Six Stages of Microdroplet Detachment from Microscale Fibers

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Six Stages of Microdroplet Detachment from Microscaled Fibres Majid Ahmadlouydarab, Ahmed A. Hemeda, and Yanbao Ma Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b03089 • Publication Date (Web): 29 Nov 2017 Downloaded from http://pubs.acs.org on November 30, 2017

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Six Stages of Microdroplet Detachment from Microscale Fibers Majid Ahmadlouydarab1,2 , Ahmed A. Hemeda2 and Yanbao Ma2∗ 1

Faculty of Chemical and Petroleum Engineering, University of Tabriz, Iran 2

School of Engineering, University of California, Merced California 95343, United States ∗

Corresponding author November 27, 2017

Abstract The detachment of droplets from cylindrical fibers is of fundamental importance for both scientific research and engineering applications. Due to the challenges to determine dynamic contact angles on the fiber surface, the process of the droplet detachment from a fiber is not well understood. In this paper, a multi-body dissipative particle dynamics (MDPD) method, a particle-based mesh-free method that can automatically capture the dynamic contact angles through direct modeling of liquid-solid particle interactions, was applied to study the detachment process of a liquid microdroplet from a cylindrical solid fiber pulled by an atomic force microscopy (AFM) tip under a constant velocity. After the validation of the numerical results through comparison with experiments in a benchmark case, the same numerical tool was applied to analyze the droplet detachment mechanisms. Based on the slope of the time history curve for the displacement of the droplet mass center, the detachment process can be divided into six stages. The change of slope in each stage can be explained from the change of surface energy. The results can greatly advance the fundamental understanding of the detachment process of microdroplets from cylindrical fibers.

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ACS Paragon Plus Environment

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Introduction

Droplets on fibers are natural phenomena observed as dew droplets on spider webs and plants. Fiber-droplet interactions are of interest for a number of scientific fields with broad engineering applications, such as liquid purification through membrane filtration [1],textile manufacturing [2], textile wetting, cleaning, and drying [3], fiber or wire coating [4], liquid aerosol filtration [5], oilwater separation [6], water collection in fog filters [7, 8], and digital microfluidics based biotechnologies [9]. A fundamental understanding of the behavior of droplets on individual fibers could potentially lead to improved engineering design in these applications. For the droplets on the fibers, there are two competing equilibrium morphologies without external forces: an axisymmetric barrel shape or a nonaxisymmetry clamshell shape [10, 11]. With external forces (such as gravity, buoyancy, drag force, and so on), the morphologies of the droplets can change dramatically [12–14]. When external drag force is greater than the adhesive force between the droplets and the fibers, the droplets will detach from the fibers. So far, there are only a small number of research works on the droplet detachment from the fibers, which are briefly reviewed below. Hanumanthu and Stebe [12] developed a theoretical model to predict equilibrium shapes and locations for axisymmetric, micro- and nano-scale droplets on conical axisymmetricl surfaces. It is predicted that there exists a critical contact angle at which the droplet detaches from the solid surface. Dawar and Chase [15] studied the droplet detachment from a fiber due to gas flow perpendicular to the fiber axis and developed an empirical correlation to calculate the average drag coefficient for the detachment. Sahu and coworkers [16] studied the droplet motion on a filament under both parallel and perpendicular air blowing conditions, and their experimental observations are elucidated in the framework of simplified models. Gauthier et al [17] studied water droplets growth and detachment on the fibrous carbon fuel cell electrode materials. Ruiter et al [11] analyzed the morphology and the stability of barrel-shaped and camshell-shaped droplets on the fibers using complementary experiments and numerical calculations. It was found that droplet detachment always occurs from the clamshell morphology in the experiments with contact angles greater than 10◦ . Mullins and coworkers [18] developed a novel fiber-droplet interfacial tension model to 2


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predict the force requirement to draw a barrel-shaped droplet away from a fiber. Hotz et al [19] measured the droplet detachment force using an atomic force microscope (AFM) and found that the measured force data could be fitted by a power law relationship. Generally, the dynamic detachment process highly depends on the fiber diameter, surface wettability and roughness, the droplet volume, and external forces [20, 21]. Especially, in AFM experiments to measure the droplet detachment force, there are dynamic contact lines on both fiber surface and AFM tip surface. It is still challenging for accurate measurements of the forces and contact angles of the droplet on the fiber surfaces [18, 19]. In addition, for mesh based numerical methods, the dynamic advancing and receding contact angles are prerequisite boundary conditions that are unknown at most time, which posts a challenge in numerical simulations. Due to the challenges in measurements and numerical simulations, the droplet detachment process from the fiber is still not well understood [22–25]. In this paper, we aims to achieve a fundamental understanding of the droplet detachment process from a microscale fiber in AFM experiments through numerical simulations using a many-body dissipative particle dynamics (MDPD) method, a mesh-free numerical method that provides a natural way to capture dynamic contact angles through particle-particle interactions.

