Micro‐computed
 tomography for the
 3D time‐resolved
 investigation of monodisperse droplet generation in a
 co‐flow
 setup

Funding information Deutsche Forschungsgemeinschaft, Grant/ Award Number: INST 212/397-1 Abstract Droplet generation in microfluidic devices has emerged as a promising approach for the design of highly controllable processes in the chemical and pharmaceutical industry. However, droplet generation is still not fully understood due to the complexity of the underlying physics. In this work, micro-computed tomography is applied to investigate droplet formation in a circular channel in a co-flow configuration at different flow conditions (Ca < 0.001). The application of an in-house developed scanning protocol assisted by comprehensive image processing allows for the time-resolved investigation of droplet formation. By tracking different droplet parameters (length, radii, volume, surface, Laplace pressure) the effect of flow conditions on droplet progression is determined. As characteristic for the squeezing regime, final droplet size was nearly independent of Ca for higher Ca tested. For lower Ca, the final droplet size increased with decreasing Ca, which points to the leaking regime that was recently introduced in the literature.


| INTRODUCTION
Process intensification via miniaturization has proven to be a suitable approach to design safe, sustainable processes in the chemical and pharmaceutical industry due to high surface to volume ratios, short diffusion lengths, and excellent mixing behavior. Furthermore, strictly laminar flows make processes tunable and controllable. Within the wide field of micro process engineering, segregated flows in small channels, such as bubble, droplet, and slug flow, have emerged as a major approach. The application of droplet flow includes, among others, the synthesis of chemicals and materials, [1][2][3][4][5] kinetic studies, 6,7 solvent extraction, 8,9 protein crystallization, 10,11 drug discovery 12,13 and delivery, 14 and cell culture. 13 Due to the wide range of applications of droplet flow and bubble flow in miniaturized equipment also droplet generators and channel geometry vary according to the application. Common droplet generation approaches are cross-flow, 15 as the T-junction, [16][17][18][19][20][21] where the continuous phase and dispersed phase meet perpendicularly, flowfocusing, [22][23][24][25] where the dispersed phase is elongated by shear forces imposed by the continuous phase, and the co-flow configuration, [26][27][28][29] where the dispersed phase is introduced to the continuous phase with a thin needle that intrudes into the main channel concentrically. Most microfluidic devices consist of rectangular channels, however, circular [30][31][32][33][34] or even trapezoidal 35,36 cross-sections are possible, too.
Despite the variety of geometric configurations, similar break-up mechanisms can be found for different geometric configurations. 37 In microfluidics, Reynolds number is generally low such that the resulting flow is strictly laminar. 37 Important parameters that affect the droplet formation and break-up in confined geometries are the capillary number Ca, flow rate ratio ϕ, and the generator geometry. 19,[37][38][39] In Equation (1) _ V dis and _ V c are the volume flow rates of the dispersed (index dis) and the continuous phase (index c). The capillary number, Equation (2), depends on dynamic viscosity η c , the mean velocity of the continuous phase v c , and interfacial tension γ.
