Drops bouncing off solid surfaces: gas-kinetic and van der Waals effects(Supplemental Material)

Mykyta V. Chubynsky,1 Kirill I. Belousov,2 Duncan A. Lockerby,3 and James E. Sprittles1

1Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom2Faculty of Physics and Engineering, ITMO University, St. Petersburg 197101, Russia3School of Engineering, University of Warwick, Coventry CV4 7AL, United Kingdom

(Dated: October 1, 2019)

The sections in the Supplemental Material contain the following: (I) simulation parameters andadditional details of the derivation of the thin-film lubrication approximation used in this work,including the justification of the assumptions, the generalization to the case when the two surfacesof the air film have different scattering properties, and the boundary conditions used; (II) the

interpolating formulae for the kinetic factors ∆{Φ,τ}{C,P} used in our computations; (III) the details

of the computational implementation; (IV) a discussion of the approach used for our drop-dropcollision simulations, their results and the comparison with those of J. Li [Phys. Rev. Lett. 117,214502 (2016)]; (V) results for the case of partially specular reflection of air molecules off the solidboundary; (VI) a discussion of van der Waals instabilities leading to contact between the drop andthe surface; and (VII) some additional details of simulation results.

I. DETAILS OF THE COMPUTATIONALMODEL

A. Simulation parameters

It is convenient to start this section by listing the pa-rameters used in our simulations, since the validity of theassumptions of our approach depends directly on theirvalues. As mentioned in the Letter, for validation pur-poses we aim to reproduce the conditions of the experi-ments of Kolinski et al. [1]. In these experiments, dropswere made of a water-glycerol mixture with kinematicviscosity 10 cSt. According to Ref. [2], this correspondsto about 60 wt% of glycerol. Based on data in thatreference, we choose surface tension γ = 6.85 × 10−2

N/m, the liquid density ρl = 1155 kg/m3 and viscos-ity µl = 1.155 × 10−2 Pa s. For air, the viscosity valueµg = 1.827 × 10−5 Pa s is used; the air density is notused in the simulations, as gas inertia is neglected, butfor the estimates of the validity of our approximationsbelow, ρg = 1.2 kg/m3 at 1 atm can be used. For thevalue of the Hamaker constant AH in Eq. (10) in the Let-ter we use 6×10−20 J, which approximately correspondsto the geometric mean [3] of the values for water-waterand mica-mica interactions found in the literature [3, 4].Given a wide spread of literature values and the fact thata water-glycerol mixture instead of pure water is used,the above value for AH may not be very accurate. How-ever, the results for the critical impact speed are notexpected to be very sensitive to the exact value due tothe steep dependences of the disjoining pressure pd onthe air film thickness h [see Eq. (10) in the Letter] andof the minimum film thickness hmin [with van der Waals(vdW) interactions “switched off”] on the impact veloc-ity V . The aforementioned parameters are used in all ofour simulations; the only parameter that changes withair pressure (aside from the air density, which, as men-tioned, is not used in the actual simulations) is the mean

free path of air molecules, which enters the kinetic param-

eters ∆{Φ,τ}{C,P} via the Knudsen number Kn (see Sec. I C

below); this mean free path is taken to be equal to 69 nmat 1 atm and is changed inversely proportionally to thepressure.

We simulate an initially spherical drop of radius R =0.8 mm placed at distance 0.08 mm above the surface,which, according to the estimates below, is sufficientlyfar that the influence of air is initially negligible. Theinitial drop velocity V is directed towards the surface;it is 0.55 m/s in the simulation that is compared to theexperimental results of Ref. [1] (see Fig. 2 in the Letter)and is varied in other simulations. Gravity is neglected.

B. Justification of the assumptions

Our model of drop impact makes a number of simpli-fying assumptions, discussed and justified below.

First, the drop is assumed spherical when it is far fromthe solid surface. This is valid when (I) its deformationas it flies through air is negligible; and (II) any shape os-cillations, normally produced when the drop forms, decayprior to impact. For (I) two conditions need to be sat-isfied. First (Ia), the shear stress in the drop should bemuch smaller than the Laplace pressure; the ratio of thetwo is the gas capillary number

Cag =µgV

σ. (1)

This is estimated below to always be small. Second (Ib),the pressure difference between the front and back sur-faces of the drop should be much smaller than the Laplacepressure. This ratio is given by the gas Weber number

Weg =ρgV

2R

σ� 1. (2)

2

This is not to be confused with the usual Weber numberWe using ρl in place of ρg that characterizes deforma-tion of the drop upon impact. For V = 0.75 m/s at 1atm (near the threshold between bouncing and contact),Weg ≈ 8.5× 10−3 and the condition (2) is satisfied veryaccurately. On the other hand, we also discuss the caseof vanishing gas-kinetic effects (GKE), which formallycorresponds to infinite gas pressure/density; however, atsome ρg Eq. (2) will no longer be satisfied. Keeping inmind that impact speeds up to ≈ 2 m/s are relevant inthat case, Weg can reach ∼ 1 already for ρg ≈ 20 kg/m3,or about 15 atm of pressure. This should be kept in mindwhen testing our predictions experimentally. As for droposcillations (II), the decay time for the slowest-decayingmode is [5]

τ =ρlR

2

5µl= 12.8 ms, (3)

while the time of flight for V = 0.75 m/s, assuming freefall, is V/g ≈ 77 ms, thus the oscillations indeed havetime to decay considerably. Note that this would not bethe case for pure water drops and also for lower speedsthat we study in the context of obtaining the regime dia-gram (see Fig. 4 in the Letter). For such cases, for com-parison to experiments, a way of producing drops withas little surface deformation as possible is required.

Second, as the drop approaches the surface, it will de-form even before contact because of air pressure buildupunder the drop. We assume that this does not happenuntil the air film is very thin, much thinner than the dropradius. The distance h between the drop and the surfaceat which significant deformation starts to occur is relatedto the gas capillary number Cag as [6]

h/R ' Ca1/2g . (4)

For V = 0.75 m/s, this ratio is indeed small, around 1.4×10−2, and remains small for all conditions of interest of ushere. This is one necessary condition for the lubricationapproximation to be applicable. Moreover, this justifiesstarting the simulation at h/R = 0.1. Of course, since the

value of Ca1/2g is small, the value of Cag that appeared

in one of the estimates above is even smaller.Third, incompressibility of air is assumed. This as-

sumption is valid if the change of air pressure is muchsmaller than the ambient value. Mandre et al. [7] esti-mate the maximum excess pressure as

p∗g ∼(RV 7ρ4

l

µg

)1/3

, (5)

which, assuming the numerical prefactor of unity, givesaround 20% of the atmospheric pressure at V = 0.75 m/s.Our simulations give a similar value for the peak excesspressure, about 15%. While this is not entirely negli-gible, it is important to remember that for most of theimpact duration, the pressure is considerably less. Forlower ambient pressures the impact speeds of interest to

us are likewise lower (see the phase diagram in Fig. 4 ofthe Letter), so the accuracy of the incompressibility as-sumption does not deteriorate, especially given the steepV dependence of p∗g. Conversely, higher speeds are ofinterest to us as well, but only at higher pressures.

Fourth, inertial effects in the gas flow are neglected. Ig-noring GKE, this corresponds to neglecting inertial termsρg(∂vg/∂t) and ρg(vg · ∇)vg, where vg is the gas veloc-ity. The speed of air in the film changes most rapidly atthe beginning of the impact, when, using estimates fromRef. [8], the film thickness reaches

h ∼ h∗ =

(µ2gR

ρ2l V

2

)1/3

; (6)

the change is from zero to

vg ∼ v∗ = V

√R

h∗(7)

over time of order

τ =h∗

V. (8)

Since v∗ is much larger than the speed at which the dropspreads laterally, the gas flow is mostly of Poiseuille char-acter, so that the relevant condition of negligibility ofinertia is

ρgv∗/τ

µgv∗/h∗2=

(ρ3gV R

µgρ2l

)1/3

� 1. (9)

This ratio is about 4× 10−2 for V = 0.75 m/s. At latertimes, the pressure gradient decreases and the gas flow ismostly of Couette character, thus vg is comparable to thedrop spreading speed. For not too high (liquid) Webernumber We, the latter is of order V . Inertial effects arethen negligible if ρgv

2g ∼ ρgV

2 is much smaller than theLaplace pressure ∼ γ/R, which is indeed the case.

