extra pressure that they provide to control logic gates and switches [8,9]. As a result, the successful design and operation of droplet logic is based upon accurate knowledge of the pressure differentials across droplets of different sizes and in various flow conditions. In addition, droplet-based single cell analysis [10,11], biochemical assays [12,13] and microparticle syntheses [14,15] present a considerable challenge for device miniaturization, since the large actuating pressure necessary to sustain the flow of an ensemble of droplets in high throughput experiments is demanding for the design of micropumps that can be integrated on the chip . Furthermore, some technologies use bubbles or droplets to disturb the laminar flow patterns that occur at low Reynolds numbers and thereby accelerate micromixing processes in the continuous phase [17,18]. The altered pressure distribution resulting from bubbles has an impact on the velocity and residence time of the other bubbles, as well as the continuous phase in the mixing units, and can therefore affect the quality and efficiency of mixing. Last, evaluation of pressure perturbations due to droplets and other emulsions in underground pores is key to enhanced oil recovery , and microfluidic devices provide an ideal platform to mimic such porous environments and measure this pressure in a laboratory.
A variety of techniques to measure pressure changes or pressure differentials in microfluidic channels have been reported. Examples include membrane-based devices, which electrically  or optically [21–24] characterize the pressure-induced deformation of a thin membrane serving as part of the channel wall, or track the displacement of microparticles due to membrane deformation . However, these methods are in general not appropriate for measuring pressure differentials due to droplets, since deformation of the membrane alters the simple rectangular geometry of the channel cross section, rendering it difficult to interpret and model results for droplets. Also, membrane-based devices are often associated with tricky or expensive fabrication techniques, such as multilayer soft lithography [20,22–25] and reactive ion etching . An alternative scheme for pressure sensing involves connecting microchannels to sealed air chambers and relating pressure to the volume of trapped air . Glass channels, however, are required in this scenario as poly(dimethylsiloxane) (PDMS), the flexible material popularly used for microfluidics, is air permeable. The need for glass channels reduces the flexibility and increases the cost. Recently, the use of external commercial pressure taps to measure extra pressure caused by droplets has been demonstrated . A limitation of this approach is that such external pressure gauges are usually not integration-friendly. That the distance between adjacent connections is several millimeters or more means that the total pressure differential can only be measured across a number of droplets, rather than across a single droplet. Finally, optical interface tracking is also employed to measure pressure fluctuations of multiphase flows [28,29]. While detecting pressure differential across single droplets has been successful , high throughput measurements might still be difficult: the geometry of these devices prohibits continuous measurement of single droplets at a high speed, since the droplets inevitably disturb the fluid-fluid interface, whose position serves as a pressure comparator.
In this work, we combine optical trapping with microfluidics and extend our preliminary results  to realize a novel method for measuring the pressure differentials across single droplets. By “pressure differential”, we are referring to the pressure change across a section of a microfluidic channel containing a droplet. It should also be noted that the quantity being measured is the “extra pressure”, i.e. the difference between the pressure differential with the droplet in the channel and that without a droplet, i.e. with only the continuous phase. Pressure is related to the displacement of the optically trapped bead measured with microscopy. In our experiments, droplets generated by an integrated nozzle on the microfluidic chip are fed continuously into a channel with a simple rectangular cross section. The extra pressure due to each droplet is measured. The geometry of the microchannels can be modified to tune the working range of the device over a large span of pressures from several Pascals to several thousand Pascals, amenable to applications in various flow conditions. This paper first addresses the working principle and fabrication of the device. We then discuss experiments with oil droplets in water and water droplets in oil. In particular, we study dependence of extra pressure on droplet size, and interpret the experimental data with reference to known mechanisms for pressure differential across droplets [7,29]. We show that our results manifest the effects of viscous stresses and capillary forces.
