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# Initial drop coalescence at a liquid-liquid interface

Yingjie Fei, Youguang Ma, Huai-Zhi Li

► To cite this version:

Yingjie Fei, Youguang Ma, Huai-Zhi Li. Initial drop coalescence at a liquid-liquid interface. AIChE Journal, In press, 10.1002/aic.18879. hal-05064574

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Submitted on 12 May 2025

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Open licence - etalab
# Initial drop coalescence at a liquid-liquid interface

Yingjie Fei¹,², Youguang Ma², Huai Z. Li¹,*

¹ University of Lorraine, CNRS, LRGP, F-54000 Nancy, France

² School of Chemical Engineering and Technology, Tianjin University, Tianjin 300350, China

* Corresponding author: huai-zhi.li@univ-lorraine.fr (Huai Z. Li)

## Abstract

The initial coalescence of drops with bulk liquid in various outer fluids was investigated, focusing on the effect of outer fluid's viscosity. The inner fluids included both Newtonian and non-Newtonian liquids. A micro-Particle Image Velocimetry (micro-PIV) system with 0.2 ms temporal and 5.2 µm spatial resolution quantified the flow field and mapped the viscosity distribution within the coalescing non-Newtonian drops. The temporary evolution of the liquid bridge width was tracked using both an electrical method with 0.8 µs resolution and an optical method with 10 µm resolution. In air, the rescaled bridge width $\Phi$ followed classical regimes with normalized time $\tau$, exhibiting a viscous regime ($\Phi \sim \tau$) followed by an inertial regime ($\Phi \sim \tau^{1/2}$). In viscous outer fluids, both two regimes are present, but a prolonged crossover regime was observed. An empirical fitting $\Phi \sim \tau \ln\tau$ is found to satisfactorily describe the whole coalescence.

**Topical Heading:** Transport Phenomena and Fluid Mechanics

**Key words:** Drop coalescence, Initial dynamics, Micro-PIV, Short spatio-temporal resolution, Scaling laws

## 1 Introduction

Drop coalescence is a common topological transition, widely observed in nature and industrial applications. In chemical and process engineering, drop coalescence plays a critical role by decreasing interfacial area and then mass and heat transfer processes, such as in
extraction & mixing efficiency, and multiphase reactors' performance.¹⁻³ In particular, the stability of an emulsion is mainly based on the suitable control of coalescence between droplets in a continuous outer fluid.⁴ On the other hand, promoting the coalescence of small oil drops into larger ones significantly accelerates oil-water separation efficiency for enhanced oil recovery (EOR) in petroleum industry.⁵ Therefore, the study of coalescence dynamics has been a hot research topic for a long time in multiple disciplines such as chemical engineering, fluid mechanics and physics.

Numerous works were dedicated to investigate these issues through theoretical, numerical and experimental approaches, aiming to reveal the mechanisms governing the coalescence dynamics.⁶⁻¹⁴ Although coalescence exhibits very complex behavior, the expansion of the liquid bridge demonstrates a typical self-similarity.¹⁵⁻¹⁷ This facilitates then the formulation of scaling laws describing the growth dynamics of the liquid bridge, including cases like non-Newtonian fluid coalescence with complex rheology or the effects of different boundary conditions.

Various regimes govern the coalescence dynamics of the liquid bridge based on a competing balance between the surface tension, viscous and inertial forces. Typically, this balance is represented by a power-law scaling for the temporal evolution of the bridge radius, where $r(t) \sim t^\alpha$. Under viscous dominated conditions, the exponent $\alpha$ is close to 1, while under inertia dominated conditions, $\alpha$ is around 1/2.¹⁰⁻¹²,¹⁸ Previous studies reveal that when the coalescing zone becomes comparable to the viscous length scale, the coalescence is mainly inertia-dominated.¹⁸ Additionally, an exact solution of the Stokes equation has led to the proposal of a scaling law with a logarithmic correction in the purely viscous regime: $r(t) \sim t \ln t$.¹¹

However, there remains ongoing debate regarding the governing mechanism involved in the earliest stage of coalescence. Paulsen *et al*. highlighted the critical role of inertial effects in the initial regime, proposing the inertially limited viscous (ILV) regime.⁷,¹⁹ Their works suggest
that the expansion of the liquid bridge should firstly pass through the ILV regime before
undergoing toward the Stokes’ viscous regime and the inertial regime. This perspective has
gained some adhesion in recent years.⁶,⁸,²⁰⁻²² In contrary, Anthony *et al.* conducted high-
accuracy numerical simulations of coalescence between two drops, which provided evidence
that the initial coalescence follows the Stokes regime exclusively, without empirical support
for the ILV regime.²³,²⁴ Moreover, the latest comprehensive review by Eggers *et al.* aligns with
this viewpoint,²⁵ further reinforcing the argument against the necessity of the ILV regime in
describing the initial stage of coalescence.

