2. Experimental procedure

The experiments were performed in a humidity and temperature-controlled chamber. During the measurements the airflow was disabled. The chamber volume ${\approx}0.75$ l was sufficiently large to ensure diffusion-limited evaporation. The binary mixtures consisted of 10 wt $\%$ 1,2-hexanediol (Sigma–Aldrich, 98 $\%$ pure), which is non-volatile, and 90 wt $\%$ water (‘Milli-Q’, resistivity 18 $\textrm {M}\varOmega \textrm {cm}$ ), which is volatile. The drops were deposited on a glass slide coated with octadecyltrichlorosilane (OTS) that is a transparent hydrophobic coating (Silberzan et al. Reference Silberzan, Leger, Ausserre and Benattar1991). The advancing and receding contact angle of a water drop on this substrate are $\theta _{{advancing}}=110.3^\circ \pm 0.6^\circ$ and $\theta _{{receding}} = 89.7^\circ \pm 1.1^\circ$ . A detailed procedure of the substrate preparation and the contact angle measurements are provided in the supplementary information.

Both drops were imaged from the side using a camera (Nikon D850) and a long-distance microscope at 7x magnification (Navitar 12x zoom). One of the drops was simultaneously imaged from below with a confocal microscope (Nikon A1R). We were able to measure the flow in the drop by adding a small amount of polystyrene fluorescent particles (microParticles GmbH, concentration = $3.8\times 10^{-3}$ vol $\%$ , diameter $= 1.14\,\mu \textrm {m}$ , $\lambda _{absorption} = 530$ nm, $\lambda _{emission} = 607$ nm) and tracking these particles with a PTV algorithm. Since the confocal microscope has a very narrow depth of field, we know exactly at what height in the drop we detect the particles. We chose to image close to the substrate ( $\approx 18\, \mu \textrm {m}$ ) such that the drop remains in focus as it evaporates. The confocal microscope is able to provide a bright field view of the drop (transmission detection) at the same time as the fluorescent image. More details of the experimental set-up and data analysis are provided in the supplementary information.

Due to the constraints of the confocal and the side view camera in combination with the humidity chamber, it was not feasible to implement an automatic dispensing system for the drops. Therefore, all drops were deposited manually using a syringe (Hamilton, model 7001) capable of accurately expelling very small volumes (syringe volume = 1 $\mu \mathrm{l}$ ). The volume of each drop was $0.15 \, \mu \mathrm{l}$ , which is the largest drop size that fits completely in the field of view of the confocal microscope. The time between the depositing of the first drop and the second drop (10 s) was much shorter than the total lifetime of the drops (4.5 min). Although precise repetitions of drop placement were not possible, the distance between the drops could be accurately determined using the side view camera in combination with the top view of the confocal after each experiment.

3. Contact line behaviour

Figure 3.

(a) The frequency different contact line behaviours were observed for 74 neighbouring drops (dark red bars) and 25 isolated drops (light grey bars). The different pinning modes are: CCA (constant contact angle, i.e. the contact line recedes on all sides of the drop), CCR (constant contact radius, i.e. the contact line is pinned on all sides of the drop) and PP (partial pinning, i.e. part of the contact line is pinned and another part of the contact line is receding). (b) Direction of movement of neighbouring drops for different initial separations and relative humidities. Since each pair of drops would overlap, we split the drops vertically around the actual relative humidity (indicated with a small bar). The drop movement is shown with a triangle pointing in the direction of the drop movement with respect to the other drop. A square is shown when a drop did move, but did not get closer or further with respect to the other drop. Drops that did not move are circles. When a drop was not in view an $\times$ is shown.

Figure 2(a–c) shows snapshots of the side view at different times of two evaporating drops. Figure 2(e) shows the bottom view of the right drop in figure 2(c). Figure 2(a) shows the drops just after they have been deposited on the substrate. The initial drop contact radius is $R_0 = 0.53$ mm and the distance between the drop edges is $d/R_0 = 1.7$ . As a visual guide, the position of the contact line has been marked by a dashed line, which extends into the next panel. The position of the contact line is also marked in the bottom view with a dashed line in figure 2(e). For the first 180 s after the drop deposit, the drop evaporates in a constant contact angle (CCA) mode. The contact line of each drop moves the same distance inwards on all sides. Up to that time, the centres of both drops remain in the same place.

