2.1 Experimental set-up

The present study focuses on head-on collisions of monodisperse droplets with a continuous liquid jet. Head-on collisions are collisions for which the trajectory of the droplet and that of the jet are in the same plane. To accurately produce such collisions, the experimental set-up sketched in figure 1(a) is used. A droplet generator (Brenn, Durst & Tropea Reference Brenn, Durst and Tropea1996) and a nozzle producing a stream of droplets and a continuous jet respectively, are fixed on micro traverses, enabling the accurate alignment of their trajectories. Different liquids are used to produce the drops and the jet which are supplied by two different pressurized tanks. Nozzle orifices are chosen to obtain droplet diameters in the range of $D_{d}=200\pm 20~\unicode[STIX]{x03BC}\text{m}$ and jet diameters of $D_{j}=290\pm 20~\unicode[STIX]{x03BC}\text{m}$. Here and in the rest of the article, subscript $d$ is used for the drop parameters or the drop liquid properties, whereas $j$ refers to the parameters and properties related to the jet and jet liquid. A signal generator supplies both the droplet generator and a stroboscopic light (LED) with the same frequency $f_{d}$ ($8000<f_{d}<28\,000~\text{Hz}$) enabling the record of frozen collision pictures. For a given collision, pictures are simultaneously recorded with two cameras (1 and 2) providing orthogonal and front views. The drops are darkly dyed producing a contrast with the clear transparent jet, therefore providing the spatial distribution of the two liquids during and after the collisions. The public domain software ImageJ (https://imagej.nih.gov/ij/) is used to post-process the pictures and calculate the collision parameters, which are shown in figure 1(b). These parameters comprise both drop and jet diameters, $D_{d}$ (180–220 μm) and $D_{j}$ (270–310 μm), respectively; the spatial periods of the drop $L_{d}$ (300–1000 μm) and of the jet $L_{j}$ (300–750 μm); the collision angle $\unicode[STIX]{x1D6FC}$ (15^°–60^°); and the velocities of the drop $\boldsymbol{u}_{d}$ (4–15 m s^-1) and jet $\boldsymbol{u}_{j}$ (3–14 m s^-1). The spatial period $L_{j}$ corresponds to the distance advected between two consecutive impacts, i.e. $\boldsymbol{L}_{j}=\boldsymbol{u}_{j}/f_{d}$. The relative velocity $\boldsymbol{U}$ between the drops and the jet is simply calculated with the formula $\boldsymbol{U}=\boldsymbol{u}_{d}-\boldsymbol{u}_{j}$ and typically ranges between 2 and 9 m s^-1. The typical uncertainties are at maximum of 5 % for the dimensions and velocities at stake. The method used to obtain the velocities is presented in appendix D.

Figure 1.

Adapted from Planchette et al. (Reference Planchette, Petit, Hinterbichler and Brenn2018b). (a) Sketch of the experimental set-up used to produce and image drop–jet collisions; (b) sketch representing the geometric and kinetic parameters of a collision.

Note that, in contrast to drop–drop collisions, the orientation of the relative velocity with respect to the jet trajectory also plays a role. Indeed, the relative velocity can be decomposed into two components $U_{\Vert }=u_{d}\cos (\unicode[STIX]{x1D6FC})-u_{j}$ in the jet direction and $U_{\bot }=u_{d}\sin (\unicode[STIX]{x1D6FC})$ orthogonal to the jet. In the present study the parallel component $U_{\Vert }$ of the relative velocity $\boldsymbol{U}$ is set to zero (practically $U_{\Vert }<0.1U$) by adjusting $\boldsymbol{u}_{j}$, $\boldsymbol{u}_{d}$ and $\unicode[STIX]{x1D6FC}$. This adjustment enables a purely orthogonal impact on the jet and reduces potential shearing effects caused by the parallel component of the relative velocity.

2.2 Liquids

The density $\unicode[STIX]{x1D70C}$, dynamic viscosity $\unicode[STIX]{x1D707}$, surface and interfacial tensions $\unicode[STIX]{x1D70E}$ are the relevant liquid properties for drop–jet collisions. The density is measured by weighing a liquid volume of 100 ml, the viscosity is determined with a glass capillary viscometer and the surface tension is determined with the help of the pendant drop method. Note that for interfacial tensions $\unicode[STIX]{x1D70E}_{d/j}$ the values are either measured with the pendant drop method (static values) or taken from the literature (dynamic values), (Takamura, Fischer & Morrow Reference Takamura, Fischer and Morrow2012; Georgiev et al. Reference Georgiev, Danov, Kralchevsky, Gurkov, Krusteva, Arnaudov, Stoyanov and Pelan2018). This only concerns n-hexadecane and is indicated by $\text{}^{a}$ in table 1 listing all liquid properties. This choice is due to the fact that the n-hexadecane/water interfacial tension is known to be sensitive to polar contaminants which play no role on short time scales (typically some minutes, to be compared to 1 ms, the maximum time scale of our observations) but influence the static measurement available via the in-house pendant drop device (P. A. Kralchevsky, private discussion 2019). Yet, no significant differences are expected, and these values are mainly used to distinguish between total and partial wetting, which is further confirmed by macroscopic observation following the evolution of an oil drop gently deposited on the surface of a liquid bath made of the aqueous glycerol solution (movies available online at https://doi.org/10.1017/jfm.2019.967).

In this study, the liquids are selected for miscibility and wettability independently of other liquid properties. Thus, for all situations, the viscosity remains similar, so that mainly the surface and interfacial tensions are varied, except for the perfluorodecalin, whose density is twice as large as for the other liquids. More precisely, we use a reference system made of silicone oil (Carl Roth GmbH, Germany) for the jet and an aqueous solution of glycerol (⩾98 %, Carl Roth GmbH, Germany) at 50 %w:w for the drops. As predicted by the measured spreading parameter $S=\unicode[STIX]{x1D70E}_{d}-(\unicode[STIX]{x1D70E}_{d/j}+\unicode[STIX]{x1D70E}_{j})\approx +14~\text{mN}~\text{m}^{-1}$, and in accordance with the literature (Ross & Becher Reference Ross and Becher1992; De Gennes, Brochard-Wyart & Quere Reference De Gennes, Brochard-Wyart and Quere2004), the silicone oil (SO 5) totally wets the aqueous solution of glycerol, denoted G5 in the rest of the article.

To study the effects of liquid miscibility, an aqueous solution of ethanol and glycerol (noted EtOH) is prepared that approaches the properties of the silicone oil. Keeping G5 drops and substituting for the jet SO 5 by EtOH provides a pair of miscible liquids whose interfacial tension is thus zero. The aqueous ethanol solution consists of ethanol (⩾99.5 %, Carl Roth GmbH, Germany), glycerol (⩾98 %, Carl Roth GmbH, Germany) and deionized water with the mass composition of 55 %, 30 % and 15 %, respectively.

To better understand the effects of the relative liquid wettability and its potential limitation in terms of encapsulation, the silicone oil is substituted by n-hexadecane (ReagentPlus 99 %, Sigma-Aldrich, USA). Alkanes are well known to provide partial wetting with aqueous solutions, which is confirmed by the evaluation of the spreading parameter $(S\approx -10~\text{mN}~\text{m}^{-1}<0)$ and by the observation of n-hexadecane lenses on top of G5. Finally, and in order to better evaluate and verify the scalings proposed later in this article, we use a fluorinated oil, perfluorodecalin (Apollo Scientific Ltd, UK). The main difference from our reference system is the oil density.

When performing the experiments the drop liquid is always coloured, the jet liquid remaining uncoloured. For aqueous solutions, the dye is Indigotin 85 (E 132, BASF, Germany). Novasolve Blue 298 (Nova Specialty Chemicals LLC, USA) is used to colour the silicone oil.

Table 1.

Properties of the liquids used in the present study.

a Indicates values taken from the literature against pure water or against decane after extrapolation to the studied system.

All experiments, including the measurements of fluid properties, are carried out at the same ambient temperature $T_{amb}=23.5\pm 1\,^{\circ }\text{C}$.

2.3 Image and shape analysis

The surface extension of each immiscible phase is obtained via detailed image analysis using the public-domain software ImageJ (https://imagej.nih.gov/ij/). Collision pictures were taken with both front (camera 2) and orthogonal (camera 1) views, see figure 2. The resolution of our imaging system is 10 μm px^-1 for general observations (regime determination, drop spacing and initial diameters), and 6 μm px^-1 for finer measurements (drop and jet deformation, drop fragmentation). Estimating the measurement uncertainty to ±1 pixel leads to ±5 % and ±3 % for the dimensions typically obtained with the general and finer resolutions, respectively. The temporal resolution of the image analysis is limited by the instants visible on the standing pictures. The time elapsed between successive images is $1/f_{d}$, giving a time resolution of 0.1 ms. To limit this sampling effect, movies are recorded using an aliasing frequency for the stroboscopic illumination. With this method, and except for the measurement of $t_{frag}$, 6 points per period are obtained, reducing the discretization to a period of 0.017 ms. Instants of maximal deformation are deduced from the temporal evolution of the quantity of interest, which is then fitted by a polynomial. The overall uncertainty of time obtained by this method is estimated to be less than 0.02 ms. Due to the difficulty in identifying accurately the drop fragmentation, the uncertainty in this measurement is in the range of ±0.05 ms.