In addition to coalescence between two drops, the coalescence of a drop at liquid-liquid
interfaces has also garnered significant attention. As early as 1960, Charles and Mason
investigated the dynamics of partial coalescence of a drop with a liquid-liquid interface²⁶. They
observed that when a drop impacts a liquid pool, it forms a liquid column, which may break
due to instability, potentially generating secondary drops. For a long time, subsequent studies
focused on topics such as the boundary conditions distinguishing partial and complete
coalescence, predictions of the radius ratio of the secondary drop to the initial drop, and the
suppression of secondary drop formation by applying external fields or modifying the viscosity
of the surrounding environment.²⁷⁻³⁰

In comparison, studies on microscopic dynamics of drop coalescence at liquid interfaces,
such as liquid bridge dynamics, remain relatively scarce. Yokota and Okumura¹⁵ investigated
the coalescence of a drop with its bulk liquid in oil within a Hele-Shaw cell, revealing a
dimensional crossover from three-dimensional viscous dynamics (~~~ *t* ~~) to quasi-two-
dimensional viscous dynamics (~~~ *t*¹/₄ ~~) in a confined geometry. Zhang *et al*.³¹ investigated the
coalescence of a pendant drop with a liquid pool by combining electrical signal and flow
visualization. They found that the liquid bridge expansion follows the classic dynamics,
exhibiting a ~~~ *t* ~~ scaling in the viscosity-dominated regime and a ~~~ *t*¹/₂ ~~ trend in the inertia-
dominated regime. Although studies on the initial stages are limited, they provide crucial
insights for guiding and optimizing various industrial applications. For instance, they are
instrumental in developing more stable and controllable emulsions, regulating the self-
assembly processes of nanoparticles and colloidal materials, and achieving controlled inkjet
printing. Thus, the investigation of initial coalescence is undoubtedly a research topic of
significant importance.

Beyond shape visualization, significant advancements have been made in observing the
internal flow fields in drops. Mohamed-Kassim and Longmire³² utilized two-dimensional PIV
at a rate of 500 frames/s to record the complete coalescence of a drop at a liquid-liquid interface.
They found that the rupture point of the thin film was often off-center and documented the
formation and inward propagation of vortex rings during drop sinking. Ortiz-Dueñas et al.³³
were the first to employ tomographic PIV to capture the complete three-dimensional velocity
field during drop coalescence, revealing multi-dimensional vortex structures and their
asymmetry.

In this work, we make use of an own developed high-speed micro-PIV, a homemade ultra-
high-speed DC electrical device, and a high-speed camera to comprehensively study the
coalescence of drops at various liquid-liquid interfaces. This investigation aims to explore the
initial drop coalescence dynamics, focusing on key aspects such as liquid bridge expansion and
internal flow fields. A variety of fluids are employed to analyze the effect of both coalescing
fluid and outer fluid properties.

**2 Materials and methods**

**2.1 Experimental setup**

The scheme of the experiment setup is displayed in Figure 1 To avoid the uneven interface
caused by the residual phase on the tank wall during the injection, the water phase is slowly
introduced from the bottom by the syringe, and the oil phase is injected from the top. The depth
of the lower layer is controlled at 15 mm, and the upper layer is 10 mm. The deeper lower layer
ensures that the experiment will not be affected by the thin liquid layer during coalescence.
The nozzle is fixed at 3.5 ± 0.2 mm above the interface in the tank, and the drops are formed
at a flowrate of 0.02 mL/min by the syringe pump. At this position, all drops are not in the
pinch-off stage of significant thinning near the nozzle when they contact the interface and
remain a pendant state.

The coalescence sequence was recorded by a high-speed camera Phantom V711 (Vision
Research, USA) equipped with a macro lens (EF 100mm f/2.8, Canon, Japan) at a speed of
60,000 fps. Parallel light can reduce the error caused by gazing reflection at the edge of the
water drop. When the camera is completely perpendicular to the tank, the deformed interface
makes it difficult to visualize. Therefore, the interface is usually observed at a little tilt angle
of the camera.

<figure><img src="image_3.png" /><figcaption>Figure 1. Scheme of the experimental setup. The tank has a square cross section (40 mm × 40 mm) and is made of transparent PMMA.</figcaption></figure>

Particle image velocimetry (PIV) is a visualization technique for measuring velocity fields
by tracking the displacement of seeding particles over a time interval in a fluid. However,
traditional commercial PIV devices have limited laser pulsation frequencies, making them
inadequate for capturing extremely fast flow fields. We developed a high-speed micro-PIV
device capable of capturing 5000 velocity fields per second, as shown in Figure 2. Two 1 mW
lasers were positioned on opposite sides of the tank, and the optical path was adjusted to ensure
proper illumination of seeding particles at the equatorial plane of the drops. Silver-coated
hollow glass spheres with an average diameter of 10 µm and a density of 1,400 kg/m³ (S-HGS-
10, Dantec Dynamics, Denmark) were used as seeding particles in this study.

In this work, the Reynolds number for seeding particles is within very low range, in the Stokes regime. The Stokes relaxation time is about 7.78 µs in NaCl solution and about 7.78 ns in PAAm solution. Since the relaxation time is much smaller than the characteristic timescale of the flow during drop coalescence, the particles can reliably follow the fluid motion and lead to reliable quantification of local velocity fields in a coalescing drop.

