2.1 Solution and substrate

The system we investigate here consists of Milli-Q water (obtained from a Reference A $+$ system (Merck Millipore) at $18.2~\text{M}\unicode[STIX]{x03A9}$ at $25\,^{\circ }\text{C}$ ), ethanol (Sigma-Aldrich; ${\geqslant}98\,\%$ ) and anethole oil (Sigma-Aldrich; trans-anethole, ${\geqslant}99.8\,\%$ ). We performed dissolution experiments in a cuvette (Hellma; inner dimensions $30~\text{mm}\times 30~\text{mm}\times 30~\text{mm}$ ), on the bottom of which a hydrophobized glass slide ( ${\approx}20~\text{mm}\times 20~\text{mm}$ ) was placed. A certain amount of water-saturated anethole was added into the cuvette, performing as the host liquid. The depth of the liquid was 7.5 mm. Water–ethanol binary drops with different volumetric concentrations of ethanol (30, 40, 50, 60 and 70 vol%) were produced in the oil through a custom needle (Hamilton; outer diameter/inner diameter 0.21 mm/0.11 mm) by a motorized syringe pump (Harvard; PHD 2000), and then directly deposited on the centre of the hydrophobized glass surface.

2.2 Emulsion/microdroplet recognition

The emulsions (nucleated microdroplets) were recognized visually. Both the w-in-o emulsion and the o-in-w emulsion had a recognizable cloudy white appearance because of the microscopic size of the nucleated microdroplets, which enables them to scatter all the colours equally. The recognition of the emulsions was processed by watching the recorded videos frame by frame. Although dissolving drops were of millimetre scale, their spherical shapes with high curvature unavoidably caused reflected light spots, which increased the difficulty of recognizing the presence of microdroplets inside the drop. Therefore, the presence or absence of emulsions was carefully determined by detecting the liquid colour variation and their movement from the recorded videos (both top views and synchronized side views). No fluorescence technique was used for microdroplet detection, in order to avoid any influence of added fluorescent materials on the spontaneous emulsification.

2.3 Micro particle image velocimetry

To investigate the flow field around the dissolving drop, we added tracking particles (Dantec Dynamics; PSP-5, diameter $5~\unicode[STIX]{x03BC}\text{m}$ , made of nylon-12) into the host liquid at a seeding density of $0.2~\text{mg}~\text{ml}^{-1}$ to perform micro-PIV measurements. As shown in appendix A.1, these particles can be considered as passive. Thanks to the low flow rate in our study, a continuous light-emitting diode (LED) light source (Thorlabs; MCWHL5) was able to provide enough volume illumination for the measurements. The light source and the camera were placed at opposite sides of the cuvette. The light passed through convex lenses before illuminating the cuvette to form a parallel light beam to increase the image contrast. At the other side, we positioned a high-speed camera (Photron Fastcam SA2, 32 GB, 50 frames per second at 2048 pixel $\times$ 2048 pixel resolution) attached with a microscope system (Infinity; model K2 DistaMax) to perform high-speed imaging. The position of the recording system was adjusted to have a focal plane crossing the droplet centre. The thickness of the focal plane is 0.02 mm. Thereby, we could obtain sharp images of the tracking particles within a cross-sectional plane of the drop. We took image pairs with an inter-framing time of 20 ms every 2 s. The obtained image pairs were first processed to reduce the noise, and then imported into PIVlab software (Thielicke Reference Thielicke2014; Thielicke & Stamhuis Reference Thielicke and Stamhuis2014) to calculate the flow field. The size of the interrogation window was taken as a 128 pixel $\times$ 128 pixel matrix, corresponding to $142~\unicode[STIX]{x03BC}\text{m}\times 142~\unicode[STIX]{x03BC}\text{m}$ . The interrogation window overlap was set as 75 %, leading to a $35.5~\unicode[STIX]{x03BC}\text{m}$ vector spacing in the calculated velocity matrix.

3.1 Characteristic states of dissolving drops

A dissolving water–ethanol (aqueous) sessile drop in anethole oil (host liquid) is displayed in figure 2. The initial ethanol concentration is 60 vol% and the initial drop volume is approximately $0.5~\unicode[STIX]{x03BC}\text{l}$ . The experimental snapshots in figure 2 (top views in the left column and the corresponding side views in the right column) present several interesting phenomena occurring during the drop dissolution, including solute plumes detaching and rising upwards from the top of the drop (arrows in figure 2 a), spontaneous emulsification inside and outside the drop (arrows in figure 2 b–d), and two oil–microdroplet rings generated by convection rolls and being suspended inside the drop (arrows in figure 2 e).

Figure 2. Experimental snapshots of a dissolving water–ethanol (v/v 40/60) drop in anethole oil experiencing different characteristic states. The first column of photos are top views and the second column are the corresponding side views. The initial drop size is approximately $0.5~\unicode[STIX]{x03BC}\text{l}$ and $t_{0}$ denotes the drop deposition time. (a) The drop begins with a transparent appearance surrounded by a clean host liquid. The arrow indicates a solute plume above the drop. (b) W-in-o emulsions (water microdroplets) suspend outside the drop. The arrows and the inserted zoom-in panel highlight the location of the microdroplets. (c) O-in-w emulsions (oil microdroplets) appear inside the drop with a preferential location around the equator of the drop (arrow, and area in between the two horizontal lines). (d) More oil microdroplets are formed and concentrate at the two sides of the equator. The w-in-o emulsions disappear. (e) In approximately 7 min, the two concentrated clusters of microdroplets evolve into two rings. The scale bar is 0.5 mm in all panels. (See corresponding supplementary movies 2 and 3 available at https://doi.org/10.1017/jfm.2019.207.)

At the beginning when the drop was deposited on a hydrophobic substrate in the oil host liquid, both the drop and the surrounding oil are transparent, except for the shadows above the drop, as displayed in figure 2(a). The shadows, indicated by the arrow in the panel, are the solute plumes, i.e. the ethanol-rich oil mixture as ethanol diffuses into the oil surrounding the drop.

In our system, there are two kinds of self-emulsification that happen: w-in-o emulsification and o-in-w emulsification. The former creates water microdroplets in the oil host liquid (w-in-o emulsion), and the latter generates oil microdroplets in the aqueous drop (o-in-w emulsion). The first appearing microdroplets are the w-in-o emulsion ones. They nucleate at a certain location in the surrounding oil, comparable to the position of the Earth’s tropic of capricorn. The emulsification normally starts within less than 5 s after the drop deposition. Figure 2(b) is a snapshot taken at 19 s to provide a visualization of the emulsion, and the inserted zoom-in panel highlights their position, indicated by arrows. The microdroplets suspend in oil for a while and then disappear. Approximately half a minute later, o-in-w emulsification sets in inside the drop, preferentially in the middle of the drop, concentrating in the region pointed at by the arrow in figure 2(c). Notably, the preferential location is different from that reported in our previous work on evaporating drops, in which droplet nucleation occurred either at the contact-line region for flat evaporating ouzo drops (Tan et al. Reference Tan, Diddens, Lv, Kuerten, Zhang and Lohse2016), or at the top of the drop for spherical evaporating ouzo drops (Tan et al. Reference Tan, Diddens, Versluis, Butt, Lohse and Zhang2017). More detailed discussion about those spontaneous emulsifications will be given in § 4.

