Damped harmonic system modeling of post-impact drop-spread dynamicson a hydrophobic surfaceRaj M. Manglik, Milind A. Jog, Sandeep K. Gande, and Vishaul Ravi Citation: Phys. Fluids 25, 082112 (2013); doi: 10.1063/1.4819243 View online: http://dx.doi.org/10.1063/1.4819243 View Table of Contents: http://pof.aip.org/resource/1/PHFLE6/v25/i8 Published by the AIP Publishing LLC. Additional information on Phys. FluidsJournal Homepage: http://pof.aip.org/ Journal Information: http://pof.aip.org/about/about_the_journal Top downloads: http://pof.aip.org/features/most_downloaded Information for Authors: http://pof.aip.org/authors

PHYSICS OF FLUIDS 25, 082112 (2013)

Damped harmonic system modeling of post-impactdrop-spread dynamics on a hydrophobic surface

Raj M. Manglik, Milind A. Jog,a) Sandeep K. Gande, and Vishaul RaviThermal-Fluids and Thermal Processing Laboratory, College of Engineering and AppliedScience, University of Cincinnati, Cincinnati, Ohio 45221-0072, USA

(Received 19 March 2013; accepted 9 August 2013; published online 30 August 2013)

The post-impact spread, recoil, and shape oscillations of a droplet impinging on a dryhorizontal hydrophobic substrate at low Weber numbers (We) (9.0 < We < 25) aremodeled as the behavior of a second-order damped harmonic system. Experimentswere conducted to capture the spread dynamics of droplets of six different liquidsimpinging on a Teflon substrate using a high speed digital visualization and imageprocessing. The selected liquids cover a wide range of viscosities and surface tensioncoefficients, and their Ohnesorge and Capillary numbers vary by three orders ofmagnitude (0.002 ≤ Oh ≤ 1.57; 0.007 ≤ Ca ≤ 7.59). High-resolution photographicimages of the post-impact spread-recoil process at different We are analyzed toobtain the temporal variations of the spread factor (ratio of liquid spread to dropletdiameter) and the flatness factor (ratio of liquid height to droplet diameter). Theseare found to be represented by the damped harmonic response of a mass-spring-damper system, where the surface tension force acts as a spring and liquid viscosityprovides the damping. Due to contact angle hysteresis, the frequency of oscillationsfor the transient flatness factor variation is slightly different from that for the spreadfactor variation. Semi-empirical correlations are developed for both the oscillationfrequency and the damping factor as a function of drop Weber and Reynolds numbers.The predictions of temporal variations of the spread and flatness factors from theseequations agree very well with experimental measurements on the hydrophobic Teflonsubstrate. C© 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4819243]

I. INTRODUCTION

The dynamics of spray droplets impinging on a solid surface, characterized by their spread,recoil, and shape oscillations, critically influence many engineering processes such as spray coating,near net-shape manufacturing, spray cooling, ink-jet printing, spray painting, and crop spraying. Notsurprisingly, a large number of experimental, analytical, and computational investigations of drop-impact dynamics, particularly for characterizing their shape and maximum spread, are reportedin the literature.1, 2 In the classical experimental study by Worthington,3 which was perhaps thefirst reported photo-visualization, drop-shape changes during the post-impact spreading of milkand mercury drops on glass were recorded using rather rudimentary imaging techniques of thetime. In the last three decades, however, relatively more detailed images of the process have beencaptured with the use of high-speed cameras for a variety of liquid-substrate combinations (see,for example, Chandra and Avedisian;4 Asai et al.;5 Mao et al.;6 Sikalo et al.;7 and Gatne et al.8).The observed drop behavior, generally described by the process of deposition, prompt or coronasplash, receding breakup, and partial or complete rebound, has been categorized into six differentpost-impact outcomes by Rioboo et al.9, 10 Also, the qualitative effects of the impact velocity, dropsize, liquid properties (density, surface tension, and viscosity), and surface wettability or substratesurface energy on the spread outcomes have been well documented.1, 10

a)Author to whom correspondence should be addressed. Electronic mail: Milind.Jog@uc.edu

1070-6631/2013/25(8)/082112/15/$30.00 C©2013 AIP Publishing LLC25, 082112-1

082112-2 Manglik et al. Phys. Fluids 25, 082112 (2013)

FIG. 1. Various stages of post-impact spread-recoil dynamics of a low-We droplet: (a) impact, (b) maximum spread,(c) change in the contact angle during recoil, (d) maximum recoil, and (e) equilibrium.

