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Abstract
Gas dynamic virtual nozzles (GDVNs) produce microscopic flow-focused liquid jets and droplets and play an important role at X-ray free-electron laser (XFEL) facilities where they are used to steer a stream of hydrated biomolecules into an X-ray focus during diffraction measurements. Highly stable and reproducible microjet and microdroplets are desired, as are flexible fabrication methods that enable integrated mixing microfluidics, droplet triggering mechanisms, laser illumination, and other customized features. In this study, we develop the use of high-resolution 3D nano-printing for the production of monolithic, asymmetric GDVN designs that are difficult to fabricate by other means. We also develop a dual-pulsed nanosecond image acquisition and analysis platform for the characterization of GDVN performance, including jet speed, length, diameter, and directionality, among others. We show that printed GDVNs can form microjets with very high degree of reproducibility, down to sub-micron diameters, and with water jet speeds beyond 170 m/s.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Gas dynamic virtual nozzles (GDVN) produce microscopic flow-focused liquid jets and microdroplets [1] that may be used for applications in the food and pharmaceutical industries. An important application of GDVNs is at X-ray free-electron laser (XFEL) facilities, where they are used to deliver hydrated biomolecules into an intense micro-focused X-ray beam in order to make ultrafast diffraction measurements. XFELs generate X-ray pulses with mJ pulse energies that cause catastrophic destruction of the liquid jet and the protein sample, but the $\sim$10-femtosecond duration ($10^{-14}$ seconds) of the X-ray pulses allows for the formation of atomic-resolution images that are free of damage through the “diffraction-before-destruction” paradigm [2]. High-speed liquid jets allow for high-frequency data collection because the liquid jet recovers prior to the subsequent XFEL pulse [3]. GDVNs are therefore likely to be of great importance to the development of mixing enzymology studies at MHz XFEL facilities, for example [4].
The idea of using GDVNs to produce microjets was first introduced in 1998 [1], and a variety of miniaturized GDVNs were developed for serial femtosecond crystallography [5]. The early methods of hand-fabricating individual GDVNs are tedious and time-consuming, and the resulting nozzle and jet characteristics vary from one GDVN to the next. This prompted efforts to develop automated and reproducible fabrication methods, such as the ceramic micro-injection molded nozzles introduced in 2015 [6]. The application of three-dimensional printing for fabricating GDVNs was also introduced in 2015 [7] and has been under development since then [8]. This paper we further develop the use of 3D nano-printing for the production of asymmetric nozzles that create reproducible flow focusing jets, high-speed jets, and also sub-micrometer jets. The jet characteristics of two different 3D printed devices are quantified, and an all-optical GDVN testing and analysis platform is presented that can measure high-speed jets with diameters less than the resolution of the imaging optics.
2. Overview of gas-dynamic virtual nozzles
2.1 Basic principles of flow-focused liquid microjet formation
An operating GDVN is shown in Fig. 1. A sheath gas indicated by the arrows focuses the liquid into a microjet through pressure- and shear-driven acceleration. By focusing the liquid with a gas rather than a solid orifice, GDVNs can deliver solutions of protein microcrystals through channels of diameter $\sim$50 $\mu$m rather than $\sim$5 $\mu$m, which enables jetting for several hours or days without clogging. Other aspects of GDVN performance that are of great relevance to XFEL studies are the jet diameter, jet length, droplet diameter, and overall stability.
There are two primary modes relating to flow focused jets: the first is the Rayleigh mode (Rayleigh 1878) [9] in which droplets form with sizes larger than the jet diameter. In the Rayleigh mode, the liquid jet breaks up into droplets due to instabilities that originate from interfacial instabilities [10]. Surface tension forces and liquid inertia initiate axisymmetric oscillations in the liquid jet. These oscillations grow as they are convected downstream of the liquid jet and finally cause the jet to break up [11]. In the second mode, namely the Taylor mode (Taylor 1963) [12,13], droplets of sizes comparable to the jet diameter or much smaller than the jet diameter are formed. The Taylor regime, or first-wind-induced regime, can be achieved by increasing the sheath gas flow rate to increase the aerodynamic effects of the sheath gas. This is the appropriate mode of operation when faster jets and smaller droplets are desired.