Yet another remarkable phenomenon is that the generated o-in-w emulsions gradually form two rings of microdroplets in the drop. The generated microdroplets first split into two groups, one above and one below the equator of the drop, which is shown in figure 2(d). The mechanism of the migration comes from two Marangoni convection rolls located separately above and below the equatorial plane (see supplementary movie 1 available at https://doi.org/10.1017/jfm.2019.207). The Marangoni flow motion is induced by surface tension gradients due to concentration variations along the interface, i.e. solutal Marangoni flow. The convection rolls drive the nucleated oil microdroplets and lead to an accumulation of these in the centre of each vortex roll, resulting in the formation of two rings of oil microdroplets as shown in figure 2(e). A more detailed discussion on the dynamics will be given in § 7. When the ethanol in the drop finally has dissolved, the solute Marangoni effect stops and the rings disintegrate (movie 1).

All these interesting phenomena happen in the first stage, when ethanol dissolving from the drop to the surrounding oil prevails. This stage takes approximately 30 min. In the second stage, the remaining water diffuses at an extremely slow speed, and no further unexpected phenomena occur. Therefore, in this paper we investigate only the first dissolving stage, up to the time the alcohol has dissolved. It is important to point out that, during the whole dissolution process, there is always a distinguishable interface between the oil host medium and the drop medium for all the experiments (drops with different water/ethanol ratios) that we performed. The sharp boundary of the drop corresponds to near-discontinuities in the gradient of the concentration–distance curve (Hartley Reference Hartley1946), which reflects the fact that the drop solution and host solution are macroscopically phase-separated at the drop–oil interface. We also stress that our experiments are very reproducible, even quantitatively.

3.2 Dissolution of drops with different initial ethanol concentrations

To quantitatively investigate the phenomena, we repeated the dissolution experiment with drops of different initial water/ethanol ratios, from 30 vol% to 70 vol% ethanol. Figure 3 shows the dissolution characteristics of the drops, including the temporal evolution of the drop volume, the variations of the contact angle $\unicode[STIX]{x1D703}$ and the footprint diameter $L$ . The annotations of the geometrical variables are available in an experimental picture (figure 3 e).

The volume evolutions of the drops are non-dimensionalized by the initial drop size $V_{0}$ to demonstrate a declining trend of the residual water volume with increasing initial ethanol concentration, as apparent from figure 3(a). That panel also reveals that indeed all drops experience two stages with two distinguishable dissolving rates, as discussed above: these two stages correspond successively to the initial stage dominated by the dissolution of the ethanol, and the subsequent slow dissolution of the remaining water. This is supported by the consistency between the initial water ratios and the drop residual volume percentage after the stage transition. The same behaviour also exists in the evaporation process of multicomponent drops (Liu, Bonaccurso & Butt Reference Liu, Bonaccurso and Butt2008; Tan et al. Reference Tan, Diddens, Lv, Kuerten, Zhang and Lohse2016). Water has an extremely small solubility in oil (immiscible), and therefore the second stage takes a much longer time than the first one (figure 3 d).

Figure 3. Morphology evolution of the dissolving drops in five different initial water/ethanol ratios (v/v 30/70, 40/60, 50/50, 60/40 and 70/30). (a–c) Temporal evolutions of the non-dimensionalized volume $V/V_{0}$ by initial drop size $V_{0}$ , contact angle $\unicode[STIX]{x1D703}$ and footprint diameter $L$ , respectively, during the dissolution. (d) For one case (v/v 30/70) the volume evolution of the whole dissolution process is displayed as the long-time behaviour of droplet volume. The shaded area on the left is shown in panel (a). (e) A recording image of a dissolving drop, with annotations of the geometrical parameters.

The evolutions of the contact angle (figure 3 b) and the footprint diameter (figure 3 c) also reveal the variation of the ethanol content in the drop. In the first 10–15 min, the contact angle increases by approximately $20^{\circ }$ , accompanied by a receding of the contact line. The increase of the contact angle is a result of the rising water concentration in the drop as ethanol is dissolving much faster than water. In this period, neither the constant contact radius (CR) mode nor the constant angle (CA) mode applies (Lohse & Zhang Reference Lohse and Zhang2015). In the second stage, the drop evolves nearly in the CR mode – there is a slightly decreasing contact angle with a stabilized contact area. During the investigated process, the drop has a very high contact angle, more than $150^{\circ }$ , because of the higher interfacial energy between the substrate and anethole compared to the energy between the substrate and the aqueous solution.

It is worth noting that, at the very beginning ( ${\sim}2~\text{min}$ ), the contact angle decreases with time and the footprint diameter increases by a small amount. To the best of our knowledge, this is the first observation of this phenomenon in dissolving multicomponent drops, although there are a few reports on this phenomenon in evaporating multicomponent drops (Liu et al. Reference Liu, Bonaccurso and Butt2008; Sefiane, David & Shanahan Reference Sefiane, David and Shanahan2008). A plausible explanation is that after deposition of the drop, the ethanol molecules in the drop tend to move towards the surface because of the lower interfacial energy between the hydrophobic substrate and ethanol compared to water, which results in the decreasing contact angle and the increasing contact area size (Liu et al. Reference Liu, Bonaccurso and Butt2008).

5.1 Idea of the model

More quantitative insight into the spontaneous emulsification is gained by theoretically analysing the multi-diffusion process of the water–ethanol drop dissolving in the host liquid. A pure diffusion model is developed for the early stage of the diffusion process, which, together with the so-called diffusion path method proposed by Kirkaldy & Brown (Reference Kirkaldy and Brown1963) and Ruschak & Miller (Reference Ruschak and Miller1972), is used to predict the appearance or absence of emulsification. Our model consists of two parts, namely one part calculating the liquid–liquid equilibrium at the interface of the two regions (§ 5.2), and the other part modelling the mass transport in the two multicomponent fluids in contact (§ 5.3). The mass transport is modelled as a one-dimensional problem with a moving interface separating the aqueous phase and the oil-rich phase, known as the Stefan problem (Crank Reference Crank1979). The liquid–liquid equilibrium of the ternary mixture is calculated by applying the condition of equal chemical potentials (fundamental thermodynamic relation) in combination with the UNIFAC model to quantify the non-ideality of the mixture.

Mass transport in liquids happens on a long time scale compared to the macroscopic phase separation at the interface, and therefore these two processes are decoupled in our model. To decouple them, the following assumptions were made. First, we assume that the equilibrium at the interface is instantaneously achieved and remains stable during the mass transport across the interface. Second, to further simplify the model, we also assume zero boundary thickness, i.e. disregarding the microscopic details of the thermodynamic equilibrium process. With these assumptions, the interfacial composition is directly given by the liquid–liquid equilibrium calculation and thereby determines the mass transport model.

Stefan problems commonly exist in many studies involving diffusion, such as heat transfer with a phase transition (thawing, freezing, melting), or moisture transport of swelling grains or polymers (Barry & Caunce Reference Barry and Caunce2008). Classic solutions to Stefan problems are given by Crank (Reference Crank1979). In § 5.3, we present definitions and the derivation of the equations with respect to our problem.