Depending on flow conditions, different regimes are distinguished. For low Ca (Ca < 0.01) interfacial stresses dominate viscous stresses and droplet formation takes place in the pressure dominated squeezing regime. 17,20,21,37,40 For higher Ca, viscous forces gain in importance and lead to droplet break-up according to the dripping or jetting regime. 17,37,38 The squeezing regime is divided into different stages. In the first stage, called the filling stage, the droplet begins to grow into the main channel and the continuous phase can bypass the droplet easily, see Figure 1(a). At the end of the filling stage, the droplet obstructs the main channel completely, except for a thin film between the droplet and the channel wall. In the second stage, the necking stage, the continuous fluid can no longer bypass the droplet as easily due to the high flow resistance imposed by the small gap between droplet and channel wall. This causes the upstream pressure in the continuous phase to rise and leads to the formation of a neck, see Figure 1(b). 17,21,38,40 As more continuous phase is pushed towards the emerging droplet the radius of the neck decreases until the droplet pinches off due to a Rayleigh-Plateau like instability. 23 Findings from the investigation of the mechanisms in the squeezing regime have led to different scaling laws for the prediction of the resulting droplet size, depending on flow conditions and geometry.
Garstecki et al. 17 proposed a simple and widely accepted scaling law for the prediction of the resulting droplet length l d,det in rectangular channels with width w c , see Equation (3). The scaling law states that the resulting dimensionless droplet length solely depends on the flow rate ratio ϕ and a geometry dependent factor α, which has to be fitted to experiments. 21 Even in the necking stage, a non-negligible portion of the continuous phase bypasses the emerging droplet/bubble. 41 The curvature at the front of the droplet K F is assumed to remain constant whereas the curvature at the rear end of the droplet K B is affected by _ V n . Equation (4) A detailed investigation of droplet formation includes the consideration of the three-dimensionality of the process. By applying microparticle image velocimetry (μ-PIV) van Steijn et al. 40 demonstrated that 3D flow in the corners of rectangular channels plays an important role in droplet break-up. Additionally, the importance of the 3D interface of droplets has already been recognized. 37 However, despite the great achievements in physical understanding of droplet formation and predicting droplet size in the squeezing regime in the past decades, the role of the three-dimensionality of droplet formation has not been cleared sufficiently. Three-dimensional droplet quantities are mostly derived from 2D photographs, applying theoretical methods, 43 as a noninvasive 3D imaging approach is missing.
In this work, we tie in with this research gap and apply X-ray based micro-computed tomography (μ-CT) for the investigation of droplet formation in circular capillaries. X-ray based μ-CT is a noninvasive measurement technique allowing high spatial resolutions (<10 μm) and overcoming the challenges related to optical imaging. 44 Despite the wide use of tomographic imaging in the fields of chemical engineering and process engineering, 45-51 the extension to mini-and microfluidics is an innovative approach. 44 One major challenge in using X-ray based tomographic imaging for dynamic processes in micro process engineering is the limited temporal resolution, which can range from a few minutes to several hours. In this work, this is met by the application of an in-house developed scanning protocol and postprocessing routine, called the multiple-select method, for time-resolved 3D imaging of periodic droplet generation. Hence, temporal resolution is enhanced significantly (≈ 0.05 s). This allows for the tracking of different 3D droplet quantities, such as volume and surface, over time.