2. Working principle
Our device accomplishes pressure measurement by combining optical trapping with microfluidics. As illustrated in Fig. 1 , a shield structure divides the microchannel into two parallel channels. At low Reynolds number, the pressure differentials across the two parallel channels are identical . We send droplets into the wider part, termed the “main channel” in this paper. In the center of the narrower and segmented “side channel”, we use an optical tweezer to trap a spherical polystyrene bead. The fluid drag force exerted on the bead is balanced by the optical restoring forces from the tweezer, giving rise to a slight displacement of the bead from the trap center. This balance is described by32,33].
The shape of the cross section of the side channel is invariant along its length. Therefore at any given point in the cross section, the local fluid velocity v is linearly dependent on the volume flow rate Q in the channel. Furthermore, Eq. (1) implies a linear relation between Δx and v. Consequently, there is a linear dependence between the bead displacement Δx and flow rate Q. The proportionality coefficient can be determined from a calibration process, which is described in a later section.
Pressure-driven flows at low Reynolds numbers in a network of microfluidic channels are analogous to voltage-driven currents in an electrical circuit . Ohm’s law states that the electric current through a conductor is proportional to the voltage difference across it via its electrical resistance. Similarly, the flow rate Q in the side channel is proportional to the pressure differential ΔP across it via its hydraulic resistance Rh:34].
In our experiments, we measure the bead displacement as droplets pass through the main channel. We then convert this to the flow rate Q in the side channel using the calibration data (discussed in a later section). We calculate the hydraulic resistance of the side channel Rh using Eq. (3). The pressure differential ΔP across the side channel, which is also equal to the pressure differential across the main channel containing the droplets, is then calculated by multiplying Q with Rh, as shown in Eq. (2).
3.1 Device fabrication and experimental set-up
Photolithography and soft lithography are currently enabling PDMS-based microfluidic devices to be fabricated with short turnaround time . Following the standard protocols of these techniques, we fabricate the master in SU8 photoresist on a silicon wafer, and replicate it in PDMS . The PDMS replica is then oxidized in an oxygen plasma chamber at 80 W for 20 seconds, and bonded to a cover glass to form the channels. The bonded device undergoes a further oxidation in oxygen plasma at 80 W for 40 minutes, creating hydrophilic functional groups (e.g. silanol groups –SiOH) on the PDMS surface [36,37]. The water-wetting nature is required for stable production of droplets with water as the continuous phase [3,6].
A microscope image of the fabricated device is presented in Fig. 2(a) . The length of the segmented side channel, denoted by the green line in Fig. 2(a), is designed to be 240 µm. The width of the side channel measures 30 µm. The main channel is 200 µm long, as denoted by the red line in Fig. 2(a), and is 60 µm wide. All channels have a depth of 40 µm. These geometries enable calculation of the channels’ hydraulic resistances by Eq. (3). The two channels have a hydraulic resistance ratio of 1: 4.7. This guarantees that droplets will only enter the main channel, and not interfere with the optical trapping in the side channel.
Water and hexadecane oil (viscosity µ = 3.0 mPa⋅s at 298 K ) with 1.5 wt% surfactant (Span 80) are injected simultaneously into the device by pressurizing the fluids in syringes (Hamilton 25 µL) connected to two inlets with syringe pumps (NE-300, New Era Pump Systems Inc., and 11 Plus, Harvard Apparatus). Polystyrene beads with a diameter of 2 µm are dispersed in the water phase. At this size, the beads will not significantly alter the profile of fluid velocity in the channels. Oil droplets in water are generated by a flow focusing structure incorporated into the chip, similar to that introduced by Anna et al. , as illustrated in Fig. 2(b). The size of the nozzle is designed to be 30 µm × 30 µm. De-ionized water flows to the nozzle concurrently from the upper and lower channels, whereas hexadecane enters from the left branch. Around the nozzle, the oil phase is squeezed by the counter-flowing water and broken into droplets . The size of droplets can be controlled by tuning the flow rates of the two phases [3,6,7]. Nevertheless, at the extremely low flow rates (several µL per hour) we are working with, the size of droplets is polydisperse, likely because of the fluctuations of squeezing forces and flow rates around the flow focusing junction. This contrasts with the monodispersity observed in some studies performed at higher flow rates .