But, before that, in § 5.2, we give a brief description of the applied equilibrium theory. The establishment of the liquid–liquid equilibria between two phases happens on a short time scale compared to the diffusion process in the adjacent bulk regions. Once the aqueous medium and the host liquid are in contact, thermodynamic equilibrium favours an oil-rich phase on one side, coexisting with a water-rich phase on the other side, i.e. a macroscopic phase separation, which is observed as a sharp boundary of the water–ethanol drop in oil. The compositions of these two phases are situated on the binodal curve (or coexistence curve) in the ternary phase diagram of the system and can be connected by a tie line (see figure 5 b).

5.2 Liquid–liquid equilibrium at the interface

The liquid–liquid equilibrium is achieved when the chemical potentials $\unicode[STIX]{x1D707}_{\unicode[STIX]{x1D6FC}}$ are the same in both phases for each species $\unicode[STIX]{x1D6FC}$ . The chemical potential is a function of pressure, temperature and mole fractions $x_{\unicode[STIX]{x1D6FC}}$ of the liquid constituents. At fixed temperature and pressure, a three-component liquid–liquid equilibrium system has only a single degree of freedom (Chu & Prosperetti Reference Chu and Prosperetti2016). The condition of equality of the chemical potentials then reduces to

(5.1) $$\begin{eqnarray}x_{\unicode[STIX]{x1D6FC}}^{\text{d}}\,\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D6FC}}^{\text{d}}=x_{\unicode[STIX]{x1D6FC}}^{\text{h}}\,\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D6FC}}^{\text{h}},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FE}$ is the liquid activity coefficient, a correction factor accounting for the non-ideality of the mixture. The superscripts indicate the region, i.e. either drop region (d) or host liquid region (h).

Figure 5. Phase separation predicted by UNIFAC. (a) Grey regions indicated homogeneous mixing, whereas phase separation is expected in the coloured regions. The blue line with the blue stars (measured data points) indicates a good agreement with the titration experiments of Tan et al. (Reference Tan, Diddens, Versluis, Butt, Lohse and Zhang2017). If the liquid is undergoing phase separation, the composition of the resulting two phases can be read off from the binodal in (b) by the colour code. As an example, we give the tie line for the pale blue region.

The mixture can only undergo a phase separation if the mixture is non-ideal, i.e.  $\unicode[STIX]{x1D6FE}_{\unicode[STIX]{x1D6FC}}\neq 1$ . Thus, the activity coefficients are fundamental to accurately describe the liquid–liquid equilibrium. Unfortunately, we have found neither sufficiently detailed experimental data for the present mixture, nor parameters for the UNIQUAC model that was used by Chu & Prosperetti (Reference Chu and Prosperetti2016). Therefore, we employed the UNIFAC model (Fredenslund, Jones & Prausnitz Reference Fredenslund, Jones and Prausnitz1975), which is more general than the UNIQUAC model, to perform the splitting of the molecules into functional subgroups. For detailed information about this model, we refer to Fredenslund, Gmehling & Rasmussen (Reference Fredenslund, Gmehling and Rasmussen1977). Owing to the better agreement with the titration experiments by Tan et al. (Reference Tan, Diddens, Versluis, Butt, Lohse and Zhang2017), we took the recent modified UNIFAC (Dortmund) parametrization (Constantinescu & Gmehling Reference Constantinescu and Gmehling2016).

In order to find the liquid–liquid equilibrium curve and the regions of phase separation, we followed the method proposed by Zuend et al. (Reference Zuend, Marcolli, Peter and Seinfeld2010). If (5.1) can only be solved trivially, i.e.  $x_{\unicode[STIX]{x1D6FC}}^{\text{d}}=x_{\unicode[STIX]{x1D6FC}}^{\text{h}}$ , or if the Gibbs free energy at the trivial solution is below the Gibbs free energy of all non-trivial solutions, the mixture remains in a well-mixed single-phase configuration. On the contrary, if there is a non-trivial solution with a lower Gibbs free energy than the trivial solution, phase separation is expected and the system will take on the solution with the globally minimal Gibbs free energy.

The data obtained by this procedure are depicted in figure 5. In figure 5(a), the region of phase separation is shown in a ternary diagram in terms of mass fractions  $m_{\unicode[STIX]{x1D6FC}}$ . In the grey regions, the liquid at the specific composition remains perfectly mixed, whereas in the coloured regions, phase separation occurs. The composition of the two resulting phases can be read off from figure 5(b) by the colour code. The system splits into an aqueous phase and an oil-rich phase whose compositions are indicated by the respective same colour. Although the UNIFAC model is in general not perfect due to its dependence on the parameter table, the comparison of the phase separation region with the titration data of Tan et al. (Reference Tan, Diddens, Lv, Kuerten, Zhang and Lohse2016) in figure 5(a) (blue stars) shows very good agreement.

5.3 Mass transport

The system under discussion is composed of three species in two different phases, i.e. it is a multicomponent and multiphase diffusion system. The diffusion process is modelled in a one-dimensional infinite space, as illustrated in figure 6. The diffusion process of each individual constituent is assumed to depend only on its own concentration gradient in the radial direction (Fick’s diffusion laws). The $m_{\unicode[STIX]{x1D6FC}}$ denote the mass fractions of the species water ( $\unicode[STIX]{x1D6FC}=\text{w}$ ), ethanol ( $\unicode[STIX]{x1D6FC}=\text{e}$ ) and anethole oil ( $\unicode[STIX]{x1D6FC}=\text{o}$ ) as a function of time $t$ and position $x$ . Again, the superscripts indicate the region, i.e. either drop region (d) or host liquid region (h). The two regions are separated by the moving interface at position $s(t)$ . The initial position of the interface is set to the origin, i.e.  $s(0)=0$ . The mass transport of each component $\unicode[STIX]{x1D6FC}$ is governed by diffusion equations, namely

(5.2a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}m_{\unicode[STIX]{x1D6FC}}^{\text{d}}}{\unicode[STIX]{x2202}t}=D_{\unicode[STIX]{x1D6FC}}^{\text{d}}\frac{\unicode[STIX]{x2202}^{2}m_{\unicode[STIX]{x1D6FC}}^{\text{d}}}{\unicode[STIX]{x2202}x^{2}},\quad -\infty <x\leqslant s(t), & \displaystyle\end{eqnarray}$$

(5.2b ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}m_{\unicode[STIX]{x1D6FC}}^{\text{h}}}{\unicode[STIX]{x2202}t}=D_{\unicode[STIX]{x1D6FC}}^{\text{h}}\frac{\unicode[STIX]{x2202}^{2}m_{\unicode[STIX]{x1D6FC}}^{\text{h}}}{\unicode[STIX]{x2202}x^{2}},\quad s(t)\leqslant x<+\infty , & \displaystyle\end{eqnarray}$$

(5.2c ) $$\begin{eqnarray}\displaystyle & \displaystyle (m_{\unicode[STIX]{x1D6FC}}^{\text{h}}-m_{\unicode[STIX]{x1D6FC}}^{\text{d}})\frac{\text{d}s}{\text{d}t}=D_{\unicode[STIX]{x1D6FC}}^{\text{d}}\frac{\unicode[STIX]{x2202}m_{\unicode[STIX]{x1D6FC}}^{\text{d}}}{\unicode[STIX]{x2202}x}-D_{\unicode[STIX]{x1D6FC}}^{\text{h}}\frac{\unicode[STIX]{x2202}m_{\unicode[STIX]{x1D6FC}}^{\text{h}}}{\unicode[STIX]{x2202}x},\quad \text{when }x=s(t). & \displaystyle\end{eqnarray}$$