<figure><img src="image_4.png" /><figcaption>Figure 2. Scheme of the micro-PIV devices.</figcaption></figure>

**2.2 Physical Properties of the Fluids**

This study investigates the coalescence of aqueous drops as inner fluid with their bulk phase
in various outer fluids, including air, mineral oil and silicone oil. The inner fluids consist of
water with 5 wt% NaCl, 5 wt% polyalkylene glycol (Emkarox) solution with 10 wt% NaCl,
0.5 wt% polyacrylamide (PAAm) solution with 5 wt% NaCl, and 2wt% PAAm solution with
10 wt% NaCl. In the experiments where the liquid bridge width was tracked via electrical
signals, the solution needed to have a certain level of conductivity. To meet this requirement, NaCl was added to all aqueous phases, and the corresponding conductivity values are gathered in Table 1. Additional fundamental physical properties, such as viscosity, refractive index, and interfacial tension, are detailed in Tables 1 and 2.

<table><caption>Table 1 Physical properties of fluids used in the experiments (20°C).</caption><thead><tr><th>Liquids</th><th>ρ<br>(kg·m<sup>-3</sup>)</th><th>η<sub>0</sub><br>(Pa·s)</th><th>σ<br>(mS·cm<sup>-1</sup>)</th><th>Refractive<br>index</th></tr></thead><tbody><tr><td>Mineral oil</td><td>838</td><td>0.028</td><td>-</td><td>1.46-1.48</td></tr><tr><td>Silicone oil</td><td>955</td><td>0.057</td><td>-</td><td>1.40-1.41</td></tr><tr><td>Water + 5 wt% NaCl</td><td>1034</td><td>0.001</td><td>99.8</td><td>1.34</td></tr><tr><td>5 wt% Emkarox + 10 wt% NaCl</td><td>1076</td><td>0.005</td><td>146.4</td><td>1.34-1.35</td></tr><tr><td>0.5 wt% PAAm + 5 wt% NaCl</td><td>1058</td><td>3.456</td><td>100.2</td><td>1.34-1.35</td></tr><tr><td>2 wt% PAAm + 10 wt% NaCl</td><td>1136</td><td>39.440</td><td>146.6</td><td>1.37-1.38</td></tr></tbody></table>

<table><caption>Table 2 Interfacial tension $\gamma$ at 20°C (expressed in mN·m$^{-1}$).</caption><thead><tr><th></th><th>Air</th><th>Mineral oil</th><th>Silicone oil</th></tr></thead><tbody><tr><td>Water + 5 wt% NaCl</td><td>71.59</td><td>55.02</td><td>41.90</td></tr><tr><td>5 wt% Emkarox + 10 wt% NaCl</td><td>59.48</td><td>20.72</td><td>21.55</td></tr><tr><td>0.5 wt% PAAm + 5 wt% NaCl</td><td>69.30</td><td>55.90</td><td>44.20</td></tr><tr><td>2 wt% PAAm + 10 wt% NaCl</td><td>66.20</td><td>54.79</td><td>45.26</td></tr></tbody></table>

The PAAm solutions are non-Newtonian and viscoelastic, behaving as power-law fluids for a wide range of shear rates $\eta = K\dot{\gamma}^{n-1}$, where K is the flow consistency index, n is the flow behavior index, and $\dot{\gamma}$ is the shear rate. The rheological laws are then:

$$ \eta = 0.26\dot{\gamma}^{-0.40} \quad (0.5 \text{ wt}\% \text{ PAAm solution}) \quad (1) $$

$$ \eta = 8.92\dot{\gamma}^{-0.63} \quad (2) $$
<figure><img src="image_5.png" /><figcaption>Figure 3. Rheological properties of the working liquids at 20°C.</figcaption></figure>

## 2.3 Basics of viscosity estimation

The instantaneous viscosity distribution within the drop can be estimated from the calculated shear rate based on the experimental rheological characterization. In the cylindrical coordinates, the mean axisymmetric shear rate in the drop can be expressed as:

$$ \dot{\gamma} = \frac{1}{2} \left( \frac{\partial u_r}{\partial z} + \frac{\partial u_z}{\partial r} \right) \qquad (3) $$

where $u_r$ and $u_z$ are the velocity components in the radial $r$ and axial $z$ directions respectively. The experimental flow fields obtained by the micro-PIV allow then to compute these derivatives in each direction:

$$ \frac{\partial u_r}{\partial z}(\text{m,n}) \approx \frac{u_{r m+1, n} - u_{r m-1, n}}{z_{m+1, n} - z_{m-1, n}} = \frac{u_{r m+1, n} - u_{r m-1, n}}{2 \Delta z} \qquad (4) $$

$$ \frac{\partial u_z}{\partial r}(\text{m,n}) \approx \frac{u_{z m+1, n} - u_{z m-1, n}}{r_{m+1, n} - r_{m-1, n}} = \frac{u_{z m+1, n} - u_{z m-1, n}}{2 \Delta r} \qquad (5) $$

Therefore, the resulting viscosity can be computed by the above-mentioned rheological laws (Eq. 1-2).

