# Emulsion-Based Single Drop Generation in a Nonconfined System

Ranim Chakleh, Noureddine Lebaz, Mahmoud Kamaeddine, Nida

Sheibat-Othman

► To cite this version:

Ranim Chakleh, Noureddine Lebaz, Mahmoud Kamaeddine, Nida Sheibat-Othman. Emulsion-Based Single Drop Generation in a Nonconfined System. Langmuir, 2025, 41 (8), pp.5504-5523. 10.1021/acs.langmuir.4c05212. hal-04957921

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Submitted on 7 Dec 2025

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# Emulsion-based single drop generation in a non-confined system

Ranim Chakleh<fnref n="1" />, Noureddine Lebaz<fnref n="1" />, Mahmoud Kamaeddine<fnref n="2" />, Nida Sheibat-Othman<fnref n="1" />*

corresponding author: nida.othman@univ-lyon1.fr

**Keywords:** single drop, capillary tube, double emulsions, shear-thinning, interfacial tension

## Abstract

Single drop generation in a non-confined system during the dripping regime is studied to evaluate the effect of operating conditions on the formation time and drop size. The drop itself is composed of a water-in-oil emulsion that rises from a nozzle into a stationary aqueous phase. This emulsion is non-Newtonian and undergoes a shear-thinning behavior, leading to an important decrease in viscosity when exposed to shear forces. Its viscosity and shear-thinning level depend on the inner water fraction, besides the oil viscosity. The viscosity appears to affect both the formation and detachment stages as opposed to what was reported in previous literature. The theoretical volume of the detached drop is calculated based on the equilibrium of the forces acting upon the attached drop, in addition to the volume added in the second stage while considering the fraction of the residual drop left attached to the capillary after detachment. This model allowed the prediction of the dynamic interfacial tension by minimizing the error between the theoretical and the experimental drop volumes.

## Introduction

Single and multiple emulsions are employed in many applications such as liquid-liquid extraction, encapsulation, combustion, inkjet printing, spray drying, or spray cooling. To better understand the phenomena taking place during emulsification, the investigation of single drop generation is of great interest. Single drop experiments are carried out in either confined microfluidic devices or non-confined media, such as capillary tubes with falling or rising drops. While research in microfluidic devices gained a more significant attention in the recent literature, studies in non-confined media remain interesting since the drop formation is

<footnote n="1">¹ Universite Claude Bernard Lyon 1, LAGEPP UMR 5007 CNRS, 69100 Villeurbanne, France</footnote>

<footnote n="2">² CPE-Lyon, Villeurbanne, France</footnote>
governed by fewer forces and the drop surface evolves freely. Such studies may serve to identify models for predicting the terminal velocity, drag coefficient, mass transfer, drop breakage, and coalescence, surface tension, or define the optimal operating conditions for a specific application¹⁻³.

Studies on a falling or rising drop from a capillary tube tip, into the air or a stationary liquid, have outlined two main stages of drop formation: a first stage (stage I) during which the drop grows and reaches equilibrium, and a second stage (stage II) where necking occurs until drop detachment (pinch-off moment)⁴. The first stage is quite long where small changes in the drop interface topology occur, while the second stage is shorter but large changes and singularities in the interface take place⁵. A more detailed four-stage drop formation model can be described in terms of the contact angle, particularly to distinguish the nozzle wetting step⁶. Also, three regimes were observed as a function of the flow rate⁷, or the Weber number⁸. The periodic dripping regime occurs at low flow rates and is characterized by the periodic production of drops with constant mass. The jetting regime is observed at high flow rates due to Rayleigh-Plateau instability where jets start to disintegrate and generate satellite drops, and the detachment point moves suddenly far from the nozzle tip (about ten times the nozzle diameter). The dripping faucet, or chaotic dripping, is an intermediate regime between the periodic dripping and jetting regimes. It takes place over a narrow range of velocities and is characterized by the formation of drops with varying mass, which is therefore difficult to model.

In this work, we focus on the formation and detachment of a single drop, rising from a capillary into a stagnant liquid in a non-confined system under dripping regime conditions. The considered drop consists of a water-in-oil (w/o) emulsion exhibiting a non-Newtonian rheology⁹,¹⁰. The emulsion drops immerge into a stationary liquid to generate a water-in-oil-in-water (w/o/w) double emulsion. The shear-thinning behavior of the drops is evaluated. Investigations of emulsion-based single drop formation are useful to provide insights into the parameters influencing the generation of double emulsions (i.e., an emulsion dispersed in a liquid). These materials have versatile applications, such as the encapsulation of active materials in pharmaceutical products or flavors in food¹¹. Most studies on emulsion-based single drop generation are however done in confined systems and usually highlight a non-Newtonian behavior which renders the prediction of the drop properties more challenging¹²⁻¹⁵. The effects of the different phases properties and the operating conditions on the drop generation characteristics in the dripping regime are examined. The studied parameters
include the emulsion drop properties (i.e., oil viscosity, inner water fraction, and inner drop size), the flow rate, the interfacial tension, the external stagnant aqueous phase viscosity, and the density difference between the drop and the external water. We also examine the drop formation time, the necking stage, and the dimensions of the drop after detachment. Finally, a model is proposed to predict the interfacial tension. In the following section, a literature review of the main studies tackling drop formation in non-confined media is presented. This section aims to highlight the forces involved during drop formation, the influence of the drop rheology, and the addition of surfactants as well as some useful correlations.

## Single drop formation studies and models in a non-confined geometry

### Single drop formation models

The size and shape of the drop during its formation are determined by the forces acting upon it$^{6,16}$ (cf. Table 1). The inertial force (cf. Eq. (1), (2)) induced by the motion of fluid in the capillary and the gravity force (Falling drop) or the buoyant force (cf. Eq.(4)) in case of rising drops tend to separate the drop from the nozzle. However, the interfacial tension force (cf. Eq.(5)) and the drag force (cf. Eq. (6)) exerted by the continuous phase oppose the drop detachment. Different expressions for the drag force are available in the literature depending on the Reynolds and the drop rigidity (cf. Eq. (7), (8)).

<table><caption>Table 1: Forces acting on the drop and useful non-dimensional numbers.</caption><thead><tr><th>Variable</th><th>Equation</th><th></th></tr></thead><tbody><tr><td>Inertial or kinetic force (parabolic velocity profile)</td><td>F<sub>k</sub> = <sup>4</sup>&frasl;<sub>3</sub>ρ<sub>d</sub>u<sub>N</sub>Q</td><td>(1)</td></tr><tr><td>Inertial or kinetic force (flat velocity profile)</td><td>F<sub>k</sub> = ρ<sub>d</sub>u<sub>N</sub>Q</td><td>(2)</td></tr><tr><td>Gravity force or Weight</td><td>W = m<sub>d</sub>g</td><td>(3)</td></tr><tr><td>Buoyancy force</td><td>F<sub>b</sub> = ΔρV<sub>d</sub>g</td><td>(4)</td></tr><tr><td>Interfacial tension force</td><td>F<sub>γ</sub> = πd<sup>W</sup>γ sin θ</td><td>(5)</td></tr><tr><td>Drag force</td><td>F<sub>d</sub> = <sup>1</sup>&frasl;<sub>2</sub>ρ<sub>c</sub>u<sub>r</sub><sup>2</sup>C<sub>d</sub>A<sub>d</sub>, u<sub>r</sub> = |u<sub>d</sub> - u<sub>c</sub>|</td><td>(6)</td></tr><tr><td>Drag force – Stoke’s law (rigid sphere)<sup>4</sup></td><td>F<sub>d</sub> = 3πμ<sub>c</sub>d<sub>d</sub>u<sub>r</sub></td><td>(7)</td></tr><tr><td>Drag force – (drop with internal circulation)<sup>6</sup></td><td>F<sub>d</sub> = 2πd<sub>d</sub>μ<sub>c</sub>u<sub>r</sub>(<sup>μ<sub>c</sub>+μ<sub>d</sub></sup>&frasl;<sub>0.66μ<sub>c</sub>+μ<sub>d</sub></sub>)</td><td>(8)</td></tr><tr><td>Capillary pressure (Young-Laplace)</td><td>P<sub>c</sub> = <sup>4γ</sup>&frasl;<sub>d<sub>nc</sub></sub></td><td>(9)</td></tr><tr><td>Capillary number</td><td>Ca = <sup>F<sub>d</sub></sup>&frasl;<sub>F<sub>γ</sub></sub></td><td>(10)</td></tr><tr><td>Reynolds number</td><td>Re = <sup>ρ<sub>d</sub> L u<sub>d</sub></sup>&frasl;<sub>μ<sub>d</sub></sub>, L characteristic length</td><td>(11)</td></tr><tr><td>Weber number</td><td>We = <sup>F<sub>k</sub></sup>&frasl;<sub>F<sub>γ</sub></sub></td><td>(12)</td></tr><tr><td>Gravity Bond number (Worthington number)</td><td>Bo or G = <sup>F<sub>b</sub></sup>&frasl;<sub>F<sub>γ</sub></sub></td><td>(13)</td></tr></tbody></table>

The predominance of a force over the others depends on several parameters such as the flow regime, the stage of formation, and the physicochemical properties of the fluid. At the end of
stage I, an equilibrium of forces is reached (i.e., for a rising drop $F_k + F_b = F_y + F_d$ or in non-dimensional form $We + Bo - Ca = 1$). Based on this equilibrium, different models were proposed to predict the drop volume and the detachment time (cf. Table 2). At low flow rates, the inertial and drag forces can be neglected. For instance, Harkins and Brown examined falling drops in air to determine the surface tension using Tate's law based on the equilibrium of only the weight and interfacial tension forces with a corrective factor accounting for the fraction of the drop that remains attached after the breakup (residual drop) ². However, Hayworth and Treybal extended this approach by considering the balance between the kinetic, buoyancy, and interfacial forces and suggested a correlation of the volume of a drop coming out in an immiscible stationary liquid ¹⁷. Null and Johnson developed a drop volume correlation based on a geometric approach ¹⁸. Rao et al. added the contribution of the viscous drag to develop models to predict the volume and the detachment time of a drop rising in a low or highly viscous stationary fluid ⁴. They considered that the drop continues to grow during the necking stage, and the transition between the stages was obtained by equating $F_b$ and $F_y$. Hence, the drop equilibrium volume ($V_{eq}$, reached at the end of stage I) differs from the detached drop volume ($V_{det}$) since the fluid continues to enter the drop during stage II ($V_n$) and part of the drop will rebound and remain attached to the nozzle after breakup ($V_r$). An improvement in the prediction of the volume entering the drop during the necking stage was later proposed by Scheele and Meister ¹⁶. More recently, this approach has been applied to aqueous two-phase systems by accounting for circulation inside the droplet in the drag force ¹⁹. The different forces acting on a drop during its formation can also be considered in more advanced modeling using for instance the two-phase flow Navier-Stokes equations with interface tracking based on the volume-of-fluid method ²⁰⁻²². Yet, this local and computationally intensive approach does not allow the consideration of dissolved species such as surfactants, and their diffusion into the liquid-liquid interface.