$D_{\unicode[STIX]{x1D6FC}}^{\text{d}}$

$D_{\unicode[STIX]{x1D6FC}}^{\text{h}}$

(5.3a ) $$\begin{eqnarray}\displaystyle & \displaystyle m_{\text{w}}^{\text{d}}+m_{\text{e}}^{\text{d}}+m_{\text{o}}^{\text{d}}=1, & \displaystyle\end{eqnarray}$$

(5.3b ) $$\begin{eqnarray}\displaystyle & \displaystyle m_{\text{w}}^{\text{h}}+m_{\text{e}}^{\text{h}}+m_{\text{o}}^{\text{h}}=1. & \displaystyle\end{eqnarray}$$

$\unicode[STIX]{x1D6FC}=\text{w}$

$\unicode[STIX]{x1D6FC}=\text{o}$

$\unicode[STIX]{x1D6FC}=\text{e}$

Figure 6. Sketch of the one-dimensional multiphase and multicomponent diffusion model with a moving interface at $x=s(t)$ separating the two regions of drop (d) and host liquid (h). The subscripts stand for the species water (w) and anethole oil (o). The mass fractions of the different species in the two different regions $m_{\unicode[STIX]{x1D6FC}}^{\text{d}}$ and $m_{\unicode[STIX]{x1D6FC}}^{\text{h}}$ are a function of time $t$ and position $x$ . The initial position of the interface $s(0)$ is defined to be at the origin.

As we focus only on the early stage of the diffusion process, the boundaries at $\pm \infty$ do not influence the dynamics in the vicinity of the interface. Therefore, the initial condition and the far-field boundary conditions are given by

(5.4a ) $$\begin{eqnarray}\displaystyle & \displaystyle m_{\unicode[STIX]{x1D6FC}}^{\text{d}}=m_{\unicode[STIX]{x1D6FC}0}^{\text{d}},\quad -\infty <x<0,\quad t=0, & \displaystyle\end{eqnarray}$$

(5.4b ) $$\begin{eqnarray}\displaystyle & \displaystyle m_{\unicode[STIX]{x1D6FC}}^{\text{h}}=m_{\unicode[STIX]{x1D6FC}0}^{\text{h}},\quad 0<x<\infty ,\quad t=0, & \displaystyle\end{eqnarray}$$

(5.4c ) $$\begin{eqnarray}\displaystyle & \displaystyle m_{\unicode[STIX]{x1D6FC}}^{\text{d}}=m_{\unicode[STIX]{x1D6FC}0}^{\text{d}},\quad x=-\infty ,\quad t>0, & \displaystyle\end{eqnarray}$$

(5.4d ) $$\begin{eqnarray}\displaystyle & \displaystyle m_{\unicode[STIX]{x1D6FC}}^{\text{h}}=m_{\unicode[STIX]{x1D6FC}0}^{\text{h}},\quad x=\infty ,\quad t>0. & \displaystyle\end{eqnarray}$$

$m_{\unicode[STIX]{x1D6FC}s}^{\text{d}}$

$m_{\unicode[STIX]{x1D6FC}s}^{\text{h}}$

(5.5a ) $$\begin{eqnarray}\displaystyle & \displaystyle m_{\unicode[STIX]{x1D6FC}}^{\text{d}}=m_{\unicode[STIX]{x1D6FC}s}^{\text{d}},\quad x=s(t),\quad t>0 & \displaystyle\end{eqnarray}$$

(5.5b ) $$\begin{eqnarray}\displaystyle & \displaystyle m_{\unicode[STIX]{x1D6FC}}^{\text{h}}=m_{\unicode[STIX]{x1D6FC}s}^{\text{h}},\quad x=s(t),\quad t>0. & \displaystyle\end{eqnarray}$$

$m_{\text{w}s}^{\text{d}}$

Following the method successfully used by Kirkaldy & Brown (Reference Kirkaldy and Brown1963), Ruschak & Miller (Reference Ruschak and Miller1972) and Crank (Reference Crank1987), we assume the movement to be proportional to $\sqrt{t}$ , which is consistent with the fact that the interface movement is driven by diffusion, i.e. 

(5.6a,b ) $$\begin{eqnarray}s(t)=\unicode[STIX]{x1D706}\sqrt{t}\quad \text{and}\quad \sqrt{D_{s}}=|\unicode[STIX]{x1D706}|,\end{eqnarray}$$

where the sign of the constant $\unicode[STIX]{x1D706}$ gives the direction of the interface movement, and $D_{s}$ can be regarded as the ‘diffusion coefficient’ of the moving interface. Thus, the solution of the governing diffusion equations (5.2) with the given conditions (5.4) reads

(5.7a ) $$\begin{eqnarray}\displaystyle & \displaystyle m_{\unicode[STIX]{x1D6FC}}^{\text{d}}={\mathcal{B}}_{\unicode[STIX]{x1D6FC}}^{1}+{\mathcal{B}}_{\unicode[STIX]{x1D6FC}}^{2}\,\text{erf}\,\frac{x}{2\sqrt{D_{\unicode[STIX]{x1D6FC}}^{\text{d}}t}}, & \displaystyle\end{eqnarray}$$

(5.7b ) $$\begin{eqnarray}\displaystyle & \displaystyle m_{\unicode[STIX]{x1D6FC}}^{\text{h}}={\mathcal{B}}_{\unicode[STIX]{x1D6FC}}^{3}+{\mathcal{B}}_{\unicode[STIX]{x1D6FC}}^{4}\,\text{erf}\,\frac{x}{2\sqrt{D_{\unicode[STIX]{x1D6FC}}^{\text{h}}t}}, & \displaystyle\end{eqnarray}$$

$x/\sqrt{t}$

${\mathcal{B}}_{\unicode[STIX]{x1D6FC}}^{1}$

${\mathcal{B}}_{\unicode[STIX]{x1D6FC}}^{2}$

${\mathcal{B}}_{\unicode[STIX]{x1D6FC}}^{3}$

${\mathcal{B}}_{\unicode[STIX]{x1D6FC}}^{4}$

(5.8a ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{B}}_{\unicode[STIX]{x1D6FC}}^{1}=\frac{m_{\unicode[STIX]{x1D6FC}0}^{\text{d}}\,\text{erf}\,{\displaystyle \frac{s}{2\sqrt{D_{\unicode[STIX]{x1D6FC}}^{\text{d}}t}}}+m_{\unicode[STIX]{x1D6FC}s}^{\text{d}}}{\text{erf}\,{\displaystyle \frac{s}{2\sqrt{D_{\unicode[STIX]{x1D6FC}}^{\text{d}}t}}}+1}, & \displaystyle\end{eqnarray}$$

(5.8b ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{B}}_{\unicode[STIX]{x1D6FC}}^{2}=\frac{m_{\unicode[STIX]{x1D6FC}s}^{\text{d}}-m_{\unicode[STIX]{x1D6FC}0}^{\text{d}}}{\text{erf}\,{\displaystyle \frac{s}{2\sqrt{D_{\unicode[STIX]{x1D6FC}}^{\text{d}}t}}}+1}, & \displaystyle\end{eqnarray}$$

