A parameter to probe microdroplet dynamics and crystal nucleation

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relative evaporation rate.Third, intensity ratios can only be used to indicate a cooperative process when evaporation rates are clearly different.Another experimental approach was proposed by Bonn et al. 12 to demonstrate the interaction between a collection of small monodisperse droplets.They used a precision balance to monitor the mass of the evaporating system.While both approaches 11,12 are elegant, they do not provide information on individual µD contraction.Our approach overcomes many of these limitations.
Here, arrays of sessile µDs are generated by a microinjector on an oil-covered 18mm-diameter glass coverslip treated so as to obtain a surface preventing µDs from spreading and coalescing (Fig. S1), as described previously 13 (additional details available in supplementary material).Two µD arrays are presented (Figs. 1 and S2), both prepared under the same conditions except for relative humidity % in room atmosphere (RH) above DMS oil and initial µD radius (R) (see supplementary material).
Both µD generation and subsequent contraction are observed under a transmission optical microscope.We stop imaging when contraction of each µD has reached completion: all the water in the µD has diffused through oil and there is a NaCl crystal at each µD position (Fig. S3 of the supplementary material).The first image just after µD generation (Fig. 1) is processed using FIJI software (Image J, NIH, USA).After thresholding and binarisation procedures, we use the "Analyze Particles" plugin to individually detect each µD position and size.A rectangular ROI is then assigned to each µD (Figs.1(b) and S4).From this ROI, comprising a single µD and its immediate vicinity, we extract the histogram of pixel grey-levels (Fig. 1(c)).We focus on its standard deviation σ, a measure of the homogeneity of pixel grey-levels. 14As each individual µD and its immediate vicinity, on each image, is reduced to a single number (σ), µD time evolution can be monitored throughout associated ROI standard deviation time evolution (ROI, σ, t), as reported for a single µD in Fig. 2(a).Each (ROI, σ, t)-plot is then processed with home-made PYTHON codes (www.python.org) to extract characteristic points (the code is available on request).The µD contraction process leading to crystal nucleation was previously described in Ref. 15, and is recalled here (Fig. S3 and video S1 of the supplementary material).Throughout the contraction process, water is leaving the µD, thus increasing NaCl concentration.This increases the µD refractive index, which leads to two characteristic stages.First, as the µD starting refractive index is lower than the DMS index, the µD refractive index will gradually increase until it matches the DMS refractive index exactly.At this matching time (t M ) the µD optically disappears and NaCl concentration can be estimated from an extrapolated relation between concentration and refractive index (tables 71 D-252). 16Second, as the contraction process proceeds, the µD refractive index will exceed the DMS refractive index: the µD gradually reappears until sudden nucleation occurs at nucleation time (t N ).Then a single crystal rapidly grows, before the µD dries completely at drying time (t D ).
Taking a single µD (assumed spherical cap, due to their small size compared to the capillary length, 2.7mm for a water droplet), with a starting concentration of 0.025 of the NaCl saturation value, the volume has to be divided by 40 before saturation concentration is reached.In terms of measurement dynamics, µD diameter explores one order of magnitude (50.2 µm at generation, 14.9 µm at saturation and in micrometer range at nucleation), volume explores two orders of magnitude (60.7 pL to <1pL), and the resulting crystal is in the micrometre range.Such spatial dynamics make it difficult to apply measurement techniques where spatial resolution is required: image resolution decreases with the number of µDs simultaneously observed.Considering magnification alone (1.3µm per pixel here), while some uncertainty on starting diameter is acceptable (2.6µm for a 50µm diameter µD, i.e ∆D/D=5% error), this no longer applies when µDs reach 14.9µm diameter.At this point, ∆D/D=17.4%,which, translated into volume, corresponds to an uncertainty of ∆V/V = 52.2%.Moreover, the size of the resulting crystal is similar to the resolution.Added to this spatial resolution issue, the vanishing refractive index difference (∆n) between continuous phase (here DMS oil) and dispersed phase (µDs) does not simplify measurements: edge contrast will fade and disappear during part of the contraction process.At a minimum, these limitations prevent the usual optical techniques from determining t M where the µD is both small and optically invisible, and under certain conditions also prevent determination of nucleation time t N .
