Contactless Manipulation of Binary Droplets via Sensing of Localized Vapor Sources

Droplet motion on surfaces influences phenomena as diverse as microfluidic liquid handling, printing and micro-organism migration. Typically, droplet motion is achieved by inducing surface energy gradients on a substrate or surface tension heterogeneities on the droplet's surface. Current configurations for droplet manipulation have, however, limited applicability as they rely on carefully tailored energy gradients and/or bespoke substrates. Here we demonstrate the contactless long-range manipulation of binary droplets on pristine substrates due to the sensing of localized water vapor sources. We show with analytical considerations that the dissipative nature of the driving forces at play, induced by tiny asymmetries in surface tension gradients, is essential to capture the vapor-sensing mechanism. We then demonstrate its versatility by printing, aligning and reacting materials controllably in space and time.


Introduction
Droplets moving on solid surfaces are at the heart of many phenomena of fundamental and applied interest in physics, biophysics, chemistry and materials science [1]. Examples include "tears of wine" due to the Marangoni effect [2], rolling droplets on self-cleaning and superhydrophobic substrates [3,4], microfluidic liquid handling [5,6] as well as enhancing heat transfer [7], sensing [8], and printing technologies [9,10]. Even microorganisms, such as Bacillus subtilis, can collectively initiate motion of macroscopic water droplets, thus inducing the colony to move over surfaces [11]. The main impediment to trigger droplet motion on solid surfaces comes from the hysteresis of the contact angle that pins the droplet's edge to the underlying substrate [12]. The generation of gradients of surface energy on this substrate is a widespread strategy to overcome the pinning of the contact line and achieve motion [13,14,15,7,16,17,5,18].
Alternatively, imbalances of surface tension can be directly induced on the droplet's surface, thus leading to its motion [8,19,20]. An emblematic, recent example of the latter mechanism is that of droplets self-sensing their evaporation profile [8,21,22]. Beyond triggering droplet motion on surfaces, achieving control over its directionality is an important step towards harnessing moving droplets for applications in, e.g., printable materials [9,10], biological assays [23,24], and microreactors [25]. However, existing techniques for droplet manipulation rely on large gradients of surface energy [13,7], on carefully engineered substrates [5,17,18], or on tailored trapping potentials [8], thus limiting control over droplet motion and its applicability on pristine substrates.
Here, we demonstrate the contactless long-range 2D manipulation of binary droplets (Co-ManD) on solid pristine surfaces. The droplets spontaneously move in response to small imbalances of surface tension gradients induced by the presence of an external localized source of water vapor. We show with an analytical model that the dissipative nature of the driving forces at play (< µN) is paramount to capture the observed dual response (from attractive to repul-2 sive) of the droplet with distance from the source. Our robust understanding of the underlying vapor-sensing mechanism allows us to showcase CoManD's adaptability in a range of potential applications in materials science.

Motion of binary droplets under an external vapor source.
In CoManD's basic configuration, a 0.5 µL binary droplet of water and propylene glycol (mole fraction of water x H 2 O = 0.95) is deposited on a clean glass slide at a distance x s from an external localized source of water vapor (Fig. 1A, see Methods and Materials). When x s → ∞ (i.e. in the absence of the source), the two-component droplet is stationary on a flat surface and, due to its ongoing evaporation, it features an apparent contact angle (θ c = 12.5 ± 0.7 • ) higher than either of the pure liquids [26]. In fact, as evaporation is faster at the droplet's edges [27] and propylene glycol (PG) is less volatile and with a lower surface tension than water, the resulting radially symmetric gradient of surface tension γ induces inward Marangoni stresses [26], which prevent spreading [28,8,26]. In the presence of the source, the droplet can instead move if the contact line is not pinned as in our case. In particular, when placed afar (x s = 2 mm, Fig.   1B), the droplet experiences an attractive force towards the source. This attractive behavior is consistent with a relative larger local increase in humidity at the droplet's edge nearest to the source, which generates motion by slowing down evaporation and comparatively increasing the value of surface tension at that edge with respect to the opposite side [8,29]. Based on these considerations on surface tension, one could naturally expect a stable equilibrium position to appear directly beneath the source's center (x s = 0), as any displacement from x s = 0 would induce a restoring force. Surprisingly, x s = 0 is an unstable equilibrium position from which the droplet gets easily repelled to an off-centered radial distance x e (Fig. 1C), where the droplet consistently settles near the end of the evaporation whether it starts from afar the vapor source or below it (Fig. 1D). Schematics of the contactless manipulation of binary droplets on solid substrates with an external localized vapor source (CoManD): a binary droplet (radius R D ; apparent contact angle θ c ) is placed at a distance x s from a needle (inner radius R s ) from where water vapor diffuses; z s is the distance between source and substrate. All experiments are performed within an environmental chamber with controlled temperature (T = 21 ± 0.5 • C) and relative humidity (RH = 50 ± 5%).

