Understanding the Early Regime of Drop Spreading - Langmuir (ACS

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Understanding the early regime of drop spreading Surjyasish Mitra, and Sushanta K. Mitra Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.6b02189 • Publication Date (Web): 11 Aug 2016 Downloaded from http://pubs.acs.org on August 22, 2016

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Understanding the early regime of drop spreading Surjyasish Mitra and Sushanta K. Mitra

∗

Micro & Nano-scale Transport Laboratory, Department of Mechanical Engineering, Lassonde School of Engineering, York University, Toronto, Ontario M3J 1P3, Canada E-mail: [email protected]

Abstract We present experimental data to characterize the spreading of a liquid drop on a substrate kept submerged in another liquid medium. They reveal that drop spreading always begins in a regime dominated by drop viscosity where the spreading radius scales as r ∼ t with a non-universal prefactor. This initial viscous regime either lasts in its entirety or switches to an intermediate inertial regime where the spreading radius grows with time following the well-established inertial scaling of r ∼ t1/2 . This latter case depends on the characteristic viscous length scale of the problem. In either case, the nal stage of spreading, close to equilibrium, follows the Tanner's law. Further experiments performed on the same substrate kept in ambient air reveal a similar trend, albeit with limited spatio-temporal resolution, showing the universal nature of the spreading behavior. It is also found that for early times of spreading, the process is similar to coalescence of two freely suspended liquid drops, making the presence of the substrate and consequently the three-phase contact line insignicant.

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Introduction From rain drops falling on leaves 13 to pollutant oil drops wetting the sh scales, 4 a common aspect for all these natural phenomena is a liquid drop spreading on a given surface. 3,59 The key question remains is that whether we understand the entire drop spreading process? Though the spreading phenomenon towards the later part, close to equilibrium, is well understood from the works of Voinov, 5 Cox 6 and Tanner, 7 the initial spreading, right after a liquid drop makes a contact with a given surface, is not yet fully understood. When any liquid drop makes contact with a surface, it spreads due to the unbalanced interfacial tension driving it to equilibrium. 1,3 Existing literature have characterized the drop spreading in air either as a two-regime process (an initial inertia dominated regime 1014 followed by a viscosity dominated regime close to its equilibrium 57,1013,15,16 ) for low viscosity liquids like water or a single viscosity dominated process for high viscosity liquids 15 like silicone oils. In case of low viscosity drops like water ( µD = 1 mPa-s) the spreading in air (i.e., drop and surface both exposed to surrounding air medium essentially rendering a low viscosity ratio of the drop and surrounding medium, µD /µS = 55) is dominated by inertia at early times as mentioned earlier while at late times close to the equilibrium, the viscous dissipation in the vicinity of the three-phase contact line becomes dominant. For such cases, the R 1/4 1/2 ) t for inertial drop contact radius r on the spreading surface follows a scaling, r ∼ ( γDA ρD

regime 10,11 (where γDA is the surface tension of drop in air, ρD is the drop density and R is the drop radius) and for the viscous regime, close to equilibrium, the drop spreads following R 1/10 1/10 the well-known Tanner's law, 7 r ∼ ( γDA ) t . The situation is very dierent for drops µD 9

of higher viscosity like silicone oils ( µD ∼ 100 − 10000 mPa-s with µD /µS = 5500 − 105 , i.e., high viscosity ratios for spreading in air). The entire spreading process is viscous dominated during which the contact radius follows a power law, r ∼ tα , where the apparent exponent α dened as α =

d (log r) d(log t)

changes throughout the spreading process from an initial value

of 0.8 to a nal value of 0.1 close to equilibrium. 15 On the other hand, spreading of a liquid drop on a substrate kept submerged in another vis2


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cous liquid is a relevant problem with manifold applications ranging from oil recovery, 17,18 design of appropriate marine surfaces 19,20 to droplet microuidics. 21,22 Only a handful of studies were performed in recent times related to spreading of drops on under-liquid substrates. 2325 Even then, for majority of those works, the dynamics of the late times of spreading where viscous dissipation dominates was the primary focus. 2325 However, spreading on under-liquid systems has a distinct advantage. There exists a characteristic length scale to describe the early times of this spreading process, lv =