However, after some more time, we see that the centres of the drops move away from each other. Figure 2(b) shows the moment when the drops start to move away from each other. Again the position of the contact line is marked with a dashed line. Figure 2(c) shows the residue of the non-volatile 1,2-hexanediol when the evaporation of water in the drop has stopped. Going from panel (b) to (c), we see that the contact line facing away from the other drop has stayed in place, while the contact line facing toward the other drop has retracted. We call this mode partial pinning (PP). During the whole evaporation, the contact angle of both drops remained the same.

In our experiments, this was not the only mode of evaporation that we observed. Figure 2(d) shows a different case of pinning-induced motion (the left drop is shown, with the other drop being on the right). Here, the drop was pinned from the start and evaporated in constant contact radius (CCR) mode. At time $t_1=102$ s the drop suddenly unpinned on one side and moved away from the other drop. This is different for the drops shown in panels (a–c), which were evaporating in the CCA mode and got pinned on one side during the evaporation. In both cases, figures 2(d) and 2(e), the result is that the drops move apart.

In total we observed four different contact line behaviours. Figure 3(a) shows the frequency of each observed mode for all the experiments. In the case of the CCR mode, the drop would not exhibit any pinning-induced motion and evaporate in CCR mode during its entire lifetime. In the case of the PP mode, the drop is partially pinned, meaning that a section of the contact line is pinned and does not move, whereas the rest of the contact line recedes inwards, resulting in a shift of the drop centre. In the majority of cases (78  $\%$ ), both the CCR and PP modes can be preceded by a period of time where the drops evaporate in the CCA mode. Meaning that the contact line recedes on all sides of the drop equally. We denote these modes as $\mathrm{CCA \rightarrow CCR}$ and $\mathrm{CCA \rightarrow PP}$ .

The drop(s) moved due to PP in 59 $\%$ of cases. Isolated drops also showed PP, although the direction in which the centre moved was arbitrary. For neighbouring drops, however, the direction was systematically away from the other drop. Figure 3(b) shows the direction in which the centres of the drops moved relative to the other drop for different relative humidities and drop separations. When the drops moved, 80 $\%$ of the cases moved further apart. Sometimes the drops would move, but they would not get any closer or further apart, i.e. they moved perpendicular with respect to each other. This happened 17 $\%$ of the cases. In one experiment, one of the drops moved towards the other drop, while the other drop moved away from the other drop. For the entire range of relative humidities and drop separations, both the (CCA $\rightarrow$ ) PP modes as well as the (CCA $\rightarrow$ ) CCR modes can be observed. Indicating there is no clear dependence on the relative humidity or drop separation.

We emphasise that the movement of the drops centres away from each other is surprising: the direction is opposite to what would be predicted based on the shielding effect. Since the local evaporation rate of the drop is suppressed near the other drop, we would expect the contact line to recede less, since less mass is lost. Furthermore, because the evaporation is highest in regions facing away from the other drop, we would expect the contact line to recede more quickly in those regions. Thus, we would expect the centre of the drop to move towards the other drop.

When we consider the effect of the flow on the contact line motion, we also expect the opposite behaviour: the flow inside an evaporating water/1,2-hexanediol drop is dominated by a Marangoni flow (as will be discussed in §§ 4 and 5) that can result in Marangoni contraction (Hack et al. Reference Hack, Kwiecinski, Ramírez-Soto, Segers, Karpitschka, Kooij and Snoeijer2021). When the two drops are close, the Marangoni flow becomes asymmetric due to the shielding effect and causes the drops to contract more where the evaporation rate is higher. Consequently, we expect the centres of the drops to move closer (Cira et al. Reference Cira, Benusiglio and Prakash2015). However, we do not observe any Marangoni contraction – during the first part of the evaporation process ( $t_0 \rightarrow t_1$ ) we observe no significant displacement of the drops. Marangoni contraction depends strongly on the wettability of the substrate. Both Cira et al. (Reference Cira, Benusiglio and Prakash2015) and Hack et al. (Reference Hack, Kwiecinski, Ramírez-Soto, Segers, Karpitschka, Kooij and Snoeijer2021) used aggressive cleaning methods to make the glass perfectly clean, which is very different from the glass substrates coated with OTS that we study.