Figure 2.

(a) Front and (b) orthogonal views of a collision showing the parameters used to estimate the maximum surface extension reached by the drop (dark phase). (c) Front and (d) orthogonal views of a collision showing the parameters used to estimate the surface extension reached by the jet envelope (clear phase). The notations are explained in the text.

The extension of the drop is tracked following the dark phase (a,b) and modelled – from the first instants of the collision until it reaches its maximal extension – by a bent disk whose unbent diameter is $D_{lam1}\approx D_{lam2}$. Here, $D_{lam2}$ is directly measured from the orthogonal view (figure 2b) whereas $D_{lam1}$ is obtained from the front view (figure 2a) as

(2.1)$$\begin{eqnarray}D_{lam1}=\frac{\unicode[STIX]{x03C0}(a+b)}{2}\left(1+\frac{3\unicode[STIX]{x1D706}^{2}}{10+\sqrt{4-3\unicode[STIX]{x1D706}^{2}}}\right).\end{eqnarray}$$

The expression of $D_{lam1}$, for which $a$ and $b$ are measured on the image, corresponds to the Ramanujan approximation of an ellipse circumference; $\unicode[STIX]{x1D706}=(a-b)/(a+b)$ is the ellipse eccentricity. Note that, by comparing $D_{lam1}$ and $D_{lam2}$ (data not shown, difference below 10 %), we verify that $D_{lam1}\approx D_{lam2}$. In reality, the thin lamella is surrounded by a toroidal rim which is ignored for simplicity. Instead, the surface of the lamella is approximated by the one of a very flat cylinder whose diameter is $D_{lam}\approx D_{lam1}\approx D_{lam2}$ and height $h$, deduced from volume conservation using the not deformed drop. This approximation has been used and validated in several drop–drop collision studies (Willis & Orme Reference Willis and Orme2003; Planchette et al. Reference Planchette, Hinterbichler, Liu, Bothe and Brenn2017). The deformed drop surface thus reads

(2.2)$$\begin{eqnarray}\unicode[STIX]{x1D6F4}_{d\,lam}=\frac{\unicode[STIX]{x03C0}D_{lam}^{2}}{2}\left(1+\frac{8}{3}\frac{D_{d}^{3}}{D_{lam}^{3}}\right),\end{eqnarray}$$

where $D_{lam}$ is evaluated either by $D_{lam1}$ or $D_{lam2}$.

Quantifying the surface of one spatial period of the deformed jet is difficult due to the complexity of the shape. To tackle this point, we estimate the surface as the sum of three contributions:

1. (i) $\unicode[STIX]{x1D6F4}_{1}=\unicode[STIX]{x03C0}L_{1}D_{j\,ext}/4$ is the area of the ellipse that forms on the impact side. The two axes of the ellipse are thus $L_{1}$ (see figure 2c, green dashed line) and $D_{j\,ext}$ (see figure 2d, red double arrow), obtained using the Ramanujan approximation with $e_{1}$, $d_{1}$ and the orthogonal view, respectively.

2. (ii) $\unicode[STIX]{x1D6F4}_{2}=\unicode[STIX]{x03C0}L_{2}D_{j\,ext}/4$ is the area of the ellipse that forms opposite to the impact side. The two axes of the ellipse are thus $L_{2}$ (see figure 2c, red dashed line) and $D_{j\,ext}$, obtained using the Ramanujan approximation with $e_{2}$, $d_{2}$ and the orthogonal view, respectively.

3. (iii) $\unicode[STIX]{x1D6F4}_{3}$ corresponds to a portion of the side area connecting these two ellipses. We approximate the side area by the slant side of a truncated cone whose large and small radii are given by $R_{1}=\sqrt{\unicode[STIX]{x1D6F4}_{1}/\unicode[STIX]{x03C0}}$ and $R_{2}=\sqrt{\unicode[STIX]{x1D6F4}_{2}/\unicode[STIX]{x03C0}}$, respectively. The height of the truncated cone is obtained by volume conservation $h_{3}=3\,(V_{j}+V_{d})/[\unicode[STIX]{x03C0}(R_{1}^{2}+R_{1}R_{2}+R_{2}^{2})]$ where $V_{j}=\unicode[STIX]{x03C0}{D_{j}}^{2}L_{j}/4$ and $V_{d}=\unicode[STIX]{x03C0}{D_{d}}^{3}/6$ are the corresponding drop and jet volumes. Only a portion of this slant area must be accounted for (see figure 2d, red continuous lines) which is equivalent to the angular fraction $\hat{\unicode[STIX]{x1D711}}=2\unicode[STIX]{x1D711}^{\circ }/360$. Thus we obtain $\unicode[STIX]{x1D6F4}_{3}=(R_{1}+R_{2})\unicode[STIX]{x03C0}\sqrt{(R_{1}-R_{2})^{2}+{h_{3}}^{2}}\,\hat{\unicode[STIX]{x1D711}}$.

Finally, the surface extension of the jet is given by

(2.3)$$\begin{eqnarray}\unicode[STIX]{x1D6F4}_{j}=\frac{L_{1}}{2}\frac{D_{j\,ext}}{2}\unicode[STIX]{x03C0}+\frac{L_{2}}{2}\frac{D_{j\,ext}}{2}\unicode[STIX]{x03C0}+(R_{1}+R_{2})\unicode[STIX]{x03C0}\sqrt{(R_{1}-R_{2})^{2}+{h_{3}}^{2}}\,\hat{\unicode[STIX]{x1D711}}.\end{eqnarray}$$

3.1 Immiscible, totally wetting liquid jet

The collision outcomes we observe using a continuous jet of SO 5 with a stream of G5 drops correspond to the four regimes described in Planchette et al. (Reference Planchette, Petit, Hinterbichler and Brenn2018b) and recalled below for completeness. The occurrence of these regimes is described in figure 3(a) using the geometric parameter $L_{j}/D_{j}$ and the drop Weber number $We_{d}$. $L_{j}/D_{j}$ was introduced in Planchette et al. (Reference Planchette, Petit, Hinterbichler and Brenn2018b) to predict the capillary fragmentation threshold. In the same study, a modified Weber number was used to describe the inertial fragmentation limit. The latter being found unsatisfactory when applied to liquids with different wetting and miscibility (data not shown), it is replaced by $We_{d}$.

1. (i) Drops in jet occur when the jet immediately engulfs the impacting drop, forming a continuous jet with a regular and periodic distribution of embedded drops, see figures 3(b) and 3(c)-A for a picture and sketch, respectively. This regime can be found for low to moderate $We_{d}$ (⩽120) and for a ratio $L_{j}/D_{j}$ lower than a critical value close to 2, see figure 3(a), full grey circles.

2. (ii) Fragmented drops in jet correspond to drops that are totally encapsulated by the jet in which they fragment, leaving the jet continuous, see figure 3(b)-B. In most cases, all drop fragments remain inside the continuous jet. In some cases, however, some of them are expelled from the jet, dragging a thin layer of jet liquid, but leaving the main part of the jet continuous, see the sketches of figure 3(c)-B. This regime can be found for higher $We_{d}$ in the range of $120\leqslant We_{d}\leqslant 180$ and for $L_{j}/D_{j}\leqslant 2$, (see figure 3a, empty blue triangles).

3. (iii) Encapsulated drops correspond to a regular stream of capsules, each of them being made of exactly one, not fragmented, aqueous drop (core) coated by a layer of jet liquid (shell), see picture and sketches of figures 3(b) and 3(c)-C. The presence or absence of satellite drops made of the jet liquid only can be observed (Planchette et al. Reference Planchette, Hinterbichler and Brenn2018a) and is reported in figure 3(a) in the form of filled or empty diamonds, respectively. This regime is found for $L_{j}/D_{j}>2$ and moderate values of $We_{d}$ (⩽135).

4. (iv) Mixed fragmentation is observed when both the drop and jet fragment and is illustrated in figures 3(b) and 3(c)-D. It is found for high $We_{d}$ and for any value of $L_{j}/D_{j}$, see figure 3(a)-D (black stars).

Figure 3.

(a) Experimental regime map for drops of G5 and a jet of SO 5. Drops in jet, grey full circles; fragmented drops in jet, empty blue triangles; encapsulated drops, green diamonds (empty without satellite, full with satellites); and mixed fragmentation, black crosses. Black empty circles correspond to the images of (b), the lines are guides for the eye. (b) Images illustrating the four regimes with following collision parameters: (A) $L_{j}/D_{j}=1.63$, $We_{d}=39.42$; (B) $L_{j}/D_{j}=1.57$, $We_{d}=167.05$; (C) $L_{j}/D_{j}=2.14$, $We_{d}=72.54$; and (D) $L_{j}/D_{j}=1.82$, $We_{d}=206.55$. (c) Sketches of the observed regimes: black, drop liquid; white, jet liquid.