µD 2 ρD γDA

(see Fig. 1) (viscous length

scale, where µD , ρD and γDA represent drop viscosity, density and surface tension in air, respectively) which ranges from 20nm for water to a few millimeters for silicone oils. Resolving this very small length scale for experiments conducted with a sessile drop resting on a given substrate kept in air poses a severe challenge due to the inherent fast spreading of the drop coupled with optical diraction limit. This can be circumvented by performing spreading experiments on under-liquid substrates, which allows a drop to spread at a slower rate than in air, thereby resolving the spatial resolution of the drop spreading process at early times.

Figure 1: (Color online) Initial stage of a liquid drop (of radius R) spreading on a substrate kept in a surrounding medium. The width of the narrow gap ζ is the dominant length scale dictating the drop spreading at this initial stage, where r is the spreading radius. Often theoretical understanding of drop spreading on a given surface has been modeled as analogous 15 to the coalescence of two freely suspended drops. 2630 For example, in case of high viscosity drops spreading in air, the power law dependence for early times has been observed to be theoretically similar 15 to the result obtained for the liquid bridge growth radius 3


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for the two coalescing drops, 26 where the bridge growth radius scales as rb ∼ − π1 tlog t. Recently, Paulsen et al. showed with an improved electrical measurement technique, 28,31,32 that drop coalescence for even low viscosity drops like water always begins in a viscous regime. 28 This suggests that there could be a possibility that even the drop spreading process could begin in a viscous dominated regime over a wide range of liquid viscosity or more precisely, for a wide range of viscosity ratios of the drop and surrounding medium. As mentioned earlier, limitation in studying drop spreading in air medium is that it is dicult to control such inherently fast dynamics resulting in inadequate spatial resolution at early times. As a result existing body of literature has only managed to study the drop spreading process for low viscosity drops from an initial inertial regime before its eventual transition to the Tanner's regime. 1013 For high viscosity drops, the viscosity ratio of the drop and the surrounding air is typically of the order of 6000 − 105 (high) and the characteristic viscous length scale is of the order of 1mm or higher. Hence, for such liquids, a spreading process is entirely viscous-dominated. 15 There is an intermediate class of liquids of viscosity between 10-50 mPa-s (yielding intermediate viscosity ratios, µD /µS ∼ 550 − 3000) for which both the initial viscous and the inertial regimes are expected to exist, but have not been captured yet due to the inherent limitations associated with small viscous length scales for these liquids, which is in the order of 10 µm. Here, through careful study of the spreading process of millimeter sized Laser oil and Dibutyl Pthalate drops on glass substrate submerged in a viscous water medium, we are able to quantify distinct initial regime(s) of this universal process where it was found that drop spreading on a given substrate in a given surrounding medium always begins in a viscous regime. This initial viscous regime either lasts in its entirety or switches to an intermediate inertial regime depending on the characteristic viscous length scale of the problem. Spreading always terminates in the Tanner's regime. 7

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Figure 2: (Color online) (a) Experimental set-up of the spreading process. (b) Time snaps of the spreading phenomenon, from side view imaging, of a Dibutyl Pthalate (DBP) drop on a boro-silicate glass substrate kept submerged in a water-lled glass cuvette. r(t) represents the spreading radius. The green dotted line represents the substrate location. (c) Corresponding bottom view images of the process. It can be seen that bottom view imaging provides better spatial clarity at very early times (rst frame to the left) compared to the respective side view image which poses a challenge in detecting the correct spreading radius. The scale bar represents 100µm.