Substrate deposition (i.e., spread-recoil oscillations without any splash, breakup, or complete/partial rebound) is the most desired drop behavior in many spray applications, and the attendantdynamics is the focus of the present investigation. During deposition, a liquid droplet impinging ona horizontal dry surface at low Weber number (We) spreads as a thin circular disk with a thick outerrim if the Ohnesorge number Oh is also low, or spreads as a thick layer if the Oh is high. The spatialcoverage of the drop reaches a maximum when part of the initial kinetic energy is consumed inviscous dissipation and the remainder is used to do work against interfacial forces. At this instance,the relatively higher advancing contact angle changes to a lower receding contact angle. During thiscontact angle hysteresis, while the drop shape changes the spread remains essentially the same. Thedroplet then recoils to form a liquid column, which subsequently collapses to spread again. Thesespread-recoil oscillations continue in time until the viscous force completely damps the motion andan equilibrium condition is attained. Typical images of these different stages in post-impact dropspread dynamics at a low We are illustrated in Fig. 1.

In characterizing the spreading process, the temporal variations of instantaneous drop spread Dand its height h are usually normalized by the drop diameter prior to impact do and represented bytheir respective dimensionless spread factor β (= D/do) and flatness factor δ (= h/do). The spreadoutcome for an impinging drop of initial diameter do and velocity V on a substrate of specifiedsurface energy, or wettability as attributed to the contact angle θ , is generally dictated by a complexinterplay of liquid properties (dynamic viscosity μ, density ρ, and surface tension σ ). These arescaled by several groupings of inertia, viscous force, and surface tension force that are described bythe following:

Weber number, We ∼ inertia

surface tension= ρV 2do

σ,

Reynolds number, Re ∼ inertia

viscous force= ρV do

μ,

Capillary number, Ca ∼ viscous force

surface tension= μV

σ= We

Re,

Ohnesorge number, Oh ∼ viscous force

surface tension× viscous force

inertia= μ√

ρσdo=

√We

Re.

Much of the literature1 has been focused on devising predictive tools for the maximum spread factor.Based on scale estimates of initial kinetic energy, viscous dissipation, and surface tension, andemploying the principle of energy conservation at the point of maximum spread, several differentcorrelations have been proposed to predict the maximum spread of the droplet.4–6, 11, 12 Roismanet al.13 developed a model for drop spread and receding by considering the drop as a thin circulardisk with a think outer rim, and its predictions agreed with their own experimental measurementsduring initial spreading. Few attempts have been made to model the post-impact transient dropspread-recoil-oscillation to equilibrium behavior, which is the objective of this paper.

While computational modeling is an attractive means of simulating the dynamic drop behavior,the process is rather challenging as it requires accurate tracking and prediction of the continuouslydeforming gas-liquid interface. Some simulations for a single drop impact have been carried out usingthe arbitrary-Lagrangian-Eulerian (Fukai et al.14), volume-of-fluid (Gunjal et al.;15 Sikalo et al.;16

082112-3 Manglik et al. Phys. Fluids 25, 082112 (2013)

FIG. 2. Schematic diagram of the experimental setup.

Sanjeev et al.17), and level-set (Ding and Spelt18) methods, but they tend to be computationally veryintensive. Moreover, simulations tend to be sensitive to the advancing and receding contact anglevariations and the attempts to model this behavior as a function of contact line velocity, along withsubstrate and liquid properties, have not been universally successful in achieving the level of accuracythat is needed for simulations. Modeling the dynamic contact angle behavior during post-impactspreading and recoil thus remains an unresolved challenge. Furthermore, typical atomization devicesproduce sprays with a spectrum of droplet diameters and impact velocities, and this would requiremultiple simulations to cover the entire range. Thus, a simple model that predicts the dynamics ofdroplet-surface interactions for the expected spectrum of drop sizes and velocities (or We and Re)would be desirable. This paper provides such a model by considering the process of drop spread andrecoil oscillations to be analogous to the behavior of a simple damped-harmonic system. The modelwas developed using experimental data for temporal variations in the spread and height of dropletsof six different liquids, covering a large range of properties (surface tension and viscosity), Webernumber We, and Reynolds number Re.