Droplet formation is often categorized into dripping and jetting instability regimes. In the dripping regime, the liquid meniscus breaks up into droplets at the nozzle orifice and a stable jet is not formed. Stable jets are formed when the liquid (flow-focused fluid) velocity is large enough such that inertial forces dominate surface tension forces at the nozzle tip [14]. Droplets form further downstream and this mechanism determines the length of the liquid jet [15]. The transition between dripping and jetting corresponds to the transition between convective and absolute instability [16,17]. A hysteresis can be observed in which a jet is sustained even when the gas flow drops below the minimum value that is needed to initiate the jet. In practice, jets containing protein samples cannot be sustained for extended periods when operating in the hysteresis regime. A third regime, known as the first-wind-induced regime, is observed when the gas flow rate is very high, and axisymmetric jetting transitions to non-axisymmetric jetting with sinusoidal oscillations [16].
2.2 Nozzle design goals
Our primary objectives in the present work are to (1) establish 3D printing methods for monolithic GDVNs that yield highly reproducible jet behaviors, and (2) develop new asymmetric nozzle designs that produce high-speed jets for data collection MHz XFEL sources, as well as sub-micrometer jets for nebulizing protein samples into nanodrops without high voltages as in electrospray devices. Our nozzle designs were inspired by our previous empirical observations in addition to published investigations into the roles of nozzle geometry and fluid properties in the processes of jet and droplet formation [18–20]. In particular, we employ the design concept proposed by Acero et al. [21] who studied asymmetric nozzles based on hypodermic syringes that produce stable jets in a wide range of operating conditions, including very low liquid flow rates.
Based on these previous studies in the literature, we varied several geometrical parameters in the GDVN design. After testing several prototypes and printing processes, we focused on two designs that can jet in a wide range of operating conditions to satisfy the requirements of XFEL experiments. Details of the two designs and their performance are discussed in the following sections.
2.3 Asymmetric nozzles for fast jets
Figure 3 shows a 3D printed nozzle design with a liquid channel that terminates with a syringe-like shape. This asymmetric design results in a Couette-type flow field in which a shear-driven liquid stream can accelerate along the surface of the syringe and reduces in size before the free-standing jet is launched from the syringe tip. The increased liquid kinetic energy helps overcome surface tension resistance, and the Couette-type flow field avoids recirculation patterns and related instabilities [21]. By placing the tip of the syringe near to the sheath gas aperture, we exploited empirical observations suggesting that jets formed in the rapid gas expansion region have increased stability at high gas pressures [22]. Our testing results suggest that these combined features allow for increased sheath gas flow rates as well as reduced liquid flow rates, which in turn enables the formation of smaller and/or faster jets (down to sub-$\mu$m diameters, and speeds of 170 m/s, in the case of water jets). Although small and/or high-speed jets are not always needed, the increased range of operating conditions increases the possibility of achieving stable jets from samples having complex viscoelastic properties.
2.3.1 Design 1
Our first nozzle design (referred to as “Design 1”) is shown in Fig. 2. The upstream end of the nozzle has two circular channels that accept 360 $\mu$m OD (outer diameter) glass capillaries that transport the liquid and gas to the nozzle. These capillaries are epoxied in place. The liquid line thickness is varied smoothly throughout the liquid flow field to minimize the possibility of clogging while maintaining a small exit orifice in order to achieve higher jet velocities. Due to the syringe shape, the liquid line tip has an elliptical cross-sectional shape with minor/major axis diameters of 40 $\mu$m and 97 $\mu$m, respectively. Testing results showed that under certain operating conditions, fast liquid water jets with velocities of higher than 170 m/sec can be produced from this design.
