Coalescence and spreading of drops on liquid pools
Varun Kulkarni, Venkata Yashasvi Lolla, Suhas Rao Tamvada, Nikhil Shirdade, Sushant Anand
CCoalescence and spreading of drops on liquid pools
Varun Kulkarni ∗ , Venkata Yashasvi Lolla , Suhas Rao Tamvada , Nikhil Shirdade, Sushant Anand ∗ Department of Mechanical & Industrial Engineering,University of Illinois at Chicago, Chicago, IL, 60607
Abstract
Oil spills have posed a serious threat to our marine and ecological environment in recent times. Containmentof spills proliferating via small drops merging with oceans/seas is especially difficult since their mitigationis closely linked to the coalescence dependent spreading. This inter-connectivity and its dependence on thephysical properties of the drop has not been explored until now. Furthermore, pinch-off behavior and scalinglaws for such three-phase systems have not been reported. To this end, we investigate the problem of gentledeposition of a single drop of oil on a pool of water, representative of an oil spill scenario. Methodical studyof 11 different n -alkanes, polymers and hydrocarbons with varying viscosity and initial spreading coefficientsis conducted. Regime map, scaling laws for deformation features and spreading behavior are established.The existence of a previously undocumented regime of delayed coalescence is revealed. It is seen that theinertia-visco-capillary (I-V-C) scale collapses all experimental drop deformation data on a single line whilethe early stage spreading is found to be either oscillatory or asymptotically reaching a constant value,depending on the viscosity of the oil drop unlike the well reported monotonic, power law late-time spreadingbehavior. These findings are equally applicable to applications like emulsions and enhanced oil recovery. Keywords : drop impact, coalescence, spreading, oil spills
1. Introduction
Oil-water interactions play a crucial role in nu-merous industries such as, pharmaceutical, cosmetic,petrochemical, agrochemical, and food processing -encompassing applications involving emulsions [1, 2,3] for drug delivery [4, 5], ointments [6, 7], paintsprays [8], salad dressings [9], pesticides [10], mi-crofluidic multiphase reactors [11, 12], material syn-thesis [13, 14], thermofluids [15], CO sequestrationand enhanced oil recovery [16]. These interactionsare also central to many problems we face today, asalient example of which are the oil spills due to leak-ages from oil tankers aboard maritime vessels. Inaddition to the large volumes spilled during such anevent, a substantial amount also enters water bodies ∗ Corresponding author
Email addresses: [email protected] (VarunKulkarni ∗ ), [email protected] (Sushant Anand ∗ ) Secondary corresponding author. Present Address: Department of Mechanical Engineering,Virginia Polytechnic Institute and State University, Blacks-burg, VA Equal contribution as second author. Primary corresponding author. in the form of impacting oil droplets (emerging fromfragmentation of leaking jets of oil). While the be-havior of spilled oil over vast regions over oceans andseas has been studied extensively [17, 18, 19], severalfacets of oil drops impacting water have remained un-explored despite drop impact being one of the mostactive areas of research [20, 21].Studies on drop impact have revealed it can becharacterized in terms of its several constituent el-ements, namely, the physical properties of the liq-uid drop [20], impact velocity [22], impact angle [23],ambient pressure [24], and the chemical/physical at-tributes of the surfaces which could be smooth/rough[22], soft/elastic [25], or a deep pool [21] or thinfilm of liquid [26]. Among these many problems,our interest lies specifically in studying how an oildroplet gently impacting on water transforms into anoil film and how its initial spreading behavior canbe understood - aspects that are critical to oil spills.The model problem we study represents a coalescenceevent involving two entities - a finite sized spheri-cal droplet and a pool of liquid representing a pla-nar surface. Also, simultaneous to the coalescence isthe spreading of the oil drop over the pool of waterunderneath which encompasses an initial time and
Preprint submitted to Journal of Colloid and Interface Science a r X i v : . [ phy s i c s . f l u - dyn ] J a n ate time behavior occurring at differing length andtime scales. While studies to date [27, 28] have con-sidered merger of droplet of one fluid with itself (inbulk form), our model problem involves an advancingthree-phase contact line comprising an oil-water-airinterface which influences the initial time spreadingbehavior dominantly - a problem which has largelyremained unanswered until now.Prior work on drop coalescence with pools of liquids(in three-phase systems) dates back to the researchby eminent scientists such as Osborne Reynolds [29]and J.J. Thomson [30] in late 1800s who showed thatsuch coalescence events proceed in stages wherein asecondary (daughter) droplet - fraction of the size ofthe original parent drop may also be produced and[30] produces vortex rings deep inside the pool. Theseinitial forays were followed by the celebrated exper-iments of A.M. Worthington in 1908 who examinedimpact of a drop of milk on a pool of water whichconfirm these observations and are detailed in theepochal monograph, The study of splashes [31]. How-ever, the exact mechanism of coalescence remainedunknown until the works of Cockbain and McRoberts[32] and Gillespie and Rideal [33] who were the firstones to examine the role of the intervening air film, itsdrainage and rupture in a two- phase system. Charlesand Mason [34] then advanced these ideas coiningthe term partial coalescence for merger that occursin stages and attributed the formation of a daughterdroplet to Rayleigh instability. These investigations[32, 33, 34, 35] in the context of gentle impact ofdroplets on a pool of the same liquid (as the drop)revolved around three major axes - understanding themechanism and forces underlying the formation of thedaughter droplet and prediction of its size; determin-ing the conditions leading to partial and completecoalescence; time for complete coalescence. However,it was not until the advent of advanced high-speedimaging in the 2000s that much of these nuances ofpartial coalescence came to light [27, 36, 28, 37]. Thekey findings from these studies, all of which were fortwo-phase systems, was that the ratio( D d /D p ) of thesize (diameter) of the daughter droplet ( D d ) to theparent drop ( D p ) and the coalescence time ( t c ) is afunction of the liquid properties given by the dropOhnesorge number, Oh p (= µ/ (cid:112) ρ p γD p ), where µ p , ρ p and γ are the liquid viscosity, density and sur-face tension respectively. Chen et al. [36, 38, 27]included the effects of gravity by introducing the non-dimensional Bond number, Bo (= ρ p gD p /γ ) andidentified three distinct boundaries for inertia, visco-capillary and gravitational regimes. Furthermore,they presented phase diagram in terms of Bo − Oh demarcating regions of partial and complete coales- cence. Similar analysis was presented by Gilet et al.