Fifth, as is standard in lubrication theory, we assumethat the contribution of the gas film to the normal stressboundary condition on the drop surface is equal to the gaspressure, neglecting what in classical hydrodynamics isthe velocity derivative term (but would be more complexwhen GKE are taken into account). At the beginningof the impact, around the time the gas pressure is at itsmaximum value p∗g, the relevant comparison is betweenµgV/h

∗ and p∗g, and the ratio of the two,(µg

ρlV R

)2/3

, (10)

is ∼ 10−3 in our conditions. During most of the rest ofthe impact, an estimate of the normal gas velocity deriva-tive is given by the inverse impact duration τ−1, since thenormal gas velocity is ∼ h∗/τ at the drop surface and 0at the solid surface ∼ h∗ away. In its turn, τ can be

3

estimated as R/V for not too high We; comparing theresulting normal stress contribution to the Laplace pres-sure gives

µgV

γ= Cag, (11)

which, as mentioned above, is small.Finally, for the gas kinetic theory results obtained for

infinite channels to remain valid in our case it is necessarythat all changes in the radial direction are small on thelength scale of the mean free path, which is indeed thecase, as even the finest features of the drop surface, suchas the kink on its edge or the perturbations growing dueto the vdW forces, are on the length scale of microns.

C. The lubrication model for the air film forgeneral scattering properties of the surfaces

As mentioned in the Letter, we are solving the stan-dard incompressible Navier-Stokes (NS) equations for theliquid drop. These equations require the normal andshear stress boundary conditions on the drop surface.This surface is divided into two parts, the part borderingthe air film (part A) and the rest of the surface (part B).How exactly the division is done is discussed below. Forpart B, the influence of the gas is ignored, thus, the nor-mal stress is assumed equal to the Laplace pressure andthe shear stress is zero. For part A, the normal stressis equal to the sum of the Laplace pressure and the gaspressure next to the drop surface pg and the shear stressis equal to the shear stress in the gas next to the dropsurface τg. The quantities pg and τg are obtained by solv-ing the lubrication equation for the air film, as discussedin the Letter and below. This equation, in its turn, de-pends on the drop parameters, namely, the radial andvertical velocities of the liquid on the drop surface andthe distance between that surface and the solid surface(i.e., the film height). Therefore, the NS equations forthe liquid and the lubrication equation for the gas needto be solved simultaneously.

The essence of the lubrication approximation is thatthe solution of the stationary problem in a flat infinitechannel (in our case, with GKE included) is assumedto be valid locally even when the width of that channelactually changes in space and time. Thus, Eqs. (1)–(4)in the Letter are assumed to be valid, with parameters

vr, h, and ∆{Φ,τ}{C,P} functions of the radial coordinate r

and time t. An equation for pg is derived from the firsttwo of these equations and the solution of it is used inthe last two of the equations to calculate τg, as well asdirectly in the boundary conditions for the drop. In whatfollows we will consider the general non-symmetric caseallowing different scattering properties of the two surfaces

and therefore all four of the kinetic factors ∆{Φ,τ}{C,P} to be

different from unity.Let Φ(r, t) be the mass flow rate through the cylindrical

surface of radius r spanning the air film. Consider now

two such surfaces, of radii r and r + δr . The mass ofgas in the volume between them is δm = 2πρgrh(r, t)δr.Assuming that the gas is incompressible (ρg = const),the rate of change of this mass is

∆m = 2πρgr∂h(r, t)

∂t∆r. (12)

From mass conservation this should be equal to the dif-ference of the mass flow rates through the two surfaces,

2πρgr∂h(r, t)

∂t∆r = Φ(r, t)− Φ(r + δr, t), (13)

which immediately gives Eq. (5) in the Letter. Aftertransformations, keeping in mind that the surface areaentering Eqs. (2) in the Letter is S(r, t) = 2πrh(r, t) andthat ∂pg/∂n is the same as ∂pg/∂r, we obtain

∂2pg∂r2

+ g1(r, t)∂pg∂r

= g2(r, t), (14)

where

g1(r, t) =∂

∂r

[ln(rh3∆Φ

P )], (15)

g2(r, t) =6µg

rh3∆ΦP

[∂

∂r(rhvr∆

ΦC) + 2r

∂h

∂t

]. (16)

In the symmetric case ∆ΦC = 1 and Eqs. (14)–(16) co-

incide with Eqs. (6)–(8) in the Letter. These equationsare solved on an interval and two boundaries with asso-ciated boundary conditions should be specified. The leftboundary is at r = 0 and the boundary condition fol-lows immediately from the smoothness of the solution atr = 0:

∂pg∂r

∣∣∣∣r=0

= 0. (17)

The other boundary condition arises from the fact thatthe air pressure outside the film should be equal to theatmospheric pressure p0. Thus, choosing a point r = r0

outside the film (which is the boundary between parts Aand B of the surface mentioned above), we specify

pg(r = r0) = p0. (18)

It is not important that the lubrication approximation isvalid all the way to r0 and moreover, the results are notsensitive to the exact value of r0 so long as at that pointthe value of h is much larger than that in the film. Thisis because outside the thin film, where h starts to growrapidly as r increases, g2 decreases faster than g1 becauseof the h−3 factor. Thus, there is a region just outsidethe film where g2 is already small, but g1 is still largeand positive; as a consequence, ∂pg/∂r decays rapidlyto nearly zero in that region and stays nearly zero as rincreases further. Therefore, the value of pg is nearlyconstant outside the film and it does not matter at whatpoint condition (18) is applied. (It is assumed that the

4

lubrication approximation is valid up to the point where∂pg/∂r becomes negligible.) To be specific, r0 can bechosen to be, for example, the maximum radial extent ofthe drop at the given time. Depending on the numericalapproach used, it may be more convenient to either solveEqs. (14)–(16) with boundary conditions (17) and (18)together with the NS equations for the drop, or use theexplicit solution as an integral:

pg(r, t) = −6µg

∫ r0

r

1

∆ΦP

[ur∆

ΦC

h2+ 2

∫ r′0r′′(∂h/∂t)dr′′

r′h3

]dr′.

(19)It may also be convenient to express ∂h/∂t in terms ofthe liquid velocity components on the drop surface:

∂h(r, t)

∂t= vz(r, t)− vr(r, t)

∂h(r, t)

∂r. (20)

II. INTERPOLATING FORMULAE FOR THEKINETIC FACTORS

The lubrication approximation reduces the gas flow lo-cally to that in an infinite channel of width h with flatand parallel walls. Such flows have been studied for awide range of Knudsen numbers

Kn =λ

h, (21)

where λ is the mean free path, by solving the ki-netic Boltzmann equation numerically, typically usingthe Bhatnagar-Gross-Krook (BGK) approximation of thecollision term (other models have been considered, butthe differences are very minor and insignificant for ourpurposes). The resulting data [9–16] and some complexinterpolating formulae with more than a dozen param-eters [17, 18] are available in the literature and can beused to obtain simple but sufficiently accurate expres-sions, which is our goal in this section.