2.3.2 Design 2
Design 2, shown in Fig. 3, has a straight liquid channel in order to be more robust against clogging than Design 1. The smaller overall volume of the nozzle also reduces the printing time by roughly 40 minutes. The elliptical cross section of the syringed liquid orifice has minor/major axis diameters of 50 $\mu$m and 146 $\mu$m, respectively (the eccentricity is larger than Design 1 due to the smaller syringe angle of 20$^\circ$). The liquid line diameter is smaller than Design 1, and testing results show that a device from this design achieves a maximum water jet velocity of approximately 140 m/s, which is approximately 25% lower than in Design 1.
3. Nozzle fabrication and assembly
3.1 3D printing with 2-photon polymerization
Since the 1990s, the availability of sub-picosecond pulsed lasers enabled broad experimental investigations into two-photon absorption phenomena and related technological applications [23–25]. Two-photon absorption cross sections increase with the square of the light intensity, which enables higher spatial resolution in laser lithography applications than is achievable with single-photon techniques. In the work presented here, we utilized a Nanoscribe Photonic Professional (GT) laser lithography printer that can achieve $\sim$200 nm resolution through two-photon polymerization (2PP) direct laser writing. This printer scans an 80 MHz pulsed laser of 780-nm center wavelength, 100 femtosecond pulse duration, and 25KW peak power within a liquid photoresist, which causes rapid polymerization upon exposure. Smaller volumes of up to approximately $300\times 300\times 300$ $\mu$m$^3$ can be printed with rapid-scanning mirrors, whereas larger print volumes utilize the scanning mirrors in conjunction with a 3-axis piezoelectric stage that translate the printed object. Once the laser polymerization stage is completed, the unpolymerized liquid photoresist is washed away during the development process.
Nozzles were composed of IP-S photoresist (Nanoscribe item no. 1907/2006 (REACH)). Depending on the design, each nozzle required 35 minutes to a few hours to print. After printing, the nozzles were developed by placing them in a bath of the developer liquid (MICROCHEM SU-8 developer (CAS Number: 108-65-6)) for 35 minutes followed by Isopropanol for 5 minutes. Following development, the 360 $\mu$m OD polyimide-coated fused silica capillaries (Molex) were attached to the liquid and gas inlets with 5-minute epoxy (Hardman double bubble red). Finally, the capillaries were fed through a 1/16-inch OD metal sleeve (IDEX U-115) and epoxied in place in order to mount the nozzles and form a vacuum seal. Figure 4 shows an optical image of an assembled nozzle. Figure 5 shows SEM images of a nozzle from the Design 2 in which only half of the nozzle was printed in order to expose the inner features.
Fig. 5. SEM images of a nozzle from Design 2. Only half of the nozzle was printed in order to reveal the internal structure.
The magnification of the objective lens used in this work was 25$\times$ with numerical aperture of 1.4. Nozzles were printed using the “dip in” mode in which the photoresist completely fills the space between the glass substrate and the objective lens. Since the scanning range of the printer with this objective is limited to 200 $\mu$m in the horizontal and vertical dimensions (where vertical is defined by the optical axis and horizontal by the plane of the substrate), the nozzle geometry is subdivided into several blocks. The laser polymerizes one block at a time in an orderly manner from bottom to top in such a way that each block is supported by the substrate or the lower block. To prevent floating blocks, cavities with sharp angles of more than about 70 degrees from the vertical are avoided in our designs [26]. The amount of laser overlap, or “stitching”, between the blocks was 2 $\mu$m for horizontal block walls and 1 $\mu$m for vertical block walls. The block walls have an angle of approximately 15$^\circ$ from the vertical in order to minimize multiple exposures by the laser beam. Each block is subdivided into horizontal planes, or “slices”, with an inter-slice distance of 1 $\mu$m.