[39]. Finally, Blanchette and Biogini [28, 37] ques-tioned the validity of Rayleigh instability [34, 35] asbeing responsible for droplet generation and insteadposited an explanation based on the development ofcapillary waves which vertically stretch the drop byfocusing energy on its summit leading to the pinch offof a daughter droplet. Compared to two-phase coa-lescence problems herein we elucidate the role of cap-illary waves in sculpting the main features of drop de-formation and pinch off in three-phase, oil-water-airsystem through previously unknown phase diagramand scaling relations dependent only on the liquidproperties.As mentioned before, the coalescence event occursconcurrently with the spreading of the oil drop onthe water surface. Much of the research so far hasfocused on coalescence without describing its conse-quences on spreading behavior. The spreading char-acteristics such as maximum spread area and spread-ing rate are vital to applications like oil recovery [16],drug encapsulation and delivery [5] besides oil spill re-mediation [40, 41]. Central to the spreading of oil ona liquid surface is the concept of an equivalent forceat the three-phase contact line due to the interfacialtensions, often described by the initial spreading co-efficient, S = σ wa − σ oa − σ ow [42, 43], where σ isthe surface/interfacial tension and the subscripts wa , oa , and ow denote the water-air, oil-air, and oil-waterinterfaces respectively. A positive value of S denotesan affinity of the liquid to spread [43] on the substratewhile a negative value implies a tendency of the massof oil to form a liquid lens [44]. For spreading oils( S >
0) a two-dimensional monolayer [43], known asprecursor film precedes the macroscopic liquid front.Studies by Huh et al. [45] and Joos et al. [46, 47]showed that its spreading is governed by a competi-tion between the interfacial tension and viscous forceswhich result in a spreading radius, r ( t ) variation withtime as, r ( t ) ∼ t / . Some spreading liquids mayalso exist in partial-wetting regime wherein they mayshow a pseudo-lens along with a thin film because oflong-range forces [48]. Regardless, for spreading oilswith a film thickness of O (10 − ) [18, 19] the radialspread has been shown to follow power law behavior, r ( t ) ∼ t n [49, 50]. Fay and Hoult [18] provide a com-prehensive summary of values for n , which are foundto lie between 3/8 and 2/3 depending on which forcesare dominant - inertial, viscous or surface tension andalso whether the film is considered axisymmetric orone dimensional.A common feature of the above-mentioned previ-ous studies is that they mainly investigated the latetime behavior which is relatively unaffected by early2ime occurrences where the coupling of the coales-cence phenomenon with spreading is evident. Theearly time spreading can affect the overall extent towhich a film can spread and has so far remained unex-plored. Moreover, understanding them might informdevelopment of oil spill mitigation measures. Conse-quently, in the present study we aim to unravel thisinitial time macroscopic behavior of oil drops spread-ing on water with the objective to contrast it withlong time behavior and highlight the differences.In summary, we identify the following key objec-tives for our work: establish scaling laws for variousfeatures of the drop as it deforms during coalescence,including the size of the daughter droplet and capturethe initial spreading dynamics which we report to bemarkedly different from the long-time behavior.
2. Materials and Methods
To consider the influence of wide ranging viscousand interfacial properties a variety of oils were cho-sen. The alkanes - Hexadecane (ReagentPlus 99%),Kerosene (Reagent Grade), Pentane (Reagent, 98%),Decane (Reagent, ≥ ≥ ≥ ≥ ® (St. Louis, Missouri, USA ). Silicone oils(SO) of varying viscosity (0.65 cSt, 1.5 cSt, 10 cSt,100cSt) were purchased from Gelest Inc. ® (Morrisville,Pennsylvania, USA) to study viscosity effects. Ta-ble 1 illustrates the properties of the liquids used in-cluding kinematic viscosity of the oil ν , interfacialtensions σ aw , σ oa , σ ow , and the spreading coefficient S , where, subscripts ow , oa , and aw denote the oil-water, oil-air, and air-water interfaces respectively.Oils may exists in different wetting states depend-ing on the value of S and Hamaker constant ( A ). Inparticular, note that pentane exists in partial-wettingregime on water [48] (see details in Section S6, SI).The density of air and water are taken to be 1.29 kg/m and 998 kg/m while their dynamic viscosi-ties are selected to be 18 . µP a − s and 1 mP a − s [52] respectively. For the oils the density is estimatedby measuring the mass of a known volume and divingone by the other (Table 1). The measurements thusobtained are within an uncertainty of 5%.The coalescence and spreading process studied inthis work occurs in ≤ s (see Fig. 2) which is in-sufficient for the impurities to be adsorbed at thewater-oil interface and influence the spreading be-havior [53]. The physical properties of the liquids thus may be equivalently expressed in terms of non-dimensional quantities such as the Ohnesorge numberof the parent drop ( Oh p ), Bond number ( Bo p ), vis-cosity ratio ( a µ r ) and ( b µ r ) and density ratio ( a ρ r )and ( a ρ r ) (see Section S2, SI, Table S1 for details onnon-dimensional quantities and Section 3.1 for defi-nition of these quantities).Deionized water was used as the bulk liquid, andwas held in a quartz chamber of dimensions 30 × × mm which was treated with Trichloro [1H, 1H,2H, 2H - Peflorooctyl silane purchased from SigmaAldrich ® to prevent the formation of a concave orconvex meniscus along the glass walls by maintaininga 90o contact angle(hydrophobic). Vapor salinizationof the glass chamber ensured any image distortionsdue to the presence of a meniscus during side viewimaging were avoided. The schematic in Fig. 1(a) shows the componentsof the experimental setup used to record videos fromthe front and observe coalescence events. Dropletspreading was captured from the top by orienting thecamera vertically towards the water surface. Needlesof sizes varying from 0.6 − µl/s to avoid any vibrations from thestepper motor housed within the syringe pump andensure consistent drop sizes for a particular needlesize. The entire apparatus was placed on an opticaltable to eliminate mechanical vibrations. All exper-iments were conducted at a temperature of 25 ± parent drops in the re-maining text) were gently deposited ( < mm/s ) ona pool of water to minimize any inertial effects. Thedepth of the pool was about 25 mm and it was largeenough to ensure that the dynamics of the overlayingoil drop was not obscured by presence of a bottomplate [61]. The drops ( R p = D p /