First of all, for nonzero Kn the gas flow depends on thedetails of the interaction of the gas molecules with thewalls. The two limits are specular reflection, for whichthe tangential component of the velocity of a moleculecolliding with the wall is preserved and the normal com-ponent is reversed, and diffuse scattering, in which casethe memory of the initial velocity is lost and the moleculeacquires a velocity drawn from the Maxwell distributionfor the wall temperature (here assumed equal to the gastemperature) and can move equiprobably in all directions(though, of course, not into the wall). To interpolate be-tween these two extremes, the accommodation coefficientα is introduced such that α = 0 for specular reflection,α = 1 for diffuse scattering and 0 < α < 1 for inter-mediate cases. We use the Maxwell model [19, 20] thatassumes diffuse scattering of a given molecule with prob-ability α and specular reflection with probability 1 − α.In our case, one “wall” is liquid and since it is “uneven”

because of thermal fluctuations and disordered (noncrys-talline), diffuse scattering (α1 = 1) is a reasonable as-sumption also confirmed experimentally (for instance,Ref. [21] gives α ' 0.9). However, the solid wall is atomi-cally smooth mica in the experiments of Kolinski et al. [1]and α2 significantly below unity is possible. While in theLetter we assume α2 = 1 everywhere, we allow α2 6= 1below and consider this case in one of the following sec-tions. To simplify the notation, we omit the subscript“2”, so α refers to the accommodation coefficient of thesolid wall, while that of the liquid wall is always equal tounity.

As is common in the literature, in place of Kn we willuse its inverse, the rarefaction parameter

δ =

√π

2Kn. (22)

The asymptotic behavior of the kinetic factors ∆{Φ,τ}{C,P} in

the limits δ → 0 and δ →∞ is known and it is desirableto ensure that the empirical formulae obey these limitsso these formulae remain accurate for arbitrarily low andhigh δ, outside the range of values for which numericaldata are available. In particular, in the hydrodynamiclimit Kn → 0 (or δ → ∞) the kinetic factors shouldapproach unity (except for α = 0, as in that case thereis, effectively, an infinite slip length). Moreover, the first-order approximation in Kn is also known and correspondsto hydrodynamic flow with Navier slip on the walls [22,23]. The slip lengths are expressed as 2λl1/π

1/2 for thediffuse (drop surface) wall and 2λl2/π

1/2 for the general(solid) wall, where [24, 25]:

l1 ≈ 1.016, (23)

l2 ≈2− αα

[1.016− 0.1211(1− α)]. (24)

Note that the slip length is infinite for specular reflection(α = 0). Resulting expressions for the kinetic factors, to-gether with the known asymptotics in the opposite limitδ → 0, are given below, separately for each kinetic fac-tor. We ensure that our fitting formulae reproduce theselimits accurately, though we do allow small deviationswhere this results in considerable simplification of theequations.

A. Mass flow rate for Poiseuille flow (factor ∆ΦP )

Solving the Stokes equation with slip gives

∆ΦP '

δ2 + 4(l1 + l2)δ + 12l1l2δ(δ + l1 + l2)

as δ →∞. (25)

This is not expected to be valid to orders higher thanO(1/δ), but it turns out to be beneficial to use the wholeexpression as the starting point, rather than just the firsttwo terms in the expansion. Note that ∆Φ

P → 1 when

5

δ →∞, except for α = 0, in which case ∆ΦP → 4. In the

opposite limit the asymptotic expression is [10]

∆ΦP ' −

6(2− α)

π1/2δln δ as δ → 0. (26)

Besides these asymptotics, there is also a relation be-tween a flow confined by two diffuse walls and that whereone of the walls is specular: since in the latter case thereis no shear stress on the specular wall, the flow is thesame as in one of the halves of the channel in the formercase. This gives

∆ΦP (δ, α = 0) = 4∆Φ

P (2δ, α = 1). (27)

To interpolate between the limits (25) and (26), wepropose the following expression:

∆ΦP ≈

δ2 + 4(l1 + l2)δ + 12l1l2δ(δ + l1 + l2)

−6(2− α)

π1/2δln

δ

(δ2 + c1δ + c2)1/2, (28)

where c1 and c2 are functions of α. Relation (27) is sat-isfied exactly if

c1(1) = 2c1(0), (29)

c2(1) = 4c2(0). (30)

The following expressions obey these relations and fit thedata to within 2% (Fig. 1):

c1 = (1.5 + 4.5α4)1/2, (31)

c2 = 0.025 + 0.075α2. (32)

In particular, for α = 1 we obtain

∆ΦP ≈ 1 +

6.072

δ− 6

π1/2δln

δ

(δ2 + 61/2δ + 0.1)1/2(33)

≈ 1 + 6.88Kn +6Kn

πln(1 + 2.76Kn + 0.127Kn2).

B. Mass flow rate for Couette flow (factor ∆ΦC)

For α = 0 the obvious solution of the Boltzmann equa-tion is plug flow, more specifically, everywhere in thechannel the velocity distribution is the Maxwell distri-bution shifted by the velocity of the diffuse wall. Sincethe mass flow rate of plug flow is twice that of shear flowwith no slip on either wall, we get for all δ

∆ΦC(α = 0) = 2. (34)

For α = 1 the result is also simple, based on symme-try considerations. In the reference frame in which thewalls move with the same speed in opposite directions,the mean flow velocity should be antisymmetric with re-spect to the middle plane of the channel and the totalnet mass flow rate in this frame is zero regardless of δ.

10-2

10-1

100

101

102

Rarefaction parameter δ

1

10

100

1000

10000

Kin

etic facto

r ∆

Φ P

α = 0

α = 0.1

α = 0.3

α = 0.5

α = 0.8

α = 1

FIG. 1. Kinetic factor ∆ΦP as a function of the rarefaction

parameter δ for several different values of the accommodationcoefficient α, obtained from the data of Refs. [10, 17], withfits given by Eqs. (28), (31), and (32).

Thus in the frame in which the solid wall is immobilethe flow should also be the same for all δ, including thehydrodynamic limit δ →∞, and thus for all δ

∆ΦC(α = 1) = 1. (35)

The high-Kn limit for arbitrary α can be derived as [9]

∆ΦC(δ → 0)→ 2− α. (36)

Finally, the solution of the Stokes equation with slip gives

∆ΦC(δ →∞) ' δ + 2l2

δ + l1 + l2≈ 1 +

l2 − l1δ

(37)

≈ 1 +(2.032− 0.1211(2− α))(1− α)

αδ,

where only the first two terms in the expansion in 1/δare retained, and this is further approximated as

∆ΦC ' 1 +

2(1− α)

αδ, δ →∞ (38)

for simplicity. An interpolating formula

∆ΦC ≈ 1 +

1− α1 + α(0.5δ + 0.25δ1/2)

(39)

satisfies all of the conditions (34), (35), (36), and (38)and fits the numerical data very well (Fig. 2).

C. Shear stress for Poiseuille flow (factor ∆τP )

First of all, it is easy to obtain the sum of the shearstress values on the two walls for any α and δ. Consider

6

10-4

10-3

10-2

10-1

100

101

102

Rarefaction parameter δ

1

1.2

1.4

1.6

1.8

2K

ine

tic f

acto

r ∆

Φ C

α = 0.1

α = 0.5

α = 0.8

FIG. 2. Kinetic factor ∆ΦC as a function of the rarefaction

parameter δ for several different values of the accommodationcoefficient α, obtained from data in Ref. [11], with fits givenby Eq. (39).

a small volume in the shape of a parallelepiped spanningthe channel, with two of the faces perpendicular to boththe walls and the direction of the flow. The total forceon this volume consists of the pressure force on these twofaces and the shear force on the faces lying on the walls.If inertia is negligible, the sum of all forces acting on thevolume should be zero. The pressure force is determinedby the pressure gradient and is independent of Kn andα. Therefore, the total shear force (and the sum of theshear stresses on the walls) should also be independentof Kn and α and thus the same as in the hydrodynamiccase (Kn→ 0). For α = 1 the two shear stresses are thesame by symmetry, so both are the same as for Kn → 0and then for all δ

∆τP (α = 1) = 1. (40)

For α = 0 the stress on the solid wall vanishes and then,since the sum is the same, the stress on the liquid surfaceis twice as large, so for all δ

∆τP (α = 0) = 2. (41)

These are the same results as for ∆ΦC [see Eqs. (35 and

(34)]. Interestingly, the limiting behaviors are also thesame: the high-Kn limit is [9]

∆τP (δ → 0)→ 2− α (42)

and Stokes equation with slip gives

∆τP (δ →∞) ' δ + 2l2

δ + l1 + l2. (43)

We therefore use the same equation as for ∆ΦC [Eq. (39)]:

∆τP ≈ 1 +

1− α1 + α(0.5δ + 0.25δ1/2)

(44)

0.01 0.1 1 10Rarefaction parameter δ

1

1.2

1.4

1.6

1.8

2

Kin

etic f

acto

r ∆

τ P

α = 0.2

α = 0.4

α = 0.6

α = 0.8

FIG. 3. Kinetic factor ∆τP as a function of the rarefac-

tion parameter δ for several different values of the accom-modation coefficient α. Data are extracted from Fig. 4(a) inRef. [9],with fits given by Eq. (44).