4. Procedures for quantifying GDVN performance
4.1 Overview of nozzle testing station
A schematic of the test station we used to characterize liquid jets is shown in Fig. 6. The station records images of liquid jets while monitoring operational parameters, including upstream sheath gas pressure, upstream liquid pressure, gas flow rate, liquid flow rate, and vacuum pressure. A double-pulsed fiber-coupled 100-ns laser with 633 nm wavelength illuminates the jet, while a high-speed camera and optical lenses gather image data. This system, combined with the image processing methods discussed below, enables accurate statistical quantification of jet characteristics such as jet velocity, jet length, jet diameter, and jet deviation angle, as well as droplet diameters. The data processing scheme described below allows for the accurate determination of sub-$\mu$m jet diameters because the calculations are based on observed droplet centroid translations that can be determined with a much better resolution than that of the optical system. This scheme is similar to other super-resolution optical localization microscopy techniques [27].
Fig. 6. Schematic of the test station. Helium gas drives the liquid jet and the pressure and mass flow rate are monitored upstream of the glass capillary that leads to the nozzle. An HPLC pump drives the liquid (water) and a flowmeter measures its volumetric flow rate. Nitrogen gas and isopropanol are used to clean and dry nozzles, particularly when running samples other than water. The nozzle is located in a small chamber at $\sim$1 mbar pressure. A pulsed $\sim$100 ns laser provides brightfield illumination for the high-frame-rate camera, and electronic delay system allows for doublets of images.
The test station setup makes it possible to easily run the GDVNs with a controllable sheath gas pressure or mass flow rate and to switch between different gases such as helium and nitrogen. The pressure of the sheath is measured at the inlet of the glass capillary. We typically test GDVNs with helium because the gas composition strongly affects jet behavior, with helium being most favorable, and because XFEL experiments rely on helium due to its low X-ray scattering cross section.
Liquids are supplied to the GDVNs by either high-pressure liquid chromatography (HPLC) pump or pressurized liquid reservoir. The liquid flow rate is measured by volumetric flow sensors. The HPLC pump can supply liquid flow rates with 0.1 $\mu$l/min accuracy. A liquid switch with 6 different entrance lines and one exit line allows for rapid switching between sample liquids or a purging gas. After each test, liquid is purged with water and/or isopropanol before purging the liquid with nitrogen gas.
The sealed vacuum chamber consists of a rectangular tube with glass windows and a quick-connect fitting that accepts various types of nozzle mountings. A dry scroll pump is used to evacuate the chamber and typically results in pressures of the order 0.1 mbar. The imaging system consists of a 10X Mitutoyo long working distance objective paired with a variable Navitar 12X UltraZoom magnifying lens, and has a working distance of 3.5 mm. The bright-field illumination source consists of a 400 $\mu$m diameter optical fiber that is re-imaged 1:1 to a $\sim$400-$\mu$m spot at the focal plane with a single bi-convex lens. At our operational wavelength $\lambda = 633$ nm, the numerical aperture of the objective is $\textrm {NA} = 0.28$, which yields a maximum resolution of $\frac {\lambda }{2 * \textrm {NA}} \approx 1.1$ $\mu$m.
4.2 Image processing and analysis
To extract the desired jet characteristics from the acquired images, we identified and separated regions in the images corresponding to the jet and droplets. Jet length and jet deviation angles were readily determined from individual images, whereas pairs of 100-ns exposures separated in time by 500 ns were used to quantify droplet speeds through Particle Tracking Velocimetry (PTV) methods [28]. By determining droplet speeds, along with measured liquid flow rates, we were able to determine liquid jet diameters at resolutions better than can be achieved by individual images.
We first subtracted the image backgrounds from the raw images (see Fig. 7(a)) due to the bright-field laser illumination. This step was important because the multi-mode optical fiber used to illuminate the jet creates a non-uniform background illumination with fluctuations due to coherent speckle. Backgrounds were estimated with a FIJI ImageJ [29] plugin that implements the rolling ball algorithm [30]. For each pixel, the average within a circle of diameter 50 pixels was subtracted from the pixel value in the original image.
Fig. 7. Illustration of image processing steps. (a) Raw image before processing steps. (b) Binary image after background subtraction and thresholding. (c) Perimeter of extracted jet region. (d) Pair of extracted droplet series separated in time by 550 ns. The droplet colors indicate matching droplets in the image pair.