The regime occurrence, presented in the form of the map of figure 3(a), summarizes the results obtained with a jet of SO 5 and drops of G5. It confirms that the dimensionless parameter $L_{j}/D_{j}$ is well suited to predicting the capillary fragmentation of the jet. The critical value of approximately 2 deviates from the theoretical value of $\unicode[STIX]{x03C0}$ found by Rayleigh. This can be explained by the effects of inertia and the presence of two phases. Both may affect the flow in the jet and thus its stability toward periodic perturbations. The second dimensionless parameter chosen for this study is $We_{d}$. Accounting for the uncertainty on the measurements of drop diameters (typically ±5 %), $We_{d}$ appears appropriate to distinguish other fragmentations from inertial origins. For the domain $L_{j}/D_{j}<2$, a first fragmentation concerning the drops is found for a critical value of approximately 120, followed by a second one for which both the drop and jet fragment around 180. For the domain where the jet is unstable ($L_{j}/D_{j}>2$), the inertial fragmentation is found around $We_{d}\approx 135$ and leads to the regime of mixed fragmentation.

This regime description and map obtained for immiscible and totally wetting jet liquid raise several important questions. First, how universal are the observed regimes? Especially, can they be obtained using any immiscible liquids or must certain wetting conditions be fulfilled? Further, how must the previous regime description be modified to account for miscible liquids? For liquid couples producing the same kinds of regimes, how universal are the observed fragmentation mechanisms and thresholds?

To answer these questions, we have systematically varied the combination of liquids and start in the next section with the extreme case of miscible liquids.

3.2 Miscible liquids

Here the drops are still made of G5, but the silicone oil has been replaced by an aqueous solution of ethanol and glycerol (EtOH).

As mentioned, the combination of miscible liquids is of great interest in view of utilizing the drop–jet collisions to achieve encapsulation, especially for medical and biomedical applications (Yarin, Pourdeyhimi & Ramakrishna Reference Yarin, Pourdeyhimi and Ramakrishna2014; Kamperman et al. Reference Kamperman, Trikalitis, Karperien, Visser and Leijten2018; Visser et al. Reference Visser, Kamperman, Karbaat, Lohse and Karperien2018). In this context, the main question concerns the final liquid distribution.

Since no more interface between the jet and drop liquid exists, the regime classification must be redefined. We propose to replace the fragmentation or not of each phase by the presence or not of each liquid in each entity (in the same as initially or in separated ones). Following this approach, we obtain the regimes described below and illustrated in figure 4; with (a) a regime map representing their occurrence, (b) pictures and (c) sketches.

Figure 4.

(a) Experimental regime map obtained with glycerol G5 as droplet and EtOH as jet. Drops in jet, grey full circles; fragmented drops and continuous jet, full purple diamonds; simple fragmented jet, red diamonds; and mixed fragmentation, black crosses. Black empty circles correspond to the images of (b), the lines are guides for the eye. (b) Images illustrating the four regimes with following collision parameters: (A) $L_{j}/D_{j}=1.97$, $We_{d}=47.0$; (B) $L_{j}/D_{j}=1.97$, $We_{d}=163.22$; (C) $L_{j}/D_{j}=2.21$, $We_{d}=56.96$; and (D) $L_{j}/D_{j}=2.2$, $We_{d}=146.25$. The black arrow indicates a discontinuity in the image. (c) Schematic representation of the observed regimes: black, drop liquid; white, jet liquid; grey, mixture of both liquids.

1. (i) Drops in jet is still observed with miscible liquids. We have decided to keep the wording despite the absence of an interface since the liquid of the drops remains very localized in the jet, similarly to the case of immiscible liquids, see figures 4(b) and 3(b)-A.

   This observation is in agreement with the comparison of $\langle x_{diff}\rangle$, the typical diffusion length during the process duration, and the drop diameter of 200 μm. Indeed, estimating the process duration by $t_{d\,osc}=\sqrt{\unicode[STIX]{x1D70C}_{d}D_{d}^{3}/\unicode[STIX]{x1D70E}_{d}}$, a fraction of the drop oscillation period (justified later), we find $\langle x_{diff}\rangle =\sqrt{2Dt_{d\,osc}}\approx 1~\unicode[STIX]{x03BC}\text{m}$ (Einstein Reference Einstein1956; Christen Reference Christen2010).

   A direct consequence of this finding is that, even with low viscosities (≈5 mPa s), it is possible to produce a continuous jet with a regular composition pattern. For comparably fast solidification processes, fibres could be obtained with a regular and controlled distribution of some compounds in a main cylindrical body showing great potential for tissue engineering (Khademhosseini et al. Reference Khademhosseini, Langer, Borenstein and Vacanti2006).

   Interestingly, both the inertial and capillary limits of this regime seem similar to the ones found for immiscible liquids, as indicated by the dashed and continuous lines of figure 8(a), found at $We_{d}\approx 120$, and $L_{j}/D_{j}\approx 2$, respectively.

2. (ii) Above the first inertial fragmentation limit, for $L_{j}/D_{j}<2$ and $We_{d}>120$, a continuous jet containing part of the drop liquid plus a stream of satellite drops made of both the drop and jet liquids can be seen, figure 4(a) purple squares. The composition of the jet and satellite drops is revealed by the dye distribution. We call this regime fragmented drops and continuous jet. Illustrations, picture and sketch, are provided in figure 4(b,c)-B. It is important to mention that the fragmentation of the droplets inside the jet does not exist, since there is no interface. Thus, what we observe here under the term fragmented drops and continuous jet corresponds only to a fraction of what is classified as fragmented drops in jet with immiscible liquids.

3. (iii) Above the capillary limit ($L_{j}/D_{j}>2$) and for low to moderate values of $We_{d}$, the jet fragments, see figure 4(a), red diamonds. Except for the absence of an interface inside the fragments, this regime, called simple fragmented jet, is similar to the encapsulated drops observed with immiscible liquids in the absence of satellite drops, see figure 4(b,c)-C.

4. (iv) For $L_{j}/D_{j}>2$ and for $We_{d}>90$ we observe that the jet fragments and that the drop liquid is found in more entities after than before the collision (see figure 4a). By analogy with immiscible liquids, we refer to it as mixed fragmentation. Note, however, that, due to the absence of an interface, the liquid distribution obtained with miscible liquids corresponds only to a fraction of the ones observed under the same wording using immiscible liquids, see figures 3(b,c) and 4(b,c)-D.

By replacing the criteria based on topology changes (fragmentation) of each immiscible phase by criteria based on the spatial distribution of the liquids (especially drop liquid present in more entities than initially), the description of the collision outcomes obtained with miscible liquids appears very similar to that found with immiscible liquids. Four main regimes separated by two fragmentation mechanisms of capillary and inertial origin are obtained in both cases.

The transitions of inertial origin found for $L_{j}/D_{j}<2$ are very similar to the ones observed with immiscible liquids. It is worth noting that, despite the new definitions, the regime occurrence remains mostly unchanged within the experimental uncertainty. To tackle the lack of an interface between the two liquids, the drop is considered as fragmented only if its liquid is found in more entities after than before the collision. Thus, the liquid distribution corresponding to fragmented drops in jet with immiscible liquids may be re-formulated for miscible liquids either as fragmented drops and continuous jet, or as drops in jet. Surprisingly, this neither leads to a reduced occurrence of fragmented drops and continuous jet, nor to a widened observation of drops in jet. Indeed, comparing the results obtained for $L_{j}/D_{j}<2$ and $120<We_{d}<180$, the split between the drop fragments and the continuous jet is as frequently observed with miscible liquids (80 %, or 12 out of 15 points, see figure 4a) as with immiscible ones (72 %, or 18 out of 25 points). Furthermore, no significant shifts in these two transitions are measured, which is far from being trivial.

Different complex and possibly coupled, at least co-existing, effects are expected to influence the fragmentation limits: the presence of Marangoni flows, the absence of Laplace pressure maxima caused by the drops in the jet, the absence of thin oil films which concentrate most of the viscous dissipation (Planchette et al. Reference Planchette, Lorenceau and Brenn2012) or more generally different energy transfer as observed for drop impacts onto liquid films for which liquid miscibility suppresses crown formation (Chen et al. Reference Chen, Chen and Amirfazli2017). The consequences of each separate effect on the fragmentation limits remain unclear. All in all, it seems that the differences observed in these two types of collisions compensate to produce the same transitions below $L_{j}/D_{j}<2$, which is quite remarkable. Additionally, one cannot exclude that, on the typical time scale of the collisions (1 ms), different liquids suddenly brought into contact may see the presence of a diffuse interface reminiscent of their composition difference, even if they are miscible (Shikhmurzaev Reference Shikhmurzaev2008).

The inertial limit found for $L_{j}/D_{j}>2$ is observed for $We_{d}\approx 90$ instead of 130 for immiscible liquids. A first possible explanation for this shift could be the change of criteria used to define the collision outcomes. The liquid distribution corresponding with immiscible liquids to encapsulated drops with satellites may appear as mixed fragmentation with miscible liquids. Indeed, due to the liquid miscibility, it is not possible to exclude traces of drop liquid in the satellites. Yet, even if the encapsulated drops with satellites were to be classified with the mixed fragmentation (see figure 3a), the transition would be found for $We_{d}$ equal to 120, far above 90.