Agreement with numerical data is good (Fig. 3). Sur-prisingly, the biggest deviation is observed for large δ forwhich Eq. (43) and therefore also Eq. (44) should workwell; it is possible that the numerical data themselves areless accurate in that range.

D. Shear stress for Couette flow (factor ∆τC)

In the high-δ limit, hydrodynamics with slip gives

∆τC(δ →∞) ' 1 +

l1 + l2δ

. (45)

Interestingly, in the opposite limit δ → 0, ∆τC is still

inversely proportional to δ, though with a different pref-actor [9]:

∆τC(δ → 0) ' π1/2

αδ. (46)

It is therefore convenient to introduce the “combined ef-fective slip length” l(δ, α) such that

∆τC(δ, α) = 1 +

l(δ, α)

δ. (47)

For α = 1 a very accurate expression, exact in both lim-its, is

l(δ, α = 1) ≈ π1/2[1.1466− 0.095 exp(−0.747δ) (48)

− 0.0516 exp(−3.724δ)].

7

10-4

10-3

10-2

10-1

100

101

102

Rarefaction parameter δ

1

10

102

103

104

105

Kin

etic f

acto

r ∆

τ Cα = 0.1

α = 0.5

α = 0.8

α = 1

FIG. 4. Kinetic factor ∆τC as a function of the rarefaction

parameter δ for several different values of the accommodationcoefficient α, obtained from data in Ref. [11], with fits givenby Eqs. (47), (49), and (50).

A simple and still quite accurate generalization to arbi-trary α is

l(δ, α) ≈ π1/2

{1

α+ c

[1.1466− 0.095 exp

(−0.747δ

c

)− 0.0516 exp

(−3.724δ

c

)]}, (49)

where

c = 1.6− 0.6α. (50)

It obeys the δ → 0 limit [Eq. (46)] exactly. In the limitδ →∞,

l→ 1.772

α+ 0.416− 0.156α; (51)

according to Eq. (45), this should be equal to l1 + l2,which is

l1 + l2 ≈1.790

α+ 0.363− 0.121α, (52)

the same as Eq. (51) to within 1% for all α between 0and 1. A comparison with numerical data is shown inFig. 4.

E. The α = 1 case and the “effective viscosities”

In the Letter our considerations are restricted to theα = 1 case. As mentioned, in this case two of the kineticfactors are trivial (∆Φ

C = ∆τP = 1) and only two nontriv-

ial factors remain, one for the Poiseuille component ofthe flow (∆Φ

P ) and one for the Couette component (∆τC).

0.01 0.1 1 10 100Knudsen number Kn

1

10

100

1000

Kin

etic fa

cto

rs ∆

Φ P,

∆τ C

∆Φ

P

∆τ

C

FIG. 5. Kinetic factors ∆ΦP and ∆τ

C as a function of theKnudsen number for diffuse scattering (α = 1), obtained fromdata in Refs. [12–16], with fits given by Eqs. (28), (31), (32)for ∆Φ

P and Eqs. (47), (49), (50) for ∆τC .

Figure 5 shows that these factors are very different; infact, the ratio of them increases slowly (logarithmically)as Kn grows. These factors enter the lubrication equa-tions in such a way that if they were the same, the flowwith GKE could be replaced by flow without GKE butwith a modified viscosity (just as Li did in the drop-dropcollision case [4]). Instead, essentially, we have two dif-ferent effective viscosities for the two components of theflow, µg/∆

τC for the Couette component and µg/∆

ΦP for

the Poiseuille component. In our simulations at 1 atm,Kn ≈ 3 is reached before the vdW-driven instability takesover; for this value, the ratio ∆Φ

P /∆τC is about 5, so an

approximation with a single effective viscosity would notbe accurate. Note also that for Kn = 3 we have ∆Φ

P ≈ 34,so GKE are very significant. For lower pressures, evenlarger Kn are reached, for which both ∆Φ

P /∆τC and ∆Φ

Pare likewise larger.

III. DETAILS OF THE COMPUTATIONALIMPLEMENTATION

The majority of the simulations, including all thosereported in the Letter, were carried out using a com-mercial software package, COMSOL Multiphysics [26](version 5.3a). Initially we used an open-source pack-age, FreeFem++ [27, 28] (version 3.51), and some of theresults reported in the SM were produced with it. How-ever, we encountered computational difficulties for highimpact speeds and found COMSOL to be superior insuch situations. For a few test cases, simulations wererun with both packages, with excellent agreement be-tween the results. While both COMSOL Multiphysicsand FreeFem++ use the finite element approach, they

8

are otherwise unrelated to each other and many of theimplementation details differ, as described below; thus,agreement between the results validates both implemen-tations.

For both COMSOL and FreeFem++ simulations, westarted with a spherical drop of radius R placed at dis-tance R/10 from the solid surface (or the symmetry planemidway between the drops in the case of drop-drop colli-sions) and moving towards the surface (symmetry plane)at speed V . Because of symmetry, for drop-drop colli-sions only one of the two drops was simulated. Given theaxial symmetry, the initial simulation domain is then ahalf-disk with a semicircular boundary.

A. COMSOL Multiphysics

Except for a few tests for drop-drop collisions, COM-SOL Multiphysics was used to simulate drop impact.Here we describe our drop impact implementation; thedrop-drop collision implementation is very similar. Belowwe use a fixed-width font for the names of COMSOLinterfaces, nodes (not to be confused with mesh nodes)and other items in the model setup.

As the basis of our simulations, we used the built-in Laminar Flow interface for a 3D axisymmetric flow,which implements a finite-element solver for the NS equa-tions. To specify stress boundary conditions on the freeboundary, we added a Boundary Stress node, active onboth the lower and upper boundaries (initially halves ofthe semicircle). Since the basic COMSOL Multiphysics(unlike the dedicated Microfluidics module) does not im-plement surface tension directly, an indirect approachhad to be used. For this, the stress on both bound-aries was specified as equal to zero, but with additionalcontributions from two Weak Contribution nodes, onefor the upper and the other for the lower boundary. AWeak Contribution node adds a specified expression tothe weak formulation of the problem used in the finiteelement approach. For the upper boundary, this expres-sion follows the standard recipe for the capillary pres-sure [29]; for the lower boundary, in addition to the cap-illary pressure term, weak contribution terms for normaland shear stress due to the gas film and the vdW dis-joining pressure (when included) are also added. Thenormal stress contribution of the gas film is equal to thegas pressure pg(r, t) and the shear stress contribution isgiven by Eq. (3) in the Letter and likewise depends onpg. The gas pressure is obtained by solving Eq. (14) si-multaneously with the NS equations for the drop. This isimplemented by using the Coefficient Form BoundaryPDE interface for the lower boundary.

As the drop deforms during the simulation, the bor-der between the lower and upper parts of the free surfacemoves. Since in our COMSOL implementation Eq. (14)for pg is always defined on the lower part, this meansthat the domain on which it is solved changes with timeand the boundary r0 of that domain on which the Dirich-

let boundary condition of Eq. (18) is applied changes aswell. This is usually not a problem, since, as mentioned,the results are not sensitive to the exact value of r0, aslong as it is far outside the air film. Even when r changesnon-monotonically along the boundary, numerical prob-lems usually do not arise, however, they are possible.To prevent them, coefficient g1 is multiplied by factor(1 − tanh(10nz))/2, where nz is the unit vector normalto the drop surface pointing outside the drop. This factoris ≈ 1 on the downward facing part of the surface and≈ 0 on the upward facing part, with a rapid crossovernear nz = 0.