We rationalise the observed pinning-induced motion by considering that the motion of the drops is strongly influenced by particles and solutes present in the drop. Due to the strong evaporation at the rim of the drop, particles or solute will accumulate at the contact line. Despite the Marangoni flow mixing the particles throughout the drop (Thayyil Raju et al. Reference Thayyil Raju, Diddens, Li, Marin, van der Linden, Zhang and Lohse2022), more particles will deposit and accumulate at the contact line where the evaporative flux is the largest (Wray et al. Reference Wray, Wray, Duffy and Wilson2021). This results in a large concentration of particles in the region facing away from the other drop, increasing the likelihood of the drop pinning there, as illustrated in figure 4. So we hypothesise that it is this preferential pinning on the side away from the other droplet that facilitates the observed repulsion of the drop centres.

So far, we have only considered a drop of water/hexanediol with a small amount of tracer particles to facilitate the PTV. To investigate the role of the tracer particles, we repeated the experiment without additionally added tracer particles. Of all the experiments with neighbouring drops (n = 14), 11 (79 $\%$ ) moved due to PP. In nine cases the drop centre moved further away from the other drop (82 $\%$ ). Although the number of experiments without tracer particles is limited, we still observe the pinning-induced motion. This means that the added tracer particles contribute little to the contact line behaviour. This observation does not invalidate our hypothesis, since we still observe very small particles or contaminants in the drop with the microscope at very high magnification. We speculate that the largest contributor of particles is from the surrounding air, which contains particulate matter of various sources (much) smaller than $1 \ \mu \textrm{m}$ such as dust, soot and aerosols (Zhang et al. Reference Zhang, Wang, Guo, Zamora, Ying, Lin, Wang, Hu and Wang2015). Future experiments in a clean room might disentangle the role of particulate matter on the contact line dynamics. However, these contaminants are simply unavoidable and uncontrollable in a standard laboratory setting or in most technological applications.

As already mentioned in the introduction, an alternative interpretation of the observed repulsion of the drop centres may be that this repulsion is a consequence of the thermal Marangoni flow: on an isothermal or nearly isothermal surface, the region around the apex of a pure drop is the coldest, due to evaporative cooling. As there is more evaporation on the drop’s side away from the neighbouring drop, the liquid there is cooler, leading to a thermal Marangoni flow away from the neighbouring drop and, thus, to repulsion. However, we think that this interpretation is not the relevant one for our case of the binary water–1,2-hexanediol droplet for several reasons. (i) It disregards solutal Marangoni flow, which dominates thermal Marangoni flow, as we discuss in detail in § 5. (ii) Thermal Marangoni flow implies a Marangoni contraction (Shiri et al. Reference Shiri, Sinha, Baumgartner and Cira2021; Kant et al. Reference Kant, Souzy, Kim, van der Meer and Lohse2024; Malachtari & Karapetsas Reference Malachtari and Karapetsas2024), which we do not observe here. (iii) If the interpretation of the observed repulsion as a consequence of the thermal Marangoni flow were correct, we would have perfect reproducibility of the effect. This is not the case, as we observe a lack of perfect reproducibility, i.e. a stochastic element, which is typical for pinning. (iv) As reported above, in the first part of the evaporation process $(t_0 \to t_1 )$ , we do not observe any significant displacement of the drops, in contrast to what one would expect if the repulsion were a thermal Marangoni effect. In contrast, if the origin of the drop repulsion is the preferential pinning at the outer side of the drop that builds up over time, this delay in the repulsion can immediately be understood. We will give further arguments in favour of our interpretation in the remainder of the paper.

Figure 4.

Schematic of the preferential PP effect. (a) Initially, the particles are distributed homogeneously throughout the drop. Due to the shielding effect, evaporation is suppressed in the area between the drops. (b) During evaporation this will result in most particles agglomerating at the side opposite the other drop due to the higher flux there. This will make it more likely that the drop pins on that side only because of the larger number of particles there. (c) The observed final position of the drop. The dashed line is the initial position of the drop.