| Experimental setup
Water droplets are generated in silicone oil using a co-flow configuration, as shown in Figure 2 Table 1. Experiments are conducted twice to ensure  Table 2). These experiments are done in triplets to ensure reproducibility.

| Scanning protocol and postprocessing
Originally, the temporal resolution of tomographic imaging is low as a whole set of projection images is required for the reconstruction of 2D projections to a 3D data set. This limits the application of computed tomography to static or very slow problems. Movement of the sample is highly undesirable as it leads to perturbations in the resulting 3D images, called movement artifacts. In this work, we present a methodology to make tomographic imaging applicable to the investigation of droplet generation in the squeezing and leaking regime. Droplet generation is highly reproducible in these stages.
Therefore, the droplet undergoes the same states during droplet formation again and again. Hereby, we define a droplet state as a certain length that the droplet has during its generation process. This is used for the so-called multiple-select method that is based on the postacquisition synchronization approach, which is also utilized in biomedical imaging. [53][54][55] The multiple-select method for the timeresolved imaging of droplet generation is depicted in Figure 3 This data set is reconstructed to a 3D data set using commercially available reconstruction software NRecon (Bruker, Billerica, MA) (4).
By repeating this for different droplet lengths l d time-resolved CT data is obtained.
For the selection of the appropriate projection for each angular position (step (2)), the projections are segmented using a pretrained convolutional neural network (CNN), the vgg16 net. 56 The vgg16 is chosen as it is a good compromise between accuracy and computation speed. 57 It consists of 16 weight layers and applies small convolutional filters (3 x 3). 56 The vgg16  proofed to be robust and accurate when testing on validation data. In test projection images, 97% of the pixels were assigned to either water, oil, cannula, tube, or background correctly. After segmentation, droplet lengths could be easily determined by applying the built-in MATLAB function regionprops. Figure 3(b) exemplarily shows a segmentation result for one X-ray projection with droplet length l d found by regionprops.
The reconstructed data sets consist of gray-value voxels, the 3D equivalent of pixels, see Figure 3(c). These voxels need to be assigned to either water, oil, cannula, tube, or background according to their gray values. The segmentation is, again, done with a pretrained vgg16 net, that was adjusted to the segmentation task by training in MATLAB in the same manner as described before.
In this work, different droplet quantities are tracked, see Figure 1.
These quantities can be divided into volumes, surfaces, and lengths/diameters. Volumes are obtained by summing voxels assigned to the desired phase and multiplying with (isotropic edge length = 12 μm) 3 . The local droplet radii r x are calculated using the cross-sectional area A cross (see Figure 1(a)) at discrete positions in flow direction according to Equation (6). The concrete value of A cross at each discrete position is obtained by summing all voxels that are assigned to the dispersed phase at the discrete position and multiplying with (isotropic edge length = 12 μm) 2 .
For the calculation of the PDMS/water surface area, the droplet is approximated as a stack of truncated cones with a height of 12 μm (spatial resolution), whose shell surfaces can be calculated easily. The 3.2 | Evolution of droplet parameters over time The remaining droplet parameters that are shown in Figure 5 In the filling stage, the droplet evolves in the radial and axial direction, as can be seen from the increase of maximum droplet radius r d,max and droplet length l d in Figure 5(d),(f). A higher flow rate of the dispersed phase (associated with higher ϕ) leads to a faster increase of the maximum droplet radius r d,max in the filling stage ( Figure 5(d)). The plot also shows a slower increase in r d,max for higher oil viscosities (associated with higher Ca). Figure 5(e) shows the evolution of the minimum neck radius r n (t). As during the filling stage no neck has been formed r n (t) corresponds to the radius directly behind the cannula and grows slowly as the droplet increases. Thereby, r n (t) is highest for the lowest oil viscosity. For the evaluation of the continuous phase, the volume that accumulates behind the neck V n (t) and the bypassing volume V b (t) are considered separately. In the filling stage, the continuous phase can bypass the droplet easily. Therefore, the amount of bypassed volume V b increases linearly ( Figure 5   The Laplace pressure Δp L can be calculated by Equation (7) 58 and is shown in Figure 7(b).

| Investigation of droplet surface and Laplace pressure
In the filling stage, the Laplace pressure difference decreases according to the increase of the droplet extension in radial and axial direction. Δp L reaches a minimum, which marks the end of the filling stage. Generally, the minimum of Δp L is lower for higher oil viscosities, thus, higher Ca. In the necking stage, the Laplace pressure difference increases again as the droplet is pushed in the downstream direction.
Δp L reaches a local maximum closely before pinch-off. It should be noted that the fluctuations visible for the curves PDMS 1cSt, ϕ = 0.667 result from noise magnification via derivation according to Equation (7).
Under the assumption that the shape of the cap of the droplet remains constant in the necking stage, dSd dVd in Equation (7) is solely affected by the forming neck. Therefore, under the assumption of a constant pressure p d in the dispersed phase, the pressure difference in the continuous phase upstream and downstream of the droplet Δp c is given by Equation (8).
As the pressure of the dispersed phase is not measured in this It should be mentioned that the calculated values V fill , V Δn , and β show large standard deviations due to the low number of data points.
However, the figure shows that the model of Korczyk et al. 18 is suitable for the prediction of the qualitative behavior of droplet generation in a co-flow configuration and circular channels.