## 2.4 Quantification liquid bridge via electrical signals
In terms of experimental techniques, one of the key challenges in studying hydrodynamic singularities is to determine the onset time of coalescence. To address the limitations of optical method, higher precision electrical detection techniques were considered. Burton et al.³⁴ were the first to correlate alternating current (AC) electrical signals with bridge width, expanding the spatial range by three orders, with bridge widths as small as 5 nm, to study near-pinch-off scaling behavior. Following this, Case and Nagel¹² applied electrical methods to measure the resistance evolution in two drop coalescence, reflecting the asymptotic regime within 10 µs.

A ultra-high-speed (up to 1.25 × 10⁶ Hz) direct current (DC) approach was developed in our group based on a modified Howland current source ³⁵ to characterize the expansion of the liquid bridge. One electrode is connected to a metal nozzle forming a pendant drop, while the other is immersed in the liquid at the bottom of the tank, with the drop acting as the resistance (R) in the load circuit as shown in Figure 4. Standard resistors (1 – 800 kΩ) were used for the voltage calibration. The U-R relationship is divided into two segments fitted separately.

For U < 6V:

$$f(R_1) = -2.93 \times 10^{-15} R^3 + 1.65 \times 10^{-10} R^2 + 9.88 \times 10^{-5} R \quad (6)$$

For U > 6V:

$$f(R_2) = -7.72 \times 10^{-23} R^4 + 1.64 \times 10^{-16} R^3 - 1.28 \times 10^{-10} R^2 + 4.59 \times 10^{-5} R + 3.73 \quad (7)$$
<figure><img src="image_6.png" /><figcaption>Figure 4. (a) Electrical circuit diagram for the ultra-high-speed DC electrical device. The resistance R represents the pendant drop. (b) calibration of the electrical voltage U to the resistance.</figcaption></figure>

Previous studies have shown that electrical signals can occur 20 to 60 µs before any visual motion of the coalescing drops is observed ³⁶. The evolution of electric resistance of the initial coalesced drop is presented in Figure 5. In our experiments, as shown in the inset of Figure 5(a), an electrical signal was detected at 16.7 µs with an estimated width $W \sim 10$ µm before the liquid bridge became observable by the camera. By adjusting the nozzle height above the liquid pool, the initial electrical signal completely matches that in Figure 5(b), whether at the air-water interface or the water-oil interface.
<figure><img src="image_7.png" /><figcaption>Figure 5. The evolution of voltage over time. (a) Illustration showing the limit of the high-speed camera, with W ~ 10 µm. (b) Electrical signals at different nozzle-to-liquid pool heights.</figcaption></figure>

During the extremely early stages, the liquid bridge is subject to strong geometric constraints, resulting in high sensitivity to resistance changes. At the moment the drop contacts the bulk phase, the resistance decreases sharply, then stabilizes as the liquid bridge widens.

The electrical conductance G is calculated as follows:

$$G = \sigma \frac{\iint_S EdS}{\int_A^B Edl} = \frac{1}{R} \qquad (8)$$

where σ is the conductivity of the material, and E is the electrical field strength. In the case of a falling drop, since the height l between the nozzle A and the liquid-liquid interface B is fixed, the conductance G primarily depends on the narrowest cross-section S of the liquid bridge through which ions move. The liquid bridge acts as a bottleneck, where the motion of ions is most obstructed in minimizing the overall conductance. Integration performed using the electrostatic software GETDP shows that when the liquid bridge is very narrow, S ~ G. Previous electrical methods directly modeled the expansion by using electric resistance R as a substitute for liquid bridge width ¹²,¹⁸,³⁷. To directly correlate the electrical method with the
optical method for the width in this work, conductance $G$ was used as an analysis tool: $W \sim G^{1/2}$.

# 3 Results and Discussion

## 3.1 Morphological evolution

Before coalescence, the micro-pump controls the drop to grow at a given feeding rate, the drop approaching velocity towards the liquid-liquid interface is monitored. When a drop approaches a flat interface with a given initial velocity, it does not coalesce immediately due to the presence of a surrounding oil film. In the present experiments, the approaching drops move at a speed between 0.02 – 0.06 mm/s. The previous studies have shown that at these velocities, the effect of the liquid oil thickness above the pool on initial coalescence can be neglected $^{18}$.

Taking the coalescence of salt water drops at liquid-liquid interface as an example in Figure 6, a visual singularity appears, followed by the formation of a liquid bridge, marking the beginning of coalescence. The moment the liquid bridge first forms is defined as $t = 0$ ms. In the initial stage of the liquid bridge expansion, where $t \le 0.5$ ms, the release of surface energy and viscous dissipation are concentrated solely within the liquid bridge. Interfacial tension is insufficient to move the entire drop at this point, so its overall shape remains largely unchanged, and only the width of the liquid bridge increases, as shown in Figure 6d. Driven by interfacial tension and inertia, the drop deforms and fluid flows downward until a liquid column is formed. Eventually, this liquid column breaks apart.
<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /><figcaption>Figure 6. Salt water drops (water with 5 wt% NaCl) coalescing with the underlying phase in outer fluid. The outer fluid in a is air, in b is mineral oil, and in c is silicone oil. (d): Evolution of the overall morphology and liquid bridge during coalescence of saltwater drop in silicone oil. (e) Schematic diagram of the intersection of a drop and a flat interface. (f) Liquid bridge widths *W* versus time *t* as drop coalesce in various environmental fluids.</figcaption></figure>

