Dynamic measurements and simulations of airborne picolitre-droplet coalescence in holographic optical tweezers.

We report studies of the coalescence of pairs of picolitre aerosol droplets manipulated with holographic optical 30 tweezers, probing the shape relaxation dynamics following coalescence by simultaneously monitoring the 31 intensity of elastic backscattered light (EBL) from the trapping laser beam (time resolution on the order of 100 32 ns) while recording high frame rate camera images (time resolution <10 μs). The goals of this work are to: 33 resolve the dynamics of droplet coalescence in holographic optical traps; assign the origin of key features in 34 the time-dependent EBL intensity; and validate the use of the EBL alone to precisely determine droplet surface 35 tension and viscosity. For low viscosity droplets, two sequential processes are evident: binary coalescence first 36 results from overlap of the optical traps on the timescale of microseconds followed by recapture of the 37 composite droplet in an optical trap on the timescale of milliseconds. As droplet viscosity increases, the 38 relaxation in droplet shape eventually occurs on the same timescale as recapture, resulting in a convoluted 39 evolution of the EBL intensity that inhibits quantitative determination of the relaxation timescale. Droplet 40 coalescence was simulated using a computational framework to validate both experimental approaches. The 41 results indicate that time-dependent monitoring of droplet shape from the EBL intensity allows for robust 42 determination of properties such as surface tension and viscosity. Finally, the potential of high frame rate 43 imaging to examine the coalescence of dissimilar viscosity droplets is discussed. 44

There are a number of methods available to study the binary coalescence of droplets. The most 58 common method is to use brightfield microscopy coupled with a camera to capture images during the 59 coalescence. For droplets with radii on the order of millimetres, an imaging frequency (frame rate) on the order 60 of 10 kHz is required to study coalescence. 9, 17-19 If one wishes to study much smaller droplets relevant to 61 processes like cloud formation (where radii are on the order of micrometres or smaller), the required time 62 resolution must be a few microseconds or lower. 8, 10 Illumination of the coalescence event to obtain a clear 63 contrast of the droplets in video images becomes increasingly challenging at microsecond exposure times. In 64 addition, high-speed cameras capable of recording sequences of images at MHz frame rates are expensive. An 65 alternative method to obtain high quality images of droplet shapes with fast time resolution and good phase 66 contrast is by polychromatic hard X-rays. 20 Fezzaa and Wang demonstrated the use of ultrafast X-ray phase 67 contrast to study the dynamics of two coalescing droplets in air with images exposed for 472 ns and acquired 68 in 3.6 μs intervals. 21 However, this method also requires a high-speed camera as well as X-rays from a 69 synchrotron. An electrical measurement of the resistance and capacitance of dilute water-glycerol droplets 70 containing salt during coalescence has been shown to indirectly provide detail about time-dependent 71 processes. 22,23 This approach can give a time resolution as fast as 10 ns but requires that the droplets be 72 anchored to separate nozzles rather than freely suspended in air. Indeed, all of the approaches discussed above 73 examine droplets supported by a substrate. 24 Kohno and coworkers have developed a method to study the 74 collisions of droplets tens of micrometres in radius pulsed from opposing piezo-driven nozzles. 25 By this 75 method, they have determined chemical reaction rates upon collision of two droplets with different 76 composition 26 and investigated shape deformations upon collision. 27 77 In previous work, we have captured two or more airborne droplets with radii 6-10 μm (corresponding 78 to volumes of just 1-4 picolitres) using optical tweezers. 8,11,[28][29][30][31][32][33][34][35] The relative positions of a pair of droplets can 79 be controlled by a holographic system through the relative positions of the two optical traps. 