Figure 5.

Flow in $\mathrm{RH} = (47 \pm 5) \, \%$ and $T = (23.8 \pm 0.3)\,^\circ \textrm {C}$ . Panels (a,b) show the traces of particles in the drop close to ( $\approx 18\ \mu \textrm {m}$ ) the substrate for the first 15 $\%$ of the total evaporation time. Panel (a) is for an isolated drop and panel (b) is for a neighbouring drop with $d/R_0=0.38$ (with the neighbouring drop on the left of the drop that is shown). (c) The azimuthal dependence of the radial velocity close to the substrate, averaged for the first 15 $\%$ of the total evaporation time and on the radial interval close to the rim, $r/R_0 = [0.8,\ 0.85]$ . The radial velocity is shown for three drop separations: $d/R_0 = \{ \infty , \, 1.04, \, 0.38 \}$ with the symbols $\{$ , , $\}$ , respectively. The other drop is located at $\theta = 0$ . The shaded area is the standard deviation of all the radial velocities included in the average. The solid lines are the azimuthal dependencies of the flux for the neighbouring drops normalised by the flux of an isolated drop: $\displaystyle {\lim _{r \to R_0}{J_1/J_0}}$ .

4. Internal flow

To further elucidate the interplay between pinning, evaporation, and solutal and thermal Marangoni flow, we have imaged the flow in the drops with PTV. Figures 5(a) and 5(b) show the trajectories of the particles close to the substrate during the first 15 $\%$ of the total evaporation lifetime. By limiting the analysis to a short time after the drop deposition, we can ignore any effect of the contact line motion, as well as the unusual segregation dynamics that occurs in the final phase of evaporation of water/1,2-hexanediol drops (Li et al. Reference Li, Lv, Diddens, Tan, Wijshoff, Versluis and Lohse2018). Figure 5(a) shows the trajectories for an isolated drop. The flow is outwards and axisymmetric. There is a stagnation point in the centre of the drop where the particles do not move. Near the rim of the drop, the particles move the most. Some particles are stuck at the contact line.

Figure 5(b) shows the trajectories for a neighbouring drop (with the neighbouring drop on the left of the drop that is shown). The flow is still outwards, but no longer axisymmetric. The stagnation point has shifted towards the other drop. We also see that the displacement of particles is much larger on the right (away from the other drop) than on the left. The evaporative flux is larger on the right of the drop as the left side is shielded by the other drop. More water evaporates on the right leading to larger concentration gradients in the drop, and consequently, larger surface tension gradients. This results in a stronger solutal Marangoni flow, which dominates a larger portion of the drop, shifting the stagnation point towards the other drop. We note that, for our case, a thermal Marangoni flow is in the same direction, though much weaker, as we will see in § 5.

To better quantify the asymmetry in the velocity in the neighbouring drop, we compute the average velocity of the particles close to the rim for the first 15 $\%$ of the drop’s lifetime. This time interval was chosen such that a sufficiently large number of particles were measured, while limiting the effects of volume loss of the drop and of the contact line motion on the measurement. We include all the particles on the interval $r/R_0 = [0.8, 0.85 ]$ . When we choose an interval closer to the rim, the results become unreliable due to the limited depth of field of the confocal microscope, which brings the flow near the water–vapour interface into focus. Choosing a smaller space and time interval is possible but will result in more noise and a larger statistical error in the data. Figure 5 shows the average radial velocity close to the rim as a function of the azimuthal angle $\theta$ in the drop for different drop separations, normalised by the average velocity of an isolated drop. The shaded region corresponds to the standard deviation for each bin.

Note that the averaged radial velocity is an average in the experimental sense, i.e. over a large number of individual partial trajectories. It is not a vertical average from the bottom of the drop to the top, which would yield only the so-called coffee stain flow, since the Marangoni flow recirculates and, thus, cancels in the axisymmetric case. In the experiment, we only measure particles trajectories close to the substrate since the confocal has a very narrow depth of field (18 $\mu \textrm {m}$ ).