Drop coalescence is affected by the fluid properties of both the drop inner liquid and the surrounding liquid. When comparing coalescence in air and in mineral oil or silicone oil (Figure 6a-c), the viscosity of the surrounding oil is found to significantly delay the expansion of the liquid bridge. The coalescence duration increases with the viscosity of the outer fluid, showing a clear trend of delay. In air, the duration is 16.5 ms, increasing to 21.5 ms in mineral oil and 27.5 ms in silicone oil. This effect is even more pronounced in non-Newtonian inner fluids. The time required for the liquid bridge to reach the same width (0.4 mm) in oil increases drastically, as shown in Figure 6f. Compared with the salt water, the time required for the polymer in air increased by 20% from 0.10 ms to 0.12 ms. However, the growth rises respectively by 72% from 0.25 ms to 0.43 ms in mineral oil, and by 77% from 0.30 ms to 0.53 ms in silicone oil. Obviously, the delayed effect of the outer fluid is more significant in non-Newtonian drops.
## 3.2 Inner flow fields and viscosity distribution

Figure 7 illustrates the evolution of the inner flow fields within a drop during coalescence obtained by the micro-PIV. In typical sessile drop coalescence at a liquid-liquid interface, the initial contact point often deviates from the apex, leading to an asymmetric flow field ³²,³⁸. However, in the present experiments, the coalescence of pendent drops is better controlled and leads to axisymmetric flow field. The drop diameters range from 2.30 to 2.54 mm. The spatial resolution of the flow fields is approximately 5.2 µm, sufficient to capture the detailed flow fields in millimeter-sized drops.

At the onset of coalescence, the velocity fields are concentrated primarily at the liquid bridge. At this stage, viscosity plays a significant role in affecting the radial velocity in both the outer and inner fluids. The higher the outer viscosity (Figure 7a and b), the quicker the dissipation of surface energy released, resulting in lower initial radial velocities. Conversely, the lower the inner viscosity (Figure 7b and c), the greater the inertial forces, leading to larger velocity fields. As surface energy drives more fluid motion, radial velocity remains dominant.

Notably, the expanding liquid bridge is constrained by the viscous outer fluid, causing fluid to flow upward along the surface, forming a recirculation ($t = 0.4$ ms). Zhang *et al.* ³¹ observed that a coalescing drop coalesces with a fluid pool in the air displays also a recirculation. From Figure 7, it is easy to find that the viscosity effect of the outer fluid amplifies the recirculation. The recirculation zone then grows rapidly, pulling more fluid into motion and promoting axial velocities. These results suggest that only significant changes in outer fluid's viscosity, with respect to the small air viscosity of $\eta = 18.5$ µPa·s, could induce such a substantial impact on inner fluid flow pattern. In addition, for Newtonian fluids, recirculation increases with the drop deformation, typically very close to the interface. For shear-thinning non-Newtonian PAAm solutions, the spatial distribution of viscosity in relation to local shear rate inside the drop draws the recirculation slightly closer to the center.
At $t = 2.4$ ms in Figure 7c, the intersection of the saltwater drop and the interface is nearly equal in width, and the deformation is significant. However, the velocity fields are mostly concentrated in the lower part of the drop, with the upper one remaining relatively stationary due to inertia. By $t = 8$ ms in Figure 7a and Figure 7b, the polymer drops show a clear velocity gradient in the upper part. This suggests that the viscoelasticity of the fluid plays a role in the upper region, pulling more fluid into the flow.

<figure><img src="image_11.png" /><figcaption>Figure 7. The velocity field inside coalescing drops. (a) Velocity field inside a coalescing drop. Polymer drops (2 wt% PAAm with 10 wt% NaCl) coalescing in silicone oil. $D_{max} = 2.54$ mm; (b) Polymer drops (2 wt% PAAm with 10 wt% NaCl) coalescing in mineral oil. $D_{max} = 2.30$ mm; (c) Salt water drops (water with 5 wt% NaCl) coalescing in mineral oil. $D_{max} = 2.48$ mm. The seeding particles are hydrophile and only present in the aqueous drops, so the velocity vectors appearing in the oil phase are located at the drops.</figcaption></figure>

Next, a quantitative analysis of the flow fields was performed in the 2 wt% PAAm solution. The maximum radial shear rate occurs at the liquid bridge, while the maximum axial shear rate
is near the drop axis. For polymer drops in mineral and silicone oils, the maximum radial shear rates are 30 - 330 and 30 - 120 s⁻¹, respectively. The maximum axial shear rates are 10 - 100 and 40 - 100 s⁻¹, respectively.

The viscosity distribution within a coalescing non-Newtonian drop is never experimentally quantified so far to our best knowledge. In this work, we successfully obtained accurate flow fields within a coalescing drop by means of the high-speed micro-PIV. Based on these high-precision experimental data, we are able to directly estimate the spatial viscosity distribution for the first time (Figure 8), shedding new insight on complex behavior within a non-Newtonian drop during very fast coalescence.