Experiments In this Article, the drop spreading is captured with adequate spatial-temporal clarity by conducting spreading experiments on under-liquid substrates. Figure 2a shows the schematic of the experimental set-up used. A custom-made contact angle measurement system was used to study the spreading of laser oil ( ρD = 1069kg/m3 , µD = 200mPa-s, γDA = 24.5mN/m,

γDW = 33.33mN/m, µD /µW = 200) and dibutyl phthalate (DBP) ( ρD = 1043kg/m3 , µD = 16mPa-s, γDA = 31mN/m, γDW = 22mN/m, µD /µW = 16) drops of radius 0.7mm on borosilicate glass surfaces kept in a water lled glass cuvette. Before each experiment, the glass

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substrates were thoroughly rinsed in ethanol, sonicated for 10mins and then cleaned with DI water. The substrates were then dried under nitrogen and put to use. This ensured that no impurities are present on the substrate during spreading experiments. A liquid drop was grown quasi-statically at the tip of a PTFE needle (D = 1mm) at a ow rate of 2-7 µL/min and were allowed to touch the cleaned boro-silicate glass substrates gently to initiate the spreading. The spreading process was captured using Photron FASTCAM UX-100 coupled with a 10X optical zoom lens with proper LED illumination. For spreading on the glass substrate kept under-water recording speeds of 8000 fps - 10000 fps proved sucient. A spatial resolution of around 1 µm/pixel was obtained using the mentioned camera-lens system at the mentioned recording speeds. As shown in Fig. 2(a), our custom-made contact angle measurement system allows imaging from both the side and the bottom view. The obtained images were analyzed in an image analysis software (ImageJ 33 ) to extract the change in spreading radius with time. The drop proles at early times of spreading obtained from side view imaging was tted to part of a circle to extract the contact radius, which is the secant of the tted circle. On the other hand, the image obtained from the bottom view was always circular with minimal deviation. The center point for such a circle, obtained by the bottom view imaging, was ascertained from the very rst image which captures the contact point of the drop-needle assembly with the substrate. Therefore, the two imaging system would generate two sets of transients for the spreading radius. The bottom view imaging was found to provide better spatial clarity at early times of spreading. The inherent problem with side-view imaging is the lack of spatial clarity at very early times. The drop appears to merge with its own reection and hence makes it dicult to pin-point the exact time instant of the contact of the drop with the substrate. Further, the contact radius obtained from side view imaging hovers around a xed value and does not show any detectable change with time for the initial few frames. The same problem was faced by Eddi et al. in their observation of drop spreading and they found their data for the spreading radius, r, as measured from bottom and side view imaging,

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matched only after r > 300µm. 15 However, since the drop spreading for under-water case is much slower than in air, the synchronization of spreading radius data obtained from bottom and side view imaging was found to be achieved for r ∼ 80µm in our present experiment. The obtained data was tted using a power-law function, r = atα , where α corresponds to the maximum root-mean-square value of the obtained t, and a is the corresponding co-ecient which depends on initial conditions and liquid properties. The exponent was evaluated by considering the evolution of the spreading radius over each decade of time. Each set of experiments were repeated 6 times to ensure consistency. The main uncertainty in the experimental data is in the the spatial resolution for very early times of the drop spreading process where an error corresponding to 1-2 pixels in demarcating the drop-surrounding water interface exists, which translates to an error of 1-2 µm in the measurement of spreading radius. Table 1: Viscous characteristic length scale ( lv = 2

2

µD 2 ρD γ

) and the corresponding transition

radius ( rCR = lv = µρDDγ ) for laser oil and DBP (both in air and under-water), and water (in air only). The drop radius for experiments under-water is 0.7mm, while that for air is 1mm. Drop Liquid Laser oil DBP Water

viscous length scale in air ( lv,a ) (µm) 1527 8 0.0014

transition radius in air (rC,A ) ( µm ) 1235 100 3

viscous length scale under-water (lv,w ) ( µm ) 1122 11 −

transition radius under-water (rC,W ) ( µm ) 886 85 −

Results and Discussions Spreading under-water

Figures 2b and 2c show the side view and bottom view time snaps, respectively, of the growth of the spreading radius for a DBP drop spreading on the glass substrate kept underwater. For DBP drops spreading on under-water glass substrates, it was found that the spreading process began in a very distinct viscous regime where the spreading radius grew 7