Dynamics of droplet's motion
To efficiently harness this mechanism, we have developed a simplified analytical model to better understand our experimental observations. In fact, this duality in the interaction between droplet and source cannot be simply explained by arguments purely based on differences of surface tension on opposing sides of the droplet, for which the droplet would always move towards the vapor source [29]. Instead, as can be seen in Fig. 2 Fig. 2A show how the position of this maximum shifts along the droplet's free surface following the position of the source, moving from the droplet's apex for x s = 0 to its contact line when the source is far away (x s = 1.2R D ). For no displacement (x s = 0), the gradient in surface tension, and hence the corresponding Marangoni flows along the free surface, are radially symmetric and pointing inward towards the droplet's apex (Fig. S2). When the source is displaced towards one edge,  Fig. 1 are shown for R s = 350 µm (Fig. S4). The solid lines represent the time evolution of the droplet's radius R D (Fig. S1B). In the simulations, the dashed lines delimit the part where corresponding experiments are available. 6 the flows in the droplet become radially asymmetric [31], strengthening under the source at first due to a steeper gradient in surface tension (Fig. S2). When the source is far away, the flows coming from the distal edge predominate instead due to the confined geometry of the droplet.
More importantly, between these two cases, the gradient in surface tension at the edge closer to the source changes directionality (Fig. S2). This change intuitively justifies the dual response of the droplet's motion to the position of the source (Fig. 1).
To formalize this intuition, we can integrate the viscous stress induced by the flows within the droplet on the liquid-solid interface to obtain a dissipative driving force in the direction of motion (Supplementary Text): This force primarily acts on the contact line, is a function of the gradient of surface tension, and has typical values < µN consistent with previous reports (Fig. S3) [8]. Figure where η is the dynamic viscosity of the mixture at a given composition and n = 11.2 is a cutoff constant (Supplementary Text). Typical experimental values of v x range between a few and a few tens of µms −1 , and are well reproduced by our model (Fig. 2C).
As the droplet's geometry and composition are changing due to evaporation ( quantitatively. Figure 3B also shows that typical measured thicknesses span at least two orders of magnitude with deposits as thin as ≈ 5 nm. For a given PVOH concentration, the resolution achievable by this printing technique in terms of average line-width depends on the characteristic length of the droplet as w ∝ V   3F and S6A demonstrate the possibility of controlling the orientation of a polycrystalline polymer, such as PEG [34], thus leading to the formation of patterns within the deposit itself with ≈ 100 nm high features (Fig. S6B). In fact, PEG quickly supersaturates in the deposit printed by the droplet (due to the evaporation of its more volatile components) and starts to crystallise as soon as a defect is generated [35]. The resulting polycrystalline phase forms physical ridges perpendicular to a moving front whose shape depends on the droplet's speed and determines the topography of the final pattern (Fig. S6): for slower moving droplets, the front matches the droplet's circular shape leaving a scallop shell pattern behind, while, as the droplet speed increases, the moving front deforms into a triangular shape which leaves a herringbone structure behind instead.
Beyond depositing material in a controlled manner (Fig. 3), CoManD also allows for redissolving previously deposited material within a second droplet, thus facilitating the controllable deposition of chemical reactions in space and time. For example, Fig. 4A shows the final deposits left behind by water/PG droplets containing a pH indicator (bromothymol blue) after they retrace previous deposits by basic water/PG droplets with varying concentrations of NaOH. As a pH indicating droplet (initially yellow) retraces a previous deposit at constant speed (here v x = 25 µms −1 ), it uptakes NaOH from the substrate, its pH increases, and its color turns from yellow to blue through green, thus printing a color gradient. The steepness of this gradient, and thus the spatial variation of the printed colors, can be regulated through the rate of uptake of the first deposited reactant (here NaOH), e.g., by varying its concentration in the first droplet (Fig.   4A).
Finally, Fig. 4B and C show the simultaneous control of different droplets with more than one vapor source, which can be useful to trigger the mixing and the reaction of materials dissolved or suspended within them in space and time. As can be seen in Fig. 4B, the coalescence of two droplets can be guided by controlling the separation distance between two vapor sources (see Methods and Materials). The flows in the resulting droplet are significantly different from the radially symmetric flows in a standard sessile binary droplet [8], and, after flowing outwards along the liquid-liquid interface between the two coalesced droplets, recirculate towards  Fig. S7. In B, the droplets contain 100 mM NaOH and a dye, either methyl red (yellow, left) or bromothymol blue (blue, right); their contents mix as evidenced by the coalesced droplet turning green. In C, the droplets contain a pH universal indicator and either 100 mM NaOH (left) or 100 mM HCl (right); upon coalescing, an acid-base neutralization reaction occurs (inset) until the coalesced droplet turns uniform as the indicator shows qualitatively. The background was subtracted from all images. Scale bars: 1 mm.
the closest source in each compartmentalized quarter (Fig. S7). As a consequence, after solute exchange has occurred at the interface predominantly via diffusion, mixing is promoted within each quarter by these flows until the coalesced droplet becomes essentially uniform in composition. Beyond mixing, chemical reactions can also be implemented by the same principle. As an example, Fig. 4C shows the acid-based neutralization reaction of NaOH and HCl: before coalescence, the two droplets are respectively very acidic (pH ≈ 1) and basic (pH ≈ 13); on coalescence, neutralization starts to occur at the interface where a gradient of pH can be observed as highlighted by the full color spectrum of the pH indicator; as the reaction proceeds, the coalesced droplet eventually reaches neutral pH uniformly.