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Numerical Method

A dissipative particle dynamics (DPD) method is a mesh-free particle based method that has been successfully applied to simulate complex fluids, especially at mesoscales that are complementary to atomic and continuum scales [26–29]. In the mesoscale regime, continuum theories often fail to capture Brownian motions, while atomistic simulation techniques such as molecular dynamics (MD) become computationally prohibitive. To tackle the mesoscale problems, DPD method tracks coarse-grained particles that are composed of a cluster of atoms or molecules. In current study, a many-body dissipative particle dynamics (MDPD) method, a modified version of DPD model is adopted in the numerical simulations. Various aspects of the DPD and MDPD models and numerical algorithms have been described in detail elsewhere [26–28]. Here, the same numerical algorithm that is described in our recent publications [28,29] is used in the current study. Therefore, 3


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we only briefly mention a few features of the numerical method here. In MDPD methodology, fluid is modeled as a group of beads. The motion of each bead is governed by Newton’s second law: − d→ ri − =→ vi dt − X −→ −→ −→ → − d→ vi mi = fi = (FijC + FijD + FijR ) dt

(1) (2)

i6=j

→ − − − where → ri , → vi and fi denote the ith bead’s position, velocity, and the total force imposed on that −→ → − bead, respectively. The three components of fi , including conservative force FijC , dissipative force −→ −→ FijD and random force FijR , are given by equations 3, 4, and 5 [26, 30]: −→ FijC = Aij ωC (rij ) + Bij (ρi + ρj )ωd (rij ) −→ − →− → FijD = −γωD (rij )(− e→ ij · vij )eij −→ FijR = ϕωR (rij )θij (δt )−1/2 − e→ ij

(3) (4) (5)

Aij and Bij are the amplitudes of attractive and repulsive forces, respectively, along with the weighing functions ωC and ωd for the attractive and replusive focres, respectively. rij = |− r→ ij |, − → − → − − → − → − → → − → − r→ ij = ri − rj , eij = rij /|rij |, and vij = vi − vj . Also, ωD , and ωR are weight functions for dissipative and random forces, respectively. θij is the Gaussian white noise with zero mean and unit variance. γ and ϕ are the amplitudes of dissipative and random forces respectively. If the dissipation parameter γ and the white noise amplitude δt satisfy the fluctuation-dissipation theorem, then the dissipative and random forces act as a thermostat [28]. This requires ωD (r) = [ωR (r)]2 and ϕ2 = 2γkB T , where kB is the Boltzmann constant and T is the temperature of the system. As microdroplets of interest in current study have diameters of a few microns and the capillary length for most of the droplet-fiber systems is around one millimeter so the gravitational force is not considered here [18, 31]. In the MDPD calculations, both the fiber and the AFM tip are constructed from three layers of frozen solid beads as shown in fig. 1. To avoid the penetration of liquid beads into the substrate as well as to satisfy the no-slip boundary condition, a bounce-forward reflection boundary condition is

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Table 1: Adopted parameters in MDPD simulations [33]. Parameters Fluid bead density viscosity surface tension System temperature Cut-off radius of attractive force Cut-off radius of repulsive force Amplitude of random force Attraction parameter Repulsion parameter Time step