Regarding the capillary fragmentation limit, we note the presence of three points at low $We_{d}$, where the jet is stable despite $L_{j}/D_{j}>2$ (values ranging from 2.1 to 2.3). We attribute this deviation to possible Marangoni flows from the undisturbed jet sections (low surface tension) to the points of drop impact (high surface tension). These flows could contribute in restoring the cylindrical shape of the jet and therefore stabilize it.

3.3 Immiscible, totally wetting drop liquid

While the effects of liquid miscibility appear to be very limited on the collision outcomes and their occurrence, one could ask about the role of liquid wettability. Indeed, the reference system was designed – using a totally wetting jet liquid – to promote the encapsulation of the drops by the jet. Thus, several questions relevant for encapsulation applications arise: Can drops be engulfed in an immiscible jet against the wetting thermodynamics? Is there a critical level of inertia to be reached? Do the drops fragment and get only partially encapsulated? Much has been done about encapsulation and its limits in the field of microfluidics. Classically, emulsions (simple or multiple) are obtained by drop formation caused by the break-up of a disperse phase in a continuous one (at one position or in subsequent ones, respectively) (Chu et al. Reference Chu, Utada, Shah, Kim and Weitz2007). Since the drop liquid is injected into the continuous phase, the effect of relative wettability between the liquids on emulsion stability is rather limited. As an alternative to the classical microfluidics approach, it was proposed to bring droplets of different liquids, dispersed in the same continuous phase, into contact to produce the encapsulation of one drop by the other (Deng et al. Reference Deng, Wang, Ju, Xie, Weitz and Chu2013). Controversial conclusions were drawn, attributing the drop engulfment to a difference of Laplace pressure. Indeed, the thermodynamics of wetting should control the final distribution of the two drop liquids (Guzowski & Garstecki Reference Guzowski and Garstecki2014). Experiments show deviations from the pure thermodynamic configuration. Drops were engulfed by each other, despite partial wetting of the two liquids. These findings were attributed to dynamic effects, and especially to the flow of the continuous phase (Deng et al. Reference Deng, Wang, Ju, Xie, Weitz and Chu2014). Similar dynamic engulfment was reported for drop–drop collisions in air using n-hexadecane and water (Wang et al. Reference Wang, Lin, Hung, Huang and Law2004) and predicted by recent numerical simulation (Wöhrwag et al. Reference Wöhrwag, Semprebon, Mazloomi Moqaddam, Karlin and Kusumaatmaja2018). Yet, only partial wetting was considered, and none of the studied geometries was close to the one of drop–jet collisions. Thus, the questions remain, and we are not aware of any investigation of this kind for in-air microfluidics. While the case of partial wetting will be studied in the next section, we focus here on the extreme case for which the drop liquid totally wets the jet, opposing their engulfment.

The droplets consist of SO 5 coloured with blue dye, whereas the jet consists of G5, which is kept transparent.

Figure 5.

(a) Experimental regime map for collisions of drops of SO 5 and a jet of G5: (A) Coated jet, full grey circles; (B) coated jet with satellites, empty blue triangles, (C) fragmented coated jet, green diamonds (empty without satellites, full with satellites). The lines are guides for the eye and the black circles correspond to the pictures shown in (b). (b) Pictures illustrating the three regimes and corresponding to (A) $L_{j}/D_{j}=1.31$, $We_{d}=100.14$; (B) $L_{j}/D_{j}=1.28$, $We_{d}=702.71$; (C) $L_{j}/D_{j}=2.02$, $We_{d}=568.94$. (c) Schematic representation of the observed regimes: dark, drop liquid (SO 5); white, jet liquids (G5).

The conducted experiments cover the same range of parameters as in the first two series ($1.0<L_{j}/D_{j}<2.2$; $2<U<10~\text{m}~\text{s}^{-1}$). The large values of $We_{d}$ ($100<We_{d}<1000$) are caused by the silicone oil properties. Qualitatively, basing our classification on the fragmentation of each phase, only three regimes can be distinguished. As shown by the pictures in figure 5(b), the liquid spatial distribution of these regimes is completely different from those found in other experiments (§§ 3.1 and 3.2). More precisely, we obtain:

1. (i) A coated jet. The jet does not fragment and is quickly recovered by a film made of the impacting drop liquid, see figure 5(b,c)-A. Note that neither the whole droplets nor part of them can be engulfed into the jet. It is the most frequently observed regime, and its occurrence extends beyond the drops in jet observed with the previous systems, see figure 5(a), full grey circles.

2. (ii) A coated jet with satellites, which corresponds to a continuous jet accompanied by a stream of small satellite droplets. Both the jet and the satellite droplets are made of a core of G5 coated by SO 5, the drop liquid, see figure 5(b,c)-B. This regime is observed for $We_{d}\geqslant 700$ ($U\geqslant 8~\text{m}~\text{s}^{-1}$) and for $L_{j}/D_{j}\leqslant 2$, see figure 5(a), empty blue triangles. For $L_{j}/D_{j}\leqslant 2$ and within the velocity range screened in these experiments (e.g. up to $U\approx 10~\text{m}~\text{s}^{-1}$) it is not possible to break up the jet. Finally, note that this regime corresponds to the fragmentation of the drops, while the jet remains continuous, making it analogous to fragmented drops in jet (§ 3) or to fragmented drop and continuous jet (§ 3.2), apart of course from a very different spatial distribution of the liquids.

3. (iii) A fragmented coated jet. The jet fragments into a regular stream of droplets, all of them being coated by a thin film of silicone oil, see figure 5(b,c)-C. The resulting number of fragments may be equal to the initial number of drops, similarly to what is observed for encapsulated drops without satellites, empty diamonds in figure 5(a). Sometimes, the fragment number is a multiple of the initial drop number, which is indicated by filled diamonds. The regime is observed for large $L_{j}/D_{j}$ and large $We_{d}$.

The absence of a capillary limit for low $We_{d}$ at $L_{j}/D_{j}\approx 2$ is remarkable, but can be explained by different factors. First, one cannot exclude that such a limit exists for a higher critical value. Furthermore, here, as in the rest of the paper, we observe the jet only over a finite distance, classifying it as non-fragmenting if it recovers a quasi-cylindrical shape. The subsequent evolution found for larger time scales is not considered. Finally, this absence may be caused by insufficient inertia or, said differently, by ineffective disturbances. Indeed, the drop spreading around the jet is expected to be very dissipative (Planchette et al. Reference Planchette, Lorenceau and Brenn2012; Chen et al. Reference Chen, Chen and Amirfazli2017; Planchette et al. Reference Planchette, Petit, Hinterbichler and Brenn2018b), and the fragmentation limit at $L_{j}/D_{j}\approx 2$ is recovered for $We_{d}>500$.

Another remarkable finding is the impossibility of forcing, even partially, the drops into the jet. In the present configuration, the capillary spreading driven by $S>0$ is facilitated by the ratio of Laplace pressures between the drops and jet: $\unicode[STIX]{x0394}p_{d\,SO\,5}/\unicode[STIX]{x0394}p_{j\,G5}=2D_{j}\unicode[STIX]{x1D70E}_{d}/(D_{d}\unicode[STIX]{x1D70E}_{j})\approx 0.85$ in contrast to the reversed situation, for which we find: $\unicode[STIX]{x0394}p_{d\,G5}/\unicode[STIX]{x0394}p_{j\,SO\,5}\approx 10$. A further increase of the relative velocity causes the fragmentation of the drops and leaves a stream of coated satellites and a coated jet. Keeping the same liquid combination, the diameter ratio $D_{j}/D_{d}$ (here ≈1.5) should be equal to 17 in order to reach the Laplace pressure ratio of the reversed configuration. This would considerably limit the encapsulation capacity of the jet and, at this stage, the use of surfactants appears more promising to tackle this challenge.

3.4 Immiscible, partially wetting jet liquid

Having revealed that total but reverse wetting prevents encapsulation of the drops, it is legitimate to ask if partial wetting can overcome this limitation. Indeed, while the addition of surfactants could suppress unfavourable total wetting, it cannot reverse it, but rather produces partial wetting conditions. Here, partial wetting is obtained using drops of the aqueous glycerol solution G5 (coloured with blue dye) with a liquid jet made of n-hexadecane (transparent). The geometric parameter $L_{j}/D_{j}$ and the drop inertia ($We_{d}$) are varied in the same ranges as in the other series of experiments. Figure 6(a) illustrates the regime map of the collision outcomes, the associated pictures are shown in (b) and sketches of the liquid distribution are provided in (c). In short, using n-hexadecane for the jet, we observe the same regimes as with silicone oil (or perfluorodecalin). More precisely, we have:

Figure 6.

(a) Experimental regime map for drops of G5 and a jet of n-hexadecane: (A) drops in jet, full grey circles, (B) fragmented drops in jet, empty blue triangles, (C) encapsulated drops, blue diamonds (empty without satellites, full with satellites) and (D) mixed fragmentation, black crosses. The lines are guides for the eye, the black circles correspond to the pictures of (b). (b) Collisions pictures corresponding to (A) $L_{j}/D_{j}=1.82$, $We_{d}=28.87$; (B) $L_{j}/D_{j}=1.44$, $We_{d}=183.07$; (C) $L_{j}/D_{j}=2.29$, $We_{d}=72.62$; and (D) $L_{j}/D_{j}=2.28$, $We_{d}=134.21$. (c) Schematic representation of the observed regimes: Black, drop liquid; white, jet liquid.