<table><caption>Table 2: Models of single drop formation based on the equilibrium of forces.</caption><thead><tr><th>Reference</th><th>Model</th><th></th></tr></thead><tbody><tr><td>Harkins and<br>Brown<sup>2</sup></td><td>Based on Tate's law: W = F<sub>y</sub><br>V<sub>det</sub> = <sup>F<sub>y</sub></sup>&frasl;<sub>&Delta;&rho;g</sub> &psi;<br>&psi;(d<sub>N</sub>/V<sub>det</sub><sup>1/3</sup>) is a correction factor accounting for the volume of the residual drop.</td><td>(14)</td></tr><tr><td>Hayworth and<br>Treybal<sup>17</sup></td><td>V<sub>det</sub> = V<sub>eq</sub> + V<sub>R</sub> - V<sub>k</sub><br>V<sub>R</sub> is the additional volume required to achieve u<sub>rise</sub> > u<sub>N</sub>, V<sub>k</sub> the equivalent<br>volume to the kinetic energy supplied by the incoming stream, and V<sub>eq</sub> is the<br>Harkins drop volume.</td><td>(15)</td></tr></tbody></table>
$$V_{\text{det}} + V_{\text{det}}^{\frac{2}{3}} \frac{0.403 \rho_d u_N^2}{\Delta \rho g} = \frac{\pi \gamma d_N}{\Delta \rho g} \psi + 0.5236 \left( \frac{18 K u_d \mu_c}{\Delta \rho g} \right)^{\frac{3}{2}} \quad (16)$$

Where $K = \frac{4.05 d_N^{0.747}}{u^{0.635} \mu_c^{0.814}}$ and $\psi = 0.655$ based on Harkins and Brown (1919).

$$V_{\text{det}} = V_{\text{eq}} + V_n \quad (17)$$

$V_{\text{eq}}$ is the Harkins volume, and $V_n$ accounts for the volume added during the necking stage:

$$V_n = Q t_n \quad (18)$$

A momentum balance is applied at the detachment time to determine $t_n$:

$$\frac{d m_d u_d}{d t} = F_b + F_k - F_d \quad (19)$$

$$u_d = \frac{dx}{dt} = \frac{c}{A} + B \left( \frac{t}{A} - \frac{1}{A^2} \right) + \left( \frac{B}{A^2} - \frac{c}{A} \right) C \exp(-At) \quad (20)$$

Rao and coworkers⁴

$$\text{With } A = \frac{12\pi d_d \mu_c}{m_d}, \quad B = \frac{\Delta \rho g Q}{m_d}, \quad C = \frac{\rho_d u_N Q}{m_d} \quad (21)$$

$$\text{and } m_d = V_{\text{eq}} \rho_d + \frac{11}{16} V_{\text{eq}} \rho_c \quad (22)$$

**Case 1:** For low $\mu_c$ or when the distance separating the residual and detached drop is not large: $t_n$ is determined by imposing $u_d = \frac{1}{2} u_N$ at the detachment moment in Eq. (20).

**Case 2:** For high $\mu_c$ or when the distance between the residual and detached drop is large:

$$t_n = \left( \frac{2Ad_{\text{eq}}}{B} \right)^{\frac{1}{2}} \text{ as Eq. (20) reduces to } u_d = \frac{B}{A} t_n \text{ (the other terms being negligible)} \quad (23)$$

$$V_{\text{det}} = \psi(V_{\text{eq}} + V_n) \quad (24)$$

$$V_{\text{eq}} = \frac{F_{\gamma} + F_{d} - F_{k}}{\Delta \rho g} \quad (25)$$

Scheele and Meister¹⁶

$V_n$ is determined using the same approach as Rao et al. (1966) but with a different drag expression:

$$V_n = k \left( \frac{Q^2 d_N^2 \rho_d \gamma}{(\Delta \rho g)^2} \right)^{\frac{1}{3}}, \text{ with } k \text{ a constant.} \quad (26)$$

$$V_{\text{det}} = \psi \left( \frac{\pi \gamma d_N}{\Delta \rho g} + \frac{20 \mu_c d_N Q}{\left(\frac{c}{\pi} V_{\text{det}}\right)^{\frac{2}{3}} \Delta \rho g} - \frac{4 \rho_d u_N Q}{3 \Delta \rho g} + 4.5 \left( \frac{d_N^2 \rho_d \gamma Q^2}{(\Delta \rho g)^2} \right)^{\frac{1}{3}} \right) \quad (27)$$

## Main observations in single drop formation studies

Several conclusions can be drawn from single drop studies in non-confined media (see Table S1 in the Supporting Information). For instance, increasing the flow rate or velocity of the dispersed phase increases the drop volume over a specific range, stabilizes it, and then causes the volume to decrease. The equilibrium drop size (end of stage I) is found to increase when increasing the nozzle diameter, the interfacial tension, or the viscosity of the continuous phase, but to decrease when increasing the difference in density between the two phases. The viscosity of the dispersed phase is found not to affect the equilibrium drop volume.

After reaching equilibrium, the liquid continues to flow into the drop, consequently, the detaching forces become more significant, and a neck (i.e., thread or filament) starts to form. This corresponds to the beginning of stage II which ends once the drop is detached. When the neck begins to shrink, the local pressure in the thread becomes high, thereby accelerating the
velocity forward and backward, which leads to filament thinning and rupture. During this stage, the drop continues to grow, but at the moment of detachment, part of it rebounds toward the tube to form a new drop. The dynamics of the pinch-off moment (i.e. duration, thread diameter, and length) were found to be affected by the dispersed phase viscosity or the viscosity ratio of both phases<sup>23</sup>. Different regimes of drop pinch-off were observed depending on the viscosity ratio: inertial regime, inertial-to-viscous transition regime, or viscous regime<sup>24</sup>. For a pending drop in air, the length of the thread increases when increasing the liquid viscosity. This prevents early breakage and allows further growth. Also, increasing the viscosity was found to increase the formation time, decrease the elongation rate, and increase the probability of satellite drop formation<sup>25</sup>. For a pending drop in liquid, the viscosity ratio also affects the shape of the thread and satellite drops. Longer and thinner threads were observed in liquids compared to air<sup>26</sup>. The pinching time was longer for higher viscosity ratios during the dripping regime, while radial oscillation and a reduction of the thread radius were observed close to the end of this regime<sup>8,27</sup>. On the other hand, increasing the flow rate increases the axial momentum of the entering liquid, consequently resulting in further elongation<sup>5</sup>.

Interfacial tension influences both drop formation and detachment (see Table S2 in the Supporting Information). Adding surfactant reduces both the surface tension and capillary pressure leading to a faster attainment of the equilibrium surface tension. Additionally, higher concentrations of surfactant shorten the duration of stage I, resulting in a smaller equilibrium drop volume. At the same time, the duration of stage II increases due to a slower neck thinning rate, which extends the thread length, decreases its diameter and increases the size of satellite drops. During stage I of drop formation in a medium containing a surfactant, the surfactant molecules adsorb on its surface while it expands. The dynamics of surfactant adsorption depends on the diffusivity of the surfactant molecules which is in turn affected by the viscosity, size of the molecules, interactions, temperature, etc. A delay in the mass transfer of surfactant molecules to the interface relative to the drop growth rate would affect the surface tension. This is particularly notable during the necking stage, where the surface expands quickly, especially around the neck. In addition, Marangoni stresses might arise due to the non-uniformity of the surfactant distribution around the pinch-off point. This may enhance the formation of satellite drops for instance<sup>5</sup>. Similarly, the viscosity plays an important role in the diffusivity and dynamics of surfactant adsorption. In the case of an emulsion drop, the surfactant present in the drop may diffuse to the interface and its
diffusivity depends on the drop viscosity as well. It is therefore important to determine the rheology of the drop and predict the shear effects in the case of non-Newtonian drops. The curvature-induced pressure during the necking stage causes the shear to vary in time and space thereby inducing an evolution of the drop viscosity if it is non-Newtonian.

Since drop viscosity affects the dynamics of the second stage, variations in viscosity may alter the mechanisms of drop necking and detachment. As a result, several studies have focused on non-Newtonian drops<sup>23</sup>. Generally, polymer solutions are used to investigate non-Newtonian effects since adding a small amount of macromolecules to a solution may render it non-Newtonian (e.g. shear-thinning or elastic)<sup>28</sup>. Differences between Newtonian and non-Newtonian fluids are observed when the non-Newtonian effect is significant or at high viscosities, and mainly in stage II (cf. Table S3 in the Supporting Information). In shear-thinning polymer solutions, during the initial stage of drop growth in air, both Newtonian and non-Newtonian fluids were found to behave similarly. However, during the necking process, the drop length was shorter for non-Newtonian fluids, with more pronounced differences at higher viscosities<sup>25</sup>. The viscosity of shear-thinning drops decreases in the thread due to the increased shear which reduces normal viscous stresses. This, in turn, increases the pressure within the thread (to maintain the difference in the surface stresses) and enhances the drainage from the thread, compared to Newtonian drops with similar zero-shear viscosity<sup>29</sup>. As a consequence, the final neck length decreases, and the frequency of pinch-off increases. Moreover, increasing the degree of shear-thinning was found to increase the contact angle<sup>26</sup>.