8,11 In order to 80 study the coalescence event, the intensity of elastic backscattered light (EBL) from the optical tweezers was 81 recorded on a fast photodiode connected to an oscilloscope, permitting indirect determination of changes in 82 droplet shape and position with a time resolution of order 100 ns. The time dependence of the EBL takes the 83 form of a damped oscillator during the relaxation of the composite droplet. Although this approach enables 84 precise, quantitative determination of both the surface tension and viscosity of the final droplet, 8, 11 the EBL is 85 droplets coalesced. The relative laser power in each trap was controlled by a half-wave plate located before 114 the SLM. 115 The important change to the experimental setup relative to that used in the past 8, 11, 28 was the 116 integration of a high frame rate camera (Vision Research, Phantom v. 7.3), which can acquire images at frame 117 rates better than 120 kHz. The droplets were illuminated with a high power LED (Thorlabs, 470 nm). 118 Additionally, the EBL (532 nm) was collected using a silicon photodetector (Thorlabs, DET 110) and recorded 119 using a low-load, 12 bit ADC resolution, 2.5 GS·s -1 sample rate oscilloscope (LeCroy, HDO 6034-MS). The 120 oscilloscope was triggered at the onset of coalescence of two trapped droplets when EBL intensity surpassed 121 a threshold value, thereby recording the full profile of intensity around this time. In all experiments, the high 122 frame rate camera was triggered synchronously with the oscilloscope. In this way, the camera images and EBL 123 could be directly compared. Due to space constraints, installation of the high frame rate camera into the setup 124 prevented the acquisition of parallel Raman spectra, which would have allowed for accurate determination of 125 droplet size, refractive index, and wavelength dispersion as in previous work. In practice, these properties were 126 not necessary for interpretation of the experiments described here. 127 High frame rate images recorded by the camera were sent to a computer operating the Phantom PCC 128 2.2 software package. Images were collected, and the contrast of these images was enhanced using the ImageJ 129 software package (v. 1.46r, http://imagej.nih.gov/ij/) in order to more clearly show the position and shape of 130 the droplets. No other image correction or manipulation was performed. Where appropriate, aspect ratios of 131 the high contrast images were determined from the ratio of the vertical (y) axis to the horizontal (x) axis (ay/ax) 132 of an ellipse superimposed onto the droplet (see Fig. 2a). 133 In the optical tweezers setup, the custom-built trapping chamber was isolated by a cover slip (Chance 134 Glass, #0 thickness) through which the objective focused the trapping beams. The traps were populated from 135 the aerosol flow generated by an ultrasonic nebulizer (Omron NE U22) containing aqueous solutions of either 136 sodium chloride (Sigma, 99.9999%) doped with a small amount of sodium dodecyl sulfate (Fisher, 137 electrophoresis grade) or sucrose (Sigma, >99.5%). The relative humidity (RH) of the trapping chamber was 138 controlled by varying the relative flow rates of dry and humidified nitrogen originating from the boil-off flow 139 of a liquid N2 dewar. The relative flow rates were controlled using paired mass flow controllers (Bronkhorst). 140 RH was measured at the outlet of the cell using a capacitance probe (Honeywell). 141 142 B. Simulations of droplet coalescence 143 A purpose-built finite-element-based computational code was used to capture the dynamics of the 144 coalescence of two drops and their subsequent oscillations. The bulk flow of the liquid is governed by the 145 incompressible Navier-Stokes equations with classical boundary conditions applied at the free-surface. The 146 complexity of the problem is such that numerical methods are required. 147 The computational framework has already been used to probe the coalescence event in a series of 148 articles that compare different models for the process, 16 establish the influence of the surrounding gas, 36 study 149 the inertia-dominated regime, 37 identify the dominant forces in the initial stages of coalescence, 38 and 150 determine how coalescing drops can jump from superhydrophobic surfaces. 39 A full description of the models 151 used, benchmark simulations confirming the code's accuracy, and a comparison to recent experimental data 152 can be found in these papers. Furthermore, a step-by-step user-friendly guide to the development of the code 153 can be found in . 40 Therefore, here we only briefly recapitulate the main 154 details. 