After subtracting image background, we converted the resulting 16-bit images to binary images using the IsoData thresholding algorithm [31], which performed best when compared against other algorithms available in FIJI ImageJ. The IsoData technique uses an iterative approach to automatically optimize the threshold selection as long as the image contains an object (jet and droplet regions in our study) and background that have different average gray levels. Figure 7(b) shows a binary image after applying IsoData to the image.
From the binary images, droplet and jet regions were identified and extracted using Fiji Imagej plugins with some customization [29,32,33]. The binary stack of images were divided into two substacks: the first substack includes only the continuous jet regions before primary breakup, with the droplets removed, while the second substack contains only the droplets that form after the jet breakup. Jet regions were identified as regions in the binary images with size larger than a threshold value and circularity lower than a threshold value. Extracted jet and droplet regions are illustrated in Figs. 7(c) and 7(d).
After extracting jet regions, we determined jet deviation angles by fitting an ellipse to each jet region. The angle of the major axis of the ellipse corresponds to the jet deviation angle. Jet lengths were quantified by the smallest rectangle that enclosed each jet region. The length of the rectangle along the jet propagation direction was taken as the jet length, but it is noteworthy that this is an overestimate because diffraction is typically recorded from a point upstream of the droplet breakup region. In practice, the effective length corresponds to the point at which X-ray surface scatter is condensed into a stable streak; i.e. we avoid exposing regions of the jet that have significant surface capillary waves.
Droplet regions were extracted with an open-source macro code that implements Hough circle transforms [34]. Afterwards, a Matlab code was used to implement Particle Tracking Velocimetry [35] of droplets in successive images to determine jet velocities. An example of extracted droplets regions from a pair of frames delayed by 550 ns is shown in Fig. 7(d). By comparing the horizontal center-of-mass coordinates of droplet pairs, and the time delay between pairs of frames, the horizontal velocities of droplets were calculated from the finite difference measurements. For each dataset with a unique set of flow conditions, three different image pairs were used to estimate droplet velocities.
5. Results of nozzle testing
We thoroughly tested nozzle designs 1 and 2 with pure water using the data acquisition and analysis procedures described above. In each case, we printed a batch of five nozzles and recorded image pairs at a variety of liquid and gas flow rates, as detailed below.
5.1 Sheath gas flow rates and pressures
The helium sheath mass flow rate in a GDVN depends on the inlet pressure along with the pressure head loss in the microcapillaries, and the nozzle aperture size and geometry. Figure 8 shows that the liquid flow rate value does not affect the helium mass flow rate, thus we can fully determine the operating condition of the nozzle by considering only the sheath gas mass flow rate and the liquid flow rates for a particular nozzle design.
Fig. 8. He pressure at the inlet of the gas capillary versus He flow rate for a nozzle of Design 1, showing that the liquid flow does not effect the gas flow in the measurements.
5.2 Jet velocities
High-speed jets of approximately 80 m/s are required in MHz XFEL serial diffraction because the intense X-ray pulses cause explosions that result in gaps in the jets, and the jets must therefore be fast enough to displace the gap before the subsequent X-ray pulse arrives [36,37]. Since liquid acceleration is very low beyond the first few hundred micrometers downstream of the orifice, even after breaking up of the jet to droplets [38,39], the calculated droplet velocities are a good approximation of the jet velocities. Figure 9 shows the jet velocities estimated by these means for nozzle designs 1 and 2. These figures show that the jet velocity highly depends on liquid and gas flow rates, and it monotonically increases with increasing sheath gas flow rate and decreasing liquid flow rate. The fastest jets are achieved when the liquid flow rate is relatively low (below approximately 5 $\mu$l/min). By comparing Design 1 with Design 2 in Fig. 9, the observed jet velocities for Design 1 are higher under similar operating conditions, which is most likely because the dimensions of the liquid line tip for Design 1 are smaller. Design 1 achieved maximum jet velocities of approximately 170 m/s, whereas Design 2 achieved maximum jet velocities of approximately 140 m/s. It should be noted that these high-speed jets were less repeatable than jets with speeds closer to 100 m/s, and that further investigations are needed in order to determine the speeds of liquids with higher viscosity and which carry microcrystals.