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Figure 3: (Color online) The spreading of DBP drops (R = 0.7mm) on an under-water glass substrate (θE = 121◦ ) is shown in curve 1. The initial (viscous) regime shows an exponent of 0.95 before switching to the intermediate (inertial) regime at the critical spreading radius rC,W = 85µm which is in accordance with the viscous characteristic length scale of the problem. Curve 2 shows the spreading of laser oil drops (R=0.7mm) on an under-water glass substrate (θE = 134◦ ) with a single initial viscous regime. For both the cases, the spreading terminated in the Tanner's regime. DW t . The best t showed an exponent of 0.95 ±0.05 (see curve 1 of Fig. 3). Here, as r = C0 γπµ D

the prefactor C0 (see Table 2) deviated from unity (commonly observed for coalescence of two hanging drops in air 26,28,29 ) suggesting the role of the surrounding liquid viscosity (or the viscosity ratio of the drop and the surrounding liquid, µD /µW = 16 in this case). Interestingly, following the initial viscous regime, an intermediate regime (see curve 1 of Fig. 3) was observed in this case which obeys a power-law growth of r ∼ tα where the exponent α = 0.49, i.e., close to 0.5. On applying the scaling law for inertial coalescence of R 1/4 1/2 two liquid drops in air, r = D0 ( γDW ) t it was found that the prefactor D0 (see Table 2) ρD

deviated from unity (suggesting the role of the surrounding viscous medium). As mentioned previously, the dominant length scale, i.e., the viscous length scale for this process can be

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characterized by the drop neck height ζ ∼ lv ∼

r2 R

just after the drop makes contact with the

substrate. 15,28 This results into a critical spreading radius value,

rC,W 2 R

= lv =

µD 2 ρD γDW

, which

would signify the crossover point from a viscous to an inertial regime. From our experiments, it was found that rC,W = 85µm (corresponding to a 0.7mm radius DBP drop spreading underwater) which matches with the theoretical value (see curve 1 of Fig. 3). This suggests that the intermediate regime observed here is an inertial regime with a non-negligible eect of the surrounding liquid viscosity evident from the prefactor. For the spreading of laser oil drops on the same glass substrate under-water (i.e., µD /µW = 200), the spreading process was found to initiate in a similar viscous regime (see curve 2 of Fig. 3) where a scaling law of close to r ∼ t (best t exponent of 0.92 ±0.02) was observed with a similar prefactor (see Table 2) deviating from unity. However, in this case, no intermediate regime was observed. The initial spreading lasted throughout the viscous dominated regime before culminating in the Tanner's regime (since for this case, rC,W = 885µm, is much larger). The spreading process never enters a possible inertial regime. It should be noted here that the exponent observed for the Tanner's regime for DBP as well as laser oil is 0.13, which is close to the 1/7 exponent commonly explained from the perspective of Molecular Kinetic Theory 34 of late times drop spreading. However, in this study, we explain this on the basis of the hydrodynamic theory 57 of drop spreading to have a consistent analysis. The non-universal nature of the prefactors in the scaling law, observed for the initial viscous and inertial regimes of DBP spreading and initial viscous regime of laser oil spreading, can be explained from similar observations previously noted in literature involving drop coalescence. 15,26,28,35 In a related study 35 involving coalescence of water-NaCl drops with oil-tetrachloroethylene mixture as the surrounding medium, a prefactor of 0.024 was observed for the initial viscous regime of the liquid bridge growth, suggesting that the prefactor can indeed vary over a wide range depending on the drop-surrounding liquid combination. It is to be noted here, that for DBP and laser oil spreading on glass substrates kept under-liquid, the equilibrium contact angles are 121◦ and 134◦ , respectively, i.e., oleophobic (any surface which is oleophilic in air

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shows oleophobicity under-water 36 ). But such oleophobicity appeared to have no eect on the scaling exponent observed for their respective spreading dynamics at initial times. This further substantiates the claim that spreading process is truly independent of the substrate at early times. 15 Table 2: The exponents and prefactors of the dierent scaling laws observed for initial underwater drop spreading. The close to equilibrium viscous Tanner's regime is excluded here (See text). Drop Liquid Laser oil DBP

viscous regime exponent, α 0.92±0.02 0.95±0.05

viscous regime prefactor, C0 0.01 0.01

inertial regime exponent, α 0.49±0.02 −

inertial regime prefactor, D0 0.027 −

Spreading in air

Figure 4: (Color online) Bottom view time snaps showing the early time spreading process on the boro-silicate glass substrate kept in air for (a) a 1mm radius water drop (shot at 20000 fps) (b) a 1mm radius DBP drop (shot at 12500 fps) and (c) a 1mm radius laser oil drop (shot at 12500 fps). r(t) represents the growth of the spreading radius with time. The scale bar represents 100 µm. 10