(5.8c ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{B}}_{\unicode[STIX]{x1D6FC}}^{3}=\frac{m_{\unicode[STIX]{x1D6FC}0}^{\text{h}}\,\text{erf}\,{\displaystyle \frac{s}{2\sqrt{D_{\unicode[STIX]{x1D6FC}}^{\text{h}}t}}}-m_{\unicode[STIX]{x1D6FC}s}^{\text{h}}}{\text{erf}\,{\displaystyle \frac{s}{2\sqrt{D_{\unicode[STIX]{x1D6FC}}^{\text{h}}t}}}-1}, & \displaystyle\end{eqnarray}$$

(5.8d ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{B}}_{\unicode[STIX]{x1D6FC}}^{4}=\frac{m_{\unicode[STIX]{x1D6FC}s}^{\text{h}}-m_{\unicode[STIX]{x1D6FC}0}^{\text{h}}}{\text{erf}\,{\displaystyle \frac{s}{2\sqrt{D_{\unicode[STIX]{x1D6FC}}^{\text{h}}t}}}-1}, & \displaystyle\end{eqnarray}$$

${\mathcal{B}}_{\unicode[STIX]{x1D6FC}}^{i}$

$\unicode[STIX]{x1D706}$

To simplify the expression of the equations, we introduce the definitions

(5.9a-c ) $$\begin{eqnarray}{\mathcal{K}}_{\unicode[STIX]{x1D6FC}}=\frac{\unicode[STIX]{x1D706}}{2\sqrt{D_{\unicode[STIX]{x1D6FC}}^{\text{h}}}},\quad {\mathcal{R}}_{\unicode[STIX]{x1D6FC}}=\sqrt{\frac{D_{\unicode[STIX]{x1D6FC}}^{\text{h}}}{D_{\unicode[STIX]{x1D6FC}}^{\text{d}}}},\quad \unicode[STIX]{x1D712}_{\text{o}}^{\text{w}}=\sqrt{\frac{D_{\text{o}}^{\text{h}}}{D_{\text{w}}^{\text{h}}}},\end{eqnarray}$$

which together with equation (5.6) are substituted into the coefficients (5.8) and the Stefan condition (5.2c ). Thereby, we obtain the coefficient constants as

(5.10a ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{B}}_{\unicode[STIX]{x1D6FC}}^{1}=\frac{m_{\unicode[STIX]{x1D6FC}0}^{\text{d}}\,\text{erf}\,({\mathcal{R}}_{\unicode[STIX]{x1D6FC}}{\mathcal{K}}_{\unicode[STIX]{x1D6FC}})+m_{\unicode[STIX]{x1D6FC}s}^{\text{d}}}{\text{erf}\,({\mathcal{R}}_{\unicode[STIX]{x1D6FC}}{\mathcal{K}}_{\unicode[STIX]{x1D6FC}})+1}, & \displaystyle\end{eqnarray}$$

(5.10b ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{B}}_{\unicode[STIX]{x1D6FC}}^{2}=\frac{m_{\unicode[STIX]{x1D6FC}s}^{\text{d}}-m_{\unicode[STIX]{x1D6FC}0}^{\text{d}}}{\text{erf}\,({\mathcal{R}}_{\unicode[STIX]{x1D6FC}}{\mathcal{K}}_{\unicode[STIX]{x1D6FC}})+1}, & \displaystyle\end{eqnarray}$$

(5.10c ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{B}}_{\unicode[STIX]{x1D6FC}}^{3}=\frac{m_{\unicode[STIX]{x1D6FC}0}^{\text{h}}\,\text{erf}\,{\mathcal{K}}_{\unicode[STIX]{x1D6FC}}-m_{\unicode[STIX]{x1D6FC}s}^{\text{h}}}{\text{erf}\,{\mathcal{K}}_{\unicode[STIX]{x1D6FC}}-1}, & \displaystyle\end{eqnarray}$$

(5.10d ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{B}}_{\unicode[STIX]{x1D6FC}}^{4}=\frac{m_{\unicode[STIX]{x1D6FC}s}^{\text{h}}-m_{\unicode[STIX]{x1D6FC}0}^{\text{h}}}{\text{erf}\,{\mathcal{K}}_{\unicode[STIX]{x1D6FC}}-1}, & \displaystyle\end{eqnarray}$$

(5.11) $$\begin{eqnarray}\sqrt{\unicode[STIX]{x03C0}}{\mathcal{K}}_{\unicode[STIX]{x1D6FC}}(m_{\unicode[STIX]{x1D6FC}s}^{\text{h}}-m_{\unicode[STIX]{x1D6FC}s}^{\text{d}})=4\text{e}^{-{\mathcal{K}}_{\unicode[STIX]{x1D6FC}}^{2}}[-{\mathcal{B}}_{\unicode[STIX]{x1D6FC}}^{4}+\text{e}^{{\mathcal{K}}_{\unicode[STIX]{x1D6FC}}^{2}(1-{\mathcal{R}}_{\unicode[STIX]{x1D6FC}}^{2})}{\mathcal{B}}_{\unicode[STIX]{x1D6FC}}^{2}/{\mathcal{R}}_{\unicode[STIX]{x1D6FC}}].\end{eqnarray}$$

We thus have a closed-form solution for the problem, given by the mass balance equation (5.11) in adjunction with thermodynamic equilibria theory. The number of mass balance equations is two, one for water ( $\unicode[STIX]{x1D6FC}=\text{w}$ ) and the other for anethole oil ( $\unicode[STIX]{x1D6FC}=\text{o}$ ), and the five unknown variables are $m_{\text{w}s}^{\text{d}}$ , $m_{\text{o}s}^{\text{d}}$ , $m_{\text{w}s}^{\text{h}}$ , $m_{\text{o}s}^{\text{h}}$ and $\unicode[STIX]{x1D706}$ . As discussed above, the values of the two compositions on the two sides of the interface are situated on the binodal curve and are connected by a tie line in the phase diagram (see figure 5 b). Owing to these equilibrium constraints, the number of degrees of freedom of the interfacial composition is one (for instance $m_{\text{w}s}^{\text{d}}$ ). Hence, there are in total two unknown variables, for instance $m_{\text{w}s}^{\text{d}}$ and $\unicode[STIX]{x1D706}$ , in the equation. By applying Newton’s method with a given initial guess, we can find roots to the two equations (5.11) (one for $\unicode[STIX]{x1D6FC}=\text{w}$ and one for $\unicode[STIX]{x1D6FC}=\text{o}$ ) and the whole problem is solved. All quantities defined in the model are listed in table 2 given in appendix C.

5.4 Diffusion coefficients

In the model, there are in total four different diffusion coefficients, namely in each of the two different regions one coefficient for one of the two species: water diffusivity in the drop medium $D_{\text{w}}^{\text{d}}$ , anethole oil diffusivity in the drop medium $D_{\text{o}}^{\text{d}}$ , water diffusivity in the host liquid medium $D_{\text{w}}^{\text{h}}$ , and anethole oil diffusivity in the host liquid medium  $D_{\text{o}}^{\text{h}}$ .

To acquire the diffusivities, we were confined to models and assumptions, since direct measurement is complicated. In the drop medium at the early stage, the trans-anethole content is negligible, so that the coefficient $D_{\text{w}}^{\text{d}}$ is assumed to be given by the mutual diffusivity in a binary water–ethanol mixture. The corresponding values were obtained by fitting experimental data from Pařez et al. (Reference Pařez, Guevara-Carrion, Hasse and Vrabec2013). The coefficient $D_{\text{o}}^{\text{d}}$ , i.e. dilute anethole in a water–ethanol mixture, is calculated in the limit of infinite dilution based on the model of Perkins & Geankoplis (Reference Perkins and Geankoplis1969), and is listed in § A.4. Both $D_{\text{w}}^{\text{d}}$ and $D_{\text{o}}^{\text{d}}$ depend on the initial ethanol content in the drop.