Fig. 9. (a) Plots of Jet velocity versus helium flow rate for different liquid (water) flow rates for 5 different nozzles from Design 1 and (b) 5 different nozzles from Design 2. (c) Plots of Jet length versus helium flow rate for different liquid (water) flow rates for 5 different nozzles from Design 1 and (d) 5 different nozzles from Design 2. Symbol colors distinguish liquid flow rates, while symbol shapes distinguish different nozzles.
5.3 Jet lengths
It is important to characterize GDVN jet lengths because the X-ray beam should be located at a distance of approximately 75 $\mu$m or more from the nozzle tip in order to avoid additional background scatter from the nozzle material. We can see in almost all cases of Fig. 9, the jet length increases with increasing liquid flow rate. At constant liquid flow rate, the jet length increases with increasing sheath gas flow rate until a maximum value is achieved at approximately 10 mg/min helium, after which the jet length is largely independent of the gas flow rate. Increasing sheath gas flow rates eventually causes a reduction in jet lengths that may be associated with the onset of whipping instabilities [40]. Since in XFEL experiments we almost always run GDVNs with sheath gas flow rates of more than 20 mg/min to be far from the jetting to dripping transition regime, liquid flow rates of roughly 5 $\mu$l/min should generally produce jets of sufficient length.
5.4 Jet diameters
Jet diameter is an important characteristic in XFEL experiments because it determines the amount of solution scatter, and also because the size of the gap that is produced when X-rays impact the jet is dependent on the jet diameter [3]. Jet diameters may be determined from jet velocities along with the liquid volumetric flow rate according to the relation $D_j = 2 \sqrt {v/\pi Q}$, where $v$ is the jet speed and $Q$ the volumetric liquid flow rate. We can see from Fig. 10 that the estimated jet diameter is mostly dependent on the liquid flow rate for He flow rates of greater than approximately 10 mg/min, while the helium sheath gas flow rate plays the secondary role. Since we almost always run the GDVNs in the operating conditions of the sheath gas higher than 20 mg/min in XFEL experiments, the liquid flow rate is the dominant factor that affects the jet diameter.
Fig. 10. (a) Jet diameter versus helium flow rate for different liquid (water) flow rates for 5 different nozzles from Design 1 and (b) 5 different nozzles from Design 2. (c) Jet angle versus helium flow rate for different liquid (water) flow rates for 5 different nozzles from Design 1 and (d) 5 different nozzles from Design 2. Symbol colors distinguish liquid flow rates, while symbol shapes distinguish different nozzles.
5.5 Jet deviation angles
Jet deviation angles are important because if a jet emits at a large angle (approximately 10$^\circ$ or more) with respect to the nozzle, the jet may collide with the sidewalls of the vacuum chamber, which causes the formation of ice pillars that grow rapidly as successive droplets freeze on top of the others. These ice pillars may grow at rates of several cm/s and thus can rapidly reach the nozzle orifice and plug the nozzle. Due to the reproducibility of 3D printed nozzles, nearly all jets formed with deviation angles of less than 5$^\circ$, as shown in Fig. 10.
5.6 Jet performance and reproducibility
The integrated plots of median jet length 9(c) 9(d), jet velocity 9(a) 9(b), calculated jet diameter 10(a) 10(b) and median jet angle 10(c) 10(d) versus He flow rate for five different nozzles from either Design 1 or Design 2 show that different nozzles from a similar design have acceptable performance consistency.