The triangular mesh on which the solution is dis-cretized is built at the beginning of the simulation andthen evolves as the drop deforms. The mesh nodes on thedrop surface and on the symmetry axis (boundary nodes)stay attached to the boundary, but are free to slide alongit; Laplacian mesh smoothing is used to evolve the rest ofthe mesh. The arbitrary Lagrangian-Eulerian approachis used to obtain the solution on the moving mesh. Allthe mesh triangles have straight edges, including thoseon the surface. The parameters of the initial mesh arespecified using two COMSOL nodes, the Size node andthe Size Expression node. The parameters specified bythe Size node correspond to the defaults for the Generalphysics Coarse mesh, except the minimum element sizeis reduced to 10−6 m and the element growth rate to 1.2.The Size Expression node specifies the expression forthe size of the elements on the free surface; this is givenby

s = s0 + s1 exp(−z0/z), (53)

where s0 = 10−6 m, s1 = 10−4 m, z0 = 6×10−4 m. Thisensures that near the bottom of the drop (z . z0) themesh is fine, but coarsens towards the top. A fine meshnear the bottom of the drop (which eventually bordersthe thin air film) ensures higher precision in that region,where the height of the drop surface above the solid sur-face needs to be accurate to a few nm; it is also necessaryto resolve fine surface features, such as the kink and theinstability waves. The typical number of mesh elementsis a few tens of thousands.

During the simulation, the drop, and with it the mesh,gets deformed significantly. Therefore in general, regularremeshing is necessary. The simulation is stopped, thedrop is remeshed and then the simulation is restarted.This has to be carried out frequently enough that thequality of the mesh remains good (all the triangles areclose to equilateral); since the rate at which the dropdeforms varies during the impact, the interval at whichremeshing is done varies as well. In addition, it wasfound that the initial stages of evolution of the kink onthe edge of the air film are sensitive to remeshing fre-quency, even when there is no significant visible distor-tion of the elements, with more frequent remeshing givingbetter agreement with the experiment of Ref. [1]. Typ-ically, a few dozen remeshings are carried out in total;fewer are needed for lower impact speeds and for the

9

FIG. 6. A typical COMSOL mesh constructed in the processof remeshing during a simulation (at t = 1.8 ms for the impactspeed of 0.55 m/s).

lowest speeds no remeshing is necessary. New meshes areconstructed in the same way as the initial one, excepta smaller z0 = 4 × 10−4 m is used, since the drop nownearly touches the solid surface, and s0 (and the mini-mum element size) is reduced to 5×10−7 m when the kinkbecomes thin in order to resolve it properly. Automat-ing this procedure will be the subject of future work. Atypical mesh built during a simulation is shown in Fig. 6.

The finite element computation uses P1+P1 basisfunctions (first-order for both velocity and pressure);surprisingly, despite theoretical considerations [30], evenwith all stabilization options turned off this did not leadto any “locking” instabilities. Newton iterations are car-ried out to solve the nonlinear problem and the MUMPSsolver is used for the linear algebraic system resultingfrom the finite-element formulation. Second-order BDF(backward differentiation formula) time stepping is usedwith free steps and the maximum time step typically 10−7

s, though at the beginning of the simulation a smallertime step (2.5 × 10−8 s or even shorter for the highestimpact speeds) has to be used to capture the rapid flat-tening of the drop surface as it approaches the solid.

B. FreeFem++

FreeFem++ solves linear elliptic partial differentialequations in 2D or 3D written in weak form using a spe-cial language with a syntax resembling the C++ pro-gramming language. We transform the NS momentumequation into a linear equation of the type solvable byFreeFem++ using backward Euler time stepping, withthe material derivative approximated as

Dv(r, t)

Dt≈ v(r, t)− v(r− v(r, t−∆t)∆t, t)

∆t. (54)

The weak form is then obtained straightforwardly. In-corporating the capillary pressure and other terms in

the boundary conditions independent of pg is likewisestraightforward, with the weak form of these terms thesame as in COMSOL. However, those terms that dependon pg pose difficulties, since there is no direct way inFreeFem++ to solve the equation for pg on the bound-ary of the domain simultaneously with the NS equationsin the bulk. For the Poiseuille component of shear stress,pg calculated at the previous time step can be used with-out any problems. However, doing the same for normalstress causes numerical instabilities, unless the time stepis extremely small. To overcome this difficulty, we use theexplicit expression for pg, Eq. (19), discretize the integralusing the trapezoid rule and substitute the resulting ex-pression for pg at the boundary nodes of the mesh intothe linear algebraic system obtained from the weak formof the NS equations. This gives a linear system for thevalues of the liquid velocity components and pressure atthe mesh nodes that does not involve pg explicitly andthus is closed and can be solved, which FreeFem++ doesby calling the UMFPACK sparse solver. This approachis facilitated in FreeFem++ by the ability to access andmanipulate directly the elements of the stiffness matrixproduced by the finite element method.

A simple form of adaptive time stepping was used, bycomparing the result of advancing the simulation by onetime step to that of advancing by two steps of half theduration. The maximum difference between the resultsover all the mesh nodes was calculated and the time stepwas decreased, typically by 10%, if it was larger than aset threshold (and the last time step re-run), or increasedby 5% if it was smaller than another, smaller threshold(in which case there is, of course, no need to re-run thelast time step). Since immediately after mesh adaptation(see below) the discrepancy would often increase dramat-ically and the time step would correspondingly decreaseby a few orders of magnitude, and then the discrepancywould start to decrease very rapidly after only a few timesteps, it was found useful to have another, much lowerdiscrepancy threshold, such that if the error drops belowthat threshold, the time step would increase much more,by a factor of two. Thus after remeshing, the time stepwould immediately drop by several orders of magnitudeand then rapidly increase again.

Simulation meshes consist of triangles with straightedges. To construct them, the meshing facility built intoFreeFem++ is used. During the simulation, all meshnodes move with the flow. Instead of using the La-grangian approach, we solve Eulerian equations, but aftereach time step, as the mesh moves, the solution is castonto the new mesh by interpolation. Mesh adaptationis carried out, typically every 100 time steps, but morefrequently at the beginning of the simulation. The meshadaptation criterion is chosen so that the mesh is keptfine near those parts of the bottom surface of the dropwhere the air film is the thinnest or the gas pressure gra-dient is the highest (typically near the edge). Since meshadaptation in FreeFem++ is decided based on functionsdefined everywhere inside the solution domain, rather

10

than just on the boundary, auxiliary Laplace problemswith appropriate boundary conditions on the lower sur-face were solved every time before the adaptation routinewas called to obtain such functions, with negligible com-putational cost. Smaller mesh sizes than in COMSOLwere used, typically around 3000-5000 triangles, whichsometimes made the simulation unstable and requiredoccasional restarting. However, the agreement betweenCOMSOL and FreeFem++ results is otherwise excellent(Fig. 7). The finite element basis functions are linear (P1)for pressure and linear with an additional cubic “bubble”function added for stability (P1b) for velocity.