2) generated by thenozzles were such that they are smaller than the cap-illary length, l c (= (cid:112) γ/ρ p g where, γ is equivalentinterfacial tension of the drop (see further in Sec-tion S3, SI), g is the gravitational acceleration and ρ p is the density of the parent drop, thus ensuringthat gravity played no role in the deformation dy-namics. To ensure that the water-air interface wasperfectly flat so that a deposited droplet does notmove towards the wall due to capillary attraction [62],the glass chamber size was made ≈
10 times largerthan typical drop sizes and the walls were silanized.3able 1: Physical properties of the oils used in the study. ρ is density, ν is kinematic viscosity, σ is interfacialtension (different from γ which is the equivalent interfacial tension defined in section 3.1), S is the initialspreading coefficient, (= σ wa − σ oa − σ ow ) where σ aw = 72 mN/m . No reference is provided for values ofdensity which are measured in the laboratory and some values for silicone oils which are directly availablein the Gelest Inc. ® (Morrisville, Pennsylvania, USA) brochure.Liquid ρ ν σ oa σ ow S ( kg/m ) ( mm /s ) ( mN/m ) ( mN/m ) ( mN/m )Silicone Oil 761 0.65 15.9 38.7 [54] 17.4Silicone Oil 853 1.5 17.8 42.5 [55] 11.7Kerosene 810 2.39 [56] 28.0 [56] 33.0 [57] 11.0Silicone Oil 935 10 20.1 43.0 [58] 8.9Silicone Oil 966 100 20.9 43.1 [58] 8.0Pentane 626 0.5 [59] 15.5 [60] 49.0 [60] 7.5Decane 726 1.27 [59] 23.83 [60] 52.0 [60] -3.83Cyclohexane 773 1.15 [59] 26.56 [60] 50.0 [60] -4.56Dodecane 746 2.01 [59] 24.91 [60] 52.8 [60] -5.71Tetradecane 756 3.06 [59] 26.6 [60] 52.2 [60] -6.8Hexadecane 770 4.46 [59] 53.3 [60] 26.95 [60] -8.25The top/side view experiments were performed inde-pendent of each other [63] since the deformation wasfound to be repeatable over a number of experimentswith an experimental uncertainty of less than 7%.In order to capture the coalescence andspreading behavior, high-speed imaging usingPhotron ® FASTCAM Mini AX camera was em-ployed. Videos were recorded at 4000 frames persecond ( fps ) at a resolution of 1024 × µs . This was deemedsufficient for capturing the entire dynamics giventhe temporal resolution was about 0.25 ms andthe typical time scale for the entire process was inexcess of 10 ms (see Fig. 2 (a),(b),(c)). A lenswith infinite focus (InfiniProbe ® TS − mm and amagnification ranging from 0 − × was attached tothe camera resulting in the arrangement providinga magnification of 1 pixel ≈ µm . An LED(Nila-Zaila ® ) illumination source was used and theemanating light was diffused using multiple diffuserplates placed between the light source and the glasschamber. D p h tip D d h max D cyl b v iiiiv iii t = 0 t = t c highspeedcameraoilsyringe tipwaterdiffuserlight source glasschamber a Figure 1: (a)
A schematic of the setup apparatusused to record videos from the front. Standard high-speed imaging was used to capture drop dynamicsduring partial coalescence. The diffuser plates areused to uniformly distribute the light from the source. (b)
The various parameters studied during partialcoalescence, namely parent drop diameter D p , dropheight at any instance in time h tip , maximum heightof drop h max , coalescence time t c , diameter of daugh-ter droplet D d , diameter of the upward cylinder D cyl .4he videos were recorded to capture the drop de-formation dynamics from the moment the parentdrop touches the pool of water to its complete co-alescence with the water.The videos were analyzedusing the open source IMAGE J software. Fig. 1(b)illustrates some of the parameters investigated duringdrop coalescence and includes ( i ) parent drop diam-eter, D p , ( ii ) height of the collapsing drop at any in-stance in time, h tip , ( iii ) the maximum height of thedrop, h max , ( iv ) diameter of the upward stretchedcylinder (jet), D cyl ( v ) the time taken for drop co-alescence, t c , and ( vi ) daughter drop diameter, D d .Similarly, Fig. 5 provides details of quantities studiedto understand the dynamics of the spreading process.
3. Results
As an oil drop gently impacts a bulk liquid (water)it levitates over the liquid surface for some momentsbefore integrating with the bulk [64, 38]. The de-lay in coalescence is due to an intervening air filmsandwiched between the oil drop and bulk liquid[64]. As the air film drains gradually it eventu-ally ruptures and leads to contact between the twoliquids[65, 35, 34]. Concurrent to the rupture ofthis air film is the initiation of capillary waves whichpropagate along the phase boundary of the drop andambient medium, deforming the drop in the process[39, 35, 34, 38, 64]. The strength of the capillarywaves, which is a measure of the energy they pos-sess is responsible for the observed drop deformationand coalescence. In these initial moments, the coa-lescence may also affect spreading of the drop overthe liquid pool. The global topological variables suchas total time for coalescence ( t c ), maximum verticalstretch, ( h max ) and the daughter droplet radius ( R d )thus strongly depend on the physical properties of theliquid drop.To probe these aspects in detail and elucidatethe key mechanisms we begin by identifying thekey dimensionless parameters and dimensional scales.Thereafter, we use this information to delineate dif-ferent regimes of coalescence by means of a regimemap followed by developing scaling laws for the freesurface features and conclude by studying their effecton the spreading kinetics. The capillary waves causing drop deformation dur-ing its coalescence with water (bulk) generate freesurface deformations (Ω) which primarily depend onthe parent drop diameter ( D p ), interfacial tension( γ ), density ( ρ b , ρ p ) and viscosity ( µ b , µ p ) of the bulk and liquid drop, velocity of impact ( U ) and the mag-nitude of gravitational acceleration ( g ). In reality, wemay have a dependence on a few more variables thanshown here because we have a three-phase system andeach of the phases - air, bulk and the ambient havetheir own density and viscosity. However, most stud-ies [36, 38, 27] thus far have been conducted on justtwo-phases. Even in our case the capillary waves es-sentially travel along the two-phase boundary of theair and the oil drop. So, to avoid repetition we con-sider only density and viscosity of the drop and thebulk with the understanding that to include the ef-fects of the ambient we would just need to change thefluid properties from bulk to the ambient. Thereforewe may now consider the drop deformation features( i - v , section 2.2) to be only a function of the 8 pri-mary variables of the problem.Ω ∼ f ( ρ p , ρ b , µ p , µ b , γ, D p , g, U ) (1)Here, γ is the equivalent interfacial tension (see Sec-tion S3, SI for details) of the drop and defined sepa-rately for spreading and non-spreading oils as , γ = (cid:26) ( σ oa + σ ow ) / S > | S | if S < π ) theorem [66] the di-mensional dependence in eqn 1 leads to, 8 − π terms which is the difference of the 8 vari-ables of our problem and the three fundamentalunits of mass, length and time. Each of these π terms is a non-dimensional group and found to be, Bo p = ∆ ρ p gD p /γ , Oh p,b = µ p,b / (cid:112) ρ p,b γD p , W e p = ρ p U D p /γ , ρ r = ρ p /ρ b and µ r = µ p /µ b , where, sub-scripts p and b refer to the parent drop and thebulk, respectively. W e p for our case for a velocity, U < mm/s is less than 5 and considered negligible.Similarly, Bo p is insignificant ( (cid:47)