Conclusions
These examples illustrate CoManD's robustness and versatility as a novel method for the con-

Supplementary Text
Figs. S1 -S7 References (  3A to E and S5 were also performed in a similar way, i.e. by moving the stage to keep the distance between droplet and source constant. All droplets were guided by holding the water vapor source at their leading edge. In Figs. 3F, 4A and S6, the stage was moved in a straight line or according to a preprogrammed pattern at a constant speed. In Figs. 3F and S6, PEG crystallisation in the supersaturated deposit was manually nucleated with a metallic needle. In Fig. 4B and C, the combination of two droplets was performed by placing one droplet under a fixed vapor source (R s = 640 µm) and a second droplet under a second identical source approximately 10 mm away, which could be moved relative to the first. By reducing the inter-source distance to 3.5 mm, the two droplets were made approach gradually until coalescence.

Profilometry
The height profiles of the deposits in Figs. 3B and S6B were measured on a DektakXT Surface Profilometer (Bruker) with a 5 µm stylus at 5 mg force. In Fig. 3B, each profile is the average of 5 different profiles, which were acquired 4 µm apart using a scan resolution of 0.5 µm/pt, then levelled and smoothed with a Hampel filter (window size: 100 data points) to remove large random spikes. The 2D profiles in Fig. S6B were obtained by acquiring 1D profiles (5 µm apart in the direction perpendicular to the scan direction) with the same parameters. Data points were levelled and then smoothed using a Gaussian-weighted moving average (window size: 30 data points along the scan direction and 4 in the perpendicular direction).

Particle Tracking Velocimetry (PTV)
The flows within the droplets in Fig. S7 were visualized by tracking 5 µm polystyrene microparticles using a custom MATLAB script. At every instant, the position of these microparticles was determined by applying a sequence of three filters to the raw images. First, a top-hat filter 22 with a 3 pixel radius disk structuring element was used to obtain a uniform background. Then, each pixel was renormalized to the 3x3 pixel window around it in order to set the centroid corresponding to each particle to a value of one. Finally, particle positions were extracted by applying a binary threshold to the processed images after excluding false positives. Trajectories were obtained from these data using a nearest neighbor algorithm between consecutive frames.