The microfluidic chip is placed in a modified optical tweezer system , as shown in Fig. 3(a) . The core part of our experimental set-up is the pair of microscope lenses sandwiching the microfluidic chip. A near-infrared laser (λ = 975 nm, emitted power P = 200.0 mW) is focused into the side channel by an oil immersion lens (Nikon, NA = 1.25, 100 ×), forming the optical trap. The fluctuation of the laser power is less than 0.5 mW, so a nearly constant trapping stiffness can be expected. The fluid velocity varies quadratically across the microchannel . Thus, to ensure consistency in measurements, the bead is always trapped at a position that is equidistant to vertical side channel walls and just touches the bottom cover glass. This position is manually controlled by the micrometers on the sample stage holding the microfluidic chip. The same lens also images the bead onto a CCD camera (DFK 21AU04, The Imaging Source), as shown in Fig. 3(b), allowing determination of bead position. Another objective lens (Nikon, NA = 0.30, 10 × ) is used for bright field imaging of the device with light from a white light emitting diode (LED) onto another CCD camera (DMK 21AU04, The Imaging Source), as shown in Fig. 3(c). Despite the fact that the image quality is adversely affected by aberration incurred by the PDMS, the resolution is sufficient for determination of the size of droplets passing through the main channel. The two cameras work concurrently at 30 frames per second, and videos from the two are synchronized by noting the frame in which the LED illumination is switched on. The position of the bead is obtained in each frame imaged by the 100 × lens by locating its centroid. Specifically, we identify pixels above a brightness threshold in the center of the bead image, and calculate their average position.
3.2 Device calibration
As mentioned in the Working Principle section, a calibration process is necessary to quantitatively establish the linearity between the bead displacement x and the pressure differential ΔP shared by the parallel channels. To do this, we fabricate a chip containing a channel that has the same cross sectional geometry as the side channel, but is 10 mm long. This chip is placed in the optical tweezer set-up, and water containing polystyrene beads with a 2 µm diameter is flowed into it. A bead is trapped with a laser power of 200.0 mW at the same position in the channel cross section as mentioned before, i.e. equidistant to vertical side channel walls and just touching the bottom cover glass. This consistency in bead position ensures that in calibration the bead position x varies linearly with the flow rate Q, and that the calibration result can be applied to subsequent pressure measurements with the bead trapped at the same position (to be discussed in the following sections). Although this trapped bead may alter the profile of fluid velocity in the channel, this effect occurs in both the calibration results and the subsequent measurements. It is therefore not expected to adversely affect the accuracy of our device. This ensures that all the parameters in calibration match those in experiments for pressure measurement. Flow rates ranging from 1 to 8 µL⋅hr−1 are applied, and position of the trapped bead is measured for each from images obtained with the 100 × objective lens. At each flow rate, the bead position is averaged over a period of 13.33 s, which corresponds to 400 video frames i.e. 400 measurements of bead position. The standard deviation of the bead position within this period of time is found to be δx = 18 nm. The results are plotted in Fig. 4 , and demonstrate the anticipated linear dependence of bead position on flow rate. A linear fit is carried out, shown as the red line in Fig. 4, and gives a slope of 37.3 nm/(µL⋅hr−1). The data quantifies the variation of bead position x with the flow rate Q in the channel. Here, the bead position x refers to the distance of the bead center from the left edge of the image cropped from Fig. 3(b) for analysis. This means that an arbitrary offset exists and that the fit is not expected to pass through the origin. It is therefore the slope, rather than the absolute bead position, that is relevant for calibration purposes. One possible concern would be that the focused laser heats the water, modifying its viscosity , and that this effect is highly dependent on flow rate due to its influence on heat dissipation. If that were the case, the relationship between x and Q would be non-linear. That the observed relationship is linear implies that the heating effect of the focused laser is not problematic.