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Such a spreading process initiating in a viscous regime should ideally be also observed for spreading studies performed in ambient air where the drop-air viscosity yields a similar ratio. Hence, we performed experiments to study the spreading of DBP, laser oil and DI water drops (R = 1mm) on boro-silicate glass substrates kept in ambient air. Figure 4 shows the early time spreading (bottom view images) behavior in terms of growth of spreading radius for water, DBP and laser oil drops on the boro-silicate glass substrate kept in ambient air. For spreading of water drops on the glass substrate kept in air ( µD /µA = 55, i.e., low), the spreading process was found to initiate in a regime where the spreading radius does not show the inertial scaling of r ∼ t1/2 . Instead, the spreading radius was found to grow as

r ∼ t0.85 i.e., an exponent signicantly higher than 0.5 (see curve 1 of Fig 5). This initial regime lasted only for a period of 0.2ms after the drop made rst contact with the substrate before transitioning into the familiar inertial regime with a scaling r ∼ t1/2 . This clearly shows that spreading for such low viscosity liquids is preceded by a regime other than the conventional inertial one. Existing literature on water drops spreading on a substrate kept in ambient air has been conventionally done with side-view imaging of the entire process. However, with bottom view imaging it is possible to overcome this problem to a certain degree, and hence our very early time data has been able to identify this dierent regime preceding the commonly observed inertial one. For water drops spreading in air, the critical spreading radius value of rC,A is 3µm which would signify a transition from the viscous (Stokes) regime to an inertia dominated regime. Our observed data for the very early times thus represents a probable crossover region between the two regimes. It should be noted here that the crossover regime observed here corresponds to only 5-6 data points in the curve, which in essence represents the fundamental problem in observing such fast spreading dynamics for low viscosity liquids in air with adequate spatial clarity. Therefore, it reinforces the fact that only way of obtaining reliable early regime of drop spreading is through our innovative under-liquid spreading experiment. In the subsequent inertial regime observed, the best t curve shows an exponent of 0.52 ± 0.02 and the prefactor C1 = 0.98, consistent

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with existing literature. 1013 Finally, the spreading culminated in the Tanner's regime with a scaling r ∼ t1/10 (the best t showed an exponent of 0.13). For DBP drops spreading on

Figure 5: (Color online) Curve 1 shows the spreading of water drops (R= 1mm) on a glass substrate kept in ambient air. The very early times data show a regime where an exponent of 0.85 is observed. This is followed by the well-studied inertial regime with an exponent close to 0.5. It is to be noted this inertial regime matches well with experiments conducted by Biance et al. 10 For DBP drops (R = 1mm) spreading in air (curve 2), a very early regime is observed where the spreading radius scales as r ∼ t (i.e., a viscous regime) followed by the inertial regime. Here also, the switch was found to happen for the critical spreading radius value of rC,A = 100µm which is in accordance with the viscous characteristic length scale. Curve 3 shows the spreading of laser oil on a glass substrate in air. The scaling exponent varies throughout the initial viscous regime (similar to the variation observed for drop coalescence of high viscosity drops in air 15,26 ) before terminating in the Tanner's regime. θE < 5◦ for all liquids spreading in air. All the curves terminated in the Tanner' s regime with an exponent close to 0.1. the glass substrate kept in air ( µD /µA = 880, intermediate), the spreading was found to γDA t (curve 2 of Fig. begin in a regime where the spreading radius grows with time as r = C1 πµ D

5), with the prefactor C1 = 1.4, i.e., of the order of unity (see Table 3). In this case, the scaling and prefactor both matches with those observed in studies of coalescence of two freely suspended drops of similar viscosity in air. 28,29,32 Here also, due to the challenges associated 12