155 The code uses an arbitrary Lagrangian Eulerian approach, so that the free surface dynamics are 156 captured with high accuracy. The mesh is based on the bipolar coordinate system, and is graded so that 157 exceptionally small elements can be placed in the region where the two droplets first touch. Consequently, and 158 in contrast to many previous works, both local and global physical scales of the coalescence process are 159 properly resolved. Triangular-shaped finite elements of V6P3 type are used and the result of our spatial 160 discretization is a system of non-linear differential algebraic equations of index two that are solved using the 161 second-order backward differentiation formula (BDF2) using a time step which automatically adapts during a 162 simulation to capture the appropriate temporal scale at that instant. The resulting equations are solved at each 163 time step using a quasi-Newton method. 164

III. RESULTS AND DISCUSSION 166
The coalescence of picolitre-droplets across a range of different viscosities and compositions was 167 studied. The experiments and simulations discussed in the following sections are grouped according to the 168 similarity in viscosity of the coalescing droplets as quantified by the Ohnesorge number, which relates the 169 viscous forces to the inertial and surface tension forces in the droplets by the relation Oh = η/(ρσa) 1/2 , where η 170 is droplet dynamic viscosity, ρ is droplet density, σ is droplet surface tension, and a is droplet radius. If Oh < 171 ~1, inertial forces initially dominate viscous forces and coalescence takes the form of damped oscillations in 172 droplet shape. If Oh > ~1, viscous forces are dominant and coalescence takes the form of a slow merging of 173 the two precursor droplets. 174 175 A. Coalescence of like-viscosity droplets with Oh < 1 176 In the experiments described here, two optically trapped 6-10 μm radius droplets are brought close 177 together to induce coalescence. In the limit of low viscosity, 41, 42 the droplet shape during and immediately 178 after coalescence takes the form of a damped oscillator with the time-dependent amplitude A(t) given by: 179 In this equation, A0,l is the initial droplet amplitude for a mode order l (which corresponds to a characteristic 185 deformation in droplet shape), τl is the characteristic damping (or relaxation) time for a given mode order, and 186 ωl is the angular oscillation frequency of a given mode order. 187 Accurate determination of properties such as droplet surface tension and viscosity relies critically on 188 the ability to infer τ and ω from a coalescence experiment. However, owing to the small size and fast timescale 189 of coalescing 6-10 μm droplets, τ and ω generally cannot be directly or precisely determined by imaging. occurring on the microsecond timescale around the coalescence event. Sequential images were captured with 198 8 μs exposure time, which required the region of interest for the camera sensor to be made very small (80 × 96 199 pixels). The droplets were doped with surfactant to decrease surface tension and oscillation frequency (ω) and 200 thereby facilitate observation of shape distortions with the camera. In the example shown in Fig. 2, the pair of 201 droplets were located at similar heights above the cover slip prior to the coalescence event, which subsequently 202 occurred transverse to the trapping beams, whereas in Fig. 3, the relative strengths of the traps were adjusted 203 such that droplets were located at different heights above the cover slip and coalescence occurred along an 204 axis parallel to the trapping beams (axial coalescence). 44 Fig. 2a shows high frame rate images (8 μs time 205 resolution) that clearly illustrate the damped oscillations in droplet shape immediately after coalescence. In addition to high frame rate imaging, the EBL measured by the oscilloscope for the same coalescence 219 event is shown in Fig. 2b. The oscilloscope provides a time resolution of ~100 ns, which is nearly a 2 orders 220 of magnitude improvement over the resolution provided by the high frame rate camera. A correspondence is 221 clear between the aspect ratio and EBL intensity. The EBL signal maxima correspond to frames from the high 222 frame rate images in which the droplet is elongated along the y-axis (high aspect ratio), whereas the minima 223 correspond to frames in which the droplet is elongated along the x-axis (low aspect ratio). Note that the optical 224 traps are located at the top and bottom of the image, so a higher EBL intensity is expected for high aspect ratio 225 droplets. The higher frequency features observed in the EBL arise from Fabry-Perot type interference 226 resonances. The change in the optical path length of the trapping beam through the droplet as it oscillates in 227 shape leads to modulation in the interference on backscattering with the direct reflection from the front face 228 of the droplet. 