1) for our case too(see Section S2, SI for further details). Equivalently,it means ( D p < (cid:96) c ), where, (cid:96) c is the capillary length(= (cid:112) γ/ ∆ ρ p g ). Note that, Oh p = Oh b (cid:0) µ r / √ ρ r (cid:1) soit suffices to use one of the variables, Oh p or Oh b .Since our study involves three fluids we use the sub-script a (for air) or b (for bulk water) as prefix to µ r and ρ r to denote the ratio of the parent drop prop-erties and the ambient fluid. For our test conditions,the density ratio, b ρ r ≈ . − . a ρ r ≈ − Oh p and a µ r . Eqn5 now transforms to the form given below,ΩΩ µ ∼ f ( a µ r , Oh p ) (3)As we will see later, eqn 3 becomes the basis for ob-taining the coalescence regime map. Non-dimensionalizing the drop deformation fea-tures would require the dimensional scale Ω µ in eqn 3to be determined. However, Ω µ is an unknown scalewith several different candidates as potential choices.These scales are determined by a balance of the threedominant forces - inertia ( I ), surface tension or capil-larity ( C ) and viscosity ( V ) considered two at a timeor all three together, each of which results in a dif-ferent length, velocity and time scale. (Section S4,SI Table S2 shows the different possibilities based onthese force balances)Although choosing D p (corresponding to the I-Clength scale) has been usually preferred in literature[36, 27, 37, 64] we explore the applicability of the I-V-C scales first proposed by Eggers and Dupont [67].These scales have recently been identified by Ganan-Calvo [68] as being appropriate for analyzing bubblebursting in a viscous liquid and show a remarkablecollapse of existing numerical and experimental data[69, 70, 68]. Similar to bubble bursting, our problemalso involves the the ascent of capillary waves alonga phase boundary comprising of air on one side andliquid on the other [69, 68], but in contrast to it theymeet at the apex of the drop (anti-parallel to grav-ity). Further justification on the choice of these scalesmay be seen in the fact that the intervening air filmdrains much faster than a liquid film in a two-phasedrop coalescence problem [64]. This implies that weneed to seek scales which are much smaller than theones used earlier thus validating the use of the I-V-Cscales. Studies like the one presented by Lai et al .[70]and Berny et al .[69] may be carried out to addition-ally bolster our claim. We restrict our attention toshowing that these scales work for our experimentaldata and validate it theoretically using simple scalingarguments. It is of significance that a µ r is constant,if we only consider the ratio of drop and air viscosityand can be dropped out of all scaling relations yield-ing a scaling dependence of the form, Ω / Ω µ ∼ Oh np ,where n is any real number.The scales obtained using the I-V-C scaling can besummarized as given in eqn 4 (derived theoretically inSection S5, SI) and will be employed as scales for non- dimensionalising parameters in the following sections. l µ = µ p ρ p γ , t µ = µ p ρ p γ , v µ = γµ p (4)Note that often we may use a simple division bylength scale l µ to prove a scaling relation, theoreti-cally even though it precludes the expected explicitinclusion of expression for various forces. This is aconsequence of the I-V-C balance which is argued tobe applicable to our case given the above considera-tions. Also, the fact that the capillary waves carvethe features on the drop surface and their strengthis directly proportional of the liquid properties [68]warrants the use of a relatively simple calculation. The discussion above has enunciated the significantrole of capillary waves in governing drop deformationand coalescence. Analysis of the high-speed imagingvideos for the 11 different oils tested (refer Table 1)in this study showed that their coalescence behav-ior after contacting the bulk water surface could beclassified into three main categories as shown in Fig.2 (a)-(d) (see Section S1, SI Video 01) arranged inincreasing order of Oh p (a measure of increasing vis-cosity of the drop).At very low Oh p << − , the capillary waves areso strong (meaning that they possess higher energy)that they constructively interfere at the apex (top-most point on the deforming drop) and stretch thedrop vertically such that a portion of the liquid masspinches off (daughter droplet) from the parent drop(top row, Oh p = 10 − Fig. 2(a)) in what is termedas partial coalescence . The process then repeats it-self until the drop completely coalesces with the bulk.At much higher Oh p ( (cid:39) Oh p = 1 Fig. 2 (c)) occurs. Interspersed be-tween these two regimes is a new regime of delayed co-alescence where the capillary waves are strong enoughto stretch the drop but relatively weak for a daugh-ter droplet to form (middle row, Oh p = 10 − Fig.2b). To the best of the knowledge of the authorsthis is the first time that the existence of this regimehas been shown to be an intermediate stage betweenregimes of partial and complete coalescence imply-ing that the transition is not sudden but gradual.Although the role of capillary waves in coalescenceof drops with planar surfaces has been well under-stood [38, 64, 37] the existence of delayed coalescence has not yet been reported like so in literature despitesome allusions [39] to it. Our findings therefore may6 a µ r ( = µ p / µ a ) Oh p −3 −2 −1 (d) Hexadecane Silicone Oil 10 cStSilicone Oil 100 cStPartial coalescenceComplete coalescenceDelayed coalescence KeroseneSilicone Oil 1.5 cSt TetradecaneDodecaneSilicone Oil 0.65 cStPentane CyclohexaneDecane (b)(c)(a) O h p � − O h p � − O h p � Figure 2: Features of drop coalescence (a)
Shows the case of partial coalescence, where the formation of anupward jet leads to the generation of a daughter droplet, Silicone oil (0.65 cSt) ( Oh p = 0.0042). (b) Showsthe case of delayed coalescence where formation of a jet is observed in the absence of a daughter dropletHexadecane ( Oh p = 0.015). (c) shows the case of complete coalescence without jetting. Silicone oil (100 cSt)( Oh p = 0.60). The existence of a new regime of coalescence termed as delayed coalescence is emphasized.