5.7 Droplet diameters
The ratio of jet diameter to droplet diameter is indicative of the mechanism of droplet formation. From the captured droplet images, we can estimate the droplet image diameters from the area $A$ of each droplet image according to the formula $D_d' = 2 \sqrt { A/\pi }$. Figures 11(b) and 10(c) show the resulting mean and standard deviations after calculating the droplet image areas of more than 3000 droplets for each flow-rate condition for a nozzle from Design 2 running with pure water. It is evident that there is a linear relationship between droplet image diameters and the jet diameters $D_j$ estimated via the two-flash image analysis results presented in section 5.4. The fitted regression line yields the formula
with an $R$-squared value of 0.98. The non-zero intercept value of 1.95 $\mu$m after curve fitting is due to the limited resolution (or “blurring”) of the optical system, which is expected because the images result from the convolution of the focal plane with the point spread function of the optics. The size of the optical point spread function is close to twice the nominal $1.1$ $\mu$m resolution of our optical system. We confirmed that the 1.95 $\mu$m offset was indeed due to the optical system by measuring the jet image diameters $D_j'$ with the same image thresholding scheme, which resulted in the relation $D_j' = 1.03 D_j + 1.86$ $\mu$m; the offset of 1.86 $\mu$m is very similar to that associated with the droplet analysis. We conclude that the actual droplet diameters are related to the actual jet diameters by the relation $D_d = 2.01 D_j$, and that this relation holds for all of the observed gas and liquid flow rates. The slope of 2.01 suggests that the Rayleigh instability is the dominant jet breakup mechanism, since it predicts a slope of approximately 1.9 [9].Fig. 11. (a) Jet diameter versus helium flow rate for different liquid (water) flow rates for a nozzle from Design 2. (b) Median droplet diameter versus helium flow rate for different liquid (water) flow rates for a nozzle from Design 2. (c) Median droplet diameter versus calculated jet diameter for a nozzle from Design 2. (d) Plot of Estimated pressure inside the nozzle versus liquid jet Reynolds number for 5 nozzles from Design 1 under different operating conditions.
5.8 Estimation of internal nozzle gas pressure
During the typical operation of a GDVN at an XFEL experiment, gas pressure measurements are made at the inlet of a $\sim$1-2 m length of capillary, which often results in a poor estimate of the internal nozzle pressure due to the significant pressure drop along the length of the capillary. Estimates of the internal pressure of the nozzle, where the liquid jets are formed, are important because we aim to design nozzles that can withstand the internal pressures with minimal wall thickness (and hence printing time), and we also wish to relate the characteristics of the liquid jet to the inertial pressure, viscous, and surface-tension forces. Since we do not have a transducer that measures pressure directly within the nozzle, we instead estimate the internal pressure via energy conservation equations.
The Reynolds and Weber numbers are important dimensionless quantities that characterize the regimes of jet and drop formation. The Reynolds number represents the magnitude of inertial forces compared to viscous forces, while the Weber number indicates the relative magnitude of inertial forces to surface tension. We calculate the liquid jet Reynolds and Weber numbers using the following formulas [19]:
where $\rho$ is the liquid density, $Q$ is the volumetric flow rate of the liquid, $R$ is the jet diameter, and $\sigma$ is the surface tension of the liquid. Measurements were made at a room temperature of 20° C, in which case purified water has the following properties [19]:Fig. 12. (a) Plot of liquid jet Weber number versus liquid jet Reynolds number for 5 nozzles from Design 1 under different operating conditions. (b) Plot of liquid jet Reynolds number versus He flow rate for 5 nozzles from Design 1 under different operating conditions, (c) Plot of liquid jet Weber number versus He flow rate for 5 nozzles from Design 1. (d) Plot of calculated sheath gas Reynolds number versus helium flow rate for 5 nozzles from Design 1 under different operating conditions. Symbol colors distinguish liquid flow rates, while symbol shapes distinguish different nozzles.