Symbol ρ µ σ kB T rc rd ϕ All Bll = Blw ∆t

Value (DPD unit) 6.10 7.41 7.30 1.00 1.00 0.75 6.00 -40.00 25.00 0.01

used on the interface between the droplet and the substrate [28,29]. Different contact angles on solid surfaces can be modeled by altering the attraction parameters Alw [28]. In addition, this attraction parameter Alw is used in MDPD for both static and dynamic wetting because the MDPD method can automatically capture dynamic contact angles through particle-particle interactions between the liquid beads and solid beads. It should be noted that there is effect on contact angles from the surface curvature [32]. Here Alw will determine the intrinsic contact angle on a flat surface (θ00 ) without the effect from surface curvature. The curvature effect on the contact angle is out the scope of this work. All parameters for MDPD calculations in current study are listed in the table 1 in DPD units, which are same as the parameters given by Arienti et. al [33]. The listed parameters in the table produces a liquid density of 850 kg/m3 and liquid-vapor surface tension of 0.026 N/m. Moreover, in table 1, Bll represents the repulsion between liquid (l) beads, while Blw represents the repulsion between the substrate wall (w) and liquid (l) beads [30, 34, 35]. In current study, to link the results from DPD domain to physical domain, one DPD length unit equals to 1 µm in physical domain. The corresponding mass unit is m = 1.39 × 10−16 kg at a particle number density of ρ = 6.1, which gives a liquid density of 850 kg/m3 . The time unit is chosen as t = 1.98 ×10−7 s so that the surface tension σ = 7.30 in the MDPD system corresponds to a liquid-vapor surface tension of 0.026 N/m. More discussions on the MDPD scaling can be found in [28, 33, 35].

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3 3.1

Results and Discussion Problem setup

Fig. 1 shows the computational domain including a fiber of radius Rf , an AFM tip of radius Rt , and a droplet of initial radius Rd . Two dimensionless parameters Rdf and Rtf are defined as: Rdf = Rd /Rf , Rtf = Rt /Rf . The center of the fiber aligns with the x axis. Zm stands for the distance between the droplet mass center and the fiber center with y = 0 and z = 0. The AFM tip is pulled in positive z direction with constant velocity of V . A capillary number is defined as Ca = µV /σ to characterize the ratio of viscous force and capillary force, where µ is liquid viscosity and σ is surface tension of the droplet. In this study, small tip velocity is chosen so that Ca is much less than 0.01, i.e., the viscous force is negligible compared with the capillary force. Furthermore, the time is nondimensionalized by µRf /σ. The numerical results are discussed in the next section.

Figure 1: Schematic of the initial configuration for 3D computations

3.2

Validation case

Mullins and coworkers [18] conducted experiments to measure detachment forces for barrel-shaped oil droplets on four different fibers with different materials, including 316 stainless steel (Rf = 4.2 µm), glass (Rf = 3.5 µm), polypropylene (Rf = 17.1 µm), and polyester (Rf = 10.2 µm). For four different materials, there are only detailed results on detachment forces vs relative displacement of droplet mass center for the 316 stainless steel fiber with Rf = 4.2 µm and Rd = 9.1 µm. 6


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Therefore, this case is chosen as a benchmark for the validation of the MDPD results. Because of experimental challenges, the contact angles on both fiber surfaces and AFM tip surface were not given. In addition, in the AFM experiments, the AFM tip radius is not specified [18]. Due to these experimental uncertainties, we mainly focus on qualitative comparison of MDPD results with experiments. In the MDPD model, the adopted parameters are θf0 0 =30◦ , Rf = 4.2 µm, and Rdf = 2.17. To study the effects on detachment forces, three different cases with different AFM tip radii, Rt = 10, 20, and 25 µm are simulated. Moreover, for the AFM tip raidus of 25 µm, three values 0 are considered in the present comparsion, i.e., θ 0 =40◦ , 70◦ , and 100◦ . For these different of θt0 t0

cases, the detachment forces are calculated by taking integration of liquid-solid interaction forces and momentum change rate of liquid particles. Numerical results of the detachment forces vs the displacement of droplet mass center (Zm /Rf ) for three different AFM tip radii are compared with 0 =70◦ . experiments as well as theoretical results by Mullins et al [18] as shown in Fig. 1a when θt0

The results of comparison are plotted in Fig. 2a. Due to the limit of theoretical model, only the early stage of the detachment process is compared. Two insets in Fig. 2 show the change of droplet shapes during the detachment process for numerical results with Rt = 10µm. In Fig. 2b, the detachment force are agian calculated for three different AFM tip contact angles with a constant 0 =40◦ of the AFM tip raidus of 25 µm. An inset figure in Fig. 2b shows the droplet shape with θt0

AFM tip which in a greement with the maximum force needed at this contact angle compared to other surfaces. The results show that there is similar trend among the numerical, experimental, and theoretical results. The oscillation in numerical results is due to Brownian motion of liquid particles in the MPDD method. For numerical results, three different AFM tip radii are considered with the same droplet volume. It shows that the detachment force is slightly higher for the case with a larger AMF tip radius. Qualitatively, there is good agreement in the detachment force curves among the numerical, experimental, and theoretical results, which demonstrates that the MDPD model can capture the dominant physics in the droplet detachment process. The detachment mechanisms of the droplet from the fiber are analyzed in the following section.