IV. DROP-DROP COLLISION SIMULATIONS

A. Lubrication approximation

When two liquid drops collide, an air film is formedbetween the drops. When the drops are identical, bysymmetry the liquid on the surfaces of the two dropsmoves with the same radial velocity, while the verticalvelocities are equal and opposite. The equality of thetwo radial velocities means that the Couette componentof the gas flow is replaced with plug flow, so the massflow rate is now

Φ = Φp + ΦP (55)

and the shear stress on the drop surface is

τg = τp + τP , (56)

where Φp and τp are the mass flow rate and the shearstress, respectively, for plug flow. For this flow compo-nent the mean speed of air molecules is independent ofz and equal to vr(r, t), even with GKE. This means thatΦp is trivial,

Φp = Sρgvr (57)

for a surface with the normal in the radial direction, andτp vanishes,

τp = 0. (58)

For the Poiseuille flow component, the expressions forboth the mass flow rate ΦP and the shear stress τP re-main the same, given by Eqs. (2) and (4) in the Letter.However, symmetry in gas scattering properties of thetwo surfaces of the air film (which are now the two dropsurfaces) is guaranteed and therefore ∆τ

P = 1. Thus,

ΦP = −Sρgh2(∂pg/∂n)

12µg∆ΦP , (59)

τP =(∂pg/∂n)h

2, (60)

where h, as before, is the film thickness (now betweenthe two drops). Note that in Eqs. (57)–(60) only one

0 500 nm

0

2

4

6

0

2

4

6T

ime,

ms

Tim

e,

ms

0 500 1000

Radial distance, µm

(a)

(b)

FIG. 7. A comparison between the colormaps showing the airfilm thickness as a function of the radial distance from thedrop axis and the time for the impact speed V = 0.55 m/sobtained with (a) COMSOL (same as the right panel of Fig. 2in the Letter), and (b) FreeFem++.

kinetic factor remains, ∆ΦP . Since the equation contain-

ing this factor [Eq. (59)] is also the only equation in thewhole problem in which the gas viscosity µg enters, fordrop-drop collisions, in contrast to drop-solid impact, itis possible to replace µg by a space- and time-dependent

11

effective viscosity,

µeffg (r, t) =

µg∆ΦP

, (61)

such that the effect of the gas flow on the drops is equiv-alent to that of the flow without GKE, but with theviscosity equal to µeff

g (r, t). This is the essence of Li’sapproach [4].

The derivation of the lubrication equation for pg is en-tirely analogous to the drop impact case. The result hasthe same form as before [Eq. (14)] and so does the ex-pression for g1 [Eq. (15)], but now

g2(r, t) =12µgrh3∆Φ

P

[∂

∂r(rhvr) + r

∂h

∂t

]. (62)

The boundary conditions are the same [Eqs. (17) and(18)] and the solution is

pg(r, t) = −12µg

∫ r0

r

1

∆ΦP

[urh2

+

∫ r′0r′′(∂h/∂t)dr′′

r′h3

]dr′.

(63)

B. Results and comparison with previous work

For a comparison with simulations by Li [4] (who, inhis turn, makes a comparison to experimental results ofRef. [31]), we have used one of Li’s sets of parameters,namely, those corresponding to tetradecane drops of ra-dius R = 167.6 µm and We = 9.33. For the numeri-cal values of the parameters (viscosity, density, etc.), seeRef. [4]. In Fig. 8, we compare our results for the time de-pendence of the air film thickness at its thinnest point tothose of Li for two values of the ambient pressure, 1 atm,when the drops bounce off each other, and 0.6 atm, whencoalescence occurs. The agreement is excellent, despitedifferences in treatment of the air film and the air outsidethe film (ignored by us, but included by Li). In the sameplot we also show the result without GKE to illustratetheir significance even at 1 atm (note the logarithmicscale of the plot), as well as that for 0.6 atm with GKE,but without vdW interactions. The latter curve coincideswith that obtained with vdW interactions included un-til immediately before the contact, but the contact itselfis avoided. The role of vdW interactions is discussed inmore detail below.

Following Li, we also compare the evolution of theshape of the drops to the experiments of Ref. [31] bysuperimposing the outlines of the drops obtained inour simulations with the experimental photos (Fig. 9).Again, excellent agreement is obtained. It should benoted that at the resolution of Fig. 9 the shape is notactually sensitive to the GKE or the details of the airfilm dynamics in general when in the bouncing regime(although these details have a critical effect on when coa-lescence occurs). Nevertheless, this test is still useful as ameans of verifying the accuracy of our computational so-lution of the Navier-Stokes equations in the liquid drops.

0 0.2 0.4 0.6 0.8Time, ms

10-3

10-2

10-1

1

10

Min

imum

film

thic

kness,

µm

no GKE, no vdW

1 atm, no vdW

1 atm, vdW

0.6 atm, no vdW

0.6 atm, vdW

Li, 1 atm, vdW

Li, 0.6 atm, vdW

FIG. 8. The thickness of the air film between two collidingdrops at its thinnest point as a function of time for tetrade-cane drops of radius R = 167.6 µm and Weber number (de-fined as in Ref. [4]) We = 9.33, both without GKE and withGKE corresponding to several different values of the ambientair pressure, with and without vdW interactions. Compar-isons to results by Li [4] are made.

FIG. 9. Simulation profiles (red and blue lines) superimposedon top of experimental photos from Ref. [31]

C. Role of van der Waals interactions

In Fig. 10 the film thickness at its thinnest point isplotted as a function of time for several different valuesof the ambient pressure, with vdW interactions switchedoff; all other parameters are as in Fig. 8. While the min-imum thickness drops very rapidly as the ambient pres-sure decreases, it never reaches zero, even for the lowestpressure of 0.2 atm, well below 0.6 atm for which coa-lescence is observed when vdW forces are included (seeFig. 8). Note that the smallest thickness for the lowestpressure in Fig. 10 is below the intermolecular distance;bouncing off after reaching such distances between dropsis, of course, completely unphysical, yet formally, it doesappear that without vdW interactions coalescence neveroccurs and, in particular, GKE alone are not sufficientto cause it. The same is observed for drop impact, withwetting never formally initiated without vdW.

12

FIG. 10. Minimum film thickness for different pressures withvdW interactions switched off. Coalescence never occurs.

D. Modes of contact

As for drop impact, there are two principal modes ofcontact, with drops touching either in the flat part of theair film [film mode; Fig. 11(a)] or at the kink on the edgeof the film [kink mode; Fig. 11(b)].

V. DROP IMPACT WITH PARTIALLYSPECULAR REFLECTION OF AIR MOLECULES

For our computations in the Letter we have made theassumption that air molecules are scattered diffusely offboth the drop surface and the solid surface (both accom-modation coefficients α are unity). However, as discussedin Sec. II, partially specular reflection off the atomicallysmooth mica surface cannot be ruled out. Here we studythe consequences of allowing α 6= 1 for the solid surface

by changing the kinetic factors ∆{Φ,τ}{C,P} accordingly (see

Sec. II).Figure 12 shows the film thickness at the thinnest point

as a function of time for the impact speed V = 0.55 m/sand several different values of α, as well as the case with-out GKE. It can be seen that the film thickness changesby more than an order of magnitude between the twoextremes of diffuse (α = 1) and specular (α = 0) reflec-tion. Even for more reasonable values of α (0.5–0.8) thechange compared to the diffuse case is quite significant.The curve with no GKE helps illustrate one more timethe overall significance of kinetic effects. Note that thesimulations in Fig. 12 were run without vdW interac-tions; while including them would have little to no effectfor higher values of α, they are expected to lead to con-tact for α = 0 and 0.1 given the typical thresholds forinitiation of instabilities (see also the next section for adiscussion of these thresholds).

0 50 100 150 200Radial distance (µm)

0

0.05

0.1

0.15

0.2

Film

thic

kness (

µm

)

(a)

0 50 100 150Radial distance (µm)

0

0.1

0.2

0.3

0.4

Film

th

ickn

ess (

µm

)

(b)

FIG. 11. Film thickness profiles at the moment of contact for(a) We = 25, p0 = 2.4 atm (film mode of contact) and (b)We = 9.3, p0 = 1 atm (kink mode of contact).

A significant decrease of the air film thickness due topartially specular reflection raises the possibility that ne-glecting this effect is responsible for the discrepancy inthe minimum film thickness between our simulations andexperiments mentioned in the Letter (see the discussionof Fig. 2 there). While at first glance this seems un-likely given excellent agreement for the bouncing-wettingthreshold, we note that this threshold is somewhat insen-sitive to details due to a rapid change in film thicknesswith the impact speed and also reasons related to vdWinteractions discussed in the next section, so cancella-tion of errors leading to near-perfect agreement cannotbe ruled out. Finding the accommodation coefficients ex-perimentally or via molecular simulations would be de-sirable.