II. EXPERIMENTAL MEASUREMENTS

As shown in a schematic of the experimental apparatus (Fig. 2), carefully controlled dropletswere generated using a precision syringe that has provision for interchangeable needles for varyingthe orifice diameter, and hence the drop size. The syringe was mounted on a vertical sliding standand could be lowered or raised in a calibrated manner so as to release the droplet from differentpredetermined heights. The hydrophobic substrate was prepared by wrapping a glass microscopeslide with a uniform layer of Teflon tape. The substrate was placed on a leveling table, which controlsits horizontal alignment as well as its perpendicularity to the syringe mount and drop trajectory. Thepost-impact drop-surface interactions were captured in real time using a high-speed high-resolutiondigital video camera system (Hi-Dcam-II version 3.0; NAC Image Technology) equipped with an8× magnification lens for close-up images. The camera was aligned at zero degrees to the substrateorientation so as to obtain a frontal view of the drop impact. A focusing, single-ended daylight PAR(parabolic aluminized reflector) lighting system (Model Lux125; ARRI Inc.) along with a glossywhite reflector was used for providing sharply contrasting and continuous lighting. With the cameraframe rate set at 2000 fps and a shutter speed of 1/4000, sequential images of the impinging dropand its post-impact spread-recoil-oscillations behavior were captured in real time high-resolutionstreaming video. A MATLAB R©-based computer program was developed to analyze the large numberof sequential images of drop deposition process in order to determine, with consistent precision,the temporal evolution of the liquid spread and height. For each case (do, V, We, and Re), ∼4000images were analyzed to determine the variation of instantaneous drop spread and height, and maptheir respective spatial-temporal evolution.

The droplet size (described by the pre-impact diameter do, and post-impact film spread D andits thickness h) was obtained on an equivalent volume-average basis. In determining the pre-impact

082112-4 Manglik et al. Phys. Fluids 25, 082112 (2013)

TABLE I. Physical properties of liquids.

Density Surface tension Viscosity Contact angle onρ (kg/m3) σ (mN/m) μ (kg/(m s)) Teflon θ (deg) We Re

Water 1000.0 72.7 1.0 121 9.03–21.2 1176–2179Acetic anhydride 1077.0 31.98 0.806 74 24.1 1711Ethylene glycol 1115.3 48.0 16.0 90 9.4–19.2 56.49–103.44:3 Aqueous-glycerin 1180.0 66.90 22.97 107 19.0 90.49Propylene glycol 1033.0 36.0 52.0 97 11.9–16.5 16.55–22.86Glycerin 1257.0 65.16 749.0 120 23.4 3.08

drop diameter, measurements were made at several different angular diameters on the drop-faceimage (digitally enhanced and enlarged) using four different images just prior to impact and theircomposite average calculated for the final value. The spread-film diameter and thickness weremeasured digitally by resolving the sharp pixel film-edge contrast in each instantaneous imageframe. The droplet velocity, and hence its Weber number, was varied and controlled by changingthe height from which the drop of a given size (function of orifice needle diameter) was released.The velocity was measured from two consecutive images just before impact by resolving thedifference between the relative drop position and the time lapse (0.5 ms precision) between their tworespective frames. Each experiment, with each drop-size and velocity combination, was repeatedthree times so as to ensure reproducibility and accuracy of measurements of the drop-impact spatial-temporal evolution. Six different test fluids (water, acetic anhydride, ethylene glycol, propyleneglycol, 4:3 aqueous glycerin solution, and glycerin) were used to produce a range of drop sizes, withdifferent surface tension and viscosities, and hence different We and Re. Their respective propertiesare listed in Table I.

The maximum uncertainty in the drop diameter and velocity measurements, based on samplepropagation of error estimation,19 was found to be ±1.38% and ±1.31%, respectively. The conse-quent uncertainty in the evaluated Weber number is ±2.35%, and ±1.90% in the Reynolds number.Finally, uncertainties in the droplet-film spread and flattening measurements are 1.47% and 3.51%,respectively. Contact angle measurements were made by a goniometer with precision of ±0.1◦.More details of the measurements methods, their precision and control, and uncertainty analysis areavailable in Ravi et al.12 and Gatne et al.20

III. DAMPED HARMONIC MODEL DEVELOPMENT

The differential equation that describes the motion of the mass in a mass-spring-damper systemand which is disturbed with an initial displacement and initial velocity is given by21

my + cy + ky = 0. (1)

Here m is the mass, c is the damping coefficient, k is the spring constant, and y is the extensionof the spring; y and y are, respectively, the second- and first-order time derivatives. To solve thissecond-order differential equation, the required conditions are provided by the initial displacementy(0) and the initial velocity y (0). The general solution for Eq. (1) is

y (t) = e−(αt / 2) [A cos (γ t) + B sin (γ t)] , (2a)

where

α = (c/m) , γ =√

ω2 − (α / 2)2, A = y (0) , and B = 1

γ

{y (0) + α

2y (0)

}. (2b)

It should be noted here that the four variables – viscous damping factor α, frequency ω, initialdisplacement y(0), and the initial velocity y (0) – uniquely define the transient variation of thedisplacement of the mass in a mass-spring-damper system. The damping coefficient determinesthe decay in the amplitude, whereas both spring constant and the damping coefficient govern the

082112-5 Manglik et al. Phys. Fluids 25, 082112 (2013)

FIG. 3. Schematic showing similarities between (a) a simple damped harmonic (mass-spring-damper) system and(b) droplet-substrate-spread dynamics.

frequency of oscillations. The task in modeling the post-impact dynamic response of a droplet is torelate α, ω, y(0), and y(0) to liquid properties and initial conditions of the droplet-substrate system.