For $d/R_0 = \infty$ , the average velocity is axisymmetric (i.e. constant for all $\theta$ ). For $d/R_0 = 1.04$ , the overall velocity is slightly lower and a clear minimum appears around $\theta = 0$ . For $d/R_0 = 0.38$ , the minimum becomes more pronounced, confirming the observations made in figure 5(b).

Next, we would also like to make a quantitative comparison with the local evaporation rate. To this end, we have to make a few simplifications, since to our knowledge, there is no expression of the local evaporative flux for multicomponent drops. However, the flux has been investigated for single component drops with a contact angle of zero, i.e. flat disks. Fabrikant (Reference Fabrikant1985) analysed the potential flow through membranes, which is mathematically equivalent to neighbouring evaporating drops. Later, Wray et al. (Reference Wray, Duffy and Wilson2020) continued the analysis and calculated the local evaporative flux $J_1$ of two identical neighbouring drops:

(4.1) \begin{align} J_1(r,\theta ) = J_0(r)f(r,\theta ). \end{align}

Here $J_0$ is the evaporative flux for an isolated drop,

(4.2) \begin{align} J_0(r) = \frac {2 D_{\textit{vap}} \Delta c_{\textit{vap}}}{\pi \sqrt {R^2 - r^2}}, \end{align}

and $f(r,\theta )$ expresses the shielding effect due to the other drop (Wray et al. Reference Wray, Duffy and Wilson2020),

(4.3) \begin{align} f(r,\theta ) = 1-\frac {F \sqrt {b^2-R^2}}{2 \pi \left (r^2 + b^2 - 2 r b \cos \theta \right )}. \end{align}

Figure 6.

The minima in the radial velocity measured between $0.8\lt r/R_0\lt 0.85$ for the first 15 $\%$ of the drop lifetime normalised by the value for an isolated drop, for various drop separations and relative humidities. The relative humidity is colour coded using the colour scale shown on the side. The bars around each data point correspond to the standard deviation of the distribution of all velocities included in the average. The solid line is the minimum in the local evaporative flux close to the rim, normalised by an isolated drop, as a function of the drop separation (4.3). The asymptotic limits of the flux are shown by the dashed lines: $4/(3\sqrt {3})$ for $d/R_0 \rightarrow 0$ and $2R_0/(\pi d)$ for $d/R_0 \rightarrow \infty$ .

The drop separation is given by $b$ , which is the distance from the centres of each drop ( $b = d+ 2 R_0$ ), where $f(r,\theta )$ is the local shielding and $F$ is the overall shielding of the flux, given by $F = 4\pi R/(\pi +2\arcsin {(R/b)})$ . Note that $R_0$ is the initial contact radius and that $R$ depends on time (i.e. $R_0 = R(t=0)$ ). Even though (4.2) and (4.3) are only valid for single component drops, this can still be a good approximation for the early stages of the evaporation since the drop mostly consists of water with only relatively small concentration differences throughout the drop.

Since the flow in the drop is driven by the evaporative flux, we can try to directly compare the measured radial velocity to the result of Wray et al. (Reference Wray, Duffy and Wilson2020). The solid line in figure 5 is the evaporative flux close to the rim for a neighbouring drop, normalised by an isolated drop, i.e. the shielding of the flux by the neighbouring drop: $f(r,\theta )$ evaluated at $r\rightarrow R_0$ . Note that no fitting parameters were used. The shape of the curve only depends on the separation between the drops, which was measured using the side view camera.

The agreement between the local flux and the radial velocity is remarkably good. We would like to emphasise that this relationship is not trivial, since we observe a strong solutal Marangoni flow in the drop, which indirectly depends on the evaporative flux. The shielded evaporative flux in combination with selective evaporation leads to the concentration gradients in the drop. This in turn leads to surface tension gradients that then drive the solutal Marangoni flow in the drop. It is very reasonable to expect that the flux and radial velocity share qualitatively the same shape, but here the agreement is also quantitative. This suggests that the relation with the local flux and velocity close to the rim is directly proportional to each other: $J\propto v_r$ .