8 Figure 2c shows the fast Fourier transform of the EBL in Fig. 2b, giving the frequency of the 229 shape oscillation and confirming that the l = 2 mode is predominately excited upon coalescence. The broad, 230 low intensity peaks at higher frequency correspond to the l = 3 and l = 4 modes. 231 3a is symmetrical with a periodically increasing and decreasing radius that is difficult to resolve from the 239 camera images. Although the shape distortion is unclear in the experimental images, it is evident from the 240 variation in EBL intensity in Fig. 3b. Qualitatively there are many similarities to the trace shown in Fig. 2b, 241 most notably the periodic changes in intensity. However, there are two key differences. First, there are fewer 242 additional features in the EBL. This difference probably arises from the fact that, in this geometry, the 243 coalesced droplet is entirely contained within one optical trap. Therefore, interference features present in the 244 transverse coalescence geometry, which results in shape oscillations that intercept the light in both optical 245 traps, are not present. Second, the fast Fourier transform of the EBL (Fig. 3c) shows that the magnitude of the 246 l = 2 mode is decreased relative to that of the l = 3 and l = 4 modes (at higher frequencies). This difference is 247 likely to be the result of the more modest distortion in shape for the l = 2 mode perpendicular to the beam path 248 for axial relative to transverse coalescence geometries. The observation of a coalescence event with an axial 249 geometry enables the existence of higher order modes to be identified in the fast Fourier transform, which are 250 not as clearly resolved in a measurement from a transverse geometry due to the dominance of the l = 2 mode 251 and additional noise from the higher frequency interference features. 252 These experimental observations were confirmed by simulating the binary coalescence of droplets 253 using the computational framework described earlier. The coalescence of two 8 μm radius droplets with η = 1 254 mPa·s, σ = 72 mN·m -1 , and ρ = 1 g·cm -3 was examined and animations showing the same coalescence 255 simulation from two orthogonal perspectives are provided in Supplementary Videos S1 and S2 of the 256 supplementary material. 47 If different assumptions for surface tension were made (e.g. using a value 257 representative of a solution containing sodium dodecyl sulfate), the frequency of the shape oscillation would 258 change, but the shape distortions observed in each plane would remain the same. Figure 4a shows the droplet 259 aspect ratio from the simulated coalescence event viewed from a plane parallel to the axis of approach 260 (analogous to the coalescence geometry in Fig. 2). From this perspective, the aspect ratios clearly follow the 261 form of a damped oscillator and are similar to the observations from Fig. 2. Figure 4b shows simulated aspect 262 ratios and droplet radius relative that of the composite droplet for the same coalescence viewed from a plane 263 perpendicular the axis of droplet approach (analogous to the coalescence geometry in Fig. 3). The droplet 264 aspect ratio is equal to 1 and unchanging, with relatively minor changes in the relative droplet radius. These 265 changes are consistent with the observations reported in Fig. 3 and corroborate our interpretation of the 266 experimental data. 267 In addition to differences in processes occurring on the microsecond timescale, the coalescence 268 geometry relative to the camera frame of reference also impacts observations on the millisecond timescale. 