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Figure 2: Variation of the detachment force by increasing the droplet mass center distance from the fiber 0 center line. Results of the detachment force for: different tip radius when θt0 =70◦ in (a), different AFM tip contact angle when Rt = 25 µm in (b). Two insets of droplet morphologies at Zm /Rd =0.34 and Zm /Rd =0.61 0 both for Rt = 25 µm shown in (a), an inset figure for AFM tip at Zm /Rd =0.74 when θt0 =40◦ in (b)

.

3.3

Analysis of droplet detachment process

To understand the droplet detachment process from a solid fiber, a baseline case with following 0 =70◦ , R = 4.2 µm, R properties was adopted; θf0 0 =40◦ , θt0 f df = 3.0, and Rtf = 9.523. Not 3 cos(θ ) − 3R2 + 1 ≥ 0 so that the chosen Rdf value satisfies the stability condition given by 2Rdf f df

that the droplet takes a barrel shape before it is get touched and pulled away from the fiber at a fixed location by an AFM tip [36]. The AFM tip is attached to the droplet and pulls it away from the fiber in the positive z direction with a constant velocity. In this case, a small capillary number, Ca = 0.001 is chosen so that the effect from the tip velocity on the detachment process is negligible. Based on the results of numerical simulation, the droplet detachment process is recorded in Supplementary Movie 01 in the Supporting Information. The detachment process is characterized by Zm , the displacement of the droplet mass center from the fiber center (see Fig. 1). It should be noted that Zm does not start at zero because the droplet cannot keep a barrel shape once the AFM tip attaches the droplet due to adhesion force between them. Fig. 3 shows a part of the time history of Zm during the detachment process and the complete time history is plotted as an inset at the up-right corner of the same figure.

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Figure 3: Time history of the normalized displacement of the droplet mass center Variation of the droplet mass center Zm normalized by the fiber radius Rf . The droplet detachment process can be divided into six stages based on the slope of the time history curve. Five representative snapshots of droplet morphology during the detachment process are plotted as insets. In stage six, the droplet is completely detached from the fiber after t > 1120.

The droplet detachment process is controlled by the interactions of three forces: (a) the adhesion force between the droplet and the AFM tip surface acting as a pulling force; (b) the adhesion force between the droplet and the fixed fiber as a drag force; and (c) surface tension from the droplet free surface. Even though the AFM tip moves away from the fiber with a constant velocity, there are dynamic changes in the droplet detachment process due to the dynamic interactions among the three forces. The dynamic force interactions also lead to dynamic changes in the droplet morphologies as well as in two contact lines on the two curves surfaces (the AFM tip and the fiber). Based on the changes of slopes in the time history curve of the displacement droplet mass center (see Fig. 3), the whole droplet detachment process can be divided into six stages. For each stage in the first five stages, a representative snapshot of morphological development of the droplet and the relative postings of the fiber and the AFM tip is shown as an inset in Fig. 3. In addition, a series of snapshots from side view along the fiber direction at different times are shown in Fig. 4. Stage one (S1) is in the range of 0 < t < 50. In this stage, the droplet wraps around the fiber