The model of Hayduk & Minhas (Reference Hayduk and Minhas1982) was used to calculate the coefficients of dilute water in host anethole oil, $D_{\text{w}}^{\text{h}}$ . The corresponding diffusivity constant $D_{\text{o}}^{\text{h}}$ is estimated according to the Stokes–Einstein equation, which states that the diffusion coefficient is inversely proportional to the size of molecules at constant viscosity. Since both ethanol and water content in the host liquid are negligible, we take $D_{\text{w}}^{\text{h}}=1.2\times 10^{-9}~\text{m}^{2}~\text{s}^{-1}$ and $D_{\text{o}}^{\text{h}}=0.66\times 10^{-9}~\text{m}^{2}~\text{s}^{-1}$ , both of which are assumed to be constant.

6.1 Concentration profiles

The mass concentration distribution of each constituent as a function of time is obtained from the model described above. Figure 7(a–c) and corresponding close-up images, figure 7(d–g), show the development of the concentration profiles for different cases of binary drops with different initial water/ethanol ratios ( $70/30$ v/v in (a), 50/50 v/v in (b) and 30/70 v/v in (c); animations of 40/60 v/v and 60/40 v/v are available in movie 5). Anethole oil, water and ethanol are labelled with yellow, blue and red colours, respectively. The vertical black dash-dotted line stands for the initial position of the interface of the aqueous water–ethanol region (left side) and the anethole oil-rich region (right side). Initially, the concentration profile of each species is a step function (first row of figure 7 a–c), which is consistent with the initial condition (5.4a,b ) of the model. Different initial ethanol contents in the drop account for the difference of the step heights in the panels.

Figure 7. Calculated results of the diffusion model. (a–c) Snapshots of the mass fraction profile at 0 s (first row) and 20 s (second row) for different cases. The vertical black dash-dotted line indicates the initial position of the interface, which separates the drop solution on the left and the host liquid on the right. Note that $m_{\unicode[STIX]{x1D6FC}}$ represents mass fractions, which are different from volume fractions. Panels (d,e) are zoom-in pictures of panels (a,c) at $t=20~\text{s}$ , showing the profile evolution. Panels (f,g) are zoom-ins of panel (e). The parameters used in the model are described in the text.

In the close-up figure 7(d), once the diffusion starts, the discontinuity in the profile at $t=0~\text{s}$ is immediately smoothed, and new turning points emerge. The dynamics of the concentration profiles is apparent from the temporal curves at $t=10~\text{ms}$ , $t=1~\text{s}$ , $t=2~\text{s}$ and so on.

The concentration gradient at the interface induces a diffusive flux for each constituent towards or from infinity (anethole in figure 7(d), ethanol and water in figure 7(e)). The step height at infinity remains unchanged due to the Dirichlet boundary condition (5.4c,d ). In the investigated Stefan problem, the interface has the freedom to shift towards the aqueous region (left side), for all the constituents in the system. After 20 s, the developed concentration profiles in different cases are displayed in the second row of figure 7(a–c). The heights of the new turning points, corresponding to the interfacial concentration $m_{\unicode[STIX]{x1D6FC}s}^{\text{d}}$ and $m_{\unicode[STIX]{x1D6FC}s}^{\text{h}}$ (ethanol, $\unicode[STIX]{x1D6FC}=\text{e}$ , in figure 7 e), are fixed during the movement of the interface, which reflects the assumption of the stability of liquid–liquid equilibrium at the interface. Figure 7(f,g) show further zoom-ins of the interface in figure 7(e).

6.2 Diffusion path theory and calculated results

The diffusion path method was proposed by Kirkaldy & Brown (Reference Kirkaldy and Brown1963), who mapped the composition of the ternary solution on its phase diagram (figure 5 b). The values of the composition along the domains are mapped in the diagram as a line, known as the diffusion path or composition path. It provides an effective way of representing the relationship between kinetic and thermodynamic aspects for a multiphase and multicomponent system.

In the spirit of the pioneering work by Ruschak & Miller (Reference Ruschak and Miller1972), we predict the diffusion-induced spontaneous emulsification by examining the geometrical relationship between diffusion paths and the binodal curve. If the diffusion path crosses the binodal curve (figure 5 b) between start and end points, supersaturation in the media is present, which induces spontaneous emulsification. It is noteworthy that the intersections of all considered diffusion paths with the binodal curve are far away from the plait point and the intersections are located at the edge of the ouzo effect region.

In a ternary system, the path is determined by any two independent constituents. Since we know the evolution of the distribution of all species, we can mathematically express their relationship. We know that the coefficients (5.10a,b ) are constant for a certain case, which implies that (5.7a ) is well determined. Therefore, (5.7a ) gives functional relationships $m_{\text{w}}^{\text{d}}=f_{1}(x/\sqrt{t})$ and $m_{\text{o}}^{\text{d}}=f_{2}(x/\sqrt{t})$ . Upon substituting them into equation (5.3a ), we obtain the mathematical expression of the diffusion path in the drop region,

(6.1) $$\begin{eqnarray}m_{\text{e}}^{\text{d}}=1-{\mathcal{F}}(m_{\text{o}}^{\text{d}})-m_{\text{o}}^{\text{d}},\end{eqnarray}$$

where ${\mathcal{F}}=f_{1}\,f_{2}^{-1}$ . In the same way, we obtain the expression of the diffusion path in the host liquid medium,

(6.2) $$\begin{eqnarray}m_{\text{e}}^{\text{h}}=1-m_{\text{w}}^{\text{h}}-{\mathcal{G}}(m_{\text{w}}^{\text{h}}),\end{eqnarray}$$

where ${\mathcal{G}}=g_{2}\,g_{1}^{-1}$ , given $m_{\text{w}}^{\text{h}}=g_{1}(x/\sqrt{t})$ and $m_{\text{o}}^{\text{h}}=g_{2}(x/\sqrt{t})$ . Notably, the mathematical expressions reveal that the diffusion path is independent of time and space. In other words, the set of compositions at a certain position for different moments and the set for a certain moment among different positions share the same diffusion path. The composition at the end point of the diffusion path lies on the binodal curve and corresponds to the interfacial composition, whereas the composition at the start point of the diffusion path indicates the composition at infinity, which is the same as the initial composition in the domain.

Figure 8. Phase diagrams showing the calculated diffusion paths in the water–ethanol–anethole phase diagram (a), in the anethole–ethanol phase diagram (b), and in the water–ethanol phase diagram (c). The oil-rich part (host liquid) of the binodal curve is labelled in blue and the water-rich part (drop region) in red. The black dotted lines are tie lines. Panels (d,e) are zoom-ins of panel (c) with increasing magnifications, showing that the diffusion path in the host liquid passes through the binodal curve when the ethanol content is high ( ${\geqslant}50~\text{vol}\%$ ). In contrast, in the aqueous phase, in all cases, there is no diffusion path crossing the binodal curve.