269 Figure 5 presents the EBL with selected images showing processes occurring during the milliseconds before 270 and after coalescence for transverse (Fig. 5a) and parallel (Fig. 5b) events. In both, the large spike in EBL at 271 time t = 0 corresponds to the moment of coalescence and the initial shape distortion. For the transverse 272 coalescence (Fig. 5a), a gradual decrease in EBL intensity is observed over the first two milliseconds after the 273 end of the shape oscillation (which occurs in between the two optical traps). From the high frame rate images 274 (inset), it is clear that this gradual decrease results from the recapture of the coalesced droplet in one of the 275 optical traps. On the other hand, for the coalescence occurring parallel to the trapping beam (Fig. 5b), the 276 droplet coalescence occurs in one of the optical traps. A gradual shift in EBL intensity is not observed after 277 coalescence because the coalescence occurs when both droplets are already confined within one of the optical 278 traps. However, the EBL intensity changes before coalescence as a droplet is gradually pulled from one trap 279 into the other, eventually inducing coalescence. This phenomenon is illustrated by the images in Fig. 5b. At 280 19 ms before coalescence, two droplets are stably trapped. Over the intervening period until coalescence, the 281 droplet located at a higher position is pulled into the axis of the adjacent laser beam and just before coalescence 282 it is almost completely obscured by the other droplet located at the beam waist. The trajectory of this droplet 283 gives rise to the slow changes in the EBL intensity before the coalescence event, similar to previous 284 observations of coalescence between a free-flowing and optically trapped droplet. 35

285
In short, the combination of high frame rate imaging and EBL allows a very precise understanding of 286 the dynamics of coalescence. First, the combination of the two approaches shows that two timescales can be 287 discriminated during coalescence in a dual optical trap. Shape distortion occurs on the microsecond timescale. 288 Migration of the coalesced droplet into one of the optical traps (coalescence transverse to the trapping beam) 289 or of a precursor droplet from one trap to another (coalescence parallel to the trapping beam) occurs on the 290 millisecond timescale. Understanding and distinguishing these two processes is essential to confidently 291 identify which portion of the EBL is relevant to the coalescence event. Second, the coalescence geometry has 292 an impact on the observed form of the EBL, and this arises due to the location of the coalescence event relative 293 to the positions of the optical traps and the axis along which the shape distortion is viewed. Correctly assigning 294 the origin of the key features in the EBL permits more confident determination of the oscillation frequency 295 and relaxation time. As will be discussed next, a full understanding of droplet dynamics in the optical tweezers 296 also allows for evaluation of the range of experimental conditions where inferring relaxation time from the 297 EBL is quantitatively appropriate. 298 299

B. Coalescence of like-viscosity droplets with Oh > 1 300
When Oh > 1, viscous forces dominate, the shape oscillations are efficiently damped, and only a slow 301 merging of two droplets is observed during coalescence. In this case, the droplet shape relaxes to a sphere with 302 time constant 48 303 = 2(2 2 +4 +3) ( +2)(2 +1) (4) 304 A straightforward method to determine the relaxation time constant is by imaging coalescing droplets and 305 examining the time dependence of the aspect ratio of the relaxing composite droplet. Unfortunately, for 306 droplets with radii on the order of micrometres and with viscosities near or a few orders of magnitude above 307 the critical viscosity for efficient damping of surface oscillations, such a measurement is beyond the 308 capabilities of most cameras as a time resolution on the order of 10 μs is required. Instead, the collection of 309 EBL after coalescence is the only means to monitor (indirectly) the relaxation in droplet shape. However, as 310 discussed previously, there are additional complications in the EBL, namely the additional interference 311 features and the millisecond scale shifts in intensity due to rearrangement of the composite droplet position in 312 the optical traps. A direct comparison of the EBL to high frame rate images allows determination of whether 313 these additional features to the EBL complicate precise quantification of the relaxation time constant. 314 Figure 6 shows coalescence events for sucrose droplets at three different RH values: 89%, 86%, and 315 82%. For each, the EBL is plotted as a function of time (left axis) along with the droplet aspect ratio from high 316 frame rate imaging as a function of time (right axis). In addition, the experimental data were fit to an 317 exponential decay using a non-linear least squares algorithm. Exponential fits to the EBL (dashed lines) and The good agreement between the measured relaxation time constants indicates that both approaches are 338 essentially equivalent. 