Besides the impact inertia, liquid viscous and liquid-gas-solid interfacial properties govern thepost-impact evolution of drop spread and its shape. Figure 3 schematically depicts the different forcesacting on the spreading liquid drop and the corresponding forces in a typical mass-spring-dampersystem. Work done by the inertia force against viscous force is responsible for energy dissipationin the system, and this has a damping effect on the drop shape oscillations during repeated spreadand recoil. In a mass-spring-damper system, the damping factor is given by the ratio of viscousdamping coefficient and mass. The corresponding factor in the droplet-substrate system is the ratioof viscous force to inertia force, which is the inverse of Reynolds number Re. Thus the viscousdamping coefficient α can be scaled and correlated as a function of Reynolds number, or

α = f (Re) . (3)

When the droplet spreads on the substrate, the liquid-air interface is stretched and part ofthe initial kinetic energy is used to do work against the action of the surface tension force. Notsurprisingly then, if we compare two liquids with nearly the same viscosity, for example, waterand acetic anhydride, the liquid with low surface tension coefficient (acetic anhydride) produces alarger spread compared to the high surface tension fluid (water).12 The surface tension force acts ina manner similar to the spring force in the damped harmonic system by arresting the spread and byinitiating recoil of the liquid to its original shape. Therefore, the surface tension coefficient affectsthe frequency of drop-shape oscillations. Once the liquid begins to recoil, similar to that in the mass-spring-damper system, the shape change goes beyond the equilibrium position by forming a liquidcolumn which increases the potential energy of the liquid. This liquid column then re-spreads on thesubstrate. These shape oscillations continue until the motion is completely damped by viscosity. Thefluid viscosity acts to slow the fluid motion and it affects the frequency of shape oscillations. In amass-spring-damper system, the frequency of damped oscillations is a function of the ratio of springconstant to mass as well as the ratio of the viscous damping coefficient to mass. The analogousfactors in the droplet-substrate system are the ratio of surface tension force to inertia force, andthe ratio of viscous force to inertia force; essentially the respective inverse of Weber number Weand Reynolds number Re. It is thus expected that the frequency of oscillation can be scaled by thefollowing functionality:

ω = f (We, Re) . (4)

The initial velocity in this scaling can be expressed in terms of the impact velocity, and the initialdisplacement can likewise be related to initial spread factor and flatness factor.

IV. RESULTS AND DISCUSSION

Experimental data for transient spread and drop-film height variations for six different liquidswere acquired, and the damped harmonic system model was developed on the basis of these mea-surements. By changing droplet size and its impact velocity, a broad spectrum of drop-impact-spread

082112-6 Manglik et al. Phys. Fluids 25, 082112 (2013)

conditions was obtained. The consequent balance of inertia, respectively, with surface tension andviscous forces varied as 9.0 < We < 25, and 3.0 < Re < 2180. Likewise, the six different liquids usedin the experiments provided a three-orders of magnitude variation in the relative viscous interactionwith surface tension forces, represented by 0.007 < Ca < 7.6, and 0.0023 < Oh < 2.0.

Referring to the spatial and temporal variation of the post-impact droplet spread-recoil behaviordepicted in Fig. 1, the resulting range of maximum drop-film spread factor was 1.284 ≤ βmax

≤ 2.533, and the corresponding minimum flatness factor range was 0.161 ≤ δmin ≤ 0.503. Moreover,by considering the truncated sphere geometry of an equal volume Q sessile drop of a given liquidand its liquid-solid contact angle θ with the hydrophobic substrate (Teflon), the long-time-scale(t → ∞) or equilibrium spread factor βeq and flatness factor δeq can be determined as follows:

For θ < 90◦:

βeq =(

Deq

do

)= 2 sin θ

do

[3Q

π (1 − cos θ)(1 + sin2 θ − 2 cos θ

)]1/3

, (5a)