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with eectively capturing such fast dynamics in air, the initial viscous regime observed was only for a very small time window (rst 7 data points in Curve 2 of Fig. 5). The spreading R 1/2 1/2 process then transitions into a very distinct regime with a scaling r = D1 ( γDA ) t , where ρD

the value of D1 is 1.19, consistent with such inertia dominated spreading regime. Here also, the transition to this inertial regime was observed to occur for a spreading radius rC,A (see curve 2 of Fig. 5) which satises the condition of viscous characteristic length scale of the problem, (ζ ∼ lv =

rC,A 2 ). R

This suggests a regime change from an initial viscous dominated

Stokes regime to an inertial regime corresponding to the viscous characteristic length scale holds true. It should be noted here that the crossover between the two regimes does not necessarily occurs at the exact value of the critical crossover radius and exhibits a small crossover period. We assume this crossover period to be viscosity dominated, similar to the case for spreading of water in air, before the actual onset of the inertial regime. At late times, the Tanner's regime was observed, as expected. Lastly, for spreading of laser oil drops on glass substrates kept in air (curve 3 of Fig. 5), the entire initial spreading dynamics was found to obey an initial viscous regime evident from the high viscosity ratio ( µD /µA = 11000), and the viscous characteristic length scale of the order of a few millimeters. For such spreading, no inertial regime was observed. Spreading began in a viscous dominated regime where the spreading radius scales as r ∼ tα , with the apparent exponent α (0.4 < α < 1.0) varying throughout the initial regime before settling into the Tanner's regime with a value of 0.08, i.e., close to 0.1. This is similar to the spreading of high viscosity drops in air, as recently observed by Eddi et al. 15 Table 3: The exponents and prefactors of the dierent scaling laws observed for initial spreading in air. The close to equilibrium viscous Tanner's regime is excluded here (See text). Drop Liquid Water DBP

viscous regime exponent, α − 0.93±0.03

viscous regime prefactor, C1 − 2

13

inertial regime exponent, α 0.52±0.02 0.55±0.02


inertial regime prefactor, D1 0.98 1.19

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Conclusions Through appropriate experimental and theoretical analyses, we conclude that spreading of sessile drops in the presence of a surrounding viscous medium (i.e., water) begins in a viscous dominated regime. The initial viscous regime makes way to an intermediate regime dominated by drop inertia when the viscous length scale of the problem is of the order favoring such a switch. A non-negligible role of the surrounding medium in dictating the spreading process is evident from the non-universal prefactor observed in those spreading regimes. Further, the corresponding spreading problem in ambient air was revisited. It was found that in air also, spreading always began in a viscous regime. However, the initial viscous regime observed in air suers from lack of adequate spatial clarity and hence can only be accurately captured for a short time period. This further reinforces the fact that under-liquid drop spreading experiments are critical to accurately capture the early regime of this universal phenomena. Similar to the under-water scenario, here also, an intermediate inertial regime was observed satisfying the condition of characteristic viscous length scale. The prefactors for the spreading regimes in air was found to be of the order of unity in accordance with existing literature. Hence, from the point of view of viscosity ratio of drop and surrounding medium, it can be inferred that spreading of sessile drops always begins in a viscous regime for a wide range of viscosity ratio of the drop and surrounding medium. For low viscosity ratios, surrounding medium plays an important role evident from the varying prefactor which is much smaller than unity. A prefactor of the order of unity is realized only when the viscosity ratio reaches the intermediate range. Only for high viscosity ratios, the growth of the spreading radius follows the Stokes ow for bridge growth rate of two coalescing viscous drops in air with logarithmic corrections. The occurrence of an intermediate spreading regime dominated by inertia is dictated solely by the viscous length scale of the problem. Finally, for all cases, close to the equilibrium, drop spreading always obeys Tanner's law.

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Acknowledgement The authors would like to thank Dr. Naga Siva Kumar Gunda, post-doctoral fellow at the MNT Lab, York University for his assistance with the experimental set-up. The nancial support from the Natural Sciences and Engineering Research Council (NSERC) in the form of Grant # RGPIN-2014-05236 and RTI-2015-472734 is acknowledged here.

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Understanding the Early Regime of Drop Spreading - Langmuir (ACS

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