Since we have moderately large values of Reynolds and Weber numbers in almost all operating conditions ($Re \geq 10$ and $We \geq 1$), we can neglect the energy sinks resulting from viscosity and surface tension [19]. Under those conditions, Calvo (1998) [1] justified the following energy conservation equation for cases in which the liquid flow rate is sufficiently large that viscous forces and surface tension may be neglected:
Here, $\Delta P$ is the pressure drop across the nozzle orifice, $\rho$ is the liquid density, and $v$ is the liquid velocity. Under the aforementioned assumption that the liquid is incompressible and that the external pressure is much smaller than the internal nozzle pressure, we may relate the internal nozzle pressure $P$ to the liquid volumetric flow rate $Q$ and the jet radius $R$: Figure 11(d) shows the estimated pressure inside the nozzle, based on Eq. (6). In this plot, we utilized jet diameter measurements made at a liquid flow rate of 20 $\mu$l/min. It is evident from Fig. 9(a) that the trend in jet speed as a function of liquid flow rate at fixed gas flow rate is asymptotic, and approaching the limit of Eq. (6). Since the velocities corresponding to a liquid flow rate of 20 $\mu$l/min differ by only $\sim$10% from those at a flow rate of 15 $\mu$l/min, we assume that these pressure estimates are reasonably accurate. We emphasize that this result suggests a simple scheme to estimate internal nozzle pressure: increase the liquid flow rate until the asymptotic behavior of Eq. (5) is observed. Although the increased liquid flow rate is needed to justify calculations based on Eq. (6), Fig. 8 shows that internal pressure is largely independent of liquid flow rate.5.9 Sheath gas Reynolds numbers
For future work, it is important to assess instabilities associated with the sheath gas under different operating conditions. For this purpose, we provide the sheath gas Reynolds number:
where $v_g$ is the gas speed, $d_g$ is the diameter of the gas orifice, $\mu _g$ is the gas viscosity, and $\rho _g$ is the gas density. Since the (measured) gas mass flow rate is $\dot {m}_g = \rho _g v_g \pi d_g^2/4$, we may instead write the sheath gas Reynolds number as6. Discussion and conclusions
We have shown that 3D printed nozzles can yield highly reproducible jet properties, which is of particular importance for XFEL serial diffraction measurements because unpredictable and misbehaved microjets are a common cause of wasted time (or completely failed experiments) at these costly facilities. We utilized an asymmetric nozzle design that would be very difficult to fabricate by other means (e.g. glass forming and injection moulding), and we demonstrated consistent jetting at speeds that are are well in excess of 100 m/s (up to 170 m/s in the data presented here), which is required for data collection at MHz XFEL pulse repetition rates [36,37]. These designs produced jets at liquid (water) flow rates down to 1 $\mu$l/min while maintaining a liquid channel ID of 50 $\mu$m or larger, which helps avoid clogging problems. We also observed jet diameters that were consistently below 1 $\mu$m (down to 0.325 $\mu$m), which is of interest for single-particle imaging experiments, fiber diffraction, or measurements from very small protein nanocrystals. Although these nozzle designs appear to improve the range of jetting conditions (as compared with symmetric designs), systematic studies of the relation between nozzle geometry and liquid jet properties are needed in order to quantitatively identify the most robust designs. The measurement system that we developed in order to characterize the performance of these printed nozzles is capable of determining jet diameters from high-speed jets that are well below the theoretical resolution of the optical system, which we achieved by employing a form of localization microscopy to the translations of droplet image centroids in pairs of images separated in time by 550 ns.
The nozzles presented here have been utilized in multiple XFEL experiments including solution scattering from membrane proteins, single-particle imaging of viruses, and crystallography measurements. 3D printed nozzles of a different form, but made from the same material using the same printing techniques, have been also been utilized in recent MHz crystallography experiments [8,37]. The systematic jet/droplet characterizations demonstrated here form a foundation for future systematic studies on different liquid samples with varying properties such as viscosity, surface tension, temperature, and particle size.
Funding
BioXFEL Science and Technology Center (1231306); National Institutes of Health (1S10OD021816); National Science Foundation (1817862).
Acknowledgments
This work is supported by the BioXFEL Science and Technology Center NSF Award number 1231306.
Disclosures
The authors declare no conflicts of interest.
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