In the early stage $t = 0.4$ ms, the viscosity in the drop near the nozzle is larger than that in the middle part. And the minimum viscosity is about 0.3 Pa·s near the liquid bridge due to the extremely extensive shear rate. As coalescence progresses, recirculation increasingly promotes high shear rate, reducing the viscosity of the main drop body to approximately 1.0 - 2.5 Pa·s, while the viscosity remains relatively high around the drop axis due to the axisymmetry. Notably, At the expanding end of the liquid bridge, the viscosity remains below 1.0 Pa·s.
<figure><img src="image_12.png" /><figcaption>Figure 8. Velocity fields and estimated viscosity η_c distribution inside a coalescing polymer drop at mineral oil - 2 wt% PAAm and 10 wt% NaCl interface.</figcaption></figure>

### 3.3 Dynamics of liquid bridges

The Ohnesorge number (Oh) is a key dimensionless parameter used to describe the coalescence dynamics.⁷,⁹,²⁷,³⁹,⁴⁰ Paulsen et al. experimentally demonstrated that the relative effect of the inner and outer fluids is closely associated with their respective Ohnesorge numbers (Ohᵢ/o).⁴¹ Accordingly, the Ohnesorge number is defined as follows:

$$Oh_{i/o} = \frac{\eta_{i/o}}{\sqrt{\rho\gamma D_{max}}} \qquad (9)$$

where ηᵢ/o is the viscosity of the inner/outer fluid, ρ is density of the higher of the two fluids, γ is the interfacial tension and D_max is the maximum drop diameter. When Ohᵢ ≪ 1, inertial forces dominate, and it is more suitable to make use of the inertial timescale tᵢ = (ρᵢD_max³/γ)⁰.⁵ for dimensionless analysis, where ρᵢ is the density of drop liquid. Conversely, when Ohᵢ > 1, viscous effect becomes more important, and the viscous timescale t_v = ηD_max/γ would be more suitable.
For the Newtonian fluids, the inner Ohnesorge number $Oh_i$ ranges from 0.0023 to 0.020. For the shear-thinning non-Newtonian fluids, if zero-shear viscosity $\eta_0$ is used in the calculation, $Oh_i$ can reach as high as 26.84, much greater than 1. However, if the experimentally determined effective viscosity $\eta_e$ is employed, the situation changes considerably. As discussed in the previous paragraphs, the local viscosity in the liquid bridge was experimentally evaluated based on the accurate measurements of flow fields by the micro-PIV. Within the timeframe considered in the present discussion, approximately 3 ms prior to coalescence, the maximum viscosity of the fluid participating in the expansion of the liquid bridge is estimated around 0.3 mPa·s, corresponding to $Oh_i \approx 0.6$. Consequently, $Oh_i$ remains less than 1 for all experiments in the present work, and the time $t$ is then rescaled by the inertial timescale $t_i$.

Furthermore, the Ohnesorge number of the outer fluid $Oh_o$, plays a critical role in determining its viscous effects unlike classical coalescence in air. Previous research has shown that when $Oh_o$ is less than 0.3, the viscosity effects of the outer fluid can be considered negligible.⁴¹ In our experiments, $Oh_o$ falls within the range of 0.047 to 0.16, indicating its minor effect on the coalescence.

In summary, although the experiments encompass both Newtonian fluids and non-Newtonian fluids, and external fluids, the overall coalescence dynamics remains in the inertia-dominated regime.

In this work, the initial coalescence was analyzed through either electrical or optical measurements. The evolution of both conductance $G$ and bridge width $W$ follows power-law relationships with time $t$, expressed as $G \sim t^\alpha$ and $W \sim t^\beta$, respectively. When the liquid bridge width increases to a certain range, the resistance isn't fully controlled by the cross-sectional area, and therefore no longer satisfies $W \sim G^{1/2}$.

For dimensionless analysis, $W$ is normalized by the maximum width of drop $D_{max}$, $\Phi = W/D_{max}$, while time $t$ is rescaled by the inertial time $t_i$, $\tau = t/t_i$. In the review of Eggers et al. ²⁵,
$t_i$ is defined by a modified density based on the sum of the inner and outer fluid densities, visibly a kind of added mass as for the motion of bubbles in a fluid. However, the experimental validation as detailed in the supplementary materials reveals that the coalescence is insensitive to density variation.

Liquid bridge growth in air

As shown in Figure 9, when the liquid bridge expands in the air, the electrical signal fails after 6.4 µs before which the exponent is $\alpha = 2.03 \pm 0.05$. Then, the exponent of visualized liquid bridge width by the high-speed camera follows $\beta = 0.58 \pm 0.06$ after 17 µs. Dimensionless scaling reveals that when drops coalesce at an air-liquid free surface, the time evolution of $\Phi$ follows consecutively the viscous regime $\Phi \sim \tau$ and inertial regime $\Phi \sim \tau^{1/2}$, consistent with results reported in the literature. 6,8,11,18,31

Intuitively, during the early coalescence, the shear rate within the liquid bridge should be high due to the sudden release of the surface energy. This implies that the higher shear-thinning effect inside the bridge leads to lower local viscosity. The dimensionless results indicate that the coalescence displays similar dynamics regardless of fluid's properties. In addition, two Newtonian fluids of different viscosity confirm also this coalescence scaling in the inertial regime. Therefore, it could be concluded that the rheological properties of non-Newtonian fluids do not significantly alter the coalescence behavior with respect to the inertia, a conclusion in agreement with the findings of Kern *et al*. 42
<figure><img src="image_13.png" /><figcaption>Figure 9. Dynamics of initial coalescence at air-liquid interfaces. (a) The conductance *G* (left Y-axis) and the width of liquid bridge *W* (right Y-axis) versus time. (b) The normalized time τ as a function of the rescaled coalescence width *Φ*, where the dashed line is the predicted power law for the viscous regime (*Φ* ~ τ), the dot-dash line is for the inertia regime (*Φ* ~ τ^0.5).</figcaption></figure>

In all figures, hollow symbols correspond to data from the electrical method, and solid symbols correspond to data from the optical method.