1. (i) Drops in jet for low to moderate kinetic energy and for $L/_{j}D_{j}<2$. It is worth mentioning that, even for very low kinetic energy (down to $We_{d}\approx 15$, the minimum reached in this study), the drop is always fully encapsulated by the jet liquid. We do not observe the ‘sticking’ regime found by Wang et al. (Reference Wang, Lin, Hung, Huang and Law2004) and predicted by Wöhrwag et al. (Reference Wöhrwag, Semprebon, Mazloomi Moqaddam, Karlin and Kusumaatmaja2018), both for drop–drop collisions. Wang et al. (Reference Wang, Lin, Hung, Huang and Law2004) observed adhesive merging in case of head-on drop–drop collisions with pure water and n-hexadecane droplets of equal diameter (≈300 μm) at a Weber number of ≈4 related to the liquid properties of pure water, which is lower than the lower value of $We_{d}$ reached here.

2. (ii) Fragmented drops in jet observed in a rather smaller area of the regime map ($170<We_{d}<230$ and $L_{j}/D_{j}<1.5$) when compared to the favourable total wetting configuration ($120<We_{d}<180$ and $L_{j}/D_{j}<2$). Beyond these limits, it is replaced by mixed fragmentation.

3. (iii) Encapsulated drops found for small and moderate $We_{d}$ and above $L_{j}/D_{j}\approx 2$ with and without satellite droplets as for a jet of silicone oil.

4. (iv) Mixed fragmentation, observed above $We_{d}=110$ for $L_{j}/D_{j}>2$, with a threshold level which increases for $1.5<L_{j}/D_{j}<2$ to $We_{d}\geqslant 130$, finishing at $We_{d}\geqslant 240$ for $L_{j}/D_{j}\leqslant 1.5$.

The results obtained with n-hexadecane show the same four regimes as with silicone oil. The existence of two types of fragmentation is also confirmed. The capillary fragmentation of the jet occurs if the geometric parameter $L_{j}/D_{j}$ is above the critical value of 2, and so is independent of $We_{d}$. This finding is very robust, since it is the same for all studied cases (miscible, immiscible, totally and partially wetting jet liquid). Indeed, the only exception concerns the totally wetting drops, for which the fragmentation is observed at the same value, but only above a certain level of inertia.

Fragmentations attributed to an excess of inertia are also present. As with miscible liquids or immiscible totally wetting liquid jet, the drops first fragment, marking the end of the drops in jet regime, followed for even greater inertia by the fragmentation of the jet, which gives place to mixed fragmentation.

Yet, in contrast to the capillary limit for which a universal critical value of 2 is observed, for the inertial fragmentations, shifts are observed in the associated values of $We_{d}$. Using silicone oil, the lower limit is found at $We_{d}\approx 120$, significantly below $We_{d}\approx 150$, the limit observed for n-hexadecane, see figures 3(a) and 6(a). These observations call for a deeper analysis of the drop fragmentation process, which is undertaken in the next section.

4.1 Drop extension

Qualitatively, the evolution of the drop can be described as follows: upon impact, it deforms and takes the shape of a bent lamella surrounded by a rim. During this phase, part of the drop kinetic energy is converted into surface energy. The resulting lamella, which can be modelled by a very flat disk (see § 2), grows in diameter until it reaches its maximum $D_{lam}^{max}$. From this point on, the interfacial tension dominates, and the drop recoils to minimize its interfacial area. This aspect is treated later.

To better understand how the drop deforms, we extract for each collision the lamella surface, $\unicode[STIX]{x1D6F4}_{d\,lam}$, as a function of the elapsed time $t$, starting from the instant of contact. The temporal evolution of $\unicode[STIX]{x1D6F4}_{d\,lam}$ is then fitted by a parabola, and the coordinates of the maximum $\unicode[STIX]{x1D6F4}_{d\,lam}^{max}$ and $t_{d\,lam}^{max}$, the time period required for the drop to reach its maximal extension, are obtained, see figure 7(a). We thus compute the maximum surface of the lamella $\unicode[STIX]{x1D6F4}_{d\,lam}^{max}$, which we normalize by the initial drop surface $\unicode[STIX]{x1D6F4}_{d0}={D_{d}}^{2}\unicode[STIX]{x03C0}$. Repeating this procedure for many collisions between drops of G5 and jets of silicone oil, n-hexadecane and perfluorodecalin, we obtain the results presented in figure 7(b) showing $\unicode[STIX]{x1D6F4}_{d\,lam}^{max}/\unicode[STIX]{x1D6F4}_{d0}$ as a function of $We_{d}$. It is remarkable to observe that the evolution of $\unicode[STIX]{x1D6F4}_{d\,lam}^{max}/\unicode[STIX]{x1D6F4}_{d0}$ is independent of the liquid employed for the jet, even though the drop gets encapsulated before reaching its maximal extension. The interfacial tension $\unicode[STIX]{x1D70E}_{d/j}$ plays surprisingly no role at this stage. Further, using $We_{d}$, all points collapse on the same straight line, which is well fitted by $\unicode[STIX]{x1D6F4}_{d\,lam}^{max}/\unicode[STIX]{x1D6F4}_{d0}=\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D6F4}_{d}}We_{d}+\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D6F4}_{d}}$, in agreement with findings obtained experimentally for drop–drop collisions with one liquid (Jiang, Umemura & Law Reference Jiang, Umemura and Law1992; Willis & Orme Reference Willis and Orme2003) or two immiscible ones (Planchette et al. Reference Planchette, Hinterbichler, Liu, Bothe and Brenn2017) and numerically for drop impacts onto solid surfaces (Wildeman et al. Reference Wildeman, Visser, Sun and Lohse2016). Here, $\unicode[STIX]{x1D6FC}_{\unicode[STIX]{x1D6F4}_{d}}=0.03$ and $\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D6F4}_{d}}=1$, predicting, as expected, no deformation for a quasi-static approach of the jet by the drop. In figure 7(c), $t_{d\,lam}^{max}$ is plotted as a function of $t_{d\,osc}=\sqrt{\unicode[STIX]{x1D70C}_{d}{D_{d}}^{3}/\unicode[STIX]{x1D70E}_{d}}$, the capillary time. For the entire range of $We_{d}$, $t_{d\,lam}^{max}$ is proportional to $t_{d\,osc}$ and not to the inertial time scale $D_{d}/U$ (see inset). The coefficient of proportionality $c$ is equal to 0.33 for silicone oil and n-hexadecane, while a slightly smaller value of 0.3 is found for perfluorodecalin. Similar findings were reported for drop–drop collisions with miscible and immiscible liquids (Planchette et al. Reference Planchette, Hinterbichler, Liu, Bothe and Brenn2017) as well as for drop impacts onto solid surfaces (Richard, Clanet & Quéré Reference Richard, Clanet and Quéré2002). This purely capillary scaling has motivated modelling approaches where the drops are treated as liquid springs (Okumura et al. Reference Okumura, Chevy, Richard, Quéré and Clanet2003; Planchette et al. Reference Planchette, Hinterbichler, Liu, Bothe and Brenn2017) and inspired strategies to control the contact time between drops and the solid surface on which they bounce (Bird et al. Reference Bird, Dhiman, Kwon and Varanasi2013).

Figure 7.

(a) Temporal evolution (blue circles, data points; dashed line, parabolic fit) of $\unicode[STIX]{x1D6F4}_{d\,lam}/\unicode[STIX]{x1D6F4}_{d0}$ for G5 drops and a n-hexadecane jet with $We_{d}=78$. The coefficient of determination $R^{2}$ is 0.9353. Panel (b)  shows $\unicode[STIX]{x1D6F4}_{d\,lam}^{max}/\unicode[STIX]{x1D6F4}_{d0}$ as a function of $We_{d}$ for G5 drops with a jet of the following: green triangles, SO 5; blue circles, n-hexadecane; and orange squares, perfluorodecalin. (c) $t_{d\,lam}^{max}/t_{d\,osc}$ as a function of $We_{d}$, inset $t_{d\,lam}^{max}$ as a function of $D/U$, same symbols as for (b).

Remarkably, these typical findings describing the drop extension are recovered in the present study, where drops impact onto a continuous immiscible jet. Thus, the drop extension alone cannot explain the differences of critical values of $We_{d}$ associated with the inertial fragmentation of the drops in the jet. Note that, for perfluorodecalin, its fragmentation is observed for $We_{d}\approx 95$, significantly below the values found for silicone oil and n-hexadecane (120 and 150, respectively). To go further, we now investigate the deformation of the jet interface, i.e. of the liquid envelope.