Figure 8(a) shows the calculated results in the ternary diagram. The mass fraction phase diagrams of two independent constituents are given in figure 8(b,c). Figure 8(b) shows the anethole–ethanol phase diagram, highlighting the water-rich regime with o-in-w emulsions, whereas figure 8(c) with zoom-ins in figure 8(d,e) shows the water–ethanol phase diagram, enhancing the oil-rich regime with w-in-o emulsions. The binodal curve is divided by the plait point into two different colours, blue for the oil-rich phase (host liquid) and red for the water-rich phase (drop region). The tie lines were calculated by the method described in § 5.2. The dot-dashed lines with different colours are the diffusion paths for different cases (same colour labelling for all figure 8 a–e). The zoom-ins in figure 8(d,e) reveal that, when the ethanol volume fraction in the aqueous phase is higher than 51.98 vol%, corresponding to a transition mass fraction $m_{\text{e}\ast }^{\text{d}}$ calculated by the model, the diffusion path has to pass through the binodal curve to meet the equilibrium points (diamonds) at the binodal. The segment of the diffusion path below the binodal is in the supersaturation condition, which indicates the appearance of w-in-o emulsions in the oil-rich side (host liquid region). In contrast, in the water-rich side (figure 8 b), all the diffusion paths meet equilibrium points but without passing through the binodal curve. Therefore, according to the model, there is no supersaturation region induced by pure diffusion processes, which predicts the absence of o-in-w emulsions in the drop region. The time independence implies that self-emulsification immediately occurs once the diffusion process starts.

6.3 Self-emulsification: comparison between model predictions and experimental observations

Figure 9. (a) Comparison between the model prediction and the experimental observations for the appearance of w-in-o emulsions outside drops. The ordinate is the initial mass fraction of ethanol in the drop and the abscissa gives the trial number. The black and white squares are experimental observation data from table 1 and denote the presence and absence of emulsification, respectively. The red dashed line is a transition value calculated by the model, above which w-in-o emulsification is predicted to happen. (b) Calculation results of the total amount of oil transported into the drop as a function of the initial ethanol content of the drop using the one-dimensional diffusion model. Different symbols present the total amount of oil at different moments. The inset gives the same data on a log–log plot.

For w-in-o emulsion, the comparison between model prediction and experimental observations is presented in figure 9(a). The vertical ordinate denotes the initial ethanol mass fraction in the water–ethanol drop and the abscissa gives the trial number, i.e. the individual experiments at this particular initial composition. The data points are from table 1: the black symbol indicates the presence of w-in-o emulsion, and the white symbol means that the w-in-o emulsion is absent. The red dashed line is the calculated transition percentage $m_{\text{e}\ast }^{\text{d}}$ , above which (blue region) the model predicts the presence of w-in-o emulsion in the host liquid, otherwise absence (grey region). The comparison reveals the reasonable predictive power. In the blue region, all data points are black, whereas in the grey region far from transition line, all the data points are white. Around the transition line, both black and white data points exist.

For the o-in-w emulsion in the drop medium, no diffusion-triggered self-emulsification was observed in experiments within half a minute (table 1), which is consistent with the model prediction. However, oil microdroplets must appear since the ethanol content in the drop is reducing over time, while the oil content is increasing. At some point, the composition in the drop region will be supersaturated and cause the ouzo effect inside the drop. However, due to the infinite domain in the model with Dirichlet boundary conditions, the model cannot account for the long-time behaviour. The onset time for the oil microdroplets is at least half a minute or more, and has a negative correlation with the initial content of ethanol in the drop as recorded in table 1.

Although we are unable to obtain the overall composition of the three-dimensional drop from the one-dimensional model, it is possible to gain more understanding on the transport of anethole oil during the early stage by calculating the total amount of anethole transported into the drop, $M_{\text{o}}^{\text{d}}$ . Its definition is given by integrating the mass fraction of anethole $m_{\text{o}}^{\text{d}}$ on the interval of aqueous region from $-\infty$ to $s$ ,

(6.3) $$\begin{eqnarray}M_{\text{o}}^{\text{d}}=\int _{-\infty }^{s}(m_{\text{o}}^{\text{d}}-m_{\text{o}0}^{\text{d}})\,\text{d}x.\end{eqnarray}$$

Then we have

(6.4) $$\begin{eqnarray}M_{\text{o}}^{\text{d}}=(m_{\text{o}s}^{\text{d}}-m_{\text{o}0}^{\text{d}})\,s+\frac{(sm_{\text{o}s}^{\text{d}}-m_{\text{o}0}^{\text{d}})s}{\sqrt{\unicode[STIX]{x03C0}}{\mathcal{R}}_{\text{o}}{\mathcal{K}}_{\text{o}}}\,\frac{\text{e}^{-{\mathcal{R}}_{\text{o}}^{2}{\mathcal{K}}_{\text{o}}^{2}}}{\text{erf}\,({\mathcal{R}}_{\text{o}}{\mathcal{K}}_{\text{o}})+1}.\end{eqnarray}$$

Figure 9(b) shows that a higher ethanol percentage in the drop, $m_{\text{e}0}^{\text{d}}$ , leads to a higher rate of transport of oil into the drop, $M_{\text{o}}^{\text{d}}$ . The inset reveals that it is an exponential growth relationship. So the difference in the oil amount in the drop caused by the ethanol content increases over time (from red solid ${\vartriangleright}$ line, at 0.1 s, to purple solid ✡ line, at 30 s). This gives a possible explanation for the experimental observation, as a higher ethanol concentration in the drop leads to more oil after the early stage of the diffusion process, which favours the occurrence of the ouzo effect inside the drop.

6.4 Model limitations

Although the model provides information on the mass transport of the multiphase and multicomponent diffusion process, as well as a good prediction of the behaviour of spontaneous emulsification, it is subject to the following limitations: (i) The diffusion model is developed as a one-dimensional model without consideration of flow motion, i.e. a pure multiphase and multicomponent diffusion process. (ii) The dependence of diffusivity on the local species concentration is disregarded and off-diagonal diffusion terms, i.e. as in the Maxwell–Stefan theory, are not considered. (iii) We apply Dirichlet boundary conditions at infinity, which implies that the model is only applicable in the early stage of the diffusion process. In the long-time limit, the finite size of the regions, in particular of the drop, will become important, since there is not an infinite reservoir for the species. (iv) The detailed process of phase separation at the interface is not considered, as we apply an instantaneous liquid–liquid equilibrium assumption.

The flow motion in the system has a big impact on the position where emulsification takes place. A strong flow rate has the capacity to prevent emulsification by changing the local concentrations or by directly dissolving the nucleated emulsion. Therefore it is necessary to investigate the fluid dynamics of the system in § 7.

7.1 Scaling analysis

In the studied system, diffusion-induced advection, in the laminar flow regime, can in turn play a significant role in the diffusion process, i.e. diffusion and advection are highly coupled in the system. The diffusion causes non-uniformities in the species distribution in both drop and host liquid media, which drives bulk flow motions (advection). The flow reorganizes the concentration field and consequently, in turn, affects the diffusion. The generated advection mainly encompasses solutal Marangoni flow and buoyancy-driven flow (Dietrich et al. Reference Dietrich, Wildeman, Visser, Hofhuis, Kooij, Zandvliet and Lohse2016), i.e. solutal natural convection, whereas thermal effects are negligible compared to the two others. As solutal Marangoni flow and buoyancy-driven flow have alternating dominance inside and outside the drop, respectively, their impacts on diffusion can be treated separately. To confirm that only one of the two respective flows is dominant in the respective domain, we perform the following scaling analysis.