Liquid bridge growth in oil

Whether a drop coalesces with a liquid pool in air or oil, the rupture of the thin air or oil film
is a critical step. Our group has previously performed investigation on the deformation of
liquid-liquid interfaces by several approaches: sedimentation solid spheres, rising gas bubbles
or rotating rods.43–45 In these studies, sedimentation solid spheres or rising gas bubbles
approaching the interface lead usually to significant deformation of liquid-liquid interface, due
to dynamical competition between gravity, inertia and interfacial tension. This deformation
leads to complex thin-film dynamics, such as interfacial tension distribution, which
subsequently affects the timing and location of film rupture.

In the present experiments, drops are always in a pendent state. In air, despite the slow
approach velocity of the pendent drop, the drop's gravity still causes noticeable deformation of
the liquid pool interface. As shown in Figure 10a, the newly formed liquid bridge in the air is slightly below the liquid surface. In contrast, the drop buoyancy in oil significantly mitigates the deformation, and the liquid bridge exists almost symmetrically to the interface (Figure 10b). Experimental observations show that in air, there is a delay between the drop's contact with the liquid pool and the onset of coalescence, as the interface deformation hinders the drainage of the air film. In this case, the initial liquid bridge width captured is approximately 50 µm. While coalescence occurs almost instantaneously upon the drop's contact with the interface in oil, and the initial width captured is about 10 µm. This peculiar difference highlights how variations in interface deformation and film properties significantly affect the coalescence, resulting in possibly different scaling behavior for the liquid bridge growth.

<figure><img src="image_14.png" /><figcaption>Figure 10. Polymer drops (2 wt% PAAm solution with 10 wt% NaCl) coalescing with the underlying phase in outer fluid. The outer fluid in a is air, in b is silicone oil. The solid blue line is the interface. The pictures do not show two drops coalescing together. The upper part is a real drop, and the lower part is the reflection of the drop on the interface.</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>Figure 11. Time evolution of the electric resistance between a drop and the underlying fluid since liquid bridge as the electric contact is established. Inset is an enlarged view from 0 to 0.02 ms.</figcaption></figure>

Figure 11 illustrates the temporal evolution of the electrical signal of the liquid bridge in different outer fluids, specifically air and oil. The electrical signal of the liquid bridge in oil fails drastically after 10.4 µs, 4 µs later than in air. When $t < 10.4$ µs the electrical resistance of the liquid bridge in the presence of oil decreases more slowly than that in air, regardless of whether the drop fluid is Newtonian or non-Newtonian. Figure 12 displays the evolution of conductance $G$ in different outer fluids. The power law exponents of the growth of liquid bridges are $\alpha = 1.94 \pm 0.11$ in mineral oil and $\alpha = 1.80 \pm 0.09$ in silicone oil respectively, slightly lower than that in air.

The evolution of the visualized liquid bridge width does not follow a single power-law relationship, two distinct growth trends are observed. Especially, the value of $\beta$ is quite similar, in both mineral and silicone oils, transitions from around 0.8 to approximately 0.5. Compared
with the classical two-stage scaling law in air, the coalescence dynamics in oil is somewhat
different.

<figure><img src="image_16.png" /><figcaption>Figure 12. The electrical conductance *G* (left Y-axis) and the width of liquid bridge *W* (right Y-axis) versus time. The outer fluids are (a) mineral oil and (b) silicone oil respectively.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Figure 13. (a) The normalized time $\tau$ as a function of the rescaled coalescence width $\Phi$ for all data, where the dashed line is the predicted power law for the viscous regime ($\Phi \sim \tau$), the dot-dash line is for the inertia regime ($\Phi \sim \tau^{1/2}$). (b) The outer fluids are mineral oil (upper) and silicone oil (bottom) respectively. In all figures, hollow symbols correspond to data from the electrical method, and solid symbols correspond to data from the optical method.</figcaption></figure>
Dimensionless analysis in Figure 13 reveals that, in oil, the early stage is controlled by the viscous regime ($\Phi \sim \tau$). After a prolonged transition, the dynamics gradually shifts to an inertial regime ($\Phi \sim \tau^{1/2}$). Notably, the entire coalescence dynamics appears to follow a $\Phi \sim \tau \ln \tau$ trend, as observed in the evolution of the liquid bridge. In the literature, Hopper$^{10}$ and Eggers et al.$^{11}$ proposed a Stokes regime with a logarithmic correction $r(t) \sim t \ln t$, which was later confirmed by the numerical simulations of high-accuracy performed by Anthony et al.$^{23}$ However, this correction is only valid at very small bridge radii ($\Phi < 0.03$), whereas our experimental data extend this trend in a larger range in the presence of an oil phase. We emphasize that the relationship $\Phi \sim \tau \ln\tau$ is just an empirical fit within the experimental range investigated in this work. It captures only the general trend: when $\tau \approx 10^{-4}$, the exponent is approximately 0.9; for $\tau$ ranging from $10^{-3}$ to $10^{-2}$, the exponent is about 0.75; and when $\tau \approx 10^{-1}$, the exponent varies around 0.5. These variations are consistent with our experimental data.