δeq =(

heq

do

)= 1

do

[3Q (1 − cos θ )2

π(1 + sin2 θ − cos θ

)]1 / 3

. (5b)

For θ = 90◦:

βeq = 1

do

[12Q

π

]1/3

, (6a)

δeq = 1

do

[3Q

2π

]1/3

. (6b)

For θ > 90◦:

βeq = 2 sin θ

do

[3Q

π cos θ(2 + sin2 θ

)]1/3

, (7a)

δeq = 1

do

[3Q (cos θ − 1)2

π cos θ(4 + 2 sin2 θ

)]1/3

. (7b)

Here, of course, the droplet volume is Q(= πd3

o/6). For the experimental results and model

development presented in this paper the equilibrium droplets were characterized by 1.029 ≤ βeq

≤ 1.861, and 0.308 ≤ δeq ≤ 0.740.

A. Predictions of transient spread variation on a Teflon substrate

When a droplet impacts on a substrate its spread factor increases from zero to a maximum value(see Fig. 1). Subsequently, the drop-film recedes and recoils, the spread factor decreases and thenoscillates about its equilibrium value that is typically less than the maximum spread. Moreover, lowOhnesorge number (Oh ∼ 0.001) droplets tend to undergo several spread and retraction oscillationswith a large change in the liquid-solid contact area during each cycle. A high Ohnesorge numberliquid (Oh ∼ 0.1 and greater), on the other hand, tends to exhibit fewer spread-retraction oscillationsand its droplet reaches the equilibrium spread in a relatively shorter time duration.

Considering an axisymmetric spread of the droplet, the rate of increase is twice the contact linevelocity which is initially close to the impact velocity. Therefore, the initial spread and its rate ofchange can be specified as

β (0) = 0; β (0) = (2V/do) . (8)

Thus, by referencing the temporal response of the spread factor to the equilibrium value, Eq. (2a)can be restated to express the β − t variations as

β = βeq + [A1 cos (γ1t) + B1 sin (γ1t)] [exp (−α1t/2)], (9a)

082112-7 Manglik et al. Phys. Fluids 25, 082112 (2013)

FIG. 4. Comparison of predictions of α1 and ω1 from Eqs. (9c) and (9d), as well as α2 and ω2 from Eqs. (11c) and (11d),respectively, with experimental data.

where

γ1 =√

ω21 − (α1 / 2)2, A1 = −βeq , and B1 = 1

γ1

{β (0) −

(α1βeq

2

)}. (9b)

The damping coefficient α1 [s−1] and frequency of oscillation ω1 [s−1], as indicated in Sec. III byEqs. (3) and (4), are functions of Re and (We, Re), respectively. These were obtained by regressionanalyses of the experimental measurements for the temporal variations of the spread factor as follows:

α1 = [0.18 + (13.85/Re) − {

29.43 ln(Re)/Re2}]

, (9c)

ω1 = [0.19 + 9.91 (Re × We)−0.625] . (9d)

The predicted variations in α1 and ω1 given by Eqs. (9c) and (9d) are compared with the experimentalmeasurements in Fig. 4, and the excellent agreement between the two is evident. Equations (9a)–(9d)thus provide the complete solution for the damped harmonic model of liquid droplet spread on ahydrophobic surface (Teflon, characterized by θ ≥ 74◦ in the present set of experiments). It may benoted that βeq can be determined a priori from measurements of a sessile drop of diameter do on agiven hydrophobic substrate.

The transient variations of the spread factor of highly viscous glycerin (Oh = 1.57) and propyleneglycol (Oh = 0.178) droplets are graphed in Figs. 5 and 6, respectively. In both cases, the droplet isseen to spread and reach its equilibrium position without undergoing any significant spread-recoiloscillations. Given the highly viscous nature of the two liquids, such behavior is not surprising.Viscous effects retard and dampen spreading, which, besides inertia, is aided by surface wetting,and as a result βmax tends to be very low. In conjunction with their respective Oh, the balance ofthese forces as expressed by the capillary number also suggests this behavior. In the case of glycerin,which has the highest Ca (= 7.59) and Oh of all the liquids considered in these experiments, thesmallest value of the maximum spread factor (∼ 1.3) is obtained. A propylene glycol droplet, with amuch smaller Ca (= 0.721) and Oh, compared to glycerin, correspondingly yields a slightly largerβmax (∼ 1.6). In both cases, however, where the post-impact drop-spread evolution is significantly

082112-8 Manglik et al. Phys. Fluids 25, 082112 (2013)

FIG. 5. Transient variation of spread factor β of a glycerin droplet (We = 23.4, Re = 3.08, Oh = 1.57, Ca = 7.59) on aTeflon substrate.

influenced by high liquid viscosity, the predictions of the mass-spring-damper harmonic systemmodel, as given by Eqs. (9a)–(9d), are seen to describe the experimental data very well.