Supplementary Text
Local Vapor Pressure at the Droplet's Free Surface The steady-state Poisson diffusion equation can be solved for a disc source of radius R s to obtain the vapor pressure of water, p H 2 O (r, θ , h(r)), at the droplet's free surface in the presence of a substrate (Fig. S1A). Assuming to a first approximation that the influence of the droplet on the local vapor pressure is negligible with respect to the source, the solution to the diffusion problem is given by [45] 2R D θ c [46], p s is the saturated partial pressure of water at the source (e.g. p s = 2.49 kPa at T = 21 • C [47]), and p RH is the partial pressure of water corresponding to the ambient relative humidity RH. To account for the presence of the substrate, p H 2 O is then calculated by adding to P H 2 O the vapor pressure P H 2 O generated by the specular image of the disc source with respect to the surface [48], i.e. centered in (x s , y s , −z s ), as Local Surface Tension at the Droplet's Free Surface At every instant t, due to the preferential evaporation of its more volatile component (water), we can assume that the bulk composition of the droplet x b H 2 O (t) changes to a new value x τ H 2 O (t) = (γ sv ) and that of the solid-liquid interface (γ sl ). In our case, S varies because of γ, while, to first approximation, we can assume that the vapor field generated by the source does not alter the contribution from the solid ∆γ s .
At any time t, the total force acting on the droplet can therefore be estimated by balancing three contributions: a driving force F d due to the non-homogenous S and γ, a dissipative force F γ associated to the gradient in γ and a viscous force F v that opposes the previous two contributions We shall evaluate the three contributions along the direction of motion x (Fig. S1A) independently in what follows. Due to symmetry in the problem, the balance of forces along the y direction is null.
For small surface tension gradients, F d x can be calculated by solving the following path integral along the contact line in polar coordinates [51]: As the water vapor does not appreciably change the contribution from the solid in our configuration (i.e. ∆γ s is constant), the solution to the previous integral is straightforward The two remaining force terms can be then calculated by evaluating the integral of the viscous stress at the liquid-solid interface to calculate the total viscous force [51]. The functional form of this viscous stress is determined by the flows within the droplet during its motion.
For small contact angles θ c , the flow patterns in the droplet advancing with a constant velocity v x can be calculated using the lubrication approximation, and they are the superposition of a Poiseuille flow induced by the pressure gradient and the Marangoni flows due to the gradient in surface tension at the liquid-air interface [51]. As dissipation is controlled by the wedge where the shear gradient is the sharpest [51], the resulting dissipative force, F v x − F γ x , along x can also 25 be calculated by evaluating a path integral along the contact line in polar coordinates, namely (8) and where η is the dynamic viscosity of the mixture determined empirically at different compositions [49] and n = 11.2 is a cutoff constant [51,8]. Interestingly, as F d x = 0, we can see how our process is entirely driven and dominated by dissipative phenomena.
Combining all force contributions, the velocity of the droplet in the vapor field generated by the source is then given at any instant t by

Time Dependence of Droplet's Geometry and Composition
The geometry of the droplet over time was estimated experimentally from videos of 0.5 µL droplets of initial composition x H 2 O = 0.95 evaporating in the presence of the source (Fig. S1).
Radius R D and contact angle θ c were measured directly in intervals of 25 s ( Fig. S1B and C).
Assuming a spherical cap geometry [46], these values were then used to estimate the droplet's volume V D as well as to define the droplet's free surface in cylindrical coordinates in time ( Fig.   S1A and D). Each set of data was fitted to a continuous function given by 5 th -order polynomials.
By assuming that no PG evaporates due to its high boiling point at 187.4 • C [52], all volume loss during evaporation can be attributed to water, so the bulk composition of the droplet, to a first approximation, can be given by: (11) where M PG = 76.   Fig. S1: Time evolution of droplet's geometry. (A) Schematic geometry of the droplet (radius R D ) and the vapor source (radius R s , saturated vapor pressure p s ) used in the model. The droplet's free surface is defined in cylindrical coordinates (r, θ , h) and a local water vapor pressure, p H 2 O (r, θ , h), is associated to each of its points; x s is the in-plane distance between droplet and source and z s is the vertical distance between source and surface. The specular image of the source with respect to the surface is also shown at −z s .