By reducing µD and its vicinity (the ROI) to a single scalar like σ, we identify the µD signal as any departure from system noise.It is in fact a differential point probe measurement approach similar to signal-to-noise ratio measurement (defined as σ ∧ 2/µ, with µ = mean of the signal), except that we only focus on standard deviation of the full grey-level distribution of pixels inside the ROI.At the refractive index matching time, t M , σ is minimum: this is the standard deviation of the noise of our system.Anywhere else (before and after t M ), the grey-level distribution of pixels contained in the ROI, and the associated σ, results from the µD contraction process via the interplay between illumination path, µD geometry (diameter, contact angle, height) and difference in refractive index, ∆n, with surrounding media.Three populations of pixels and their associated grey-level distribution can be distinguished, each contributing to the signal σ emerging from the ROI.(1) Those in the vicinity of the µD, fixed by the system noise.(2) The brightness decay zone, extending from µD edge toward centre, fixed by ∆n and the µD geometry, both determined by Brewster and total internal reflexion angles relative to illumination path.(3)-The central µD area, where light absorption occurs along the direct light path.The balance of pixel grey-level distributions associated with these populations gives rise, throughout µD contraction, to the measured signal and its evolution: σ-curves (Fig. 2(a)).
The variation of σ in the vicinity of t M is related to ∆n: it decreases before t M , and increases afterwards.This makes t M a global minimum on the σ-curve, easy to find numerically.In a previous paper we observed a constant µD contraction rate for pure water and in different salts, 17 thus from t M and NaCl concentration in the µD at that time we can easily deduce the linear contraction rate of each µD in the array, which is proportional to the reverse of t M .
At t N , the smooth evolution of σ throughout the contraction process is hindered by the nucleation and fast growth of a crystal inside the µD.This clear and sudden jump of σ to higher values is easy to determine with simple thresholding procedures.Then, the nucleated crystal will grow until the µD completely dries, t D .Note that precise determination of t D requires more sophisticated algorithms than this simple minimum determination and thresholding, and will not be addressed here.
The information obtained here on each individual µD from (ROI, σ, t)-plots is sparse compared to the usual approaches (side-views systems) [17][18][19] where, with the help of highly resolved images, geometrical µD parameters (diameter, contact diameter, contact angle, height) are monitored.However, our approach overcomes the acknowledged limitations to the study of collective contraction effects, 12,18,20 such as the low number of droplets in the patterns (limiting statistical validity) and their large size (possible convection).For instance, for the 527 µDs in Array 1, we plot in Fig. 2(b) the t M cumulative distribution frequency (CDF).Despite the monodisperse µD starting sizes, 13 they do not contract at the same rate, which might appear to be due solely to their different local pinning conditions.However, when µD position in the array is taken into account in plotting line-by-line t M -CDF (Fig. 2(c)), the main reason for such a wides contraction rate distribution is revealed: collective effects, i.e. neighbour-dependent contraction rates.This results from the instantaneous water concentration field and subsequent gradients a µD will experience, fixed both by itself and by neighbouring µDs.Comparing the general trend from one µD line to another (Fig. 2(c)), we see high similarity in the shape of their respective CDFs.This results from a spatial scale common to all lines: the µD-to-µD first-neighbour distance within a line is uniform throughout the array (at generation time).Only local pinning due to impurity will modify this uniformity.The shift in t M -CDF mean values for each line comes from another spatial scale, line-to-line distance, by far larger and less uniform than µD-to-µD first-neighbour distance within a line.External array lines (1 and 9) only have a single neighbouring line (respectively 2 and 8), which explains why they statistically contract faster.For the inner lines, the position of t M mean values is a function of its distance from the two neighbouring lines, and line 5 is the most representative of this behaviour: it is the fastest of the inner lines, being the farthest from neighbouring lines.