339 Figure 6b shows the coalescence of sucrose droplets at 86% RH, which is considerably more viscous 340 (ηest = 350 mPa·s). Note the change in timescale relative to Fig. 6a. In this case, relaxation occurs over hundreds 341 of microseconds and tens of images are recorded that show the relaxation in droplet shape. Again, for the EBL, 342 the higher order features on the decreasing side of the trace correspond to a Fabry-Perot type resonant 343 condition. Nonetheless, very good agreement exists between the relaxation time constants fit from the EBL 344 (52±4 μs) and the droplet aspect ratios (48±4 μs). 345 Figure 6c shows the coalescence of sucrose droplets at 82% RH, which corresponds to droplets with 346 a viscosity about an order of magnitude larger than those studied at 86% RH (ηest = 6400 mPa·s). In this case, 347 relaxation occurs over several milliseconds and it is clear that the agreement between the fit obtained from the 348 EBL (360±30 μs) does not agree well with that obtained from the droplet aspect ratios (870±60 μs). The reason 349 for this relates to the timescale of droplet recapture into the optical traps, which also occurs over milliseconds. 350 As a result, two different processes are occurring that give a more complicated EBL, and separating them 351 becomes quite challenging. The two processes are evident in the experimental images of Fig. 6c. Initially (see 352 image at 661-678 μs) the newly-coalesced droplet is located between the two optical traps, which are located 353 at the top and bottom of the image. As the coalescence progresses, the droplet gradually relaxes to a sphere, 354 but the droplet position shifts upwards over the same time period as it migrates to the upper trap (image at 355 2113-2130 μs). Although these are two relatively simple processes to distinguish in the images, they convolute 356 the EBL, giving a relaxation time constant that is smaller than that determined from the droplet aspect ratios. 357 In short, these observations indicate that once coalescence times last for more than a millisecond, EBL is no 358 longer an effective approach to quantitatively infer changes in droplet shape. 359 In the experimental setup used here, the droplet radius was not directly measured, so it is not possible 360 to precisely determine the droplet viscosity using Eqs. (2) or (4). However, in a typical experiment droplets 361 are usually 6-10 μm radius. Assuming the two precursor droplets are both 8 μm radius and droplet surface 362 tension and density are equal to that of pure water (72 mN·m -1 , 1 g·cm -3 ), the droplet viscosity can be estimated. 363 Note that these are estimations but would likely be within approximately a factor of 2 of the true viscosity. 364 We used these approximate viscosities along with the assumed radius, surface tension, and density to simulate 365 the expected droplet dynamics at those viscosities. The simulations were accomplished using the 366 computational framework described earlier and provide a way to directly compare the experimentally observed 367 aspect ratios to an idealized coalescence event for droplets of the same properties. 16 Images of simulated 368 droplets are provided below the images of experimentally-observed droplets at the same time period after 369 initial coalescence. Across all three experiments, there is clear agreement between the experimental and 370 simulated droplet shapes, providing confidence in the measurements. A more quantitative comparison between 371 simulations and experiments is discussed next. 372 Figure 7 illustrates the timescales over which droplet viscosity can be determined using both 373 approaches simultaneously. Figure 7a shows the case of a low viscosity droplet (Oh << 1), where shape 374 oscillations are evident. The circles represent droplet aspect ratio maxima derived from high frame rate images 375 and the triangles represent the maxima in EBL intensity. The corresponding dotted lines give the best fit 376 exponential decay to the experimental data. The solid line gives the simulation results. In Fig. 7b the circles 377 represent droplet aspect ratios from the high frame rate imaging, dotted lines give the best fit exponential 378 decay, and solid lines give the simulation results. Fig. 7b compares the relaxation timescales of all three 379 droplets examined in Fig. 6. Figure 7 demonstrates that the time range required for droplet shape relaxation 380 can span several orders of magnitude, even over a relatively small range in RH. 381 Droplet aspect ratios extracted