4.2 Jet or envelope extension

Detailed information about the estimation of the jet surface was given in § 2.1. After all relevant geometric parameters, see figure 2(c,d), are extracted from the recorded pictures, the surface area is calculated according to (2.3). For each collision, the calculated jet surface area $\unicode[STIX]{x1D6F4}_{j}(t)$, which corresponds to the area of the jet portion of length $L_{j}$, is plotted as a function of time. After normalization of $\unicode[STIX]{x1D6F4}_{j}(t)$ by its initial value $\unicode[STIX]{x1D6F4}_{j0}=\unicode[STIX]{x03C0}D_{j}L_{j}$, the data points are fitted by a third-order polynomial function, similar as for the drops. For details, see appendix C. This type of function provides good fits for all liquids within a defined time period around the maximum extension of the jet ($t_{j\,max}\pm 0.2~\text{ms}$). We thus extract for each curve the value of the local maximum, which provides $\unicode[STIX]{x1D6F4}_{j\,max}/\unicode[STIX]{x1D6F4}_{j\,0}$. The time instant $t_{j\,max}$ is obtained by fitting the evolution of $D_{j\,ext(t)}/D_{j}$, which follows the same temporal evolution as $\unicode[STIX]{x1D6F4}_{j\,(t)}/\unicode[STIX]{x1D6F4}_{j\,0}$, but subject to less measurement noise. See appendix C for a direct comparison. Having done this fitting for many collisions involving different jet liquids, we test our results against appropriate scalings.

As expected, scalings based on $We_{d}=\unicode[STIX]{x1D70C}_{d}D_{d}U^{2}/\unicode[STIX]{x1D70E}_{d}$ or $We_{j}=\unicode[STIX]{x1D70C}_{j}D_{j}U^{2}/\unicode[STIX]{x1D70E}_{j}$ fail to bring all $\unicode[STIX]{x1D6F4}_{j\,max}/\unicode[STIX]{x1D6F4}_{j0}$ measurements to a single line, shown in the insets of figure 8(a). Instead, we observe a linear variation of $\unicode[STIX]{x1D6F4}_{j\,max}/\unicode[STIX]{x1D6F4}_{j0}$ with either $We_{d}$ or $We_{j}$ for each liquid pair; the slope and the constant varying between silicone oil, n-hexadecane and perfluorodecalin. The surface scaling with $U^{2}$ points to a typical capillary-inertial dominated process. Yet, the slope and constant variations seem to indicate that at least one term – most likely the capillary one but possibly the inertial one – is wrongly evaluated by $We_{d}$ and $We_{j}$. A careful analysis of the collision calls for a modified Weber number based on the ratio of the kinetic energy scaling as $\unicode[STIX]{x1D70C}_{d}{D_{d}}^{3}U^{2}$ and the surface energies opposing the deformation. These energies are composed of both the drop and jet contributions, namely $\unicode[STIX]{x1D70E}_{d}{D_{d}}^{2}+\unicode[STIX]{x1D70E}_{j}S_{j}$. Here, $S_{j}$ is the typical jet surface which is deformed by the collision. We evaluate this surface by $S_{j}=D_{j}D_{d}$. Here, $D_{d}$ is used to estimate the typical length of the jet section impacted by the drop. Note that $L_{j}$ is too large to correctly represent this length ($L_{j}>D_{j}\approx 1.5D_{d}$). Thus, we obtain for the modified Weber number:

(4.1)$$\begin{eqnarray}We_{d+j}=\frac{\unicode[STIX]{x1D70C}_{d}D_{d}U^{2}}{\unicode[STIX]{x1D70E}_{d}+\unicode[STIX]{x1D70E}_{j}D_{j}/D_{d}}.\end{eqnarray}$$

The experimental data $\unicode[STIX]{x1D6F4}_{j\,max}/\unicode[STIX]{x1D6F4}_{j0}$ are plotted against $We_{d+j}$ in figure 8(a). Independently of the jet liquid used, all points align along the same curve that can be well approximated by

(4.2)$$\begin{eqnarray}\unicode[STIX]{x1D6F4}_{j\,max}/\unicode[STIX]{x1D6F4}_{j0}=\unicode[STIX]{x1D6FC}_{d+j}We_{d+j}+\unicode[STIX]{x1D6FD}_{d+j},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FC}_{d+j}=0.023$ and $\unicode[STIX]{x1D6FD}_{d+j}=0.61$ (grey line). We notice that $\unicode[STIX]{x1D6FD}\neq 1$, evidencing the validity limit of this scaling for very small inertia. Similar deviations were already observed for binary immiscible drop collisions and for ternary ones (Planchette et al. Reference Planchette, Lorenceau and Brenn2012, Reference Planchette, Hinterbichler, Liu, Bothe and Brenn2017). While, to our knowledge, they have not been discussed in the literature, they could originate from the change of capillary energy due to the coalescence or encapsulation of the drops. Similarly, for large kinetic energies, small deviations are observed, which could be due to a transition toward a different regime of deformation, in which thinner and wider structures may arise. Indeed, where such lack of linearity was reported for viscous binary drop collisions (Willis & Orme Reference Willis and Orme2003), no explanation was provided. Finally, one cannot exclude the deformation being purely inertial. Expressing, instead of energy balance, the momentum conservation, an alternative scaling can be obtained which is also found to be very satisfying, see appendix B. Scalings derived from momentum conservation were successfully used in drop collisions studies, see for example Jiang et al. (Reference Jiang, Umemura and Law1992) and the stretching separation limit established for binary drop collisions. At this stage of our study, we do not have any strong argument to surely determine the most appropriate analysis.

Figure 8.

(a) Value of $\unicode[STIX]{x1D6F4}_{j\,max}/\unicode[STIX]{x1D6F4}_{j0}$ (full symbols) as a function of $We_{d+j}$. The grey line corresponds (4.2). The insets represent $\unicode[STIX]{x1D6F4}_{j\,max}/\unicode[STIX]{x1D6F4}_{j0}$ as a function of $We_{j}$ (top) and $We_{d}$ (bottom). (b) $t_{j\,max}$ (empty symbols) plotted against the oscillating time period of the drop $t_{d\,osc}$ (left) and of the jet $t_{j\,osc}$ (right). (c) $t_{j\,max}$ versus $t^{\ast }=AU/\unicode[STIX]{x1D70E}_{j}$. The top inset evidences the linear variation of $t_{j\,max}$ with $U$, the bottom inset shows the variations of $a$, the slope obtained for each jet liquid, as a function of $1/\unicode[STIX]{x1D70E}_{j}$. For all graphs: drops of G5 with a jet of SO 5, green triangles; n-hexadecane, blue circles; and perfluorodecalin, orange squares.

The time scale $t_{j\,max}$, at which the envelope reaches its maximum extension, identified with the one for which $D_{j\,max}$ is maximum, is more difficult to apprehend. Looking at $t_{j\,max}$, we notice that neither $t_{d\,osc}=\sqrt{\unicode[STIX]{x1D70C}_{d}{D_{d}}^{3}/\unicode[STIX]{x1D70E}_{d}}$ nor $t_{j\,osc}=\sqrt{\unicode[STIX]{x1D70C}_{j}{D_{j}}^{3}/\unicode[STIX]{x1D70E}_{j}}$ represent well the experimental data, see figure 8(b). A modelling attempt is made by considering that the jet deforms as long as the kinetic pressure $p_{k}\propto U^{2}$ working on a typical section $\unicode[STIX]{x1D6F4}$, considered as fixed for constant $D_{d}$ and $D_{j}$, is sufficient. Balancing $p_{k}\unicode[STIX]{x1D6F4}$ with $\unicode[STIX]{x1D70E}_{j}(Ut_{j\,max})$, the capillary force corresponding to the jet deformation, provides

(4.3)$$\begin{eqnarray}t_{j\,max}=p_{k}\unicode[STIX]{x1D6F4}/\unicode[STIX]{x1D70E}_{j}U\propto U/\unicode[STIX]{x1D70E}_{j}.\end{eqnarray}$$

We thus define $t^{\ast }=AU/\unicode[STIX]{x1D70E}_{j}$, where $A$ is a constant with the dimension kg m^-1, and plot the experimental data $t_{j\,max}$ against $t^{\ast }$ with $A=1500~\text{kg}~\text{m}^{-1}$ in figure 8(c). The inset representing $t_{j\,max}$ as a function of $U$ demonstrates the linear variation (slope $a$ for each liquid). The second inset shows that the relation between $a$ and $1/\unicode[STIX]{x1D70E}_{j}$ is according to the prediction of (4.3). Thus, despite the experimental uncertainty on $t_{j\,max}$, the agreement with (4.3) is excellent and only some slight deviations can be observed for collisions with fluorinated oil and large velocities. In the latter case, other effects could take place that we have neglected.

It is important to note that $t_{j\,max}$ is in all cases significantly larger than $t_{d\,lam}^{max}$. Indeed, for all liquids, we have reported $t_{d\,lam}^{max}\approx 0.3\,t_{d\,osc}$, (figure 7c), while $t_{j\,max}>0.3\,t_{d\,osc}$ as evidenced in figure 8(b) for all investigated collisions. We also notice that the delay between $t_{j\,max}$ and $t_{d\,max}$ varies between the three jet liquids, in agreement with the finding of (4.3). For a given impact velocity, this delay increases from n-hexadecane, to silicone oil and to perfluorodecalin. Further, for a given liquid, the delay increases with $\boldsymbol{U}$.