The characteristic velocity of the solutal Marangoni flow $U_{M}$ scales like

(7.1) $$\begin{eqnarray}U_{M}\sim \unicode[STIX]{x0394}\unicode[STIX]{x1D6FE}/\unicode[STIX]{x1D707},\end{eqnarray}$$

where $\unicode[STIX]{x1D707}$ is the dynamic viscosity (§ A.5), and $\unicode[STIX]{x0394}\unicode[STIX]{x1D6FE}$ is the interfacial surface tension difference caused by compositional variations. The characteristic velocity associated with the buoyancy-driven convection $U_{B}$ is obtained by balancing the viscous dissipation rate within the convection roll with the rate of gain of potential energy due to gravity (Tam et al. Reference Tam, von Arnim, McKinley and Hosoi2009), i.e. 

(7.2) $$\begin{eqnarray}\int \unicode[STIX]{x1D707}(\unicode[STIX]{x1D6FB}u)^{2}\,\text{d}V\sim \unicode[STIX]{x0394}\unicode[STIX]{x1D70C}gU_{B}\ell ^{3},\end{eqnarray}$$

where the left side scales as $\unicode[STIX]{x1D707}\ell (U_{B})^{2}$ . Here $\ell$ is the characteristic length scale and $g$ is the gravitational constant. Hence, the characteristic velocity of the buoyancy-driven flow $U_{B}$ scales as

(7.3) $$\begin{eqnarray}U_{B}\sim \unicode[STIX]{x0394}\unicode[STIX]{x1D70C}g\ell ^{2}/\unicode[STIX]{x1D707}.\end{eqnarray}$$

With (7.3) and (7.1), we can estimate the ratio of buoyancy-driven flow to Marangoni flow by

(7.4) $$\begin{eqnarray}\frac{U_{B}}{U_{M}}\sim \frac{\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}g\ell ^{2}}{\unicode[STIX]{x0394}\unicode[STIX]{x1D6FE}}.\end{eqnarray}$$

We can note that the ratio is a Bond number, which measures the importance of gravitational forces compared to interfacial surface tension. The Bond number is proportional to $\ell ^{2}$ , which indicates that the relative importance of the two flow mechanisms has a strong dependence on the spatial scale of the domain.

We can estimate the ratio inside the drop by taking the drop radius $R\sim 0.75~\text{mm}$ as the length scale $\ell$ . Since the interfacial surface tension is composition-dependent and varies during drop dissolution, we estimated the difference $\unicode[STIX]{x0394}\unicode[STIX]{x1D6FE}\sim 12.1~\text{mN}~\text{m}^{-1},$ as half of the interfacial surface tension between pure water and anethole oil (Tan et al. Reference Tan, Diddens, Versluis, Butt, Lohse and Zhang2017). The density difference is selected as the biggest density difference between water and ethanol, i.e.  $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D70C}_{\text{w}}-\unicode[STIX]{x1D70C}_{\text{e}}$ ( ${\sim}211~\text{kg}~\text{m}^{3}$ ). The estimation shows that $U_{B}\sim U_{M}/10$ , which implies that Marangoni flow inside the drop is prevailing. Outside the drop, the length scale $\ell$ is selected as the host liquid depth, $\ell =7.5~\text{mm}$ . Then the estimate of the velocity ratio becomes $U_{B}\sim 10U_{M}$ , which indicates that buoyancy-driven flow is dominating outside the drop.

7.2 Flow motions and its influence on spontaneous emulsification

Taking the understanding of the flow motion in the system from the scaling analysis, we disregard the Marangoni flow in the following discussion about the hydrodynamics in the host liquid (large-scale domain), whereas the buoyancy effect is neglected inside the drop (small-scale domain). The hydrodynamics outside and inside the drop are discussed successively.

As the water–ethanol drop dissolves, ethanol diffuses into the surrounding anethole oil. Then, buoyancy starts to play a role, as ethanol is less dense than water and anethole oil, i.e.  $\unicode[STIX]{x1D70C}_{\text{e}}<\unicode[STIX]{x1D70C}_{\text{o}}$ and $\unicode[STIX]{x1D70C}_{\text{e}}<\unicode[STIX]{x1D70C}_{\text{w}}$ ( $\unicode[STIX]{x1D70C}_{\text{w}}=998~\text{kg}~\text{m}^{-3}$ , $\unicode[STIX]{x1D70C}_{\text{e}}=787~\text{kg}~\text{m}^{-3}$ and $\unicode[STIX]{x1D70C}_{\text{o}}=988~\text{kg}~\text{m}^{-3}$ at $22\,^{\circ }\text{C}$ ). The surrounding ethanol-rich oil floats up in the form of a solute plume, causing an upwelling flow. Simultaneously, fresh oil liquid far from the drop replenishes to achieve mass continuity, as sketched in figure 1(a). So a convection flow outside the drop forms as a consequence, which is clearly observed through PIV measurements (figure 1 b). In the region next to the tropic of capricorn of the drop and not far from the drop surface, there is weak flow, enclosed by the yellow circle in the close-up image. This indicates a subtle influence from convection on the diffusion process in this region, whereas in other regions, the refresh oil brought by the relatively strong flow prevents the formation of local supersaturation. This is the reason that accounts for the appearance of w-in-o microdroplets in a certain position with the weak flow rate.

The generation of the buoyancy-driven convection affects the distribution of the diffusion rate along the drop surface. The convection flow brushes away the diffused ethanol next to the drop surface and varies the concentration distribution of ethanol in the surrounding oil. Around the equator of the drop the concentration boundary layer is thin, due to the intense inflow of bulk oil without ethanol (figure 1 b): the normal ethanol concentration gradient outside the liquid–liquid interface $(\unicode[STIX]{x2202}_{r}c^{\text{e}})_{side}$ has a steep slope. At the top of the drop, the ethanol concentration gradient $(\unicode[STIX]{x2202}_{r}c^{\text{e}})_{top}$ may also increase, but only to a lower extent, as the replenishing oil is already contaminated with ethanol (and water) during its travel along the drop surface. Owing to the boundary condition, the replenishment of fresh oil caused by the convection is suppressed near the corner of the drop, where the ethanol concentration gradient $(\unicode[STIX]{x2202}_{r}c^{\text{e}})_{C.L.}$ is much less affected. Therefore, the buoyancy-driven convection predominantly enhances the diffusion flux near the drop equator rather than that above the drop or in the contact region and, as a consequence, the radial gradients of ethanol concentration along the drop surface obey the relationships

(7.5a ) $$\begin{eqnarray}\displaystyle & \displaystyle (\unicode[STIX]{x2202}_{r}c^{\text{e}})_{side} & \displaystyle >(\unicode[STIX]{x2202}_{r}c^{\text{e}})_{top},\end{eqnarray}$$

(7.5b ) $$\begin{eqnarray}\displaystyle & \displaystyle (\unicode[STIX]{x2202}_{r}c^{\text{e}})_{side} & \displaystyle >(\unicode[STIX]{x2202}_{r}c^{\text{e}})_{C.L.}.\end{eqnarray}$$

$(\unicode[STIX]{x2202}_{r}c^{\text{e}})_{side}$