13

0 1 2 3 4 5 6Time (msec)

10

100

1000M

inim

um

air f

ilm h

eig

ht

(nm

)

no kin. eff.

diffuse

α = 0.8

α = 0.5

α = 0.1specular

α = 0.3

FIG. 12. Dependence of the air film thickness at its thinnestpoint on time for V = 0.55 m/s and different values of theaccommodation coefficient α. For this calculation, vdW in-teractions were not included.

VI. INSTABILITIES LEADING TO CONTACT

A. General considerations

As discussed in the Letter, vdW interactions play a sig-nificant role immediately before contact; in fact, withoutthem contact does not occur (see Fig. 10, as well as Fig. 3in the Letter). The vdW interactions act to amplify smallfluctuations in free surface shape that, from a physicalperspective, could be generated by thermal noise; de-scribing this process would in principle require fluctuat-ing hydrodynamics [32, 33]. However, even within de-terministic simulations vdW instabilities are still able tobe initiated, due to numerical noise. As this noise hasno relation to the underlying physics and, in addition,axial symmetry is imposed, even though surface pertur-bations starting from random fluctuations do not have tobe axisymmetric, the theoretical analysis below (assum-ing thermal fluctuations as the initiator) is important asa means of verifying the simulation results. Good agree-ment (despite the theoretical analysis making its own as-sumptions) is ultimately due to the fact that relevantquantities (such as separation before contact) are insen-sitive to the initial fluctuation amplitudes because of fastexponential growth (making, e.g., the time to grow to acertain size only logarithmically dependent on the initialamplitude) and strong dependence of the relevant growthrates on the separation distance.

B. Instabilities of flat stationary films

As the starting point of our analysis of vdW-driven in-stabilities in drop impacts and drop-drop collisions, weconsider a thin flat gas film of thickness h between ei-ther two semi-infinite volumes of liquid (for drop-dropcollisions) or a semi-infinite volume of liquid and a solidsurface (for drop impact). Consider a perturbation of theinterface(s) with wave vector k and amplitude bk (for thecase of two fluid-fluid interfaces, it is assumed that theirperturbations are in antiphase, as this leads to the fastestgrowth, and the amplitudes are the same for simplicity).Assuming that bk is small so the linear approximation isvalid, the amplitude of the corresponding normal stressjump across the interface is

pk = akbk, (64)

where ak is a factor that depends on the absolute valueof k. The sign of the jump is chosen so that the pertur-bation will decay and thus is stable if ak < 0 and willgrow and thus unstable if ak > 0. Factor ak has two con-tributions. The stabilizing Laplace pressure contributionis proportional to the curvature of the interface and inlinear approximation gives a −γk2 term. The destabiliz-ing vdW disjoining pressure contribution is determinedby the local film thickness and thus is k-independent inFourier space. According to Eq. (10) in the Letter, thechange in the disjoining pressure upon changing the filmthickness from h to h+ δh is

δpd ≈AH

2πh4δh. (65)

Since the amplitude of δh is either |bk| when one of theinterfaces is solid or twice that when both interfaces arefluid, we get

ak =

{2aH − γk2 (collision),

aH − γk2 (impact),(66)

where

aH =AH

2πh4. (67)

An equation for the growth rate (albeit a rather com-plex one) can be written down in the general case [34, 35].This equation can, of course, be solved numerically for aparticular set of parameters, but this by itself does notgive much insight. However, the problem can be simpli-fied considerably and analyzed by making use of the factthat there are some small parameters and some effectsmay be negligible for specific conditions of interest.

In our consideration, we will make a number of as-sumptions, which will be verified for the range of pa-rameters of interest to us here after obtaining the re-sults. First, the usual lubrication approximation assump-tions are made, in particular, the film is assumed to bemuch thinner than the wavelength of the perturbation

14

(kh� 1) and inertial effects in the gas film are neglected.Second, pressure gradients in the liquid are neglectedcompared to those in the gas film so the gas pressure isequal to the normal stress jump given by Eq. (64). Third,the gas flow is Poiseuille, with the Couette/plug compo-nent negligible and the gas-liquid interface serving as ahard wall evolving very slowly as the instability grows.Gas film dynamics then decouples from the liquid dy-namics, which simplifies the consideration significantly.

There are only minor differences (a few numerical fac-tors) between the cases when one of the interfaces is solid(thus is flat and does not evolve) and when both are fluid,so we consider both cases at the same time. We useEq. (59) for the mass flow rate and consider flow throughthe boundaries of a volume spanning the film and smallin other two directions, applying mass conservation, sim-ilarly to how Eq. (14) was derived, but neglecting vari-ations in h and without assuming axial symmetry. Weobtain

dh

dt=h3∆Φ

P

12µg∇2pg. (68)

Considering a mode with wave vector k and keeping inmind that the gas pressure is the same as the pressurejump given by Eq. (64), we then get

hk = −h3k2ak∆Φ

P

12µgbk, (69)

where hk is the rate of change of the amplitude of thefilm thickness mode with the wave vector k. The rateof change of the amplitude of the interface mode, bk, isrelated to hk as either hk = −2bk when both interfaceschange (drop-drop collision) or hk = −bk when only oneinterface changes (as the other is solid; drop impact).Thus,

bk =

h3k2ak∆Φ

P

24µgbk (collision),

h3k2ak∆ΦP

12µgbk (impact).

(70)

Then the growth rate of interface modes with wavenum-ber k is

σk =bkbk

=

h3k2ak∆Φ

P

24µg(collision),

h3k2ak∆ΦP

12µg(impact).

(71)

Note that the result for impact is similar to that used byKolinski [36] to estimate the growth rate of the instabilityin his drop impact experiments, but differs by a factor∆ΦP /4. Kolinski uses the theory of Brochard Wyart and

Daillant [37] for a liquid film (thus no GKE; factor ∆ΦP )

bordering a vacuum, rather than a viscous fluid (factor1/4). Modes with σk > 0 (or, equivalently, ak > 0)are unstable; they have k < (2aH/γ)1/2 (collision) ork < (aH/γ)1/2 (impact). Using Eqs. (66) and (67), weobtain for the largest growth rate

σmax =

h3a2

H∆ΦP

24γµg=

A2H∆Φ

P

96π2γµgh5 (collision),h3a2

H∆ΦP

48γµg=

A2H∆Φ

P

192π2γµgh5 (impact),(72)

for

k = kmax =

{(aH/γ)1/2 = [AH/(2πγh

4)]1/2 (collision),

[aH/(2γ)]1/2 = [AH/(4πγh4)]1/2 (impact).

(73)

C. Application to drop-drop collisions and dropimpacts and comparison to simulations

When the distance between the two interfaces changeswith time (as is the case for drop collisions and impacts),we can still apply the above theory by assuming that itgives the correct instantaneous growth rate for a partic-ular mode at a given moment. That is, the amplitude ofthe mode with wave vector k is

bk(t) ∝ exp

(∫ t

max(σk(hmin(t′)), 0)dt′), (74)

where the form of the integrand ensures that the ampli-tude of the thermal capillary waves stays does not decaywhen there is no instability. We use the minimum filmthickness hmin as the value of h, because the instabilitygrows the fastest and contact occurs where the film is thethinnest.

We define the critical air film thickness hc as the mini-mum film thickness at the moment of contact in a thoughtexperiment run in parallel in which vdW interactionsare “switched off”, but all other conditions are identi-cal. (Alternatively, consider the original experiment withvdW interactions on, but ignore or smooth out the “wig-gles” associated with the instability and find the mini-mum thickness at the time of contact.) Noting a verysteep dependence of the maximum growth rate σmax onh, it is the modes with k around kmax(h = hc) that growthe most by the time h reaches hc; indeed, modes withsmaller k grow for a longer time, but the growth rateis significantly lower. For a mode with k = kmax(hc),growth occurs (σk > 0) when h < 21/4hc ≈ 1.2hc; basedon our simulations, we can estimate the time it takes hto decrease from 1.2hc to hc as roughly 10% of the to-tal collision time, or around 1 ms. Only modes within arelatively narrow range of k around kmax(hc) grow sig-nificantly; those with at least 60% of σmax are betweenk ≈ 2kmax/3 and k ≈ 4kmax/3. For simplicity, we assumethat these modes have their growth rate equal to σmax

and ignore the rest of the modes.