from the simulated droplet shape are represented in Fig. 7 by the solid 382 lines. Examples of the simulated droplet shape are given in Fig. 6. The simulated results are shifted in time to 383 align with the experiments, as the moment of initial contact between the two precursor droplets does not 384 necessarily correspond with the trigger threshold on the oscilloscope. A comparison of experimental and 385 simulated droplet aspect ratios indicates that they agree well, validating the approach of using droplet aspect 386 ratio to infer droplet relaxation time and indirectly validating (for lower viscosity droplets) the use of EBL to 387 infer the relaxation time. For the lowest viscosity droplet (Fig. 7a), a minor disagreement between the high 388 frame rate images, EBL, and the simulation is evident, but in fact this disagreement is rather small. For the 389 images, it arises from the rapid changes in droplet shape during the ~8 μs period of exposure. For the EBL, the 390 mismatch is much smaller and originates from broad maxima resulting from the interaction of the EBL from 391 the two traps. Across all studied coalescence events, if additional processes were occurring in the droplet that 392 impact the relaxation, the result would be a mismatch between simulation and experiment. Therefore, the 393 agreement between the two indicates that the dynamics of coalescence are well understood for the precision 394 required in these experiments. 395

C. Coalescence of dissimilar-viscosity droplets 397
The preceding discussion provides comparisons between simultaneous measurements of the EBL from 398 coalescing droplets and high frame rate imaging and demonstrates that consistent information can be gained 399 by both approaches. However, the EBL alone is not always sufficient to fully understand the coalescence event. 400 As illustrated in the discussion of Fig. 6, if the coalescence timescale is of the same magnitude as the timescale 401 for the composite droplet to migrate into one of the optical traps, then extraction of the coalescence timescale 402 from the EBL becomes challenging and may indeed be ambiguous. Even more complex is the interpretation 403 of the EBL during the coalescence of two droplets of different viscosity. Figure 8a shows an example of 404 coalescence between two droplets of very different composition and viscosity. Only images are shown because 405 the EBL in this experiment is not informative. First, a sucrose solution was nebulized into the chamber to 406 capture a sucrose droplet. Next, a sodium chloride solution was briefly nebulized into the chamber to capture 407 a second droplet. As a result, the lower droplet in the image is composed of sucrose and a small amount of 408 sodium chloride, whereas the upper droplet contains only sodium chloride. Both droplets were equilibrated to 409 55% RH and then coalesced, with the progress monitored by high frame rate imaging. Assuming binary 410 component droplets containing one solute and water, the sodium chloride droplet is estimated to have a 411 viscosity of ~5×10 -3 Pa·s and the sucrose droplet a viscosity about 10 4 Pa·s, a difference of about seven orders 412 of magnitude. It is likely the viscosity of the sucrose droplet is less than 10 4 Pa·s due to the addition of a small 413 amount of sodium chloride while trapping the second droplet. Nonetheless, the viscosity can be expected to 414 be far higher than for a pure sodium chloride droplet. For comparison, Fig. 8b shows the coalescence of two 415 sodium chloride droplets equilibrated to 55% RH and Fig. 8c shows the coalescence of two sucrose droplets 416 equilibrated to the same RH. 417 From Fig. 8a, it is apparent that the coalescence of dissimilar viscosity droplets does not proceed in 418 the same manner as coalescence of two droplets of similar chemical composition and viscosity. Initially, after 419 coalescence, a composite droplet with a phase-separated structure is formed. As the coalescence progresses, 420 the sucrose droplet (which does not contain much water) gradually dissolves into the sodium chloride droplet 421 (which retains a substantial amount of water) until a fully coalesced, spherical droplet is formed after about 422 3000 μs. Note that the shift in the centre of mass of the coalescing droplets during the period of the coalescence 423 is due to the recapture of the composite droplet into the upper optical trap. By contrast, the coalescence of two 424 sodium chloride droplets at the same RH is complete within only a few microseconds (Fig. 8b), whereas the 425 coalescence of two sucrose droplets at the same RH is very slow, with little observable progress over 120 000 426 μs (Fig. 8c). Instead, the timescale of coalescence more closely approximates that of sucrose equilibrated to 427 82% RH (Fig. 6c). This experiment demonstrates the potential for high frame rate imaging of coalescing 428 droplets to enable quantification of dissolution kinetics, which are important to understanding powder 429 properties, 49 coating qualities, 50 and cloud droplet formation. 