Single drops composed of a suspension (i.e. dispersion of particles in a liquid) may also exhibit a non-Newtonian rheology<sup>30</sup>. Besides causing an increase in the apparent drop viscosity, and probably non-Newtonian effects, the particles may accumulate in the thread and affect the dynamics and geometry of the drop and the thread<sup>31</sup>. Given that the particles in suspensions are non-deformable unlike the inner droplets in the emulsion drops considered in the present work, which may lead to different behaviors (e.g., bulging the thread), we will not discuss suspension drop studies in detail.

Emulsion drop formation has primarily been studied in confined systems. In an example of emulsion drop formation in a non-confined system, the formation of yield stress water-in-oil (w/o) emulsion drops in air resembled that of Newtonian fluids, albeit with a different geometry<sup>32</sup>. Yet, single drop formation of a shear-thinning o/w emulsion in non-confined systems has not been addressed in the literature and represents therefore the focus of this work.
## Materials and experimental methods

Water-in-oil (w/o) emulsions are injected through a capillary tube immersed in a stationary aqueous phase (water, with or without glycerol). The formation and detachment of the w/o drops are monitored using a digital camera and the effect of various parameters on these stages is investigated.

### Materials

Silicone oils (SO) of different viscosities (cf. Table 3) were purchased from Sigma-Aldrich (Germany). Abil EM 97s from Evonik and Tween® 20 from Sigma-Aldrich were used as surfactants in the organic and external stationary aqueous phases, respectively. Glycerol (Fisher Scientific) was used to enhance the viscosity of the external aqueous phase without affecting its rheology (the external phase remains Newtonian).

<table><caption>Table 3: Physical properties of the silicone oils (SO) at 25 °C.</caption><thead><tr><td>Material</td><td>Density (kg m<sup>-3</sup>)</td><td>Viscosity (mPa·s)</td></tr></thead><tbody><tr><td>SO v5</td><td>913</td><td>5</td></tr><tr><td>SO v10</td><td>930</td><td>10</td></tr><tr><td>SO v20</td><td>950</td><td>20</td></tr></tbody></table>

### Preparation of the inner emulsion and external aqueous phase

The w/o emulsions consisted of a 0.1M NaCl aqueous solution dispersed in the organic phase (silicone oil containing 2 wt. % of Abil), with the aqueous phase accounting for 40 wt.% of the emulsion ($\phi_w$ = 40 wt.%). This was done using an IKA T 18 digital ULTRA-TURRAX® at 8,000 rpm (unless otherwise indicated) for 4 min (cf. Figure 1a). The emulsions were generated at 25±2 °C. The viscosities of the oil ($\mu_o$) and the external phase ($\mu_w^{ex}$, or $\mu_c$) were varied. The water content was varied in the produced emulsions by diluting the prepared emulsion ($\phi_w$ = 40 wt.%) in the same silicone oil-Abil mix to generate emulsions with $\phi_w$ = 30, 20, 10, and 5 wt. %, so that the produced emulsions conserve the same inner droplet size distribution.

Salt is typically added during the formulation of double emulsions to increase the osmotic pressure within the inner drops, hence counterbalancing the Laplace pressure that promotes the release of inner drops, thereby ensuring the stability of the double emulsion³³. In a previous work, the optimal NaCl concentration SO-based w/o/w double emulsions was found to be 0.1 M ¹⁰. Higher concentrations caused swelling of the double emulsion due to the diffusion of the external water whereas lower concentrations led to a significant release of inner drops. In the present work, we ensure that the change of drop volume during its
formation on the capillary occurs only due to growth with no swelling or release taking place. Although the type of salt and its concentration may slightly influence the surfactant spreading and interfacial tension, in this range they are not assumed to affect the drop formation stages.

The external stationary phase consists of a mixture of ultrapure water and Tween 20, with a concentration varying between 0 and 30 times the critical micellar concentration (CMC = 0.074 g L⁻¹). To vary the viscosity of this solution, various amounts of glycerol were added.

The undertaken single drop experiments are listed in Table 4 where the emulsions are denoted as W (wt. % water content)-v(oil viscosity (mPa s)). The density difference between the dispersed and the stagnant phase (Δρ = ρ<sub>w</sub><sup>ex</sup> - ρ<sub>d</sub>) is indicated in the same table. It is worth mentioning that the addition of surfactant was found not to greatly affect the density or viscosity of the emulsions. The density of the dispersed phase was calculated using the Miller and Mann model<sup>34</sup> which assumes the emulsion to be quasi-uniform:

$$ \rho_d = \phi_w \rho_w + (1 - \phi_w) \rho_o \quad (28) $$

<table><caption>Table 4: Emulsion-based single drop experiments</caption><thead><tr><th>Exp #</th><th>Dispersed phase Wφ<sub>w</sub>(%)-vμ<sub>o</sub>(mPa·s)</th><th>Stagnant phase</th><th>μ<sub>w</sub><sup>ex</sup> (mPa·s)</th><th>[Tween 20]</th><th>Δρ (kg/m<sup>3</sup>)</th></tr></thead><tbody><tr><td>1</td><td>W5-v10</td><td>water</td><td>1.00</td><td>1 CMC</td><td>63.60</td></tr><tr><td>2</td><td>W10-v10</td><td>water</td><td>1.00</td><td>1 CMC</td><td>60.30</td></tr><tr><td>3</td><td>W20-v10</td><td>water</td><td>1.00</td><td>1 CMC</td><td>53.60</td></tr><tr><td>4</td><td>W30-v10</td><td>water</td><td>1.00</td><td>1 CMC</td><td>46.90</td></tr><tr><td>5</td><td>W40-v10</td><td>water</td><td>1.00</td><td>1 CMC</td><td>40.20</td></tr><tr><td>6</td><td>W10-v5</td><td>water</td><td>1.00</td><td>1 CMC</td><td>75.60</td></tr><tr><td>7</td><td>W10-v20</td><td>water</td><td>1.00</td><td>1 CMC</td><td>44.60</td></tr><tr><td>8</td><td>W10-v10<sup>†</sup></td><td>water</td><td>1.00</td><td>1 CMC</td><td>60.30</td></tr><tr><td>9</td><td>W10-v10<sup>††</sup></td><td>water</td><td>1.00</td><td>1 CMC</td><td>60.30</td></tr><tr><td>10</td><td>W10-v20</td><td>water</td><td>1.00</td><td>30 CMC</td><td>44.60</td></tr><tr><td>11</td><td>W10-v20</td><td>water</td><td>1.00</td><td>0</td><td>44.60</td></tr><tr><td>12</td><td>W10-v20</td><td>water +30% glycerol</td><td>2.50</td><td>1 CMC</td><td>123.5</td></tr><tr><td>13</td><td>W10-v20</td><td>water +45.7% glycerol</td><td>5.00</td><td>1 CMC</td><td>164.8</td></tr><tr><td>14</td><td>W10-v20</td><td>water +57.81% glycerol</td><td>10.0</td><td>1 CMC</td><td>196.7</td></tr><tr><td>15</td><td>W0-v10 (without Abil)<sup>†††</sup></td><td>water</td><td>1.00</td><td>0</td><td>68.10</td></tr><tr><td>16</td><td>W0-v10<sup>†††</sup></td><td>water</td><td>1.00</td><td>0</td><td>68.10</td></tr><tr><td>17</td><td>W0-v10 (without Abil)<sup>†††</sup></td><td>water</td><td>1.00</td><td>1 CMC</td><td>68.10</td></tr><tr><td>18</td><td>W0-v10<sup>†††</sup></td><td>water</td><td>1.00</td><td>1 CMC</td><td>68.10</td></tr><tr><td>19</td><td>W0-v10<sup>†††</sup></td><td>water</td><td>1.00</td><td>30 CMC</td><td>68.10</td></tr></tbody></table>

† Emulsion generated at 4,000 rpm

†† Emulsion generated at 16,000 rpm

††† Pure SO v10 with or without Abil

Experimental setup

The experiments were carried out in a 110 mm×47 mm×157 mm rectangular chamber made up of transparent quartz plates (cf. Figure 1 b-c). A stainless-steel needle (of inner diameter d<sub>N</sub><sup>i</sup>=210 µm and outer diameter d<sub>N</sub><sup>o</sup>= 413 µm) was inserted through a rubber cap tightly fixed
at the bottom of the chamber. The diameter of the needle was selected to be large enough so that the passage of the inner water droplets (with a maximum diameter of 3 µm) is allowed without deformation. The wall effect of the chamber can be neglected as the generated drops have diameters of only a few millimeters, which are negligible compared to the width of the chamber.

In each experiment, the corresponding stationary phase was allowed to settle in the chamber to eliminate any entrained air bubbles. Consequently, the previously prepared w/o emulsions were injected through the needle to generate single emulsion drops into the continuous phase. A PSD/4 syringe pump purchased from Hamilton and equipped with a syringe of capacity 500 µL was used and the flow rate was varied automatically between 0.023 and 0.6 mL min⁻¹.