Interestingly, there is good superimposition of both outer lines (1 and 9) CDFs: despite being generated at different times and different spatial locations on the substrate, they still present the same CDF.This validates the µD generation method and confirms the homogeneity of the PMMA substrate: while local pinning still occurs, we observe enough µDs to make it appear as noise on the CDF shape shown by both lines.To resume, the difference between two CDFs comes from differences both in µD diameters and local pinning, but the statistical approach allows us to identify a common CDF.The plots of Array 2 t M -CDF show the same trends (Figs.S5a and S5b of the supplementary material).
Moreover, our approach makes it possible to measure nucleation induction times of each µD in the array.Nucleation induction time is the time over which a system can withstand supersaturation, constant or increasing, 21 before nucleation occurs.In the context of nucleation studies, accurate nucleation induction time measurement has always been difficult to achieve.The main experimental difficulty comes from the stochastic aspect of the nucleation process: we do not know where, when and how many such events occur.Moreover, the size of the critical nucleus is in the nanometre range, requiring either measurement systems addressing the nanoscale (in-situ electron microscopy, 22,23 AFM, 24,25 laser confocal microscopy 26 ), or highly resolved current measurement (using an external localized DC electric field with a nanoelectrode 27 ).Because of these difficulties, such small-scale experiments are not performed routinely and their quantitative relevance is still to be confirmed.Consequently, most experiments address the microscale, for instance using turbidity measurements. 28owever, this is quite a large system compared to the supposed size of critical clusters.The price to pay is that assumptions on the nucleation mechanism have to be made in order to model both growth of individual nucleated clusters and their possible interactions (like Ostwald ripening) until the crystal can be detected. 29,30Here, we present experiments in which the statistical approach is guaranteed through the large number of identical crystallisers (µDs) that can be monitored simultaneously.All µDs will lead to nucleation of only one crystal 31 and because of the fast growth rate (greater than 200µms -127 ), the time required for the newly formed nuclei to grow to a detectable size is negligible with regard to the induction time. 32Thus, these experiments allow us to measure real nucleation induction times without needing to make assumptions on the nucleation mechanism.
In the (ROI, σ, t)-plots, nucleation of a crystal inside the µD is seen through a sudden and large increase in σ (Fig. 2(a)).In Fig. 3(a), we plot the line-by-line CDF of t N for the 527 microdroplets of Array 1.To take into account each µD individual contraction rate 17 which is proportional to the reverse of t M , we normalise t N relative to this rate: t N is divided by t M .The line-by-line CDF of normalised nucleation times is shown in Fig. 3(b), and across lines we observe a narrow and similar distribution of the measured normalised induction times.Such reproducibility in measurements already represents a step forward in experiments addressing nucleation mechanisms.µD-contraction can be represented as the sum of two contributions: a vertical water flux toward oil interface with atmosphere, and lateral water fluxes from µD to µD, both responsible for the collective effect.Any change in the experimental conditions (R, D, h, RH) will affect this balance, as demonstrated by Array 2. In practice, for Array 1 µD-contraction, every (ROI, σ, t)-plot presents the same characteristic behaviour: a minimum is reached at t M and σ values increase until there is a sudden jump corresponding to crystal nucleation (hereafter these curves are called type 1).For Array 2 µD-contraction (Fig. S6 and video S2 (supplementary material) for a movie of the microdroplet dynamics) and t N -CDF curve (Fig. S5c of the supplementary material), evolution of µDs seems similar to Array 1 µDs.However, a closer look at Array 2 (ROI, σ, t)-plots shows that some µDs present (ROI, σ, t)-plots with additional features compared to type 1 σ-curve characteristics (Figs. 4  and S7).For two neighbouring (Fig. 4) µDs, there is a temporal correlation between crystal nucleation in µD#641 (showing type 1 σ-curve) and a sudden drop in the σ-curve for µD#640.After this drop, σ reaches a minimum value and then increases again, until there is a sudden jump corresponding to crystal nucleation.σ-curves presenting such a correlated oscillation with nucleation in a neighbouring µD will hereafter be called type 2. The advantage of using (ROI, σ, t)-plots is clear here: both Array 1 and Array 2 would be mistakenly considered equivalent without this tool.