51 Moreover, high frame rate video imaging of the 430 coalescence of particles that initially appear phase separated can give information about the transition to 431 equilibrium morphology and allow the determination of properties such as the surface tensions of the two 432 initial droplets. 14, 52-54 433 434

IV. CONCLUSIONS 435
In this work, the dynamics of droplet coalescence in a dual optical trap were investigated using a high 436 frame rate camera capable of time resolution <10 μs, EBL from the trapping laser, and simulations of droplet 437 coalescence dynamics. Examination of the coalescence dynamics of low viscosity droplets (η ≈ 10 -3 Pa·s, Oh 438 << 1) simultaneously by both high frame rate imaging and EBL resolves key processes during coalescence. 439 Coalescing droplets are pulled out of their respective optical traps at the moment of coalescence, as capillary 440 forces are much stronger than the traps' optical forces. Two processes occur on different timescales for low 441 viscosity droplets: droplet shape distortion and relaxation to a sphere occur on the microsecond timescale, 442 whereas droplet migration into one of the optical traps occurs on the millisecond timescale. High frame rate 443 imaging allows for clear delineation of these two processes and unequivocal assignment of the different 444 observed features in the EBL. The specific geometry of the coalescence impacts the magnitude of the observed 445 shape oscillation, which can be resolved through the EBL. As droplets become more viscous (Oh > 1), the 446 surface oscillations are immediately damped and coalescence is simply the merging of two droplets. We show 447 that the relaxation time can be directly determined from the time-dependent change in droplet aspect ratio from 448 the high frame rate images. Additionally, the relaxation can be inferred from the decay in the EBL. If relaxation 449 is sufficiently fast (<~2 ms), the two approaches are equivalent. However, if the droplet is sufficiently viscous 450 (e.g. >~1 Pa·s) the convolution of the relaxation in droplet shape with recapture of the coalesced droplet into 451 an optical trap hinders a quantitative determination of relaxation time by EBL. In these cases, droplet imaging 452 provides a clearer, more quantitative approach. Simulations of droplet coalescence permitted validation of both 453 approaches and showed that the fundamental physical processes during coalescence are well understood. This 454 result indicates that the extracted relaxation times and oscillation frequencies using the holographic optical 455 trap approach are robust and implies that the approach allows for straightforward determination of both surface 456 tension and viscosity. Lastly, study of the coalescence of dissimilar viscosity droplets highlights the timescale 457 of transition to equilibrium shape and suggests it may be possible use holographic optical tweezers to elucidate 458 dissolution kinetics. liquid crystal on silicon spatial light modulator to create two optical traps whose relative positions can 537 be adjusted. Elastic backscattered light (EBL) is directed to a photodetector and recorded by an 538 oscilloscope. Brightfield illumination is accomplished with a blue LED and is directed to a high frame 539 rate camera. 540 Aspect ratios are also reported for each image after the coalescence time. b) EBL collected after 544 coalescence (left axis, time t = 0 corresponds to the moment of coalescence) and droplet aspect ratios 545 (ay/ax) determined from high frame rate imaging (right axis). c) Fast Fourier transform of the EBL gives 546 the frequency of the shape oscillation. 547  Fig. 2) and b) from a plane 554 transverse to the axis of coalescence (analogous to the experimental observations shown in Fig. 3). 555 Time-dependent droplet aspect ratios are plotted in a) and b). Relative change in radius is plotted in b). 556 Simulated droplet properties are: η = 1×10 -3 Pa·s, σ = 72 mN·m -1 , and ρ = 1 g·cm -3 . 557