A fast-digital camera (Mini UX, Fastcam Mini) was placed facing the front quartz plate for image acquisition and backlighting was provided by a LED panel laid behind the chamber. The camera resolution was set to 1,280×1,024 pixels. The sequence of drop formation, growth, and detachment was recorded using an image acquisition software (Photron) at a rate of fps = 1,000 for 2 s. Prior to recording, several drops were allowed to form to ensure the establishment of a steady flow. The images were post-processed in FIJI software, through which the dimensions and velocity of the drops were determined. The experimental volume of the drop was determined by assuming that it exhibits an axial symmetry around the axis aligned with the capillary tube ¹,²⁵. Hence, the total drop volume can be obtained as the sum of the volumes of infinitesimal cylinders of height 1 pixel each. This was realized in FIJI via a macro. Figure 1d depicts a representation of a segmented drop where the width and length of each rectangle represent the height and diameter of the cylinder, respectively. The speed of the drop $u_d(t_{i+1}) = \frac{y_c(t_{i+1})-y_c(t_i)}{\Delta t}$ was determined from the images by tracking the position of the drop centroid ($y_c$), at each time step where $\Delta t = \frac{1}{fps}$
<figure><img src="image_2.png" /><figcaption>(b)</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>(c)</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>(d)</figcaption></figure>

<figure><img src="image_5.png" /></figure>
Figure 1: (a) Schematic of the preparation setup of the primary w/o emulsion, (b) schematic of the experimental setup, (c) photos of the experimental setup, and (d) an example of a segmented drop.

## Characterization of the w/o emulsions

The interfacial tension of the w/o emulsions at equilibrium was measured with KRÜSS DSA 10 MK2 pendant drop tensiometer. Their viscosity was measured at 25 °C with an Anton Paar MCR 302 rheometer equipped with a cone-plate configuration of diameter 49.977 mm, angle 0.994°, and truncation of 102 µm. This truncation was chosen so that the internal drops of water were neither squeezed nor deformed. The measurements were done over shear rates ranging between 1 and 1,000 s⁻¹. The size of the inner droplets was measured using Dynamic Light Scattering (NanoZS Malvern) after importantly diluting the w/o emulsions in the corresponding silicone oil.

## Results and discussion

### Characterization results of the w/o emulsions

#### Inner droplet size distribution (DSD)

The influence of the oil viscosity and mixing intensity on the size distribution of the inner droplets of the prepared emulsions is depicted in Figure 2. An increase in the oil continuous phase viscosity gives rise to narrower DSDs and smaller water droplets (cf. Figure 2a). This observation is a result of the increased shear rate induced by the more viscous continuous phase³⁵. In reality, the viscosity of the continuous phase affects the drop size in the viscous sub-regime, while it has no effect in the inertial sub-regime³⁶. In the current experiments, the droplet breakage during the formation of the w/o emulsion is taking place in the viscous sub-regime³⁷ since the largest drop diameter was found to be below the Kolmogorov length scale (η):

$$ \eta = \left( \frac{\nu^3}{\varepsilon_{\text{eff}}} \right)^{\frac{1}{4}} \qquad (29) $$

Where $\nu$ is the kinematic viscosity and the mean effective energy dissipation ($\varepsilon_{\text{eff}}$) is given by ³⁸:

$$ \varepsilon_{\text{eff}} = \frac{\varepsilon_c}{(1 + \phi_w)^3} \qquad (30) $$

where $\phi_w$ is the dispersed phase fraction (here water) and $\varepsilon_c$ is the energy dissipation in the continuous phase obtained from the power equation:
$$ \varepsilon_c = \frac{N_p \omega_R^3 d_I^5}{V} \qquad (31) $$

where $\omega_R$ is the rotation speed, $d_I$ the diameter of the impeller, $V$ the volume of the medium and $N_p$ is the power number which varies between 1.7-3 for similar geometries$^{39}$.

Figure 2b shows that increasing the homogenization intensity leads to the formation of smaller droplets, with a narrower DSD. The obtained average droplet sizes (with $\mu_o=10$ mPa.s, $\phi_w=10\%$) are of $d_{32}^w=0.92, 0.68$, and $0.45$ µm for mixing intensities of 4,000, 8,000, and 16,000 rpm, respectively. In fact, increasing the intensity of mixing increases the energy dissipation and the shear rate, and therefore droplet breakage. Regarding the effect of $\phi_w$, as the emulsions of various water content were obtained by diluting the same mother emulsion of 40 wt.% water, similar DSDs are obtained with different $\phi_w$.

<figure><img src="image_6.png" /><figcaption>Figure 2: DSD of the w/o emulsions: a) impact of the oil viscosity (continuous phase) (Exp. 6, 2, 7), and b) impact of the mixing intensity (Exp. 9, 2, 8).</figcaption></figure>

## Viscosity of the emulsions

Figure 3 shows the variation of the apparent viscosity of the prepared emulsions as a function of the shear rate ($\dot{\gamma}$) for different water fractions ($\phi_w$), oil viscosities ($\mu_o$), and mixing intensities (i.e., droplet size ($d_{32}^w$)). The figure shows that the viscosity of the emulsions is high at low shear rates, while it decreases and plateaus as the shear rates increase, revealing a shear thinning behavior$^9$. The viscosity curves were fitted to the Oswald-de Waele power law (Eq. (32)), where $K$ stands for the flow consistency index and $n$ is the flow behavior index. The corresponding fitting parameters are listed in Table 5. The identified values of the flow behavior indices $n$ are smaller than one, with lower $n$ corresponding to higher degrees of shear thinning.
$$ \mu_{\text{eff}} = K \dot{\gamma}^{n-1} \qquad (32) $$

<figure><img src="image_7.png" /><figcaption>Figure 3: Viscosity of the emulsions as a function of the shear rate: impact of a) the internal water fraction (Exp. 1-5), b) the oil viscosity (Exp. 6, 2, 7), and c) the inner droplet size (Exp. 9, 2, 8). The red curves stand for the power law fitting curves.</figcaption></figure>

As expected, increasing the water fraction results in emulsions with higher viscosities and higher degrees of shear-thinning, whereas highly diluted emulsions (i.e., $\phi_w$=5-10 wt.%) manifest almost a Newtonian behavior with $n$ approaching 1 (cf. Figure 3a and Table 5). This can be explained by the fact that the water droplets form aggregates that immobilize the oily continuous phase in between, thus increasing the effective viscosity at low shear rates $^{40}$. This is also due to the increased hydrodynamic interaction between droplets at high $\phi_w$ $^{41}$. Furthermore, the attraction between the surfactant molecules on the droplet surface and the oil phase may cause a hydration effect where the oil phase molecules adhere to the droplet surface. This, therefore, reduces the distance between the droplets and consequently increases the van der Waals forces which prompts their aggregation $^{42}$. However, at higher shear rates the aggregates break down and the viscosity decreases towards the Newtonian level.

The trend in Figure 3b-c, highlighting the influence of the oil viscosity, is correlated with the droplet size (observed in Figures 2a-b). We can observe an increase in the emulsion viscosity with the increase of the oil viscosity since the droplet size decreases. However, varying the inner drop size from 0.45-0.92 µm (using agitation speeds between 4,000-16,000 rpm, Figure 2b) has a negligible effect on the viscosity (cf. Figures 3c).

<table><caption>Table 5: Power law fitting parameters for the viscosity curves.</caption><thead><tr><td>Exp. name</td><td>K</td><td>n</td></tr></thead><tbody><tr><td>W5-v10</td><td>14.70</td><td>0.96</td></tr><tr><td>W10-v10</td><td>20.11</td><td>0.92</td></tr></tbody></table>
<table><tr><td>W20-v10</td><td>63.50</td><td>0.82</td></tr><tr><td>W30-v10</td><td>128.72</td><td>0.75</td></tr><tr><td>W40-v10</td><td>189.51</td><td>0.75</td></tr><tr><td>W10-v5</td><td>13.54</td><td>0.92</td></tr><tr><td>W10-v20</td><td>43.22</td><td>0.91</td></tr><tr><td>W10-v10<sup>†</sup></td><td>23.20</td><td>0.92</td></tr><tr><td>W10-v10<sup>††</sup></td><td>25.63</td><td>0.88</td></tr></table>

## Interfacial tension

Figure 4 shows the effect of various parameters on the time evolution of the interfacial tension (γ) of the w/o emulsions. Preliminary tests with pure silicone oil with 10 mPa.s were undertaken to evaluate the effects of the hydrophilic and hydrophobic surfactants (cf. Figure 4a). For purely oil droplets, adding the hydrophobic surfactant (Abil, at 2 %) to the oil leads to an important decrease in γ, while adding the hydrophilic surfactant (Tween 20, at 1 CMC) in the continuous phase has a weaker effect since adding both surfactants render comparable results as Abil alone. However, when the concentration of Tween 20 is increased to 30 CMC, its effect becomes more significant which indicates that 1 CMC does not fully saturate the medium and the drop surface. This effect is also seen when drops of an emulsion (i.e., W10-v20) are considered (cf. Figure 4b).

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>
<figure><img src="image_10.png" /><figcaption>Figure 4: Time evolution of the interfacial tension: effect of the a) type of the surfactant in pure oil drops (Exp. 16-19), b) concentration of Tween (Exp. 11, 2, 10), c) internal water (Exp. 18, 1, 2, 3), d) oil viscosity (Exp. 6, 2, 7).</figcaption></figure>

The effect of the internal water fraction ($\phi_w$) on the interfacial tension of an emulsion is presented in Figure 4c. A slight increase in $\gamma$ is observed when increasing $\phi_w$ from 0 to 10%, but afterward, no effect is observed between $\phi_w$ = 10% and 20%. It is to be emphasized that this is an indirect effect of $\phi_w$ due to a simultaneous change in the surfactant concentration and number of inner droplets when changing $\phi_w$. In fact, the amount of hydrophobic surfactant is calculated in the used formulation with respect to the oil phase. Hence, when increasing $\phi_w$, a smaller amount of Abil is used while more inner droplets are present, which results in a higher surface tension. Also, the dynamics of surfactant adsorption might depend on the apparent viscosity of the emulsion drop, which increases when increasing $\phi_w$, yet this does not influence the equilibrium value$^{43}$. Regarding the effect of the oil viscosity of the emulsion drop (cf. Figure 4d), no difference was observed between 10 and 20 mPa.s, but a slightly lower surface tension was measured when $\mu_0$= 5 mPa.s. This is due to the faster diffusion of surfactant (Abil) in a less viscous medium ($\mu_0$= 5 mPa.s), thereby leading to lower interfacial tension values$^{44}$.