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Figure 4: Droplet morphologies from side view along the fiber direction at different times. (see Fig. 4 with t = 35). It is slowly pulled up by the AFM tip until the bottom of the droplet reaches the bottom of the fiber. Then, the detachment process enters stage two (S2) in which the droplet starts to rupture at the bottom. Consequently, two wet frontiers are generated near the bottom of the droplet and both wet frontiers gradually recede from the bottom of the fiber (see Fig. 4 with t = 94) when the droplet is continuously pulled up. Due to the stronger adhesion force associated with the receding of the two wet frontiers at the contact line between the droplet and the fiber surface in stage two compared with that in stage one, the displacement of the droplet is slower in S2 than that in S1, i.e., the slope in S2 is flatter than that in S1 (see Fig. 3). Stage two ends at t ≈ 100 when the two frontiers of the droplet reach to the upper half surface of the fiber followed by stage three (S3). In stage three, the two wet frontiers of the droplet continue to recede toward the top ridge of the fiber. Comparing the displacement in S3 with S2, the slope in S3 is slightly higher than that in S2. The possible reason is due to different trends of wetting in S3 and S2. In S2, the two frontiers of the droplet are located on the lower half of the fiber, the droplet tends to wet the surface even without external driving force. In S3, the two frontiers of the droplet moves on the upper half of the fiber surface, the droplet tends to recede from the fiber surface without external driving force. Consequently, the detachment process is slightly faster in S3 compared with that in S2. Figure 4 with t = 140 shows the droplet morphology near the end of

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S3. Stage three ends at about t ≈ 150 when the wetting area on the AFM tip reaches a maximum, which is followed by stage four (S4). In S4, the droplet starts to move backward to the fiber surface and the displacement of the droplet mass center decreases even though the AFM tip continues to move away from the fiber with a constant velocity. In this stage, the wetting area on the AFM tip decreases while the wetting area on the fiber surface increases. Stage four ends at t ≈ 175 followed by stage five (S5) in which the droplet is continuously stretched until it is detached from the fiber surface at t ≈ 1120. In stage six (S6), the droplet is detached from the fiber (t > 1120). The fourth figure in Fig. 4 shows the droplet morphology in S5 at t = 247. The last two figures in Fig. 4 shows the droplet morphologies before (t = 1087) and after the detachment (t = 1205), respectively. Noting that the shape and contact angle of the droplet in contact with the AFM tip is in an excellent agreement with prior work in Stage 6 [37–39]. To get better understanding of the physics in the detachment process, we conduct free surface energy analysis. This free surface energy for the tip-droplet-fiber setup is given by [40–42]:

E = σAd + (σtd − σt )Atd + (σf d − σf )Af d

(6)

where A is interfacial area and σ is surface tension. “d” stands for the droplet, “f ” stands for the 0 , Eq. 6 can fiber, and “t” stands for the AFM tip. Using Young-Laplace equation,(σt − σtd ) = σθt0

be re-written as: 0 E = −σ(Atd cos(θt0 ) + Af d cos(θf0 0 ) − Ad )

(7)

0 and θ 0 ) as well as air-droplet surface tension (σ) are constants while the Contact angles (θt0 f0

interfacial area A is a variable. At the equilibrium state, one needs to minimize the Eq. 7 and 0 ) + dA cos(θ 0 ) = dA .” In the MDPD simulations, solve numerically the equation “ dAtd cos(θt0 fd d f0

interfacial area A is proportional to the number of liquid particles. Therefore, we focus on the analysis of the numbers of liquid particles on the three interfaces: fiber-droplet (NF ), air-droplet (ND ), and tip-droplet (NT ) interfaces, respectively. The changes of liquid particle numbers during the detachment process are shown in Fig. 5. For comparison, the normalized liquid particle numbers

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on the fiber-droplet interface (NF 0 ) and the air-droplet interface (ND0 ) in absence of the AFM tip are also plotted in Fig. 5. Here, the numbers of liquid particles are normalized by Nt0 , total liquid particle numbers (Nt0 = NF 0 + ND0 ) on the fiber surface without the AFM tip. As shown in Fig. 5, ND (on the air-droplet surface) goes through a slow decrease in S1 first, then a slight increase in S2, a sharp increase in S3, a sudden decrease in S4, and a slow increase in S5. When the detachment happens at t ≈ 1120, there is a sudden decrease in ND before ND reaches a constant after the detachment. NT (on the AFM tip surface) keeps increasing in S1, S2 and S3, then sharply decreases in S4, and gradually decreases in S5. When the detachment happens, there is a sudden jump in NT . Then NT reaches a constant after the detachment. Similarly, NF (on the fiber surface) also shows a non-monotonic behavior: it keeps decreasing in S1, S2 and S3, then it keeps almost constant in the first half of S4, suddenly increases in the second half of S4 followed by a gradual decrease in S5, and eventually decreases to zero after the detachment. Generally, when the a liquid droplet bridges two hydrophilic surfaces, a small disturbance to the equilibrium status will lead to back and forth oscillation of the droplet between the two surfaces until the oscillation motion dies down due to viscous dissipation. In this study, such oscillation takes place once in S3 and S4, but it quickly dies down at the end of S4 partially due to the high viscosity and partially due to the motion of the AFM tip. In addition to interfacial area A, as eq. 7 indicates, both solid substrates wettability properties affect the interfacial morphology. For example, our numerical results indicate that the major effect of the fiber wettability (θf0 0 ) is on the stage S3 and S4 of the detachment process. As shown in fig. 6(a), decreasing the contact angle on the fiber surface moderates the maximum fluctuation amplitude (Dm ) of the droplet mass center. A reduction in θf0 0 not only reduces Dm , but also eliminates the minor fluctuations in Dm . Hence, there is a high competition among the liquid-solid interfacial viscous dissipation forces, air-liquid induced curvature forces, and pulling forces imposed by the tip, so any change in θf0 0 will change the liquid-solid interfacial viscous dissipation and will affect the detachment process. It is interesting to know that our results show a linear relation between Dm and θf for the studied cases as shown in fig. 6(b). Note that, thorough analysis 0 , and indicates significant effects of other key parameters including capillary number, Rdf , Rtf , θt0