(see Section S1, SI Video 1 for videos showing coalescence behavior in the three cases). (d) The regimemap delineates the regions of partial, delayed, and complete coalescence as a function of the viscosity ratio( a µ r = µ p /µ a ) and the parent drop Ohnesorge number ( Oh p ). The scale bar in each row corresponds to alength of 1 mm.be succinctly classified into three distinct coalescenceregimes − ( i ) Partial coalescence (jetting producinga daughter droplet), ( ii ) Delayed coalescence (jettingwithout daughter droplet) and, ( iii ) Complete coales-cence (no jetting). The terminology of jetting refersto the abrupt change in slope of the drop contourfrom zero to infinity ( t = 3 ms , frame 3 Fig. 2(b) andframe 2, Fig. 2(a)) as against a more gradual changeback to zero (see Section S1, SI Video 01).In section 3.1 we showed that the different dropmorphologies can be conveniently represented in theterms of non-dimensional parameters a µ r and Oh p .Upon plotting our experimental data using these vari-ables, regions delimiting the partial, delayed andcomplete coalescence were clearly observed. A lineof best fit drawn including these points gives rise toa linear scaling dependence of the form, Oh p ∼ µ r .Region 1 (circles) represents the regime of partial co-alescence, where both the a µ r and Oh p are relativelylow. At higher a µ r and Oh p , delayed coalescence isobserved, represented by the triangle plots in region2. Region 3 represents the regime of complete coa-lescence, where the capillary wave propagation is suf-ficiently damped. The diamond plots in this regionrepresent drops with relatively high a µ r and Oh p .Note that replacing the ambient medium with a dif- ferent fluid will likely produce lines parallel to theexisting data points, although it will not have anyeffect on the slope of these lines. In the sections thatfollow we quantify features which are manifestationsof these regimes and determine their dependence onliquid (oil) properties. The foregoing exposition has identified jetting orvertical stretching of the drop as one of the differen-tiating factors between complete and partial/delayedcoalescence. Even though visually apparent, an un-ambiguous quantification in terms of a drop defor-mation feature to distinguish different regimes isparamount. To achieve this goal, we consider themaximum height of the deformed drop ( h max , Fig.1(b)-(iii), Fig. 2 (a),(b),(c) second column), a nat-ural choice, which also serves as an indicator ofthe strength of the capillary waves [39]. To iden-tify h max for each of the three regimes we plot thevariation of h tip /D p with time, t as shown in Fig.3(a). We immediately notice that for complete co-alescence the droplet height ( h tip ) never exceeds thedroplet diameter ( i . e . h tip /D p <
1) while delayed andpartial coalescence are marked by vertical stretch-ing ( h tip /D p > h tip /D p for partial and delayed coales-cence is immaterial as the generation of the daughterdroplet followed by a cascade of self-similar events islucid enough to differentiate between the two regimes.On this note it is pertinent to point out thatthe horizontal portion in the temporal variation of h tip /D p (Fig. 3(a)) for partial coalescence ( ) is anindicator of the period for which there is no changein h tip even though a the air film has ruptured. Thisis characteristic of partial/delayed coalescence wherethe nature of the capillary waves is such that for abrief period until they reach the bulk air/drop in-terface they do not cause any visible change in h tip and increase the total time for coalescence. Con-trastingly, in the complete coalescence regime h tip decays rapidly thereby reducing this time (discussedat length later in Section 3.6). h max , representing the end of the vertical elonga-tion, hence is a valuable measure and strongly de-pendent on the drop’s viscosity as revealed by ouranalysis so far. Using the I-V-C length scale ( l µ ) de-rived in the previous section it may be expected thatthe scaled data for h max when expressed in termsof Oh p would collapse for all liquids giving rise to asimple power law relation which can then be extendedto a wide range of fluids. Fig. 3(b) precisely showsthis dependence where with an increase in Oh p , themaximum height of the drop decreases and can bemathematically expressed as, h max l µ ∼ Oh − p (5)Based on our observations we see that the maximumheight of the drop is of the order of the diameter ofthe parent drop ( h max ∼ D p ). Non-dimensionalizing h max and D p with the length scale( l µ ) and recogniz-ing that ρ p γR/µ p = Oh − p we obtain the relationshipbetween h max and Oh p which matches exactly withthe experimental values as shown in eqn 5. In thenext two sections we delve deeper into mechanism ofdaughter droplet formation. As noted above vertical stretching is critical to theformation of the daughter droplet. This process not only involves extension of the liquid mass upwardsbut also horizontal thinning at the point where itjoins the bulk [36, 64, 34]. Bestriding the free sur-face boundary are capillary waves shaping the dropcontour as they move and elongate the drop. Theirjourney can be divided into three stages - (Fig. 3(c))( i ) From incipience until they constructively interfereat the apex ( ii ) Tip reversal which ends with a uni-form cylindrical element ( iii ) Collapse of the cylin-drical element (of length, L cyl and diameter, D cyl forming a single daughter droplet. Once a daughterdroplet is produced stages i to iii are repeated againuntil the final droplet coalesces with the bulk formingan immiscible liquid film. We observed a cascade ofas many as 6 daughter droplets which decreased to1 at higher Oh p . Cases where one or more dropletsformed were very few and not considered in the cur-rent analysis.Against the background it must be noted thatdroplet ejection from a jet has usually been explainedas a consequence of capillary (Rayleigh-Plateau) in-stability commonly seen in dripping faucets and liq-uid atomization [71]. It is a matter of debate whetherthe same mechanism governs droplet formation in jetsformed due to capillary waves as reported in burstingbubbles [68] and partial coalescence [28]. For our caseit is evident that the coalescence process is controlledby the competition between the horizontal and verti-cal rates of collapse and when the rate of horizontalcollapse is greater than that of vertical collapse, adaughter droplet is produced. The daughter dropletgeneration is thus directly dependent on the strengthof the capillary waves meeting at the apex of the dropdetermined by the viscosity of the drop. Fig. 3(e)shows the scaling dependence of the non-dimensionaldaughter droplet radius on Oh p displaying a powerlaw exponent of − . V d , ( ∼ D d ) is equal tothe volume of this cylindrical mass V cyl , ( ∼ L cyl D cyl )before pinch-off ensues. L cyl is the length of the cylin-der before its collapse begins (Fig. 1) and is equal to h max . This implies that the scaling for L cyl and h max should be the same. On similar lines, D cyl scales as D p which amounts to D d ∼ h max D p . Using the I-V-Clength scale we obtain the following scaling relation. D d l µ ∼ Oh − p (6)8 R d / l µ Oh p (×10 -3 )
10 ms 9 ms 6.5 ms 5 ms 3.75 ms 3.25 ms 3 ms 1 ms 0 ms S T A G E S T A G E S T A G E -3 -2 -1 h t i p h m a x h m a x / l µ Oh p h t i p / D p t ( ms ) (d)(e)(c)(a)(b) Oh p (×10 -3 ) Complete coalescenceDelayed coalescencePartial coalescence HexadecaneSO 10 cStSO 100 cStKeroseneTetradecaneSO 0.65 cStSO 1.5 cStPentaneDodecaneCyclohexaneDecane D cy l / l µ Figure 3: (a)
Variation of h tip /D p with time, t . A value of greater than 1 means the droplet undergoes partial or delayed coalescence and exhibits jetting . (b) Experimental non − dimensional maximum height( H max ) of the drop as a function of Oh p (c) Three stages prior to pinch-off of a daughter droplet in partial coalescence:
Stage 1 - incipience of capillary waves concluding with the crashing of the waves at the apex,
Stage 2 - tip reversal and gradual horizontal thinning of the drop,
Stage 3 - Formation of cylindrical liquidentity and its final collapse to produce a daughter droplet. Blue arrows show the movement of the capillarywaves along the drop/air interface . Red block arrows indicate the direction of the stretching (vertical)and thinning (horizontal). (d)
Experimental scaling for non-dimensional D cyl with Oh p . Noticeably, it isthe same as the for non-dimensional h max . (e) Experimental scaling for non-dimensional daughter droplet( D d ) with Oh p . Coefficient of determination, R for the power law fits is ≥ ± D d /l µ ∼ Oh − . p . Predictably, theradius of the daughter droplet decreases with an in-crease in Oh p . With increased viscous dissipation,the capillary waves are dampened significantly, re-sulting in a smaller daughter droplet size. The scal-ing relation obtained agrees well with previous stud-ies [64, 27] where it is suggested that the ratio of theradius of the daughter droplet to the radius of theparent drop remains constant at low Oh p ( (cid:47) . D d ∼ D p which we show for low Oh p can be scaled using l µ and can also be obtainedusing our scaling arguments. When a liquid drop comes in contact with the bulkliquid and starts to spread, it undergoes a change inshape due to the capillary waves straddling its sur-face, the final outcome of which is the formation ofa lens, flat thin film or pseudo-lens (see Section S6,SI for more details). The time interval between themoment the intervening air film ruptures to the for-mation of the liquid film is defined as the coalescencetime, t c . Alternatively, in terms of spreading coales-cence time ( t c ) may be envisaged as time that thedrop takes to show steady state behavior (refer sec-tion 3.7). For this calculation, time t = 0 is iden-tified by the moment when the capillary waves aregenerated and is marked by formation of a promi-nent crest. To understand clearly the instant of timewhen coalescence is considered complete we refer toFig 2 (a), (b), (c) for estimating t c . For partial, de-layed and complete coalescence this is approximately19, 10 and 8 ms respectively (Fig. 2(a)). In the fi-nal coalesced state the drop assumes the shape of aflat liquid film few 10 µm in thickness ( S >
0) or alens approximately 100 µm in thickness at its center( S <
0) (see Section S1, SI Video 01).Fig. 4 shows the experimentally measured coa-lescence time non-dimensionalized using the I-V-Ctimescale and plotted against Oh p as given by thescaling relation below (7). From Fig. 4, it is evi-dent that the coalescence time decreases with an in-crease in Oh p , confirming the dominant role of capil-lary waves in delaying the vertical collapse, and con-sequently coalescence of the drop. To theoreticallyderive the scaling relation obtained experimentallyin Fig. (4), the coalescence time t c can be consideredto scale with the ratio of the maximum height of thedrop and the capillary wave velocity V wave which isthe velocity with which capillary waves move alongthe drop/air interface. For a viscous fluid/air inter-face it scales as γ/ρ p D p [68] which results in the scal- -3 -2 -1 Oh p t c / t µ Figure 4: Variation of coalescence time( t c ) with Oh p .This is the time required for complete coalescencefrom initial rupture of air film to the first instant offormation of a lens ( S <
0) and thin film (
S > R for the power law fit is ≥ − ± V wave /V µ ∼ Oh p (verified experimen-tally in Section S7, SI). Note that as counter-intuitiveas it may seem the scaled velocity of the waves doesincrease with increasing viscosity. Mathematically,this may be attributed to the scaling, V µ (= γ/µ p )which increases with lowering of viscosity. Physicallyspeaking, V wave /V µ = µ p V µ /γ or the wave capillarynumber, Ca wave . It is the relative measure of viscousto capillary forces. A higher value of Ca wave signi-fies higher dissipation due to viscosity which is thecase for higher Oh p . Interestingly, the scaling alsosucceeds in highlighting the role of viscosity wherethe raw experimental values (see Section S8, SI Fig.S3 (b)) are of the same order of magnitude and failto exhibit a discernible trend. Continuing further weemploy the scaling relations for h max and V wave andobtain the following scaling for the non-dimensionalcoalescence time, t c t µ ∼ Oh − p (7)The coalescence time ( t c ) has special significancefrom the standpoint of spread of an oil spill as itmarks the boundary of the topological transition froma spherical drop to a planar liquid thin film or alens. Most of the oil spill mitigation measures haveconsidered only spreading and dynamics when oil isspilled as a film or has taken that form. The mea-surement for t c presented here can serve to determinethis boundary when the spreading kinetics changes10ramatically. In the next section we investigate thespreading of the drop in the time( t c ) that it coalesceswith the bulk. In the context of oils spills, the coupling of thecoalescence process and the spreading behavior is ofimmense significance. Most literature has focused onvery late time spreading without studying the initialmoments of the coalescence process. As such the scal-ing laws obtained in this late time spreading regimeare monotonic and power law [18, 19]. The early timespreading behavior for our case can vary based on itsphysical properties with profound implications on thefinal state of the oil slick formed.If the drop is very viscous it damps the capillarywaves substantially and monotonically spreads to amaximum size and forms a liquid film. If on the otherhand, the drop has low viscosity, the inertial forcedue to the capillary waves and the equivalent interfa-cial tension force compete with each other, resultingin an oscillatory spreading behavior (see Section S1,SI Video 02). To further understand the behaviorof different oils during this transient spreading, wetracked the instantaneous radius of the spreading liq-uid drop r ( t ) at any instant of time is plotted andshown in Fig. 5(b). The montage in Fig. 5(a) for oilswith low Oh p (= 0.0038) (Hexadecane) is presentedas an example to demonstrate the inter-connectivitybetween coalescence and spreading behavior. A lowviscosity (Hexadecane) drop shows an oscillatory be-havior, while a drop with higher viscosity (SO 100cSt) monotonically spreads to a maximum radius. Toanalyze the most important moments during spread-ing we looked closely at the features of oscillatoryspreading as indicated by the markers at different in-stants in time in Fig. 5(b). Note that we only con-sider the radial growth of the macroscopic film andnot the precursor, as may be the case for spreadingoils( S >
S <
0) such ascenario typically does not arise as there is no mi-croscopic precursor (see Sections S2, S6 SI for moredetails).