Since the profile of fluid velocity for laminar flow in the channel only depends on the total flow rate and on the geometry of its cross section and does not vary with its length , this result also applies to the much shorter side channel used in the droplet experiments. The quantitative relationship we obtain between bead displacement Δx and flow rate Q enables calculation of pressure differential ΔP from measured bead position by combining this calibration result with Eqs. (2) and (3), as described in the Working Principle section.
4. Results and discussion
4.1 Extra pressure of hexadecane droplets in water
In this section, we demonstrate the measurement of the extra pressure due to hexadecane droplets in water. Water is supplied to the device with a syringe pump at a flow rate of 6.0 µL⋅hr−1, and hexadecane at 2.5 µL⋅hr−1. Hexadecane droplets are produced at the integrated flow focusing nozzle (Fig. 2(b)). This pattern of droplet generation is robust, a result of the channel walls being hydrophilic after the plasma treatment. As droplets flow through the main channel, the computer records the images obtained by the two CCD cameras through the two lenses at a frame rate of 30 Hz. The droplets are of varied sizes, presumably as a result of natural fluctuations in the squeezing and shear forces at the flow focusing nozzle.
The motion of the bead, trapped at the aforementioned position, equidistant to the vertical side walls and just touching the bottom cover glass, in the side channel, is monitored by analysis of the images obtained by the 100 × lens (e.g. Fig. 3(b)) using centroid detection, as noted previously. Figure 5(a) plots the variation of bead displacement with time over a 6-second interval. Here the bead displacement is offset by defining its zero as follows. Using synchronized videos from the 10 × lens (e.g. Fig. 3(c)) over this 6-second span, we identify intervals where only water is present in the main channel. The average bead position is found for these intervals. This position is then taken as zero displacement. Using the calibration result described in the previous section together with Eqs. (2) and (3), the variation of bead displacement is converted to the variation of extra pressure, shown in the right axis in Fig. 5(a). From the definition of bead displacement above, an extra pressure of zero indicates the case where only water flows through both the main and side channels. A positive extra pressure indicates a larger pressure differential across the parallel channels compared with the referenced value in the water-only case, and a negative extra pressure means a smaller pressure differential.
Video frames imaged through the 10 × lens (e.g. Fig. 3(c)) enable us to determine the length of each droplet passing through the main channel. Synchronization of this video with the data in Fig. 5(a) provides an opportunity for us to analyse the dynamics of pressure fluctuation in detail and its correspondence to the events in the main channel. For instance, this analysis reveals that in Fig. 5(a), the time interval between 4.73 s and 4.90 s (i.e. the peak denoted by a green arrow) corresponds to a single droplet whose length is 165 µm. The extra pressure during this droplet’s presence in the main channel takes an average of 5.35 Pa. We find that most pronounced peaks in Fig. 5(a) correspond to single droplets traversing in the main channel. Negative extra pressures are observed before and after many peaks. This is attributed to the fact that the droplet temporarily blocks the inlet or the outlet of the side channel before its entry into, or exit from, the main channel. This blockage lowers the flow rate in the side channel, and yields a pressure differential smaller than that in the standard water-only case. Occasionally, two closely-spaced droplets arrive at the parallel channel structure. In this scenario, the bead displacement no longer represents the extra pressure caused by the droplet in the main channel, since a second droplet in the vicinity may partially block the side channel and interfere with the flow. This results in a wider peak or some irregularly shaped structure in Fig. 5(a), e.g. the plateau around 2.83 s (denoted by a yellow arrow). These events are readily identified, however, from the video recorded using the 10 × lens.
Compiling the measured size and extra pressure for each droplet, we obtain the relation between extra pressure and droplet length, plotted in Fig. 5(b). The droplets range in size from ~75 µm to ~185 µm. The extra pressure they cause ranges from ~2.5 Pa to ~6.3 Pa. From Fig. 5(b), it can be seen that there is a positive correlation between the two quantities. The lack of a definitive theory concerning the extra pressure of droplets in rectangular channels makes it difficult for us to compare our data rigorously with theoretical predictions. Instead, we will refer to some basic intuition and a simple model in the literature to qualitatively discuss the factors affecting the extra pressure of droplets.