Figures 7 and 8 depict the temporal variation of spread factors for moderate Oh (∼ 0.01)liquid droplets, namely, 4:3 aqueous glycerin (Oh = 0.048) and ethylene glycol (Oh = 0.042). Forboth these liquids, relatively stronger recoil with a large change in the liquid-solid contact areais observed after the initial spread. After the first retraction, the droplet re-spreads to nearly orclose to its equilibrium spread factor value. The maximum spread factors, βmax ∼ 1.64 and 1.74,respectively, for aqueous glycerin and ethylene glycol are also somewhat greater in magnitude thanthose for propylene glycol and glycerin droplets. Symptomatic of the effects of reduced surfacetension and/or viscosity, as described by lower Oh and Ca (= 0.21 and 0.19, respectively, for 4:3aqueous glycerin and ethylene glycol), a strong drop-recoil from the maximum spread is observedbut the subsequent oscillations are minimal. Nevertheless, the efficacy of the damped harmonicsystem model of Eq. (9) in describing and predicting the measured β − τ behavior is evident fromthe Figs. 7 and 8.

The dynamic spread factor variation with We ∼ 20 for acetic anhydride and water droplets,which have three-orders-of-magnitude smaller Oh (= 0.003 and 0.002, respectively) compared toglycerin, is graphed in Figs. 9 and 10, respectively. Both water and acetic anhydride have very lowviscosity (one-to-two orders of magnitude lower) compared to all other liquids considered in these

FIG. 6. Transient variation of spread factor β of a propylene glycol droplet (We = 16.5, Re = 22.9, Oh = 0.178, Ca =0.721) on a Teflon substrate.

082112-9 Manglik et al. Phys. Fluids 25, 082112 (2013)

FIG. 7. Transient variation of spread factor β of a 4:3 aqueous glycerin droplet (We = 19.0, Re = 90.5, Oh = 0.048, Ca =0. 21) on a Teflon substrate.

FIG. 8. Transient variation of spread factor β of an ethylene glycol droplet (We = 19.2, Re = 103.4, Oh = 0.042, Ca =0.19) on a Teflon substrate.

FIG. 9. Transient variation of spread factor β of an acetic anhydride droplet (We = 24.1, Re = 1711, Oh = 0.003, Ca =0.014) on a Teflon substrate.

082112-10 Manglik et al. Phys. Fluids 25, 082112 (2013)

FIG. 10. Transient variation of spread factor β of a water droplet (We = 21.2, Re = 2179, Oh = 0.002, Ca = 0.01) on aTeflon substrate.

experiments. Also, acetic anhydride has very low surface tension (∼ 32 mN/m). The combinationof low viscosity and surface tension facilitates large spread of the acetic anhydride droplet, and ithas the highest value of βmax (∼ 2.53). The second-order damped harmonic model of Eq. (9) mapsthis behavior rather well (Fig. 9), but with a small over-prediction of βmax and subsequent recoil. Onthe other hand, water has a much higher surface tension, which coupled with low viscosity resultsin strong post-maximum-spread recoil followed by multiple spread-recoil oscillations. The largedifference in surface tension of acetic anhydride and water manifests in a significant difference intheir equilibrium droplet spread values (βeq ∼ 1.86 and 1.67, respectively). Moreover, for the waterdroplet, the model prediction of maximum spread and oscillation frequency agrees rather well withexperimental data (Fig. 10), although the extent of recoil is under predicted.

B. Predictions of transient flatness factor variations on a Teflon substrate

The post-impact behavior of a water droplet on a Teflon substrate exhibits very strong recoilthat leads to the formation of a liquid column. At high Weber numbers, the column breaks upor fractures to form one or more smaller droplets that fall back onto the substrate.22 For low We(< ∼20), the water column height may exceed the diameter of the droplet or the flatness factorexceeds unity (δ > 1).12, 20 After the initial maximum spread, however, the transient variation of δ

for a water droplet shows that the behavior essentially follows a damped harmonic motion. We havetherefore considered the variation of the flatness factor δ from its minimum value δmin (hmin /do), orfrom the beginning of the first recoil in our model. We also note that our model, Eq. (9), predictsrather well both the maximum spread factor βmax and the corresponding time as presented in Sec.IV A. For droplets with Ca > ∼0.1, the shape of the liquid layer at maximum spread can beapproximated as a circular disk and based on mass conservation the corresponding minimum heightof the liquid or minimum flatness factor can be obtained. For lower capillary number droplets, theliquid thickness can be estimated by considering the drop shape as a thin disk with a thicker toroidalring at the periphery as described in Roisman et al.13 Based on such evaluation of the experimentaldata, the range of minimum flatness factor was 0.161 ≤ δmin ≤ 0.503.