Collective effects influence µD contraction rates via the instantaneous water concentration field 11,12,18,20, and this  µDs due to nucleation in one µD.When a crystal nucleates in a µD (µD#641, Fig. 4) its rapid growth abruptly decreases the Na + and Cl -concentrations in the droplet.This suddenly increases the chemical potential of water inside the µD (toward the chemical potential of pure water).As a result, there is a sudden increase in water flux from this µD toward its surrounding µDs when they are close enough (µD#640, Fig. 4).These µDs without crystals will grow and thereby decrease their solute concentration.As a consequence, both their ∆n from surrounding oil and their σ will drop to lower values, as shown by oscillations present on type 2 σ-curves.
Now that we have identified this type 2 behaviour, it is clear that it needs to be taken into account when crystallisation is studied.The hypothesis that each µD can be considered independent of the others can now be tested before assuming that a true statistical view of nucleation is obtained.While collective effects during µD contraction can easily be corrected through normalisation relative to t M (Fig. 3(b)), this is not the case for collective effects due to crystal nucleation: the independence required for statistical induction time measurements of contracting µDs is not respected.Although we expected to find interactions, the mechanism shown here is the opposite of those suggested. 10n fact, the nucleation of a crystal in a µD does not trigger but delays nucleation in neighbouring µDs.Moreover, CDF shape is not significantly different regardless of whether nucleation-mediated interactions occur (Fig. S6c of the supplementary material) or not (Fig. 3(a)).(ROI, σ, t)-plots appear to be a useful tool both to determine µD independence and to provide data of good quality for nucleation studies.
The method presented in this paper allows us to simultaneously monitor the contraction of a statistically relevant number of µDs of the same size.The full time-sequence of images is processed to characterise µD dynamics.Every individual µD image is reduced to a scalar standard deviation (σ), and µD-contraction is monitored using standard-deviation time-evolution plots.
It is shown that this approach not only makes it possible to measure nucleation time but also that µDs interact via water diffusion.It is also shown that this effect decreases the nucleation rate, contrary to previous findings.Our "σ approach" can be compared to recording an order parameter in phase transition.This makes it ideal for studying the dynamics of systems where images are the primary outputs.
See supplementary material for microdroplet generation, time sequences of microdroplet dynamics of Array 1 and 2, CDF of characteristic times of Array 2 and time evolution of standard deviation of neighbouring µDs 622 to 641 of line 8 of Array 2. Video S1 and S2 for movies of the microdroplet dynamics of Array 1 and 2 respectively.
FIG. 1. Array1 (a) immediately after generation, bottom line (line 9) is the most recently generated, 527 µDs with initial droplet radius of 25.1µm ±0.8 (for line 9), with a volume of 60.7pL assuming a contact angle of 130 • , 17 (b) µDs with the ROI of a magnified area of (a) (line 6 from the top) at t=133s after the generation of Array 1 (see video S1 (supplementary material) for a movie of the microdroplet dynamics) and (c) grey level histogram for the ROI of µD 294 at t=270s.

FIG. 3 .
FIG. 3. (a) Line-by-line CDF of t N : for the 527 µDs of Array 1, and (b) vs normalised nucleation time line-by-line.
work .Moreover, a temporal correlation is observed between two neighbouring FIG. 4. Time evolution of standard deviation of neighbouring µDs 640 and 641 of line 8 of Array 2. Insert is a zoom in the nucleation zone.