| Investigation of relative leaking strength
It can also be described in terms of η = f(V d,det , V d,fill , ϕ, V Δn ) or by η = f(β, ϕ, Ca) as shown in Equation (10), where V Δn is the difference between neck volume closely before pinch-off at pinch-off time t po and at the end of the filling stage t fill , see Equation (11).
In this work, V d,det , V d,fill , and V Δn can be evaluated directly using the presented tomography approach, which allows for the calculation of η = f(V d,det , V d,fill , ϕ, V Δn ) from experimental data. The results are presented as points in Figure 9. Additionally, Figure 9 shows the calculated values for η calculated using η = β ϕ Ca as solid lines for β as obtained by Korczyk et al. 18 18 suggests that in circular channels leaking is less relevant at the same Ca Á ϕ compared to rectangular channels and that the transition from squeezing to leaking takes place at lower Ca Á ϕ.
The relative leaking strength can be used to calculate the ratio between the fluid bypassing the droplet and the total volume flow rate of the continuous phase by using Equation (12).
From the master curve fitted to own experimental data in the range 4.7 Á 10 −5 < Ca Á ϕ < 8.6 Á 10 −4 the proportion of the continuous phase that bypasses the emerging droplet is 37% > _ Vb _ Vc > 3%. As described previously, leaking is stronger for lower values of Ca Á ϕ .
Yao et al. 42 reported the ratio _ Vb _ Vc for a rectangular channel and complete wetting in the range 1.5 Á 10 −4 < Ca Á ϕ < 6.1 Á 10 −4 . The portion of bypassing liquid was found to be approximately in the range 25% > _ Vb _ Vc > 11% for the emerging droplet. 42 In the same range of Ca Á ϕ, the master curve fitted to own experimental data shows 17 % > Ca Á ϕ > 5% for the emerging droplet. The portion of continuous phase bypassing the droplet is lower by approximately a factor of two for the circular channel tested in this work compared to the rectangular channel tested by Yao et al., 42 which supports the thesis that leaking is less relevant for circular channels. In contrast to circular channels, the emerging droplet does not cover the crosssectional area of a rectangular channel uniformly. As the emerging droplet aims to minimize energy via surface minimization the corners of a rectangular channel remain uncovered by the dispersed phase. The portion of the cross-sectional area for the flow of bypassing liquid is larger compared to circular tubes, which results in lower flow resistance. Therefore, the importance of leaking is lower in circular channels than in rectangular channels, which is proved in this work.

| SUMMARY AND OUTLOOK
Micro-computed tomography was applied for the investigation of highly periodic droplet formation in a co-flow needle-in-capillary arrangement (d i ≈ 1.58 mm) at low capillary numbers. Water droplets were generated in the co-flow configuration with silicone oil as the continuous phase. A scanning protocol followed by a postprocessing procedure, the multiple-select method, was developed. By applying the multiple-select method the temporal resolution for X-ray-based micro-computed tomography was enhanced to 0.05 s. The timeresolved 3D data generated using the multiple-select method allowed for the determination of different droplet quantities (length, diameter, surface, volume, Laplace pressure difference). which means that leaking is less relevant in circular channels than in rectangular channels. Additionally, the transition from squeezing to leaking was found to be at lower values Ca Á ϕ for circular channels (Ca Á ϕ = 3:0125 AE 1:4 ð Þ Á 10 − 5 ) than for rectangular channels (Ca Á ϕ = 7:4 AE 0:3 ð Þ Á 10 − 5 ) as found by Korczyk et al. 18 The analysis of the evolution of droplet surface over droplet volume supports the assumption that droplet evolves in a series of equilibrium states. The amplitude of pressure fluctuation in the continuous phase upstream of the emerging droplet was found to be in the range of 9.7-33.4 Pa. It was found that the fluctuations decrease as the capillary number increases, which is supported by findings in the literature. 16 With this work, we prove the suitability of the presented