The pressure differential across a droplet originates from viscous dissipation and capillary forces, if no surfactant is present [7,29]. The former comes from the fluid pressure induced by viscous stresses of the dispersed phase. For hexadecane droplets in water, an increase in length substitutes water for a more viscous fluid, hexadecane, leading to more viscous dissipation in the channel and generating a larger extra pressure. Simple models for this pressure differential ΔPbody make the assumption that the droplet body fills the entire channel cross section, and that flow in both phases is laminar [7,29]. Under these assumptions, ΔPbody can be approximated by using the formulae for single-phase laminar flows.
The second contribution to pressure across a droplet is primarily an effect of Laplace pressure . The non-uniform thickness of lubrication films between a droplet in motion and channel walls gives rise to asymmetric caps . Intuitively, the upstream cap is “pushed” by the continuous phase and thus becomes flatter, i.e. less curved. This implies that the curvature-dependent pressure jump across the upstream interface is smaller and cannot fully compensate for the pressure drop across the downstream interface. Consequently, the capillary forces at droplet caps always yield a positive extra pressure ΔPcaps across the droplet. Our intuition is that an increase in droplet length leads to enlarged difference in curvature between the two caps, as a stronger pushing force from the continuous phase might be necessary for a longer droplet, and such force is primarily exerted on the upstream cap. However, the variation between the curvatures of the caps of the hexadecane droplets seems too small to be detected from the microscope images we obtain. Later in the paper we show data on the differences in curvature between the end caps of droplets of different lengths, but these are water droplets in hexadecane moving at higher speeds.
The picture above only represents a complete description of the mechanisms when no surfactant is present in either phase. However, we add surfactant into the dispersed phase. This facilitates generation of hexadecane droplets by reducing the interfacial tension between the two phases. Surfactant is known to alter the mechanism described above and provide additional contribution for pressure differential across a droplet [7,42]. This considerably complicates the interpretation for experimental data, and essentially prevents us from comparing our results rigorously and quantitatively with theories.
4.2 Device sensitivity
The previous results (Fig. 5) demonstrate the capability of the device we introduce for measuring extra pressures of ~10 Pa and below. Before discussing the next set of experiments, we would like to evaluate the sensitivity in these measurements. We relate pressure differential to bead displacement. Theoretically, the uncertainty in pressure measurement comes from uncertainties in determination of the position of the trapped bead. Factors contributing to such uncertainties in position detection include Brownian motion of the bead, motion blur due to a finite exposure time, fluctuation of laser power etc . The combined uncertainty of those factors can be evaluated experimentally: we trap a bead in non-flowing water at the same laser power of 200.0 mW, and record the time variation of bead position. The result shows a standard deviation in bead position of δx0 = 6.9 nm. Such variation corresponds to an uncertainty of δP0 = 0.26 Pa for experiments with hexadecane droplets in oil. This δP0 can be regarded as the intrinsic sensitivity of our device.
In droplet experiments, however, slight fluctuations in flow rate exist, amplifying the uncertainty in the pressure measurements. The finite number of frames during the passage of one droplet also contributes to the uncertainty. Data in Fig. 5(a) can be used to coarsely evaluate the combined effect of all sources of error when we plot Fig. 5(b). We find that in those data, the standard deviation in bead position for water-only situations is δx = 13 nm within 7 frames, the typical duration for the passage of a droplet in the main channel. This corresponds to an uncertainty of δP = 0.49 Pa.
The ability of device to work in the low pressure range (several Pascals) with high sensitivity is unique compared with other existing techniques, which usually work in the range of hundreds of Pascals and above [21–28]. Furthermore, this sensitivity can be easily improved by either increasing the frame rate or taking measures to stabilize the flow rates.
4.3 Modified design and experimental protocols for measuring extra pressure of water droplets in hexadecane
In this section and the next, we reverse the two phases, dispersing water droplets in a continuous hexadecane phase. We use our device to measure the pressure perturbations caused by water droplets. In addition, we run the experiments at higher flow rates.