A similar fitting trend has also been observed in the experimental data of Paulsen et al.,$^{41}$ with a detailed analysis provided in the Figure S2 (Supplementary Materials S1). Certainly, since studies on coalescence in an outer fluid remain quite limited and experimental data are scarce in the literature, further experimental, numerical and theoretical investigations are still required to elucidate physical mechanisms behind such a logarithmic fitting.

## Conclusion

Three cutting-edge techniques were employed to investigate the initial coalescence of a drop with its bulk liquid in different outer fluids, with a focus on the effect of the outer fluid viscosity.

A micro-PIV technique, developed with a temporal resolution of 0.2 ms and a spatial resolution of 5.2 µm, successfully quantifies the inner flow fields inside the coalescing drop and allows then to estimate the spatial viscosity distribution during coalescence in non-Newtonian drops for the first time. The inner flow fields show that the viscosity effect in the outer phase significantly enhances recirculation in the drop. Afterwards, the evolution of the
width of the liquid bridge was tracked using two distinct methods: DC electrical signals with a
resolution of 0.8 µs and visualization by high-speed camera with a resolution of 17 µs. The
results obtained confirm that Ohᵢ below 1 in all experimental conditions, justifying then the use
of the inertial timescale for time scaling. Since Oh₀ < 0.3, the viscosity effect of the outer fluid
has a negligible impact on the coalescence dynamics.

In air, the rescaled width $\Phi$ versus normalized time $\tau$ follows the classical description, exhibiting a viscous regime $\Phi \sim \tau$ and an inertial regime $\Phi \sim \tau^{1/2}$. However, in viscous outer fluids, the viscous drag slows down the liquid bridge expansion, extending the coalescence phenomena to allow more accurate investigation at the same time. The coalescence unfolds both the viscous and inertial regimes. It is worth noting that a distinct and extended crossover stage is observed between these two regimes, indicating a gradual rather than abrupt transition compared to air. A suitable fitting $\Phi \sim \tau \ln\tau$ is found to effectively describe the throughout coalescence dynamics and could provide further facilities for industrial applications involving different scales and regimes. But this fitted correlation only represents an overall tendency without physical mechanisms involved. Further experimental, numerical and theoretical investigations are still essential to gain new insight into the potential physical mechanisms behind such a logarithmic fitting.

## Acknowledgments

The fellowship followed by the China Scholarship Council to Y. Fei is greatly acknowledged.

## Data availability statement

The numerical data from Figures 3 - 6, 9, 11, 12, and 13 are available as a .zip file in the Supplementary Material. This .zip file also includes the main parameters of PIV pre- and post-processing, as well as the raw data used for viscosity prediction that are described in Figures 7
and 8. Additional data that support the findings of this study are available on request from the
corresponding author.

Nomenclature

<table><tr><td>D<sub>max</sub></td><td>Maximum drop width, m</td></tr><tr><td>f</td><td>Frequency, Hz</td></tr><tr><td>G</td><td>Electrical conductance, S</td></tr><tr><td>R</td><td>Electrical resistance, Ω</td></tr><tr><td>S</td><td>Narrowest cross-section</td></tr><tr><td>t</td><td>Time, s</td></tr><tr><td>t<sub>i</sub></td><td>Inertial timescale, s</td></tr><tr><td>t<sub>v</sub></td><td>Viscous timescale, s</td></tr><tr><td>U</td><td>Electrical voltage, V</td></tr><tr><td>W</td><td>Width of the liquid bridge, m</td></tr></table>

Greek letters

<table><tr><td>α</td><td>Exponent in the power law</td></tr><tr><td>β</td><td>Exponent in the power law</td></tr><tr><td>γ</td><td>Interfacial tension, mN·m<sup>-1</sup></td></tr><tr><td>γ̇</td><td>Shear rate, s<sup>-1</sup></td></tr><tr><td>η</td><td>Viscosity, Pa·s</td></tr><tr><td>η<sub>0</sub></td><td>Zero shear viscosity, Pa·s</td></tr><tr><td>η<sub>e</sub></td><td>Estimated viscosity, Pa·s</td></tr><tr><td>ρ</td><td>Density, kg·m<sup>-3</sup></td></tr><tr><td>σ</td><td>Electrical conductivity, S·m<sup>-1</sup></td></tr></table>
$\tau$ Normalized time

$\Phi$ Rescaled width of liquid bridge

Dimensionless groups

Oh Ohnesorge number, $Oh = (\eta/\rho\gamma D_{\max})^{0.5}$

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