The different figures highlight a variation of the interfacial tension with time, which emphasizes the fact that the equilibrium surface tension ($\gamma_\infty$) cannot be used unless the time of drop formation exceeds the surfactant adsorption time. The time for the interfacial tension to plateau depends heavily on the tested parameter and can exceed 30 s. However, this device allows the generation of a static drop, and the time evolution depicted in these figures represents the duration required to reach the equilibrium values when measured using the tensiometer. So, it does not consider the flow rate effect on both the volume and viscosity of the dispersed phase given the shear-thinning character of the drops. It is important to note that it was not possible with the available tensiometer device to measure the time evolution of the
interfacial tension of emulsions in water containing glycerol, because the drops were not able to hold over the tip of the capillary due to the important reduction in $\gamma$. Similarly, the interfacial tension could not be measured for emulsions with $\phi_w$ = 30 and 40 wt. %, as the formed drops had elongated shape and their contour could not be captured by the tensiometer software.

Qualitative observations of the forming drops and satellite drops

Examples of drops forming at the tip of the capillary tube and subsequently detaching are illustrated in Figure 5. The drops are not represented at the same scale, as the objective is to provide a “qualitative” representation of the effect of various parameters on the drop formation. A quantitative analysis of those results is presented in the subsequent sections. Figure 5 A-D represent drops of pure silicone oil with or without surfactants, while the remaining figures depict drops of emulsions generated at a flow rate of $Q = 0.5$ mL/min. The incorporation of water droplets inside the oil drop alters the optical properties, as reflected by the emulsion drops (cf. Figure 5 E-J) which appear opaque in contrast to the oil drops (cf. Figure 5 A-D), which remain clear.
<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /></figure>
<figure><img src="image_13.png" /><figcaption>Figure 5: Example of drop sequences forming at the tip of the nozzle at $Q=0.5$ mL min$^{-1}$.</figcaption></figure>

All drops exhibit neck formation prior to their detachment, except for those made of pure SO v10 in pure water without surfactant (cf. Figure 5A). In this case, the drop barely extends, revealing therefore only a short thread before detachment. However, the introduction of a surfactant such as Abil (cf. Figure 5B), Tween 20 (cf. Figure 5C), or both (cf. Figure 5D) leads to the formation of a neck. This also results in a drop of a more pear-like shape in stage I
(first column of each figure) as opposed to the spherical shape observed in the case without surfactants. These observations highlight the influence of surfactants on the neck dynamic. Surfactants also affect the volume of the residual drop left attached to the nozzle after detachment, which is negligible in the absence of surfactants. In fact, Surfactants reduce the Laplace pressure, hence slowing down the neck thinning process<sup>44</sup>. This, consequently hinders the detachment and promotes the neck extension further. Notably, Abil (cf. Figure 5B) or a combination of both Abil and Tween 20 (cf. Figure 5D) results in longer necks compared to the use of Tween solely (Figure 5C). These findings are consistent with the observations in Figure 4a, further emphasizing the role of surfactants in modulating neck formation and detachment dynamics.

Similarly, emulsion drops also exhibit neck formation. Altering the dispersed phase viscosity by varying either the water content (cf. Figure 5 E-F) or the oil viscosity (cf. Figure 5 E, H), as well as altering the external water properties by adjusting either the concentration of Tween 20 (cf. Figure 5 G-I) or the external water viscosity (cf. Figure 5 H, J) do not qualitatively affect the drop formation stages. A comparison of drops of emulsion (cf. Figure 5 E) and drops of pure oil in the presence of both surfactants (cf. Figure 5 D) reveals that the shear-thinning behavior of the emulsion has a minimal influence on the different stages of drop formation, as the same behavior is observed in either case.

The following presents a qualitative description of the formation of satellite drops. Overall, the size of these satellite drops is small and not accurate to measure with the available setup. Satellite drops are not observed when the drops are generated in pure water, where only short threads are obtained (cf. Figure 5A). This absence is expected since satellite drop formation typically depends on the length of the neck<sup>45</sup>. The same behavior is observed in the case where the drops of emulsions are generated in pure water (cf. Figure 5J). However, the introduction of surfactants in the case of oil drops (cf. Figure 5 B-D) results in the formation of small satellite drops at flow rates Q ≥ 0.2 mL/min. These small drops remain attached to the detached drops (see for instance Figure 5D). During the necking process, the neck thins, leading to pinch-off. Afterward, the part of the thread attached to the drop undergoes a contraction just below the detached drop, hence resulting in a small drop that remains connected to the larger drop.

For emulsion drops where surfactant is present in both the dispersed and continuous phases, the thread often undergoes two successive pinch-offs at the end of stage II (i.e., at $t = t_f$): The drop first breaks near the lower part of the thread then closer to the drop, leading to the
formation of satellite drops at flow rates $Q \ge 300$ mL/min. Small satellite drops are observed for drops with low viscosities. However, increasing the drop viscosity, such as in drops with $\phi_w = 40$ wt. %, suppresses the formation of satellite drops, where after the detachment, the thread rebounds and merges with the detached drop. This behavior is also expected since higher dispersed phase viscosity dampens the thread surface instabilities.

The size of satellite drops increases with the external water phase viscosity. In this case, those are observed at flow rates as low as 0.023 mL/min. At higher surfactant concentrations (i.e., Tween 20 = 30 CMC), satellites are only observed at high flow rates ($Q = 600$ mL/min) (see Figure 5I and Figure S1 in the Supporting Information). This observation can be attributed to the surfactant dynamics. At high concentrations, the thread thinning slows down significantly due to the abundant presence of surfactant molecules, which cover the elongated region and prevent thinning in different regions. At low flow rates, the surfactant molecules have sufficient time to distribute uniformly, hence stabilizing the thread and preventing the formation of satellite drops. However, at high flow rates, the short formation time causes inhomogeneous surfactant distribution leading to thinning at different regions, and hence the formation of satellite drops⁴⁶.

## Time of drop formation on the capillary

The total drop formation time ($t_f$) is calculated as the period elapsed between the detachment of two consecutive drops. Figure 6 shows the impact of the various tested parameters on the drop formation time. Because of the low error values, the error bars can be barely seen on those graphs. As can be seen, $t_f$ decreases exponentially when increasing the flow rate. This is due to the enhanced inertial force that promotes drop growth and causes the force equilibrium to be reached earlier, causing therefore a faster drop detachment. Figures 6a-c show that $t_f$ increases with the viscosity of the dispersed phase (either when increasing $\phi_w$ or $\mu_o$, or to a lower extent when decreasing the inner droplet diameter). In fact, $\mu_d$ helps maintain the wholeness of the drop, and is known to mainly affect drop formation during stage II. It also enhances the stability of the drop by damping the surface oscillations⁵,⁴⁵. However, the effect of the viscosity becomes negligible as the flow rate increases, where the inertial force becomes predominant. Also, as we deal with a shear thinning emulsion, increasing the flow rate, and accordingly the shear rate to which the drop is subjected, causes a decrease in the viscosity followed by a decrease in the detachment time.
Regarding the viscosity of the external stationary phase, Figure 6d shows that increasing $\mu_w^{ex}$ between 1 and 5 mPa.s leads to a decrease in $t_f$, then no effect is observed at a higher $\mu_w^{ex}$. In fact, increasing the outer phase viscosity results in an increase in the drag force that helps keep the drop attached ⁴. However, the buoyancy force also increases when glycerol is added to the external phase, which promotes drop detachment (see $\Delta\rho$ in Table 4). In addition, the glycerol may lead to a decrease in the interfacial tension ⁴⁷⁻⁴⁹. The last two factors cause a decrease in $t_f$. Therefore, these effects cancel out each other in some cases.

Figure 6e shows that the addition of the hydrophilic surfactant to the external phase decreases $t_f$. This is attributed to the reduction in $\gamma$ (cf. Figure 4b), which consequently favors the drop detachment ⁴³. However, quite similar formation times were observed when the concentration of Tween was increased from 1 to 30 CMC.

<figure><img src="image_14.png" /><figcaption>Figure 6: Variation of the drop detachment time with the flow rate: Effect of (a) the internal water fraction (Exp. 18, 1-5), (b) the oil viscosity (Exp. 6, 2, 7), (c) the size of inner droplets (Exp. 8, 2, 9), (d) the external water viscosity (Exp. 7, 12-14) and (e) the concentration of the hydrophilic surfactant in the external water phase (Exp. 11, 7, 10).</figcaption></figure>

## The drop-necking process

Figure 7 shows an example of a drop before and after detachment. The dimensions of the drop after detachment are represented by its width $w_d$, length $l_d$, and volume $V_{det}$. The volume of
the attached drop just before detachment is denoted $V_{at}$, and the part of the drop that remains attached to the nozzle after detachment is $V_r$. The length of the thread at the detachment moment ($L_{br}$) represents the distance between the capillary tip and the point where the detachment takes place. All volumes were calculated from the images using the segmented method described in section 0.