The first observation one can make from Fig. 5(a)is that a low viscosity drop spreads to a maximumradius r max ( (cid:78) ) due to inertia provided by pinch-offdrop, and subsequently recoils to a minimum radius r min ( (cid:70) ) before the oscillatory behavior eventuallydampens to form a stable oil lens ( (cid:7) ) on the surfaceof water. The difference in radii of these extremepoints ( r max − r min ) is the extent of recoil (∆ r , seeSections S2, SI) and the time interval between these two instances is the recoil time ( t rec ). These recoilcharacteristics are discussed further in this section.Irrespective of the final spreading behavior (seeSections S2 and S6, SI), the inertial force impartedto the drop as the air film ruptures causes the initialspread to a maximum size. Therefore, the maximumradius attained by the drop r max is an indicator ofthe inertial force imparted to the drop, ultimately de-termining the extent of spread of the drop. Opposingthis inertial force is the viscous dissipation in the dropdue to its inherent viscosity besides the restoring sur-face tension force which is oscillatory. It is expectedthat with an increase in the viscous dissipation, themaximum radius of the drop would decrease. The-oretically, the maximum radius ( r max ) attained bythe drop is of the order of the diameter of the parentdrop ( r max ∼ D p ). Using the I-V-C length scale l µ tonon-dimensionalize the quantities, we obtain a scal-ing law for r max in terms of the properties of the dropas r max /l µ ∼ Oh − p . This is in close agreement withthe experimental data provided in Fig. 5(c) wherethe maximum radius attained by the oil lens is non-dimensionalized using l µ and plotted against Oh p .The restoring force responsible for the retraction ofthe drop to its minimum radius, r min is provided bythe combination of interfacial tensions σ oa and σ ow .Therefore, a high initial driving force propelling thedrop to r max does not necessarily result in the largerspread of the drop. During this first cycle of spreadand retraction the extent of recoil ∆ r (= r max − r min )is indicative of the driving (rupture induced) andrestoring interfacial tension force. A higher restor-ing force results in the reduction of the final lens size,thereby reducing the steady state, late time spread( r ss ). Theoretically, the extent of recoil too, is of theorder of the parent drop diameter (∆ r ∼ D p ), re-sulting in the scaling law, ∆ r/l µ ∼ Oh − p obtainedby non-dimensionalizing ∆ r with l µ and using previ-ously established definition for Oh p . This lies reason-ably close to the experimental data in Fig. 5(d) whichdisplays a dependence on the Ohnesorge number as,∆ r/l µ ∼ Oh − . p . Another associated metric, the re-coil time t rec is an important feature as it affects thetime it takes for the drop to reach a steady state andhence governing the late time spreading behavior ofthe drop. This is one of the primary reasons for in-creased coalescence times for partial coalescence sincea large value of t rec implies longer time to achieve co-alescence. The scaled recoil time, t rec using the I-V-Ctime scale is plotted against Oh p in Fig. 5(e) display-ing the scaling law t rec /t µ ∼ Oh − p .11 .7.2. Initial droplet spreading behavior The spreading behavior is highly dependent on thephysical properties of the oils. Experimental resultsin Fig. 5(a) show that drops with a higher Oh p display a monotonic increase in the contact radius, r ( t ) as they spread while drops with comparativelylow Oh p show a non-monotonic behavior. In orderto mathematically model the spreading kinetics wedraw an analogy to the forced response of a secondorder mechanical system. Our motivation being thatsuch behavior has been observed in drops impactingliquid surfaces [72, 73] and therefore, it is not un-reasonable to expect such a description to work forour case too. To develop these ideas we consider amechanical spring-mass-damper system as shown inFig. 5(f) with a block of mass m ( kg ), a spring withspring constant k ( N/m ), and a dashpot with viscousdamping coefficient c ( N − s/m ).To determine the value of these forces as relevantto our case, in the mechanical spring-mass-dampersystem analogue we examine closely the oil drop be-havior as it approaches the bulk liquid. Drainage ofthe intervening air film establishes contact betweenthe drop and bulk, and a step force of magnitude F acts along the oil-air phase boundary after the rup-ture of the air film. This initial step force ( F ) isa consequence of the rupture of the air film and thecapillary wave that is generated. The magnitude ofthis step force is proportional to difference in pressureacross the film given by ( σ oa + σ aw ) D p [74]. The dy-namic viscosity of the drop resists any deformationand slows down the rate of spread of the drop. Theratio of viscosity of the drop and air is much greaterthan the ratio of viscosity of the drop and the bulk( a µ r >> b µ r ) for most of our oils (see Section S2,Table S1, SI), and hence the viscosity of the drop ismuch more significant than the viscosity of the bulkin damping the spread of the drop. Inertia of thedrop aids the spreading while the equivalent interfa-cial tension γ provides the restoring force. We com-bine these various elements to write a second orderlinear differential eqn for the contact radius of thedrop at any instant of time, r ( t ) as, m d rdt + c drdt + kr = F (8)Where, the initial radius r (0) = R p equal to the ra-dius of the parent drop, and initial velocity v (0) = 0are the initial conditions required to solve for r ( t ).In eqn 8, the mass of the drop m is given as m = ρ p πD p /
6, the coefficient of viscous damping force c is written as µ p D p , and the spring coefficient due toequivalent interfacial tension k can be expressed as k = γD p .As the inertial, viscous, and surface tension forcesare all equally dominant, we consider the appropriateI-V-C length and time scales to transform the equa-tion of motion (eqn 8) into its non-dimensional form.For the sake of brevity, the dimensionless variablesfor time (cid:101) t and instantaneous radius (cid:101) r are defined as (cid:101) t = t/t µ and (cid:101) r ( t ) = r ( t ) /l µ . Recognizing the pre-viously defined quantities m , c , and k , we write thenon-dimensional form of eqn 8 for the spreading ofthe drop as given below. Oh − p d (cid:101) rd (cid:101) t + Oh − p d (cid:101) rd (cid:101) t + (cid:101) r = (cid:101) F (9)Here, Oh p = µ p / (cid:112) ρ p γD p is the Ohnesorge num-ber of the parent drop and (cid:101) F = F /γD p is the di-mensionless form of the force obtained by normalizingthe initial step force F with the equivalent interfa-cial tension force γD p acting on the drop at the timeof contact.The solution of the non-homogeneous ordinary dif-ferential equation (ODE) (eqn 9) is given by (cid:101) r ( t ) = A e s t + A e s t + (cid:101) F , where s and s are the roots ofthe characteristic equation of the system, and A and A are constants obtained from the initial conditionsfor displacement and velocity (See Section S9, SI).The characteristic equation of the system is writtenas s + 2 ζω n s + ω n = 0, where ω n is the undampednatural frequency given by ω n = (cid:112) k/m , and ζ is theviscous damping ratio given by ζ = c/c cr . c cr is thecritical damping value given as c cr = 2 √ km .The roots of the polynomial, given by s , = − ζω n ∓ ω n (cid:112) ζ −