We are motivated to do this for several reasons. First, our device has great flexibility in the design of its geometry and the pressure range over which it operates. We demonstrate how the device design can be readily modified to accommodate higher flow rates, and achieve measurement of a pressure differential above 1 kPa. Second, water (the dispersed phase) has smaller viscosity than hexadecane (the continuous phase). This viscosity contrast tends to decrease the pressure differential across the channel containing a water droplet in hexadecane, while capillary forces always contribute positively to extra pressure. Thus by performing experiments on water droplets in hexadecane, we can differentiate the effect of viscous dissipation in droplet body from that of capillarity around droplet caps, as these mechanisms affect the pressure differential in different ways. Last, the relative strength of viscous to capillary forces can be expressed by the dimensionless capillary number Ca = µV/σ , where µ is the fluid viscosity of the more viscous phase, V is the characteristic fluid velocity, and σ is the interfacial tension between the two fluid phases. ΔPbody is shown to linearly depend on Ca, whereas ΔPcaps is found to vary as Ca2/3 [7,42]. This indicates that ΔPbody grows at greater rate with Ca than ΔPcaps does, and is therefore expected to be the more important factor at larger Ca values. In a given channel, a higher flow rate results in higher fluid velocities and a larger capillary number. Higher flow rates should therefore enable us to better manifest the viscous effects and the viscosity contrast of the two phases.
Optical trapping of a bead in hexadecane in the side channel is feasible only when the flow velocity is sufficiently low. With the channel cross sectional area taken into consideration, this corresponds to a flow rate of several microliters per hour. To maintain a high flow rate (needed for a large capillary number) in the main channel and to simultaneously enable optical trapping in the side channel, a larger ratio of hydraulic resistances between the two channels must be realized, so that the flow rate in the side channel is only a very small fraction of the total. In order to do this, we elongate the side channel and fold it into a serpentine shape, as shown in Fig. 6(a) . Figure 6(b) is a microscope image of part of the fabricated parallel structures. Limited by the field of view, this image only shows some of the upward and downward sections of the side channel. The length of the main channel in this modified device is 440 µm, and the length of the serpentine side channel measures 7.2 mm. The widths of the main and side channels are 50 µm and 20 µm, respectively. All the channels are 30 µm in depth. Using Eq. (3), we calculate that the ratio of the hydraulic resistance of the side channel to that of the main channel is ~100: 1, with the assumption that the two convey the same single-phase flow. This large hydraulic resistance ratio allows for optical trapping and pressure measurements at high flow rates. Despite the altered geometry, the device works under the same principle: the bead displacement Δx is proportional to the pressure differential ΔP across the main and side channels.
The channel’s inner surfaces need to be hydrophobic or lipophilic for robust generation of water droplets in hexadecane [3,6]. We adopt the sol-gel approach developed by Abate et al.  to make the channel walls hydrophobic. Briefly, we prepare a mixture of 1 mL tetraethylorthosilicate (TEOS), 1 mL methyltriethoxysilane (MTES), 0.5 mL (heptadecafluoro-1,1,2,2-tetrahydrodecyl) triethoxysilane, 2 mL trifluoroethanol and 1 mL 3-(trimethoxysilyl)propyl methacrylate as the sol-gel solution. 0.5 mL of this solution is combined with 0.9 mL methanol, 0.9 mL trifluoroethanol, and 0.1 mL aqueous HCl, pH 2, and heated at 85 °C for ~2 minutes with intermittent shaking. The mixture is then diluted with methanol with a 1: 5 ratio of sol-gel to methanol. The mixture is flowed into the channels immediately after plasma bonding, and then we heat the device at 180 °C to evaporate the sol-gel. The remaining sol-gel coated on channel walls render them highly hydrophobic without appreciably modifying the cross sectional geometry.