Since, in all investigated collisions, the drop has already started to recoil while the jet keeps extending, it is legitimate to compare the previous time scales, $t_{j\,max}$ and $t_{d\,lam}^{max}$, to the instant $t_{frag}$ at which the drop fragments. Is the drop fragmentation driven by the extension or by the recoil of the drop or jet?

4.3 Drop pinch-off, on the importance of recoil

To investigate the drop fragmentation, its mechanisms and kinetics, pictures recorded with camera 1, providing the so-called orthogonal view, are used. Thanks to the dye present in the drop liquid only, we can follow the evolution of the drop and observe its potential fragmentation.

Figure 9.

(a) Schematic representation of a drop–jet collision at $t_{d\,lam}^{max}$, the instant of drop maximal extension (lamella); $t_{j\,max}$, the instant the jet reaches maximal extension (the red dashed lines mark the drop maximal extension) and $t_{frag}$, the instant of drop fragmentation (if $\unicode[STIX]{x1D701}=L_{max}/D_{d}>\unicode[STIX]{x1D701}_{c}$, introduced thereafter). (b) Schematic representation of a drop–drop collision at $t_{max}$, the instant of drop maximal extension and $t_{frag}$ the instant of fragmentation (when $\unicode[STIX]{x1D701}>\unicode[STIX]{x1D701}_{c}$). The sketches show for each instant a front (right) and orthogonal (left) view of the system.

To understand the evolution of the drop recoil within the jet, it may be useful to recall results obtained with drop impacts on solid surfaces or drop–drop collisions. In the latter situations, the drop recoil is isotropically directed toward the centre of the lamella, forming a cigar whose axis is aligned with the one of the lamellae, see figure 9(b). This cigar may fragment following a pseudo Rayleigh criterion, giving rise to two drops of equal size. For drop–jet collisions, the constraint exerted by the jet interface on the encapsulated drop dramatically modifies the recoil geometry, as sketched in figure 9(a). While the drop keeps forming a (bent) lamella which relaxes to produce an elongated drop, the axis of the elongated drop (along its length $L_{max}$) is found to be perpendicular to the collision plane, i.e. perpendicular to the orientation it would take in the absence of the jet. When the drop fragments, it occurs on both extremities of this elongated drop according to a pinch-off process, resulting in a main drop and two smaller satellite drops.

Figure 10.

(a) Collision between G5 drops and SO 5 jet at $We_{d}=178$ and $L_{j}/D_{j}=1.33$ recorded with the orthogonal camera. Schematic illustration of $L_{max}$, $D_{lam}^{max}$, $t_{d\,lam}^{max}$ and $t_{frag}$. (b) Time scales of the collision with: $t_{d\,lam}^{max}$, full colour symbols: $t_{j\,max}$, empty colour symbols; and $t_{frag}$, black colour symbols as a function of $We_{d}$. All time scales are normalized by $t_{d\,osc}$ and the jet liquids are: blue, n-hexadecane; green, silicone oil; and orange, perfluorodecalin. (c) The normalized length of the elongated drop $L_{max}/D_{d}=\unicode[STIX]{x1D701}$ as a function of $We_{d}$ using full symbols for drops in jet and open ones for fragmented drops in jet or mixed fragmentation. The small graphs illustrate the slope $c$ of the curves subjected to $\unicode[STIX]{x1D70E}_{j}$ and $D_{d}\unicode[STIX]{x1D70E}_{d}-D_{j}\unicode[STIX]{x1D70E}_{j}$ as well as the constant $d$ as a function of $\unicode[STIX]{x1D70E}_{d/j}-\unicode[STIX]{x1D70E}_{j}$.

To quantify this process, especially its kinetics, we report in figure 10(b) the instant of fragmentation normalized by the drop oscillation time, $t_{frag}/t_{d\,osc}$, as a function of $We_{d}$. For comparison we also plot $t_{d\,lam}^{max}/t_{d\,osc}$ and $t_{j\,max}/t_{d\,osc}$. To give insight into the fragmentation threshold, and thus to its mechanism, we also report the normalized length of the elongated drop $L_{max}/D_{d}=\unicode[STIX]{x1D701}$ as a function of $We_{d}$, illustrated in figure 10(c). Here also, the data are presented for three jet liquids (silicone oil, n-hexadecane and perfluorodecalin), the drop liquid remaining unchanged (G5). We observe that, independently of the jet liquid, the drop fragmentation takes place after the maximal extension of both the drop and the jet, $t_{d\,lam}^{max}$ and $t_{j\,max}$, approximately after $3\,t_{d\,lam}^{max}$ except for a few points obtained with n-hexadecane for which the fragmentation occurs even later, around $6\,t_{d\,lam}^{max}$ (see fifth column in figure 9). Note that, since the exact instant of fragmentation can neither be exactly determined from the pictures nor deduced from a fit (in contrast to $t_{d\,lam}^{max}$ and $t_{j\,max}$), error bars have been added to $t_{frag}$, figure 10(b). They correspond to the last instant at which the drop is definitely seen as not fragmented and to the first instant where the drop can confidently be observed as fragmented. Yet, it is worth commenting the finding that $t_{frag}\approx 3\,t_{d\,lam}^{max}\approx 0.9\,t_{d\,osc}$. The late fragmentation cases observed with n-hexadecane will be discussed at the end of this paper. First, beside the presence of different encapsulating liquids, and, more importantly, beside the changes adopted by the recoiling drop, this result is similar to the one obtained for drop–drop collisions using a single liquid (Willis & Orme Reference Willis and Orme2003), see also figure 9(b). This seems to indicate that the kinetics of the drop recoil remains dominated by the typical oscillation time $t_{d\,osc}$ of an isolated drop and that the pinch-off is driven by an excessive extension of the recoiling drop, as for drop–drop collisions with two and three drops, using one or two liquids (Willis & Orme Reference Willis and Orme2003; Planchette et al. Reference Planchette, Lorenceau and Brenn2012; Hinterbichler, Planchette & Brenn Reference Hinterbichler, Planchette and Brenn2015; Planchette et al. Reference Planchette, Hinterbichler, Liu, Bothe and Brenn2017). This hypothesis can be further tested by looking at figure 10(c), where $\unicode[STIX]{x1D701}$ is plotted as a function of $We_{d}$ using full symbols for drops in jet outcomes and open ones for fragmented drops in jet or mixed fragmentation. While for each liquid pair the data are aligned along different lines, the transition between full and empty symbols is found at the same critical value of $\unicode[STIX]{x1D701}$ ($\unicode[STIX]{x1D701}_{c}\approx 3$) for all liquids. Thus, attributing the pinch-off of the encapsulated drop to an excessive elongation appears relevant. The critical value is close to the ratio of disturbance wavelength to filament diameter separating the unstable from the stable ranges of wavelengths, as predicted by Rayleigh’s liquid jet stability analysis. This break-up mechanism and critical value were mentioned for other liquid systems deformed by collisions (Planchette et al. Reference Planchette, Hinterbichler, Liu, Bothe and Brenn2017). However, in the present study, the shape of the elongated drop strongly differs from a cylinder, evoking a different break-up mechanism. A closer inspection of the drop at fragmentation shows dumbbell-shaped drop endings. As already mentioned, these bulbs may then pinch-off, leaving a main drop and two smaller ones, strongly recalling the end-pinching mechanism introduced by Stone, Bentley & Leal (Reference Stone, Bentley and Leal1986) and Stone & Leal (Reference Stone and Leal1989b). We note that the critical value of 3 is slightly lower than 4, the value found in the absence of flow for the end-pinching mechanism (Stone et al. Reference Stone, Bentley and Leal1986). Further experiments (data not shown) indicate that increasing or decreasing the viscosity ratio between the drop and jet phases leads to larger values of $\unicode[STIX]{x1D701}_{c}$ as expected for the mentioned pinch-off mechanism. Thus, the discrepancy is most probably caused by complex flows and complex drop shape. While flows within the encapsulated phase may delay the pinch-off (Hoepffner & Paré Reference Hoepffner and Paré2013), it was reported that flows in the encapsulating phase lead to break-up for smaller drop extension (Stone & Leal Reference Stone and Leal1989a). In the current process, the jet envelop is subjected to capillary recoil, which generates non-negligible flows. Combined with the drop–jet geometry, this recoil further influences the shape of the encapsulated drop which differs from the cylinder considered in Stone et al. (Reference Stone, Bentley and Leal1986), Stone & Leal (Reference Stone and Leal1989b). Indeed, the lamella is drained into two opposite bulbs separated by a central part. The corresponding local curvatures are expected to generate additional capillary instabilities and could therefore cause the pinch-off to occur for lower values of $\unicode[STIX]{x1D701}_{c}$.

It is also remarkable to observe that, for each liquid, $\unicode[STIX]{x1D701}$ seems to vary linearly with $We_{d}$, the slope and constant varying from liquid to liquid. This indicates that the elongation of the drop during its recoil phase has in all cases an inertial origin. Finally, it can also be noticed that the values measured above $\unicode[STIX]{x1D701}_{c}$ are more dispersed than those found below. This can be understood as a consequence of the drop pinch-off that may slightly affect the measurement of $L_{max}$, adding noise to the data.