As the droplet reaches its post-impact maximum spread and minimum height, the contact linevelocity and the rate of change in height become zero. Thus, because our the damped harmonicsystems model begins at the instant of minimum drop-film height, for the subsequent dampedoscillatory motion the initial conditions are given by

δ (0) = (hmin/do) , δ (0) = 0. (10)

082112-11 Manglik et al. Phys. Fluids 25, 082112 (2013)

FIG. 11. Contact angle hysteresis during initial recoil stage of a droplet with fixed spread but changing flatness.

By referencing the time-dependent response to the equilibrium condition (as was done in the previouscase) the variation of flatness factor can be obtained as

δ = δeq + [A2 cos (γ2t) + B2 sin (γ2t)][exp (−α2t/2)

], (11a)

where

γ2 =√

ω22 − (α2/2)2, A2 = {

δ (0) − δeq}, and B2 = 1

γ2

[δ (0) + α2

2

{δ (0) − δeq

}].

(11b)We further note that the damping factor α2 can be related to the ratio of the amplitude betweensuccessive peaks, whereas the frequency ω2 can be related to the time difference between successivepeaks. These are once again functions of Re and (We, Re), as noted in Eqs. (3) and (4), and theywere obtained from a regression analysis as follows:

α2 = [−6.66 + 5.29Re1/3 − 0.24Re2/3]−1

, (11c)

ω2 = exp[−1.44 + 1.173 (Re × We)−0.25 + 5.78 (Re × We)−0.5

]. (11d)

The consequent predictions for α2 and ω2 are seen to be in excellent agreement with the experimentaldata graphed in Fig. 4.

It should be noted here that subsequent to a droplet’s maximum spread, contact angle hysteresisis observed during each ensuing spread-recoil oscillation. The advancing contact angle changes toa lower receding contact angle, and the shape of the drop and its drop flatness factor also changesin this process; the spread factor, however, remains the same during this wetting hysteresis. This isevident from the annotated photographic images of a typical droplet at three different instances inits spread evolution on the Teflon substrate depicted in Fig. 11. Here, as the drop begins to recoil,the contact angle changes from advancing to receding, Figs. 11(a)–11(c), and h (or flatness) changesbut D (or spread) remains constant. As a consequence, the frequency of oscillations for the flatnessfactor is expected to be somewhat different from that for the spread factor.

The predicted time-dependent evolution of flatness factor δ given by Eq. (11) for a glycerindroplet is compared with the experimental data in Fig. 12. The excellent agreement between thetwo is self-evident, and it provides another validation of the damped harmonic system model foraccurately predicting the flatness factor response of a very high viscosity fluid droplet. The Oh(= 1.57) for this droplet is about two-to-three orders of magnitude higher than that of a comparablewater droplet (Oh = 0.002), and as seen in Fig. 5 it undergoes only one recoil after the initial spread.Correspondingly, there is a monotonic increasing progression of δ in time from its minimum valueto its equilibrium condition, and significant shape oscillations are absent thereby reflecting the roleof high glycerin viscosity in the fully damped variation of δ.

Figures 13–15 graph the transient variations in flatness factor δ from its minimum value fordrops of propylene glycol (Ca = 0.72), 4:3 aqueous glycerin (Ca = 0.21), and ethylene glycol(Ca = 0.19), respectively. The values of capillary number in each case indicate that the effect ofsurface tension, relative to viscosity, is an order of magnitude and more greater than that in anequivalent glycerin droplet (Ca = 7.59). As a consequence, δ − τ plots exhibit multiple shapeoscillations before the drop attains an equilibrium shape; larger number of undulations is observed

082112-12 Manglik et al. Phys. Fluids 25, 082112 (2013)

FIG. 12. Transient variation of flattening factor δ of a glycerin droplet (We = 23.4, Re = 3.08, Oh = 1.57, Ca = 7.59) on aTeflon substrate.

with decreasing Ca. Moreover, of the three liquids, the capillary number for propylene glycol isthe largest and correspondingly the dimensionless time required to damp its oscillatory motionis shorter (τ ∼ 10). Whereas the aqueous glycerin and the ethylene glycol droplets have aboutthe same capillary number, and their equilibrium state is attained in nearly the same time frame(τ ∼ 15). Once again, the excellent match of predicted behavior with the experimental measurementsfor δ − τ for each of propylene glycol, aqueous glycerin, and ethylene glycol droplets provides aself-evident validation of the damped harmonic model.