1, are highly dependent on thedamping ratio ζ and so we distinguish between theroots for an underdamped system with 0 < ζ < ζ >
1. For anunderdamped system, the roots are s , = − ζω n ∓ iω n (cid:112) − ζ where i = √−
1, while for an overdampedsystem, the roots are given by s , = ω n ( − ζ ± (cid:112) ζ − (cid:101) r ( t ) = e − ζω n t ( C u cos ( ω d t )+ C u sin ( ω d t ))+ (cid:101) r ss (10) (cid:101) r ( t ) = e − ζω n t ( C o cosh ( ω ∗ t ) + C o sinh ( ω ∗ t )) + (cid:101) r ss (11)where (cid:101) r ss represents the non-dimensionalized steadystate radius of the film after it has stabilized (seefigure 5). The constants C u , C u , C o and C o areobtained from the initial conditions (at t = 0, d (cid:101) rdt (0)= v = 0, r (0) = R p ) and are given by, C u = C o = R p − r ss l µ , C u = v o + ζω n C u ω ud l µ and C o = v o + ζω n C o ω od l µ .Further, the natural frequency and the damping co-12 kc F o R p F o Oh p =0.61 r / l µ t / t µ r / l µ t / t µ Oh p =0.039 (×10 ) (×10 ) (a) (b) (c)(e)(d) (g)(f) (h) HexadecaneSilicone Oil 10 cStSilicone Oil 100 cStPartial coalescenceDelayed coalescenceComplete coalescence KeroseneSilicone Oil 1.5 cStTetradecaneDodecaneSilicone Oil 0.65 cStPentaneCyclohexaneDecane4.5 ms 19 ms0 ms 15 ms4.5 ms0 ms R p R p r ( t ) r max ( t ) max S i de T op ∆ r / l µ Oh p −2 −1 t r e c / t µ Oh p −2 −1 -2 −1 r m a x / l µ Oh p SO 100 cStHexadecane r ( mm ) t (ms) Figure 5: Features of drop spreading (a)
Side and top views of the coalescence and spreading for low Oh p (= 0.038, Hexadecane, delayed coalescence ). (b) Variation of the spreading radius, r ( t ) for oils with low Oh p (= 0.038, Hexadecane) and relatively high Oh p (=0.610, Silicone Oil 100 cSt). The symbols indicatedifferent stages in the spreading of a drop. (cid:78) - maximum radius r max , (cid:70) - minimum radius r min , (cid:7) - steadystate radius r ss (c) Experimental scaling of the maximum radius of the film ( r max /l µ ∼ Oh − p ) and, (d) Extent of recoil ∆ r/l µ (where, ∆ r = r max − r min ) as a function of the Oh p (∆ r ∼ Oh − / p ). Viscosityreduces values of r max and r min and hence their difference. Similarly, (e) shows the scaling for recoil time t rec as a function of Oh p ( t rec /l µ ∼ Oh − p ). It is difference between instants of time corresponding to (cid:70) and (cid:78) (f ) Spring-mass-damper analogue of the spreading drop. The radius of the drop at every time instant r ( t ) is measured to determine the features of spreading. The direction of the draining air film is shownby pink arrows. (g) , (h) The solution of the reduced order spring-mass-damper model in comparison withthe experimental data of the spreading drop. Red lines denote theoretical predictions while the data pointsrepresent experimental measured values. Both (g) underdamped ( Oh p = 0.026) and (h) overdamped ( Oh p = 0.61) responses of the model match well with the experimental data showing 95% statistical confidence.Coefficient of determination, R for the power law fit is ≥ (c) , (d) and (e) have an uncertainty within ± Oh p as ω n = Oh p and ζ = Oh p /
2. The terms ω ud and ω od representthe damped frequency and overdamping coefficientfor the underdamped and overdamped systems re-spectively. They are defined as ω ud = ω n (cid:112) − ζ and ω od = ω n (cid:112) ζ −
1. The solution for eqn 9 isfound for the underdamped and overdamped casesusing the Oh p of the drop and the initial conditions(see Section S9, SI) and plotted alongside the experi-mental data in Fig. 5(g),(h). O (1) prefactors rangingbetween 3.46 and 6.87 are used in the estimation ofthe damping frequency ( ω d ). The theoretical resultsmatch the experimental data closely with 95% sta-tistical confidence. The spreading of the drops canthus be predicted accurately using the response of asecond order mechanical system.
4. Discussion and Conclusions
To summarize, in this work, we show that the co-alescence behavior of an oil drop in a three-phasesystem depends highly on the physical properties ofthe drop and can be characterized by the Ohnesorgenumber of the parent drop, Oh p , namely ( i ) Partial( Oh p ≈ − ), ( ii ) Delayed ( Oh p ≈ − ), and ( iii )Complete coalescence ( Oh p ≈ − ). We also findthat initial time spreading follows an oscillatory be-havior before reaching its final shape. An analyticalmodel based on the forced response of a second ordermechanical system is shown to be predict the spread-ing behavior of the drop well.For the first time, the effect of coalescence onspreading is investigated using both experiments andtheory. To show that the experimental results maysuitably be extended to a wide range of oils, a novelapplication of the I-V-C scale is pursued. Throughthese implementations the significant role of the dropviscosity and initial spreading coefficient on the coa-lescence and spreading behavior is hypothesized.While delayed coalescence has been observed fortwo-phase oil-water systems before [39], here we haveshown, its existence as an intermediate regime be-tween partial and complete coalescence in a three-phase system and unreported heretofore. In con-trast to previous works [27, 64], mainly involvingtwo-phases we have also shown that the dimensionalscales obtained via an inertial, viscous, and capillaryforce balance (I-V-C) are the most suitable scales foranalyzing different features generated by the mov-ing capillary waves such as maximum drop height,apex velocity, and coalescence time, leading up tothe generation of the daughter droplet for this kindof three-phase drop-interface system. This is a signif-icant advancement from previous works [36, 38, 27] which have not analyzed spreading accompanied bycoalescence and are restricted to two-phase systems.Furthermore, our results show that the initial spread-ing of the drop differs markedly from the late timebehavior [19, 45], and is influenced by Oh p .Our manuscript provides the foundation for fur-ther studies on oil-spreading dynamics. For ex-ample, future work may focus on studying the co-alescence and spreading behavior in presence ofsurfactants/nanoparticles in oil/water/both media.The difference in monotonic versus non-monotonicspreading of different oils could inform developmentof oil-water separation techniques. Consequently, weanticipate that the results of this work can guide stud-ies on oil-water interactions specifically aiding deter-mination of control parameters and optimization forapplications such as targeted drug delivery, produc-tion of emulsions, enhanced oil recovery, dispersion ofoil during a spill and measures to mitigate its spread.
5. Acknowledgment
SA acknowledges funding support from NSF (EA-GER Award no. 2028571), UIC College of Engineer-ing and Branco Weiss Fellowship.
6. Declaration of interests
The authors declare no conflict of interest.
Appendix A. Supplementary Information
Supplementary information to this articlecan be requested by contacting the authors [email protected] or [email protected].
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