To further substantiate the relation between the flux and the radial velocity, we investigate the minimum of $J$ and $v_r$ as shown in figure 5(c). Figure 6 shows $J$ and $v_r$ at $\theta = 0$ as a function of the drop separation for various relative humidities. The solid line is the minimum in the local evaporative flux for neighbouring flat drops. The discontinuous lines indicate the asymptotic limits for small and large drop separations. The data points are the minima in the measured average radial velocity in neighbouring drops, the colour indicates the relative humidity. The bars indicate the spread of the distribution using the standard deviation of all measured velocities included in the average.

Despite the large spread of the distribution for the measured radial velocity, the agreement between the velocity and the flux is very good. For very small drop separations, we find that the velocity deviates from the flux. This is because we are limited in how near we can measure to the contact line. For very large drop separations, we also observe a deviation. This is because the magnitude of the fluctuations in the average radial velocity are similar to the minimum in the radial velocity. For all intermediate drop separations, the minimum in the flux and velocity match.

5. Direct numerical simulations on the relative importance of solutal and thermal Marangoni flow

We now want to further elucidate the role of solutal and thermal Marangoni force on the flow in the drop. We do so by direct numerical simulations, as it is then possible to fully turn off either the solutal Marangoni forces or the thermal ones, or even both. Unfortunately, it is not easily feasible to directly simulate two neighbouring droplets – at least not without major simplification – since it is numerically impossible to fully resolve the required three-dimensional mesh due to the prohibitively large computation cost this would require. Therefore, in this section we focus on isolated droplets only, which we can numerically resolve by making use of the axisymmetry. For evaporating isolated water–1,2-hexanediol droplets, this axisymmetry is given during most of the evaporation time; only in the final phase it may be broken as then the surface tension of the water–1,2-hexanediol mixture non-monotonously depends on the 1,2-hexanediol concentration, cf. Diddens, Dekker & Lohse (Reference Diddens, Dekker and Lohse2024). In this section we focus on the effect thermal Marangoni has when solutal effects are present. The case of purely thermal Marangoni flow is less relevant for our experiments. We therefore discuss it only in Appendix A. For this case, we find that the flow pattern is quite different.

5.1. Numerical methods

Figure 7.

Snapshot at $t$ = 15 s of a numerical simulation of an isolated water/hexanediol drop evaporating on a glass substrate in the same conditions as the experiment shown in figure 5 ( $c_{0,{hexanediol}} = 10 \,\mathrm{wt\,\%},\ RH = 50\,\%,\ T = 24\,\mathrm{^\circ C},\ \theta = 50^\circ , \ V = 0.15\,\mathrm{\mu l}$ ). The simulations are axisymmetric and include thermal effects. The left half shows the temperature and the right half shows the concentration of either water vapour in the gas phase of hexanediol in the drop. The arrows indicate the local evaporative flux and the trajectories of tracer particles show the flow in the drop.

Numerical simulations are done using pyoomph (source code: https://github.com/pyoomph/pyoomph, documentation: https://pyoomph.readthedocs.io), which is a python multi-physics finite element framework based on oomph-lib (http://www.oomph-lib.org/) and GiNaC (http://www.ginac.de/). A detailed description of the used model can be found in Diddens (Reference Diddens2017).

Figure 7 shows a numerical snapshot after 15 s of evaporation. Similar to Diddens et al. (Reference Diddens, Tan, Lv, Versluis, Kuerten, Zhang and Lohse2017) we simulate the whole domain (gas, substrate and liquid) including the thermal effects of the whole domain (i.e. also of the substrate), as shown on the left-hand side of figure 7. The substrate is modelled after the glass slides used in the experiments and has the thermal properties of glass and a thickness of 170 $\mu \textrm {m}$ . The ambient temperature is set to $24\mathrm{^\circ C}$ , the relative humidity to $RH = 50\,\%$ , mimicking the conditions of the experiment shown in figure 5. The right-hand side of figure 7 shows the concentration of water vapour in the gas phase. The only method of water vapour transfer is diffusion and natural convection is ignored. The arrows indicate the local evaporative flux of water into the gas phase. The drop itself is pinned (i.e. CCR mode). Inside the drop we consider both convection and diffusion of hexanediol using physical properties of water/1,2-hexanediol (Li et al. Reference Li, Lv, Diddens, Tan, Wijshoff, Versluis and Lohse2018). We add tracer particles with a trail to visualise the flow inside the drop. The particles are only visual and are not coupled to the flow. We can study the effect of solutal and thermal Marangoni flow by making the surface tension independent of composition and/or temperature at the liquid–gas interface of the drop.