At this stage of our analysis, one question remains: What fixes the slope and constant of the linear variation of $\unicode[STIX]{x1D701}$ with $We_{d}$? Or in different terms, how do we predict the evolution of $\unicode[STIX]{x1D701}$ to deduce the corresponding value of $We_{d}$ marking the first inertial fragmentation?

4.4 Consequences on the inertial fragmentation limit

We extract from figure 10(c) the slope $c$ and the constant $d$ for each curve, so that, for a given liquid pair, the evolution of $\unicode[STIX]{x1D701}$ with $We_{d}$ is well reproduced by $\unicode[STIX]{x1D701}=cWe_{d}+d$. We attribute the term in $cWe_{d}$ to the relative deformation and recoil of the drop and jet and thus expect both $\unicode[STIX]{x1D70E}_{d}$ and $\unicode[STIX]{x1D70E}_{j}$ to influence the value of $c$. This interpretation is tested in figure 10(c). The larger $\unicode[STIX]{x1D70E}_{j}$, the smaller $c$, indicating that the lateral extension of the drop is limited by large values of $\unicode[STIX]{x1D70E}_{j}$. Excellent agreement is found by comparing $c$ to $\unicode[STIX]{x1D70E}_{d}D_{d}-\unicode[STIX]{x1D70E}_{j}D_{j}$, where $D_{d}=200~\unicode[STIX]{x03BC}\text{m}$ and $D_{j}=300~\unicode[STIX]{x03BC}\text{m}$. Beside the tentative character of our analysis, the expression $\unicode[STIX]{x1D70E}_{d}D_{d}-\unicode[STIX]{x1D70E}_{j}D_{j}$ seems to confirm the competitive character of the drop and jet deformation and recoil. The constant term $d$ is independent of the deformation magnitude (independent of $We_{d}$) and is thus attributed to the relative deformability of the jet and the drop caused by the relative Laplace pressures. Indeed, whatever the deformation amplitudes are, the interface of the encapsulated drop and the one of the jet envelope follow each other, so that, at first order, the curvatures coincide. Thus, the difference between the Laplace pressure jumps at the jet and at the encapsulated drop is expected to be proportional to the difference of interfacial tensions $\unicode[STIX]{x1D70E}_{d/j}-\unicode[STIX]{x1D70E}_{j}$. This is in very good agreement with the experimental measurements, as shown by the graph representing $d$ as a function of $\unicode[STIX]{x1D70E}_{d/j}-\unicode[STIX]{x1D70E}_{j}$. Thus we obtain

(4.4)$$\begin{eqnarray}\unicode[STIX]{x1D701}^{theo}=2.9\times 10^{-2}(1-\unicode[STIX]{x1D70E}_{j}D_{j}/\unicode[STIX]{x1D70E}_{d}D_{d})We_{d}+3.5(\unicode[STIX]{x1D70E}_{d/j}-\unicode[STIX]{x1D70E}_{j})/\unicode[STIX]{x1D70E}_{d}.\end{eqnarray}$$

Figure 11.

Regime maps with $L_{j}/D_{j}$ and $\unicode[STIX]{x1D701}^{theo}$ as scaling parameter. G5 is used as the drop liquid and as jet liquid, from (left to right), silicone oil, n-hexadecane and perfluorodecalin. Full circles, drops in jet; empty diamonds, encapsulated drops; empty triangles, fragmented drops in jet; and black crosses, mixed fragmentation.

Here, the surface and interfacial tensions have been normalized by that of water, and the dimensions by the drop diameter $D_{d}$. The numerical factors 2. 9⋅10^-2 and 3.5 are found empirically from the graphs of figure 10(c) using all points below the drop fragmentation threshold, i.e. for $40<We_{d}<160$. Their validity cannot be determined with currently available data. Further experiments involving other liquid pairs should be performed to address this point. These models (4.4) are used in figure 11, where ($L_{j}/D_{j}$; $\unicode[STIX]{x1D701}^{theo}$) regime maps are plotted for, from left to right, silicone oil, n-hexadecane and perfluorodecalin. For all jet liquids, the first inertial fragmentation (vertical continuous line) is found for a critical value of $\unicode[STIX]{x1D701}^{theo}=3$, in agreement with our analysis. The other capillary limit remains unchanged and is found for $L_{j}/D_{j}\approx 2$ (dashed lines) for all jet liquids.

Thus, to account for different surface and interfacial tensions, we propose replacing the inertial-capillary parameter $We^{\ast }$ introduced in Planchette et al. (Reference Planchette, Petit, Hinterbichler and Brenn2018b) or $We_{d}$ by $\unicode[STIX]{x1D701}^{theo}$. In contrast to $We^{\ast }$ or $We_{d}$, this enables us to account for the competition between the drop and jet inertial extension via $(1-\unicode[STIX]{x1D70E}_{j}D_{j}/\unicode[STIX]{x1D70E}_{d}D_{d})We_{d}$. Further, the term $(\unicode[STIX]{x1D70E}_{d/j}-\unicode[STIX]{x1D70E}_{j})/\unicode[STIX]{x1D70E}_{d}$ renders the relative deformability of the jet envelope and the encapsulated drop during recoil. This parameter has been built based on a detailed analysis of drop–jet collisions produced by strongly varying the interfacial tension as well as the jet surface tension and density. By testing this parameter against the three immiscible jet liquids of this study (silicone oil, n-hexadecane and perfluorodecalin) we demonstrate its robustness for predicting the first inertial fragmentation. We recall that, in this study, liquids were chosen to have similar viscosity and viscosity ratio, which is the reason why they do not appear in $\unicode[STIX]{x1D701}^{theo}$. Indeed, we expect viscous losses to strongly modify the amplitude of the drop and jet extensions (Planchette et al. Reference Planchette, Petit, Hinterbichler and Brenn2018b), the critical aspect ratio of pinch-off (Stone et al. Reference Stone, Bentley and Leal1986) and, more generally, the kinetics of deformation. Additionally, strong variations in the size ratio between the drops and jet may significantly modify the qualitative evolution of the drops and jet, which constitutes another limit to the present analysis. These aspects go beyond the scope of this study and will be presented in a forthcoming article.

Finally, let us come back to the few late fragmentation cases observed for n-hexadecane, for a moderate $We_{d}$, i.e. between the point where the drops start to fragment and $We_{d}\approx 185$. Note that, above this value of $We_{d}$, the kinetics observed for other liquids is recovered, and fragmentation occurs around $3\,t_{d\,lam}^{max}\approx 0.9\,t_{d\,osc}$. For lower $We_{d}$, the drop fragmentation does not happen around $0.9\,t_{d\,osc}$, but around $1.6\,t_{d\,osc}$ and later. By carefully looking at the pictures taken by both the front and side cameras, we notice that the first elongation of the drop still occurs around $3\,t_{d\,lam}^{max}\approx 0.9\,t_{d\,osc}$, but without leading to immediate pinch-off. Instead, the drop further recoils and extends in the orthogonal direction. The fragmentation takes place during this second orthogonal elongation. The absence of fragmentation during the first elongation could be first interpreted as the result of internal flows (Stone et al. Reference Stone, Bentley and Leal1986). Yet, this does not happen for other liquids, while the internal flows are expected to be similar (for similar deformation at least). Thus, another reason could be the fast recoil kinetics of the liquid system. Indeed, for n-hexadecane, the jet surface tension $\unicode[STIX]{x1D70E}_{j}=26.5~\text{mN}~\text{m}^{-1}$ is larger than for silicone oil and perfluorodecalin ($19.5~\text{mN}~\text{m}^{-1}$ and $13~\text{mN}~\text{m}^{-1}$, respectively). Higher surface tension leads to a faster contraction of the jet envelopes. For moderate excess of kinetic energy (moderate $We_{d}$), and thus moderate elongation $L_{max}/D_{d}$, the jet recoil kinetics may be too fast and may prevent the encapsulated drop pinching off. Instead, the drop fragmentation occurs during the second elongation, whose kinetics may be slowed down. Assuming that, after the first drop recoil, the drop behaves as a spring whose constant is fixed by its interfacial tension $\unicode[STIX]{x1D70E}_{d/j}$, we find that the oscillation period of the encapsulated drop is modified by a factor $\sqrt{\unicode[STIX]{x1D70E}_{d}/\unicode[STIX]{x1D70E}_{d/j}}=1.16$, providing $t_{d\,osc}^{\ast }=1.16\,t_{d\,osc}$. The fragmentation can therefore be expected for $t_{frag}^{\ast }=0.3(3\,t_{d\,osc}+2\,t_{d\,osc}^{\ast })\approx 1.6\,t_{osc}$, see figure 9(a). This value is in good agreement with the experimental results (see figure 10b, dashed-dotted line). This interpretation is also coherent with the previous analysis and would explain why this phenomenon is observed only for n-hexadecane and moderate $We_{d}$. Note that we cannot exclude further causes for the kinetic changes. While viscous friction between the two phases may not be a good candidate (n-hexadecane is slightly less viscous than silicone oil and perfluorodecalin), a stronger Laplace pressure jump may change the internal flows, making the pinch-off possible only during the second elongation.