<figure><img src="image_15.png" /><figcaption>Figure 7: Images of a drop just before detachment and after detachment.</figcaption></figure>

Figure 8 shows that the thread length, $L_{br}$, increases when the flow rate increases. Indeed, when increasing $Q$, the inertial force increases compared to the capillary force due to the higher axial momentum and pressure. This results in more fluid building up and a longer thread<sup>22</sup>. Figures 8a-c show that $L_{br}$ increases as the viscosity of the dispersed phase increases, either because of increasing $\phi_w$ or $\mu_o$, and again to a lower extent when decreasing the inner droplet diameter. This can be due to the viscous force that retards the thread breakup by counterbalancing the capillary pressure and dampening the oscillations responsible for neck breakup<sup>46</sup>. Regarding the effect of the external phase viscosity, $L_{br}$ appears to decrease as $\mu_c$ increases (cf. Figure 8d). In fact, as the viscosity ratio decreases, $L_{br}$ decreases which is aligned with literature observations (Zhang and Stone 1997). Also, the non-Newtonian behavior of the drop might play a role where the increased shear stress imposed by the viscous outer phase may lower the drop apparent viscosity, and consequently accelerates its rupture. Indeed, the complex structure and rheology of the drop lead to a nonlinear behavior, where sometimes increasing a parameter (e.g., outer viscosity) does not necessarily lead to a
decrease in the length. Finally, Figure 8e shows the effect of the hydrophilic surfactant addition on $L_{br}$. Increasing the surfactant concentration in the stationary phase increases slightly $L_{br}$. Indeed, the addition of the surfactant reduces the surface tension and the capillary pressure which allows the thread extension. Yet, it can be noticed that at $Q = 0.6$ mL/min, the thread length decreases abruptly. This seems to be the onset of jetting as reflected by the cylindrical jet that starts to form (see Figure S1 at $t_f = 60$ ms in the Supporting Information). As reported by Utada et al.⁵⁰, jetting occurs at high $Ca$ of the external phase (Eq. (10)) and $We$ of the dispersed phase (Eq. (12)) which is the case at $Q = 0.6$ mL min⁻¹. In this case, the high inertial force favors the formation of a long thread, while the high amount of surfactant retards the neck thinning, thereby extending the thread. In addition, the shear-thinning behavior of the drops implies a reduction in the viscosity at high flow rates and hence favors the thinning. The newly created section of the thread is initially devoid of surfactant molecules, and the short formation time causes surfactant accumulation to become uneven. This inhomogeneity shifts the pinch-off location toward the regions with weaker surfactant coverage⁴⁶ which is reflected by the sudden decrease in the breakage length. Further investigations at higher flow rates are required to confirm the transition into the jetting regime and to better understand the dynamics involve.

<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /></figure>

<figure><img src="image_18.png" /></figure>

<figure><img src="image_19.png" /></figure>

<figure><img src="image_20.png" /></figure>
Figure 8: The length of the thread plus the new drop as a function of the flow rate: Effect of a) the internal water fraction (Exp. 18,1-5), b) the oil viscosity (Exp. 6,2,7), c) the mixing intensity (Exp. 8,2,9), d) the external water viscosity (Exp. 7,12-14), e) the surfactant concentration (Exp. 11,7,10).

## Dimensions of the detached drop

The variation of the aspect ratio of the detached drop ($AR = w_d/l_d$, Figure 7), representing the ratio of the drop width to its length just after detachment, is shown in Figure 9. Only the effect of the inner water fraction is depicted given that similar qualitative results were obtained when varying the other parameters. It can be seen that, under the tested conditions, the detached drops are not spherical but rather ellipsoidal with the major axis being the horizontal axis.

<figure><img src="image_21.png" /><figcaption>Figure 9: Effect of the internal water fraction on the variation of the aspect ratio of the detached drop with the flow rate (Exp. 18, 1-5).</figcaption></figure>

Figure 10 depicts only the effect of the various tested parameters on the width of the detached drop since similar trends were found for the length. Given that the aspect ratio is comparable for the different cases, an increase in the width implies an increase in the drop size (see also Figure S2 in the Supporting Information). It can be seen that the drop size increases with the flow rate for all cases until reaching a quasi-constant value at high flow rates. The viscosity of the dispersed phase is found to increase the detached drop size (cf. Figures 10a-c). Previous studies have shown that the first stage of drop formation is governed primarily by the buoyancy and interfacial forces and to a lower extent the drag force, whereas the second stage is affected by the drop viscosity $^{4,21,22}$. Indeed, a more viscous thread would remain attached for a longer period, thereby allowing further growth of the drop. Besides, increasing the oil
viscosity is correlated with an increase in the density followed by a decrease in $\Delta\rho$ (see Table 4) and consequently the buoyancy force. Similarly, increasing $\phi_w$ causes an increase in the apparent drop density and, hence, a decrease in the buoyancy force. This contributes to increasing the duration of stage I and the drop volume.

Figure 10d shows that increasing the viscosity of the external stationary phase decreases the drop size. In the current study, the viscosity of the external phase was varied by adding glycerol to the water, which leads to a decrease in the surface tension and an increase in $\Delta\rho$ (therefore increasing the buoyancy force). These factors foster the detachment of the drop which results in a decrease in the drop volume. On the other hand, the increase of $\mu_w^{ex}$ results in an increase in the drag force favoring the increase of the formation time. However, due to the influence of the other correlated parameters, the drop detached with a smaller volume when increasing $\mu_w^{ex}$.

The influence of the presence of the hydrophilic surfactant in the stationary phase and its concentration is represented in Figure 10e. Assuming that the addition of surfactant (at most 2% of the external phase) to the stationary phase does not greatly alter its viscosity and density, one can conclude that the interfacial force is the only affected force. It can be seen that the incorporation of the surfactant results in a decrease in the drop size, whereas increasing the concentration above 1 CMC does not lead to a noticeable variation despite the fact that the surface tension tends to decrease (cf. Figure 4b). It is expected that a decrease in the interfacial tension would promote the drop detachment, as revealed previously by the total formation time (cf. Figure 6e), up to a certain limit beyond which the equilibrium of interfacial tension should be attained <sup>51</sup>. Yet, the graph of the temporal evolution of the interfacial tension shows discrepancies between 1 and 30 CMC (cf. Figure 4b). This could be due to the equilibrium surface tension not being reached within 40 s of measurement using the pendant drop technique.

<figure><img src="image_22.png" /></figure>

<figure><img src="image_23.png" /></figure>

<figure><img src="image_24.png" /></figure>
<figure><img src="image_25.png" /><figcaption>Figure 10: The variation of the width of the detached drop with the flow rate: the effect of (a) the internal water fraction (Exp. 18, 1-5), (b) the oil viscosity (Exp. 6, 2, 7), (c) the mixing intensity (Exp. 8,2,9), (d) the external water viscosity (Exp. 7, 12-14) and (e) the hydrophilic surfactant concentration (Exp. 11, 7, 10).</figcaption></figure>

The experiments reveal that in most cases a residual portion of the drop ($V_r$) remains attached at the nozzle tip upon detachment (cf. Figure 7). This remaining volume was found to increase with the volume of the generated drop (cf. Figure S3 in the Supporting Information). The fraction of the remaining volume with respect to the total volume of the attached drop at the moment of detachment is given by $1 - \psi = V_r/V_{at}$ (cf. Figure 11). First, one can observe that this ratio does not exceed 6% in most cases except for those involving a highly viscous continuous phase (cf. Figure 11d), where it remarkably increases up to 11%. Second, the general trend emphasizes the insensitivity of the remaining drop volume fraction, $1 - \psi$, to the dispersed phase viscosity (cf. Figure 11 a-c). However, the non-Newtonian nature of the dispersed phase seems to affect the values of $1 - \psi$ as a larger fraction of the drops remained attached in the case of Newtonian fluids ($\phi_w = 0\%$) compared to non-Newtonian fluids ($\phi_w \neq 0\%$) (cf. Figure 11a). Small amounts of surfactant in the external phase appear to have a negligible effect on $1 - \psi$, but larger quantities exhibited a more evident influence leading to an increase in the remaining fraction (cf. Figure 11e). Finally, except for the cases with high $\mu_w^{ex}$ and [Tween 20], the remaining drop fraction increases with the volumetric flow rate until reaching a quasi-constant value.
<figure><img src="image_26.png" /><figcaption>Figure 11: Remaining drop fraction ($1 - \psi = V_r/V_{at}$): the effect of (a) the internal water fraction (Exp. 18, 1-5), (b) the oil viscosity (Exp. 6, 2, 7), (c) the mixing intensity (Exp. 8,2,9), (d) the external water viscosity (Exp. 7, 12-14) and (e) the hydrophilic surfactant concentration (Exp. 11, 7, 10).</figcaption></figure>

## Identification of the interfacial tension at equilibrium based on the drop volume

In this section, a method is proposed to determine the interfacial tension at the equilibrium stage. To accomplish this, the theoretical volume of the formed drop was computed by adaptation of the models of Rao et al.⁴ and Scheele and Meister¹⁶ (Table 2), and fitted to the experimental values.

The final volume of the attached drop before detachment ($V_{at}$) is made up of two volumes: the volume formed at the equilibrium stage ($V_{eq}$) and that added during the necking stage ($V_n$):

$$V_{at} = V_{eq} + V_n = V_{eq} + Qt_N \quad (33)$$
Since, upon detachment, a portion of the drop remains attached to the capillary tube, the volume of the detached drop ($V_{det}$) is given by:

$$V_{det} = \psi V_{at} \qquad (34)$$

Under the investigated conditions, the value of $1 - \psi$ was estimated from the images to be around 6% (see previous section). Its exact estimation is used in the model.

### Equilibrium drop volume ($V_{eq}$)

First, the equilibrium drop volume ($V_{eq}$) is calculated at the end of the static stage, where all forces acting upon the drop balance:

$$F_b + F_k - F_\gamma - F_d = 0 \qquad (35)$$

The expressions of these forces are given in Table 1. In Eq. (5), $d_N^w$ is equivalent to the outer nozzle diameter ($d_N^o$) and we consider the contact angle $\theta$ to be equal to 90°. In this model, $\gamma$ is not known in view of its dynamic character. Hence, the main objective of the model is to predict $\gamma$. Here, the drag force expression is selected in a way to fit our experimental observations. As high shear rates ranging between 422 and 1 100 s⁻¹ are exerted at the nozzle, the final viscosity measured at $\dot{\gamma} = 1 000$ s⁻¹ is considered given that beyond 400 s⁻¹ the viscosity becomes almost insensitive to the shear rate. At this shear rate, Figure 3 shows that the highest $\mu_d^\infty$ reached was of 50 mPa.s (for W40-v10) while the lowest $\mu_w^{ex}$ equals 1, so the viscosity ratio is always $\mu_d/\mu_w^{ex} < 50$. Also, the Reynolds number is very low at all the considered flow rates (Re<5). The behavior of the drop cannot therefore be assimilated to that of a rigid sphere (as in Rao et al. work⁴), and internal circulations are supposed to take place⁵². Accordingly, the expression of the drag force with internal circulation (Eq. (8)) is used¹⁹. In this equation, $u_r = u_d$ because the external phase is stationary. The velocity of the drop at the pinch-off moment is determined experimentally based on the taken image sequence.