5.2. Results from direct numerical simulations

Figure 8.

Velocity and temperature in the drop at $t$ = 15 s with (a) no Marangoni flow, (b) only solutal Marangoni flow, and (c) both solutal and thermal Marangoni flow. The left half shows the temperature in the drop. The right half shows the velocity magnitude on a logarithmic scale. The trail of the tracer particle is 3 s long. The radius and height of the drop are $R = 0.576\ \mathrm{mm}$ and $H = 0.254\ \mathrm{mm}$ .

Figure 9.

(a–c) The radial velocity in the drop at different heights of the numerical simulations shown in figure 8. The grey area is the interval over which the experiments have been averaged ( $0.8\lt r/R_0\lt 0.85$ ). (a) No Marangoni flow, (b) only solutal Marangoni flow, and (c) both solutal and thermal Marangoni flow. (d) The average numerical radial velocity averaged over the same interval as the experiments for the different drop heights. Note that the average radial velocity measured experimentally in figure 5 for an isolated drop is $v_0 = 15.8 \ \,\mu \textrm{m}\,\textrm{s}^{-1}$ .

To better understand the role of surface tension gradients due to the local composition and temperature gradients, we start with a constant surface tension before adding solutal and thermal effects. Figure 8(a) shows the velocity and temperature in the drop without any Marangoni flow. Since the drop is pinned, we observe a replenishing flow toward the contact line (i.e. coffee stain flow), although the particle trails are not sufficiently long to adequately show the direction of the flow. The average velocity magnitude is 0.514 $\,\mu \textrm{m}\,\textrm{s}^{-1}$ . The temperature in the drop is overall 2.26 $\mathrm{K}$ colder than the ambient temperature due to evaporative cooling at the liquid–gas interface. Although the evaporative flux (and therefore the cooling) is higher near the contact line, this is actually the hottest part of the drop with the drop apex being the coldest part. This is due to the conductive substrate that supplies heat to the drop. In general, it is known that both the thermal properties and dimension of the substrate and the liquid drop will determine the temperature distribution in the drop (Ristenpart et al. Reference Ristenpart, Kim, Domingues, Wan and Stone2007; Dunn et al. Reference Dunn, Wilson, Duffy, David and Sefiane2009; Diddens et al. Reference Diddens, Tan, Lv, Versluis, Kuerten, Zhang and Lohse2017; Wang et al. Reference Wang, Karapetsas, Valluri and Inoue2024).

Next, we make the surface tension dependent on the local composition (but not yet on temperature) as shown in figure 8(b). We find that the flow is recirculating through the drop and that the average velocity magnitude is increased significantly to 26.3 $\,\mu \textrm{m}\,\textrm{s}^{-1}$ . Despite the large change in velocity, the average temperature as well as the temperature distribution is virtually identical compared with the case without Marangoni flow. Calculating the Péclet number using the average velocity, the radius of the drop and the thermal diffusivity $\kappa$ of water, we find that

(5.1) \begin{equation} {\textit{Pe}}_T = \frac {L v}{\kappa } = 1.06 \times 10 ^ {-7}. \end{equation}

This means that the heat transfer in the drop is dominated by conduction and is unaffected by the flow in the drop. Additionally, the source of heat (conduction through the glass slide) and the sink of heat (evaporative cooling at the liquid–gas interface) are also unchanged. Therefore, the temperature in the drop is the same with or without solutal Marangoni flow.

Finally, we also make the surface tension dependent on the local temperature as shown in figure 8(c). We find that the temperature distribution is identical to the previous cases and that the flow in the drop looks almost the same as the case with only solutal Marangoni flow. With thermal Marangoni, the average velocity magnitude in the drop is increased to 29.8 $\,\mu \textrm{m}\,\textrm{s}^{-1}$ , a difference of less than 12 $\%$ . Additionally, the direction of the thermal Marangoni flow is in the same direction as the solutal Marangoni flow, since it increases the velocity, meaning that we would expect that thermal effects enhance the attractive effect the drops have due to Marangoni contraction.