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Figure 5: Variation of normalized numbers of liquid particles on the interfaces: NF (at the fiber-droplet interface), ND (at the air-droplet interface), and NT (at the AFM tip-droplet interface). NF 0 and ND0 are the normalized number of particles on the fiber-droplet and vapor-droplet interfaces in absence of the AFM tip. The particle numbers are normalized ty total liquid particle numbers (Nt0 = NF 0 + ND0 ) on the fiber surface without the AFM tip.

Figure 6: (a) Variation of the droplet mass center (Zm ) in time when fiber surface bears different contact angle. The droplet mass center position Zm is normalized by the fiber radius Rf . The maximum fluctuation amplitude (Dm ) also has been shown on the figure. (b) Variation of Dm with variation of the contact angle of the fiber’s surface (θf0 0 ).

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AFM tip’s shape on the proposed microdroplet detachment mechanism. Theses influences and its results are out of the scope of the present work and they may be reported elsewhere.

4

Summary and Conclusions

The detachment process of a droplet from a cylindrical fiber has not been simulated before this study due to unknown dynamic contact angles that are prerequisite conditions for most mesh-based numerical methods. In this paper, a mesh-free multi-body dissipative particle dynamics (MDPD) method is applied to study the detachment process of a microdroplet from a cylindrical fiber driven by an AFM tip. In the MDPD method, the liquid-solid interactions are described by three different forces for particle-particle interactions. The intrinsic contact angle (θf0 0 ) on a flat plate surface is determined by a single attraction parameter (Alw ) to characterize the attraction force between a liquid particle and a solid particle. Because the particle size in the MDPD is at least one hundred times smaller than the radius of the solid wall surface curvature, the same attraction parameter (Alw ) can be used for wall surfaces with different curvatures but with the same solid materials. In addition, the same attraction parameter (Alw ) can be used for both static and dynamic liquidsolid interactions. In other words, the MDPD method can automatically capture dynamic contact angles through liquid-solid particle interactions. It was demonstrated that the droplet detachment process can be successfully simulated by the MDPD method. The qualitative agreement in the detachment force between the numerical results and experiments demonstrates the validity of the numerical approach. The droplet detachment process is characterized by the displacement of the droplet center (Zm ). Surface energy is analyzed in order to understand the physics in the droplet detachment process. It was found that the detachment process can be divided into six stages based on the slope change in the time history curve of the displacement of the droplet mass center (Zm ). The change of slope in each stage can be explained from the change of surface energy. Droplet oscillation between the AFM tip surface and the fiber surface is observed. Parametric studies of the detachment process with effects from intrinsic contact angles, the fiber radius, the droplet size, and the AFM tip size will be the future work.

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Associated Content

Supporting Information A movie is included to show the droplet detachment process. The Supporting Information is available free of charge on the ACS Publication website at DOI:

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Author Information

Corresponding Author ∗ E-mail:

[email protected]

Author Contributions MA and AAH performed numerical modeling numerical simulations. MA, AAH and YM analyzed the data. All authors contriguted to the discussion and writing of this work.

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Acknowledgement

We acknowledge financial support from startup grant from UC Merced.

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Six Stages of Microdroplet Detachment from Microscale Fibers

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