### Added drop volume during the necking stage ($V_n$)

To determine the time of necking necessary to predict the additional volume of fluid being introduced into the drop during the necking stage ($V_N$), a momentum balance is established (Eq. (19)). It represents a first-order linear differential equation of momentum with time, and its solution is represented in Eqs. (20)-(21) (if we assume a constant mass of the drop during the necking stage and with the condition $u_d = 0$ at $t = 0^4$. In our case, the term $(\frac{B}{A^2} - \frac{C}{A})C$ was found to be negligible, hence, Eq. (20) reduces to:
$$ u_d = \frac{dx}{dt} = \frac{c}{A} + B \left( \frac{t}{A} - \frac{1}{A^2} \right) \qquad (36) $$

Which gives after another integration over time (with the condition $x = 0$ at $t = 0$):

$$ x = \left( \frac{c}{A} - \frac{B}{A^2} \right) t + \frac{B}{2A} t^2 \qquad (37) $$

By assuming $x = d_{eq}$ when $t = t_n^4$, since we are dealing with highly viscous systems (case 2), the following equation is obtained, and $t_n$ is identified as its positive root:

$$ \left(\frac{c}{A} - \frac{B}{A^2}\right) t_n + \frac{B}{2A} t_n^2 - d_{eq} = 0 \qquad (38) $$

The expressions for A, B, and C are depicted in Eq. (21) which highlights the dependency of these parameters on the mass of the attached drop. A general representation of the mass (Eq. (22)) is represented in Eq. (39) where the mass of the drop is assumed to be equal to the mass of the static drop in addition to the virtual mass or added mass represented in the second term on the right-hand side. It is necessary to add this term to account for the acceleration of the two phases with respect to each other, i.e. the inertia of the continuous phase⁵³.

$$ m_d = \rho_d V_{eq} + \kappa \rho_c V_{eq} \qquad (39) $$

$$ \kappa = \frac{\alpha_0}{2-\alpha_0} \qquad (40) $$

$$ \alpha_0 = abc \int_0^\infty \frac{d\lambda}{(c^2+\lambda)\sqrt{(a^2+\lambda)(b^2+\lambda)(c^2+\lambda)}} \qquad (41) $$

The coefficient $\kappa$ (Eqs. (39)-(40)) depends on the shape of the drop and the direction of its movement,⁵⁴. In Eq. (41), a, b, and c are the dimensions of the ellipsoid with c being aligned with the direction of the drop movement. In this study, it was assumed that the drop is symmetric around the z-axis which implies $a = b$. At the moment of detachment, it was found that $a = b < c$ (oblate spheroid). However, the ratio of $\frac{a}{c} = \frac{b}{c} \sim 0.9$, so for the sake of simplification, it is assumed that $a = b = c$, thus $\kappa = 0.5$.

## Model implementation and validation

The interfacial tension was obtained by minimizing the mean relative error between the theoretical and experimental $V_{det}$ using `fmincon` function in MATLAB®. Model validation is done by comparing the predicted formation time (i.e. $t_f = t_{eq} + t_N$) and the total experimental formation time (not used by the model). Figure 12 shows that an acceptable agreement is obtained between the model and the experimental results, with a maximum relative error below 15% in most cases.
<figure><img src="image_27.png" /></figure>

<figure><img src="image_28.png" /></figure>

<figure><img src="image_29.png" /></figure>

<figure><img src="image_30.png" /></figure>

<figure><img src="image_31.png" /><figcaption>Figure 12: Experimental versus theoretical total drop formation time. Symbols represent the experimental results while the dashed lines represent the model: the effect of (a) the internal water fraction (Exp. 18, 1-5), (b) the oil viscosity (Exp. 6, 2, 7), (c) the mixing intensity (Exp. 8,2,9), (d) the external water viscosity (Exp. 7, 12-14) and (e) the hydrophilic surfactant concentration (Exp. 11, 7, 10).</figcaption></figure>

The effect of the various tested parameters on the predicted $t_{\text{eq}}$ and $t_N$ is shown in Supporting Information (cf. Figures S4-S5). It can be seen that $t_{\text{eq}}$ undergoes a similar trend as $t_f$. This suggests that $t_{\text{eq}}$ and consequently $V_{\text{eq}}$ ($V_{\text{eq}} = Qt_{\text{eq}}$) are both sensitive to the viscosity of the dispersed phase. This differs from literature observations tackling low viscosity Newtonian Drops, where the viscosity was considered not to affect stage I<sup>21,22</sup>. This suggests that, for non-Newtonian drops (or here emulsion drops), the internal viscous forces have an influence on the drop formation not only during the necking stage but also during the static stage, particularly at low flow rate (cf. Figure S4 in the supporting Information).

Although both stages are affected by $\mu_d$, the sensitivity of $t_N$ (accordingly $V_n$) to $\mu_d$ is much higher than that of $t_{\text{eq}}$. The decay of $t_N$ with the flow rate is quasi-linear (except for the experiments conducted with glycerol) (cf. Figure S4 in the supporting Information), unlike $t_f$ and $t_{\text{eq}}$ which decay exponentially. In the presence of glycerol in the external water phase, the values of $t_N$ were found to decay exponentially and plateau around a constant value at high Q.
Increasing the amount of glycerol in the external phase, and consequently, $\mu_w^{ext}$ (between 1 and 5 mPa.s), ends up with a slight decrease in the necking time. This can be attributed to the decrease of the interfacial tension, and consequently a decrease in the capillary pressure, responsible for accelerating the drop detachment. Yet, increasing $\mu_w^{ext}$ beyond 5 mPa.s does not influence $t_N$ (similarly to the previous observations tackling the effect of $\mu_w^{ext}$ on the other studied characteristics).

An example highlighting the order of magnitude of the predicted values of $t_{eq}$, $t_N$ and $t_f$ (Exp. 5) is shown in Figure 13 (a similar qualitative behavior is observed in the other experiments). These results were also represented through the ratio $\chi_n = \frac{t_N}{t_f}$ (cf. Figure S6 in the Supporting Information). It can be noticed that, at low $Q$, the drop undergoes most of its growth during stage I ($t_{eq} \gg t_N$ and $\chi_n$ is low). However, as $Q$ increases, both stages contribute significantly to the final drop volume ($\chi_n \to 40-50\%$). For the systems with high $\mu_w^{ex}$, $\chi_n$ reaches a maximum value of only 20%, so the duration of the first stage remains predominant. This could be explained by the fact that drops growing in highly viscous media are exposed to high shear rates, resulting in a reduction in their apparent viscosity and thereby negating the non-Newtonian character of the drop.

<figure><img src="image_32.png" /><figcaption>Figure 13: Formation, equilibrium, and necking time (Exp. 5).</figcaption></figure>

### Predicting the interfacial tension

Figure 14 shows that the predicted values of the interfacial tension increase with the flow rate. This behavior can be attributed to the surfactant not having sufficient time to adsorb on the surface of the drop due to the lowered formation time$^{46}$. The same effects previously
manifested in Figure 4 are also observed in Figure 14. The model was able to capture the effect of the addition of glycerol on the interfacial tension. The interfacial tension seems to increase with the addition of glycerol but remains independent of the added amount (cf. Figure 14d). This can be explained by the fact that in the presence of glycerol, the formation time is short and hence the surfactant molecules may not have enough time to adsorb to the interface. Moreover, the diffusion of surfactant molecules is lowered in a higher viscosity medium, which may lower its adsorption. As mentioned earlier, one cannot rely on the equilibrium surface tension unless the time to attain equilibrium is shorter than that of drop formation and this is not the case for most experiments except for those conducted at low Q. In these cases, it can be seen that the predicted interfacial tension is close to the equilibrium interfacial tension measured by the tensiometer which validates our model. On the other hand, the maximum values predicted by the model are quite higher than the initial interfacial tension values measured using the pendant drop method. The observed discrepancies might arise due to the time lapse between the generation of the drop in the pendant drop apparatus and the start of recording during which the interfacial tension may potentially drop. In addition, this can be due to the effect of the flow rate which can affect the size of the generated drop and hence the surfactant coverage. In fact, the pendant drop method involves generating a drop and then tracking the evolution of the interfacial tension of the stagnant drop which, therefore, does not simulate a real case where the drop surface evolves and moves dynamically.

<figure><img src="image_33.png" /></figure>

<figure><img src="image_34.png" /></figure>

<figure><img src="image_35.png" /></figure>
<figure><img src="image_36.png" /><figcaption>Figure 14: Evolution of the interfacial tension with the flow rate: the effect of (a) the internal water fraction (Exp. 18, 1-5), (b) the oil viscosity (Exp. 6, 2, 7), (c) the mixing intensity (Exp. 8,2,9), (d) the external water viscosity (Exp. 7, 12-14) and (e) the hydrophilic surfactant concentration (Exp. 11, 7, 10).</figcaption></figure>

A representative example showing the magnitude of the forces acting upon the drop at the end of stage I (at equilibrium) and at the moment of detachment (end of stage II) is represented in Figure 15 (the other experiments showed qualitatively similar trends). The figure shows that the drag force can be neglected during the formation stage. Also, under the tested conditions (i.e., dripping regime), the inertial force is lower than the buoyancy and interfacial forces. In addition, once equilibrium is reached, the buoyancy force becomes the dominant force.