To compare the numerical results more quantitatively with the experimental results, we evaluate the radial velocity numerically at the same heights as in the experiment. The depth of field of the confocal is approximately 18 $\mathrm{\mu m}$ when the top surface of the substrate is perfectly in focus (see supplementary information). Figure 9(a–c) shows the radial velocity at different heights for all three cases shown in figure 8.

Without Marangoni flow, as previously mentioned, we see the outward replenishing flow that diverges close to the contact line (figure 9 a). With only solutal Marangoni enabled (figure 9 b), we see that the radial velocity has a maximum and becomes negative due to the recirculation (only the positive part is shown in the figure). With both solutal and thermal Marangoni flow enabled (figure 9 c), we see no change except in the magnitude of the flow. The shaded region corresponds to the interval over which the experiments have been averaged ( $0.8\lt r/R_0\lt 0.85$ ) and the numerical averages for each height on the same interval are shown in figure 9(d).

The experimental radial velocity for an isolated drop is $v_0 = 15.8 \,\mu \textrm{m}\,\textrm{s}^{-1}$ , which is significantly higher than the numerical radial velocity without Marangoni flow (0.515 $\,\mu \textrm{m}\,\textrm{s}^{-1}$ ). This indicates that the flow in the experiments is dominated by Marangoni flow. However, the experimental radial velocity is not as high as the numerical radial velocity with solutal Marangoni flow (52.6 $\,\mu \textrm{m}\,\textrm{s}^{-1}$ ). A similar deviation between experiments and numerics was found for the evaporation of water and glycerol drops by Thayyil Raju et al. (Reference Thayyil Raju, Diddens, Li, Marin, van der Linden, Zhang and Lohse2022).

Based on the numerical simulations we conclude that thermal Marangoni plays a minor role compared with solutal Marangoni flow. Qualitatively, thermal and solutal Marangoni flow have the same effect since the direction of the thermal Marangoni flow is the same as the solutal Marangoni flow.

6. Summary, conclusions and outlook

In summary, we have experimentally studied evaporating neighbouring binary droplets, employing direct optical observations, confocal microscopy and PTV. Our model droplets consisted of a water–1,2-hexanediol binary mixture, which is a standard model system for inkjet printing of multicomponent droplets. Due to the preferential evaporation of the droplets at the side away from the other droplets, we (and all colleagues we had asked) expected the centres of the drops to move towards each other. However, in contrast, they move apart. We hypothesise that this behaviour is a consequence of the particles or contaminants in the drop being deposited on the sides away from the other drop, leading to preferential pinning on those sides, with all consequences for the overall flow dynamics in the droplets. We also discuss an alternative interpretation of the observed repulsion, namely that it is a consequence of thermal Marangoni flow developing for two pure evaporating droplets on a isothermal surface (cf. the numerical study of Malachtari & Karapetsas Reference Malachtari and Karapetsas2024), but give arguments why we think that this interpretation is not applicable in our case.

We have moreover shown that the flows in the neighbouring droplets become asymmetric due to the symmetry breaking by the other droplet, as expected, but the radial velocity close to the rim of the drop is still in good agreement with (4.3), i.e. the local evaporative flux at the rim for a single component drop. Finally, with the help of direct numerical simulations for an evaporating isolated, axisymmetric binary droplet, we have elucidated the relative contributions of the replenishing flow and the solutal and thermal Marangoni flow, with the latter one playing only a minor role.

Though the model system consisting of water–hexanediol binary droplets is commonly used in the research of inkjet printing, it by far does not feature all possible effects. Different multicomponent liquid compositions would lead to different behaviours and in the case of soluble or insoluble surfactants the situation becomes even more complicated. So our work is only the beginning, but given how counter-intuitive the observed behaviour was, we thought it was worth reporting our observations and giving a hypothesis for their origin for our case of neighbouring water–hexandiol binary droplets. The relevance of such evaporating multicomponent neighbouring droplets in technological applications is huge, way beyond inkjet printing, and mastering their deposition and how they dry is key for most of these applications.