We do not add surfactant to either phase in order to reduce the complexity of the origins of pressure across droplets. However, the large interfacial tension (~41 mN/m)  between pure water and hexadecane makes it rather difficult to generate droplets in a controllable manner. To overcome this problem, we add 25 wt% ethanol into water, reducing the interfacial tension to σ = ~17 mN/m. This value is estimated via linearly interpolating the data reported by Steegmans et al . We use a T junction nozzle, similar to that introduced by Thorsen et al. , to generate water droplets in hexadecane, as shown in Fig. 6(c), since we find that for our device the T junction water droplet-maker has better performance than a flow focusing structure. The size of the nozzle is enlarged to 60 µm × 60 µm to obtain larger droplets [3,6,7]. In Fig. 6(c), hexadecane is flowing from the left to the right; water arrives from the upper channel and is broken into droplets at the nozzle. We use syringes with a larger volume (Hamilton 250 µL) to pump the fluids at high flow rates. To capture the dynamics of pressure variation, we also replace the previously used cameras with faster ones (Grasshopper, Point Grey for low magnification microscopy, and acA2000-340km, Basler for high magnification microscopy). The emitting power of the trapping laser is increased to 300.0 mW. A calibration process similar to that described for optical trapping in water is carried out in a channel whose cross section matches that of the modified side channel. The calibration curve is plotted in Fig. 6(d). Each data point is the averaged bead position during 4 s, which contains 3600 video frames i.e. 3600 measurements of bead position. The standard deviation of the bead position during this period of time is δx = 19 nm. A linear fit of the data in Fig. 6(d) shows a slope of 104 nm/(µL⋅hr−1).
4.4 Extra pressure of water droplets in hexadecane
In this section we present experimental data on the extra pressure resulting from water droplets in hexadecane. The experiments start by injecting hexadecane and water at a total flow rate of 100.0 µL⋅hr−1. This gives an average flow velocity of v = 18.5 mm/s in the main channel (if we ignore the sharing of flow rate in the side channel), and a capillary number Ca = 0.0033. Under this condition, the water droplets in hexadecane we obtain are highly monodisperse in size. To obtain a range of droplets of various sizes, we modify the flow rates of the two phases, but maintain a total flow rate of 100.0 µL⋅hr−1 to ensure that the capillary number is constant. As before, the trapped bead is imaged by the 100 × lens onto the bottom camera, and the oil droplets in the main channel are imaged by the 10 × lens onto the top camera. The videos recorded by the cameras are used to determine the bead displacement and droplet length throughout the experiment. As before, averaged bead position is converted to extra pressure using the calibration results. Next, we fix the total flow rate at 80.0 µL⋅hr−1 and repeat the measurement process described above, thereby obtaining the extra pressure vs droplet size for a capillary number of 0.0026. Last, we fix the flow rate at 50.0 µL⋅hr−1, and obtain data at a capillary number of 0.0016. These results for three different capillary numbers are summarized in Fig. 7 . We find that extra pressure is no longer monotonic with droplet length. The extra pressure is initially positively correlated to length for smaller droplets. When droplets are further lengthened, the extra pressure then decreases with droplet length. For example, at Ca = 0.0033, the extra pressure increases with droplet length when droplets are shorter than a threshold value of ~240 µm (denoted by the orange arrow in Fig. 7), but is negatively correlated to length for droplets longer than that.
The same trend is observed for the other two capillary numbers. From Fig. 7 it can also be seen that the threshold length shifts to larger values when capillary number decreases, but this observation looks less definitive than the non-monotonic behavior of extra pressure. These results are found to be qualitatively consistent with those observed by Vanapalli et al .