<figure><img src="image_37.png" /><figcaption>Figure 15: Example of the forces acting on the drop (a) at the equilibrium (end of stage I) and (b) just at the moment of detachment (end of stage II) (Exp 7).</figcaption></figure>

Conclusion

In this study, the formation and detachment of liquid drops rising in a stationary liquid from a capillary tube were investigated. The used dispersed phase consists of water-in-oil (w/o) emulsions (i.e., droplets of water dispersed in oil), while the external phase consists of water
(with or without glycerol as viscosity enhancer). To mimic the conditions prevailing during the formation of a w/o/w double emulsion (i.e., w/o emulsion dispersed in water), hydrophobic and hydrophilic surfactants were added to the dispersed and external phases, respectively. The dispersed phase was injected through a syringe pump at a flow rate ranging between 0.023-0.600 mL min⁻¹. The various stages of drop formation and detachment were captured using a fast digital camera and the obtained image sequences were used to gain insights into the dimensions of the formed drop and its velocity. The effect of the physical properties of the drop, i.e., the oil viscosity (5-20 mPa.s), the internal water content (0-40 wt.%), and the size of the internal water drops were assessed. The influence of the external phase properties (i.e., the viscosity and surfactant concentration) were also examined. Besides, drops of pure oil (with and without surfactant) were generated to highlight the effect of the surfactant on one hand, and the difference between the Newtonian and the shear-thinning behavior on the other hand.

All the experiments whether conducted with pure oil or with emulsions showed that the drop formation happens over 2 stages: stage I ends when the forces acting upon the drop balance (i.e., buoyancy, kinetic, interfacial tension, and drag) and a thread (neck) starts to form, while stage II extends from the instant when a neck appears up to the moment when a primary drop breaks and completely separates from the capillary. However, for the experiment conducted with pure silicone oil in pure water, the formed neck was less pronounced and the generated drop was nearly spherical as opposed to the emulsion drops or drops containing surfactant where the drop was found to be ellipsoidal. It was noticed that the duration of stage I governs the total time of formation at low flow rates, but both stages contribute significantly to the drop volume at higher flow rates.

The viscosity of the dispersed phase was found to affect both stages, with a greater influence on the second stage. This resulted in an increase of the formation time and consequently, the drop volume. However, increasing the viscosity of the external phase or introducing a surfactant into the external phase led to a decrease in both the time of formation and drop volume.

Furthermore, the drop volume was found to increase with the volumetric flow rate but to plateau beyond Q = 0.2 mL min⁻¹. After breakup and separation, a residual drop remained attached to the capillary and its volume was found to be negligible (less than 6 % of the total volume of the attached drop) except for the cases where the drops emerged in high-viscosity media (this volume reached 11 %). In addition, the fraction of the remaining drop was found
to be nearly insensitive to the viscosity of the dispersed phase but to the character of the drop since the Newtonian drop (oil + surfactant) was found to leave behind a larger residue compared to the drops of emulsions made up of the same oil.

The two-stage macroscopic force balance was used in this study to determine the interfacial tension by fitting the theoretical detached drop volume to the experimental values. The model was able to predict the increase of the interfacial tension with the increase of the flow rate that is due to the reduced surfactant adsorption. In addition, the model showed an increase in the surface tension with the increase of the drop viscosity which ensures a reduction in the diffusivity of surfactant molecules in highly viscous media.

The current study focused on assessing the effect of the different parameters on the formation and detachment of drops of emulsions (non-Newtonian drops) in Newtonian media under the dripping regime. Further investigations can be carried out at higher flow rates to better understand the transition from dripping to jetting regime. Moreover, the effect of the same parameters on the rising and terminal velocity of these drops can be investigated. In addition, the current study can be extended to monitor the formation of drops in non-Newtonian media since such situations can be encountered in biochemical and drug delivery applications.

## Acknowledgment

This work was funded by Agence Nationale de la Recherche under the grant agreement ANR-22-CE51-0006-01 (DEPMod Project).

## Supplementary material

Supporting Information contains a literature summary of the effects of the operating conditions on single drop formation and detachment, as well as some experimental observations that supports the conclusions of the paper.

## Nomenclature

<table><tr><td>A<sub>d</sub></td><td>Cross-sectional area of the drop (m<sup>2</sup>)</td></tr><tr><td>C<sub>d</sub></td><td>Drag coefficient (-)</td></tr><tr><td>d<sub>d</sub></td><td>Detached drop diameter (m)</td></tr><tr><td>d<sub>l</sub></td><td>diameter of the impeller (m)</td></tr><tr><td>d<sup>i</sup><sub>N</sub></td><td>Nozzle inner diameter (m)</td></tr><tr><td>d<sup>o</sup><sub>N</sub></td><td>Nozzle diameter (m)</td></tr><tr><td>d<sup>w</sup><sub>N</sub></td><td>Wetted nozzle diameter (m)</td></tr><tr><td>d<sub>nc</sub></td><td>Neck curvature diameter (m)</td></tr></table>
<table>
  <tr>
    <td>F<sub>b</sub></td>
    <td>Buoyancy force (N)</td>
  </tr>
  <tr>
    <td>F<sub>d</sub></td>
    <td>Drag force (N)</td>
  </tr>
  <tr>
    <td>F<sub>k</sub></td>
    <td>kinetic force (N)</td>
  </tr>
  <tr>
    <td>F<sub>σ</sub></td>
    <td>Interfacial tension force (N)</td>
  </tr>
  <tr>
    <td>g</td>
    <td>Gravity acceleration (m s<sup>-2</sup>)</td>
  </tr>
  <tr>
    <td>K</td>
    <td>Flow consistency index (-)</td>
  </tr>
  <tr>
    <td>L<sub>br</sub></td>
    <td>length of the thread at the detachment moment (distance between the capillary tip and the point where the detachment takes place)</td>
  </tr>
  <tr>
    <td>l<sub>d</sub></td>
    <td>Length of the detached drop (m)</td>
  </tr>
  <tr>
    <td>m<sub>d</sub></td>
    <td>Drop mass (kg)</td>
  </tr>
  <tr>
    <td>n</td>
    <td>Flow behavior index (-)</td>
  </tr>
  <tr>
    <td>N<sub>p</sub></td>
    <td>Power number (-)</td>
  </tr>
  <tr>
    <td>P<sub>c</sub></td>
    <td>Capillary pressure (N/m<sup>2</sup>)</td>
  </tr>
  <tr>
    <td>Q</td>
    <td>Dispersed phase flow rate (m<sup>3</sup> s<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>t<sub>eq</sub></td>
    <td>Equilibrium time-Duration of stage I (s)</td>
  </tr>
  <tr>
    <td>t<sub>f</sub></td>
    <td>Total time of formation (s)</td>
  </tr>
  <tr>
    <td>t<sub>N</sub></td>
    <td>necking time-Duration of stage II (s)</td>
  </tr>
  <tr>
    <td>u<sub>c</sub></td>
    <td>Continuous phase velocity (m s<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>u<sub>d</sub></td>
    <td>Dispersed phase velocity (m s<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>u<sub>N</sub></td>
    <td>Velocity inside the capillary (m s<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>u<sub>r</sub></td>
    <td>Relative velocity (m s<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>u<sub>w</sub></td>
    <td>Velocity at the capillary wall (m s<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>V</td>
    <td>Volume of the medium (m<sup>3</sup>)</td>
  </tr>
  <tr>
    <td>V<sub>at</sub></td>
    <td>Attached drop volume at the detachment moment (m<sup>3</sup>)</td>
  </tr>
  <tr>
    <td>V<sub>det</sub></td>
    <td>Detached drop volume (m<sup>3</sup>)</td>
  </tr>
  <tr>
    <td>V<sub>eq</sub></td>
    <td>Equilibrium drop volume (m<sup>3</sup>)</td>
  </tr>
  <tr>
    <td>V<sub>n</sub></td>
    <td>added drop volume during stage II (m<sup>3</sup>)</td>
  </tr>
  <tr>
    <td>V<sub>r</sub></td>
    <td>Remaining (residual) drop volume attached to the nozzle after detachment (m<sup>3</sup>)</td>
  </tr>
  <tr>
    <td>w<sub>d</sub></td>
    <td>Detached drop width (m)</td>
  </tr>
  <tr>
    <td>W</td>
    <td>Weight (N)</td>
  </tr>
</table>

Greek Letters

<table>
  <tr>
    <td>γ</td>
    <td>Interfacial tension (N m<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>γ̇</td>
    <td>Shear rate (s<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>ε<sub>eff</sub></td>
    <td>mean effective energy dissipation (W kg<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>ε<sub>c</sub></td>
    <td>energy dissipation in the continuous phase (W kg<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>θ</td>
    <td>Contact angle (°)</td>
  </tr>
  <tr>
    <td>λ</td>
    <td>Viscosity ratio (-)</td>
  </tr>
</table>
<table><tr><td>μ<sub>c</sub></td><td>Continuous phase viscosity (Pa·s)</td></tr><tr><td>μ<sub>d</sub></td><td>Dispersed phase viscosity (Pa·s)</td></tr><tr><td>μ<sup>ex</sup><sub>w</sub></td><td>Water continuous phase viscosity (Pa·s)</td></tr><tr><td>ρ<sub>c</sub></td><td>Continuous phase density (kg m<sup>-3</sup>)</td></tr><tr><td>ρ<sub>d</sub></td><td>Dispersed phase (i.e., emulsion drop) density (kg m<sup>-3</sup>)</td></tr><tr><td>ν</td><td>kinematic viscosity (m<sup>2</sup> s<sup>-1</sup>)</td></tr><tr><td>ω<sub>R</sub></td><td>rotation speed (tr s<sup>-1</sup>)</td></tr><tr><td>ψ</td><td>volume ratio correction factor (remaining drop fraction)</td></tr></table>

Dimensionless groups

<table><tr><td>Bo</td><td>Bond number (-)</td></tr><tr><td>Ca</td><td>Capillary number (-)</td></tr><tr><td>Re</td><td>Reynolds number (-)</td></tr><tr><td>We</td><td>Weber number (-)</td></tr></table>

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