The changes in flatness factor with time for an acetic anhydride droplet, which has very lowviscosity and correspondingly low capillary number (= 0.014), are presented in Fig. 16. The predictedflatness factor variation of the damped harmonic model is once again seen to agree rather wellwith experimental data, particularly for the first two oscillations. Although the model continues topredict the amplitude correctly in the subsequent oscillations, the agreement with their frequency isweaker. The frequency appears to change beyond τ ∼ 10, at which point, it may be recalled fromSec. IV A (Fig. 9), the spread factor β attains its equilibrium value and the liquid-solid contactline does not move thereafter. Hence, the changes in flatness factor beyond τ > 10 reflect dropletheight modulations during the subsequent shape oscillations without any change in the spread factor.The viscous dissipation is low during this process, compared to the spread-recoil prior to τ < 10,and both damping and surface tension forces determine the frequency of oscillation of the damped

FIG. 13. Transient variation of flattening factor δ of a propylene glycol droplet (We = 11.9, Re = 16.5, Oh = 0.21, Ca =0.72) on a Teflon substrate.

082112-13 Manglik et al. Phys. Fluids 25, 082112 (2013)

FIG. 14. Transient variation of flattening factor δ of a 4:3 aqueous glycerin droplet (We = 19.0, Re = 90.5, Oh = 0.048, Ca= 0.21) on a Teflon substrate.

FIG. 15. Transient variation of flattening factor δ of an ethylene glycol droplet (We = 19.2, Re = 103.4, Oh = 0.042, Ca =0.19) on a Teflon substrate.

FIG. 16. Transient variation of flattening factor δ of an acetic anhydride droplet (We = 24.1, Re = 1711, Oh = 0.003, Ca =0.014) on a Teflon substrate.

082112-14 Manglik et al. Phys. Fluids 25, 082112 (2013)

FIG. 17. Transient variation of flattening factor δ of a water droplet (We = 21.2, Re = 2179, Oh = 0.002, Ca = 0.01) on aTeflon substrate.

harmonic system. As such the change in frequency of oscillation beyond τ > 10 may be due to thechange in damping effects.

Of all the liquids considered in these experiments, water has the highest surface tension and verylow viscosity. Because of this combination of properties, which typically produced droplets with Ca∼ 0.01 and Oh ∼ 0.002, there is a strong post-impact recoil of the drop after the initial spread, asseen in the δ − τ signature in Fig. 17. The stronger surface tension force tends to inhibit spreadingafter the initial recoil and accentuates the subsequent recoil and shape oscillation evolution priorto reaching equilibrium. This tends to make the flatness factor variation asymmetrical about theequilibrium position, which perhaps would explain the somewhat weaker agreement between themodel predictions and experimental measurements, as compared to the fit with results of the otherliquids. To remedy this and to better accommodate the amplitude variation, perhaps a non-linearspring needs to be considered in the mass-spring-damper system modeling.

V. CONCLUSIONS

The temporal variations of dimensionless spread (spread factor β) and dimensionless height (flat-ness factor δ) during post-impact spread-recoil oscillations of a liquid droplet on a dry hydrophobic(Teflon) substrate at low Weber numbers (9.0 < We < 25) were modeled as a second-order dampedharmonic system. The model was based on experimental data obtained with high speed videographyfor droplets of six different Newtonian liquids that covered a wide range of interfacial properties(0.002 ≤ Oh ≤ 1.57; and 0.007 ≤ Ca ≤ 7.59). Using the analogy between the mass-spring-damper ofa second-order harmonic system and inertia-surface tension-viscous forces of the droplet-substratesystem, the spread-recoil oscillation frequency was correlated with Weber and Reynolds numberand the damping factor was correlated with the Reynolds number. The spread and flattening factorvariations predicted by this model agree well with experimental data, and this is especially close forOh > ∼0.01 and Ca > ∼0.1. The proposed model provides a simple method to predict post-impactspread dynamics on a hydrophobic substrate at low Weber number (We < ∼20).

It may further be noted that the reported experiments are for the class of hydrophobic substrates(low surface energies) typically represented by a Teflon (polytetrafluoroethylene or PTFE) surface.Substrates with high surface energy (or hydrophilic surfaces) would exhibit different dynamicresponses of the liquid droplets and associated correlations for α and ω.

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