We interpret the results in Fig. 7 by referring again to the aforementioned effects of viscous dissipation and capillary forces. For water droplets in hexadecane, viscous dissipation tends to result in a smaller pressure differential across the main channel, because the dispersed phase is lower in viscosity. On the other hand, capillary forces always contribute a positive pressure. We believe that capillary forces are primarily responsible for the increase of extra pressure with length for shorter droplets. To be more specific, below the threshold length, we suggest that an increase in size results in the continuous phase exerting a stronger pushing force on the droplet, which in turn results in a larger difference between the curvature of upstream and downstream caps. The Laplace pressure around the caps therefore gives a larger positive contribution to the pressure across the longer droplet. This mechanism can be directly observed from the microscope images of the droplets. Figure 8(a) displays images of four droplets of different lengths. In the Fig., the flow is from the left to the right with Ca = 0.0033. It can be clearly observed that the upstream cap becomes flatter and less curved as the droplet length increases. To quantify the curvature of the caps we fit a circular arc to the image of water/hexadecane interface. The radius of curvature of the cap in the image plane is then approximated by the radius of this arc. As shown in Fig. 8(b), the radius of curvature of the upstream cap increases with increasing droplet length, whereas the downstream cap has a nearly constant radius of curvature. The data of Fig. 8(b) confirms our physical interpretation, but is unfortunately insufficient for a quantitative estimation of the extra pressure contributed by capillary effects. This is because at any point on a curved interface, two radii of curvature exist. A quantitative evaluation of Laplace pressure would therefore require knowledge of both principal radii.
Moreover, it can be noted in Fig. 8(b) that the radii of curvature of both caps vary little with droplet length when the length is above a certain value. We believe that within this range, capillary forces cease to play the major role, and viscosity contrast dominates the relation between extra pressure and droplet length. For such droplets, elongation in length does not generate much more difference between the curvatures of the caps, but brings more water with lower viscosity into the main channel. Accordingly, an increase in length within this range primarily yields less viscous dissipation in the channel, and thereby results in a decrease in extra pressure, as observed in Fig. 8(a). It is worth noting that although the critical length above which curvature no longer changes significantly in Fig. 8(b) (~370 µm) does not seem to match the threshold length in Fig. 7 (e.g. ~240 µm for Ca = 0.0033), it still captures the basic physics governing the behavior of extra pressure for highly elongated droplets.
To summarize this section, we demonstrate the operation of our device at higher pressures and larger capillary numbers, and measure the extra pressure of water droplets in hexadecane under such conditions. Our results exhibit a competition between viscous and capillary forces. Such a competition gives rise to the observed non-monotonic relation between extra pressure and droplet length.
In this paper we have introduced a novel technique for pressure measurement in microfluidics by incorporating optical trapping. Our device is governed by a simple working principle, and can be readily fabricated with standard approaches. We employ the device to measure the extra pressure caused by hexadecane droplets in water and water droplets in hexadecane. We provide physical interpretation for the data obtained. The results demonstrate the effects of both viscous dissipation and capillary forces.
The experimental results demonstrate the high sensitivity and broadly adjustable pressure range of the device. These are a consequence of the fact that the channel geometry can be readily modified. In particular, our device can be used in the low pressure range of several Pascals, which might be difficult to access using other techniques. Inherent to the operating principle is the fact that tuning the trapping stiffness is a means for varying the dynamic range of the device. This can be done by adjusting the emitting power of the laser, or via other techniques, e.g. integration of plasmonic optical trapping structures into the microfluidic channel [47–49]. The latter could also facilitate the trapping of sub-micron particles, suitable for pressure measurement in channels with smaller geometries.
Understanding the extra pressure brought about by micro-droplets is important to various devices and topics, such as droplet-based microfluidic logic, droplet-facilitated biochemical assays, syntheses and mixing, and extraction of residual oil in reservoirs. Furthermore, the simple design, easy fabrication and tuneable pressure range could expand applications of our technique to other areas. For example, our device should be able to measure the pressure required to squeeze cells through constrictions. This information may be of use in medical diagnostics . Finally, we anticipate that multi-trap optical tweezing , possibly with microfabricated diffractive lenses [52,53], could be used, thereby enabling pressure measurements to be performed simultaneously in multiple channels and a higher data throughput.
This work was supported by the Advanced Energy Consortium via the Bureau of Economic Geology at the University of Texas at Austin.
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