As discussed above, in contrast to simulations, phys-ically the unstable modes start as thermal fluctuations.This is the point of view we adopt below; comparison tosimulations then allows us to confirm the insensitivity ofthe results to the precise details of the initial fluctuations.For a two-dimensional surface (such as the surface of adrop) the mean-square displacement of a given point onthe surface due to thermal fluctuations, caused by modeswith wavenumbers between k1 and k2 (with the rest of

15

the modes “filtered out”), is [38]

〈z2〉 =kBT

2πγlnk2

k1, (75)

which for water at T = 300 K and k2/k1 = 2 gives〈z2〉1/2 ∼ 0.1 nm. As this should grow to an ampli-tude large enough to span the air film (roughly 10-50 nmin thickness) during the above-mentioned 1 ms interval,the estimated growth rate is σmax ' 5 × 103 s−1. Us-ing Eq. (72) for drop impact, we get an estimate for thecritical film thickness,

hc '(

A2H∆Φ

P

192π2γµgσmax

)1/5

. (76)

The characteristic wavenumber of the instability is kmax,which can be calculated using Eq. (73) with h = hc.While the above estimate of σmax is crude, it is adequatefor our purposes, since the dependence of hc on it is veryweak and the dependence of kmax is not too strong either.

Comparing our calculations to simulations, we firstnote that AH , γ, and µg are material parameters andremain the same throughout out study. The growth rateσmax also remains roughly the same as long as our esti-mate is valid. Only the kinetic factor ∆Φ

P varies, whichagain emphasizes the qualitative importance of GKE.While the dependence on ∆Φ

P is weak, this factor canvary by several orders of magnitude. Generally, as theambient pressure decreases, ∆Φ

P increases and with it, hcis expected to increase as well. This is indeed observed inFig. 13, from which hc can be estimated as the minimimfilm height without vdW interactions at the same time atwhich contact occurs when these interactions are present.More quantitatively, without gas kinetic effects (formallyfor p0 → ∞), ∆Φ

P = 1 and Eq. (76) gives hc ≈ 12.5 nm;with GKE, for finite ambient pressures ∆Φ

P itself dependson h and iteratively, one finds hc ≈ 25 nm for 1 atm and≈ 38.5 nm for 0.1 atm, which is close to the numeri-cal values seen in Fig. 13. The wavenumber at 1 atm iskmax ≈ 4×105 m−1, which corresponds to the wavelengthof about 15 µm compared to about 10 µm numerically (avisual estimate based on the inset of Fig. 1 in the Letter).The growth rates in the cases presented in Fig. 13, cal-culated numerically by taking the logarithmic derivativeof the difference between the minimum air film thicknessvalues without and with vdW interactions, increase withtime as expected, since the film thickness decreases, butare generally between 3×103 and 104 s−1 in the relevantinterval, i.e., similar to each other and to our estimatefor σmax.

We also note that under the assumption that σmax isindependent of the impact speed V , the critical height forinstability hc is predicted to be V -independent as wellwhen the ambient pressure is kept constant. This pre-diction is tested in Fig. 14, where minimum film heightswith and without vdW interactions are plotted for severaldifferent impact speeds, but the same ambient pressurep0 = 0.1 atm. We find that hc indeed remains roughly

0 1 2 3 4

Time, ms

100

101

102

103

104

105

Min

imum

film

thic

kness, nm

0.1 atm, 0.2 m/s

1 atm, 0.85 m/s

∞, 2 m/s

38.525

12.5

FIG. 13. Dependence of the air film thickness at its thinnestpoint on time for several different pressures and impactspeeds, with (solid lines) and without (dashed lines) vdWinteractions. The theoretical predictions for the critical filmthickness hc are shown with dashed horizontal lines.

constant (and similar to the above prediction) for mod-erate speeds, but is lower for higher speeds. For thesehigher speeds hmin drops rapidly below our estimate ofhc, so, obviously, the instability has to grow much fasterand our V -independent estimate of σmax is too low. Thecases presented in Fig. 13 were chosen to avoid this situ-ation.

For drop-drop collisions, Eq. (76) still applies, exceptthe factor 192 in the denominator is replaced by 96. Forour test cases considered in Sec. IV B (tetradecane dropsof R = 167.6 µm at 0.6 and 2.4 atm with We = 9.3and 25, respectively), the collision time scale is an orderof magnitude shorter than the above estimate for dropimpact and the decrease in the film thickness is morerapid relative to that time scale, particularly at 0.6 atm,meaning that the growth rate σmax should be more thanan order of magnitude larger than in the drop impact caseconsidered above (although this is alleviated somewhatby the fact that the surface tension of tetradecane is lowerthan that of the water-glycerol mixture, so the initialthermal fluctuations are larger). Assuming σmax = 105

s−1, we obtain hc ≈ 21 nm for 0.6 atm and hc ≈ 16 nmfor 2.4 atm, in excellent agreement with the numericalresults (25 and 17 nm, respectively). The value of kmax

in the latter case (for which the film mode of contact isobserved, thus the instability wavelength is well-defined)is predicted to be ≈ 2.1×106 m−1, giving the wavelengthof about 3 µm, which also agrees with the simulations.The observed growth rates in the relevant time intervalare roughly 4× 104 and 1.2× 105 s−1 .

As a final comment, we note that an approach some-times used in the literature when studying drop coales-cence (e.g., [39]) is to ignore vdW interactions but in-

16

0 1 2 3

Time, ms

100

101

102

103

104

105

Min

imu

m f

ilm

th

ickn

ess,

nm

0.2 m/s0.313 m/s0.34 m/s0.36 m/s0.37 m/s0.38 m/s0.4 m/s0.45 m/s

38.5

FIG. 14. Dependence of the air film thickness at its thinnestpoint on time for the ambient pressure p0 = 0.1 atm and sev-eral different impact speeds, with (solid lines) and without(dashed lines) vdW interactions. The theoretical predictionfor the critical film thickness hc is shown with dashed horizon-tal lines. The mode change from kink to film occurs between0.36 and 0.37 m/s (see Fig. 4 in the Letter), with no significantchange in hc.

troduce a threshold critical distance below which coales-cence initiation is assumed. As Eq. (76) and our exam-ples illustrate, the critical film height hc can vary con-siderably; besides the kinetic effects, it also depends onAH , γ, µg, as well as the collision time scale via σmax.Therefore, assuming a constant hc is not necessarily ap-propriate; Eq. (76) (or its analog for drop-drop collisions)provides a more accurate estimate of the critical approachdistance.

D. Validity of the assumptions

As mentioned, in our consideration we have made anumber of assumptions. We now check our treatment forself-consistency by verifying these assumptions using ourresults.

First, we check the long wavelength assumption,kmaxhc � 1. For drop impact, for p0 → ∞, p0 = 1atm and p0 = 0.1 atm we get hc ≈ 12.5, 25 and 38.5nm and kmax ≈ 1.7 × 106, 4.2 × 105 and 1.8 × 105 m−1,respectively, so the values of kmaxhc are ≈ 0.021, 0.011and 0.0069. For the two drop-drop collision cases we haveconsidered (p0 = 0.6 atm, We = 9.3 and p0 = 2.4 atm,We = 25), the values of kmaxhc are ≈ 0.026 and 0.034,respectively. All of these values are well below unity.

Second, we have neglected pressure gradients in theliquid compared to those in the air film. Based on airmass conservation, we have obtained a relation between

the Laplacian of the air pressure and the speed at whichthe interface moves [Eq. (68)]. A similar equation canbe obtained from liquid mass conservation, with a fewdifferences: h is replaced by the only length scale in theliquid, 1/k; there is a minus sign, as the liquid flows tothe regions that the air escapes from; the pressure changeis considered in the direction parallel to the interface; thenumerical factor is expected to be of order unity. It isassumed here that inertia is negligible in the liquid, whichis verified later. We get

dh

dt∼ − 1

µlk3∇2‖pl. (77)

Comparing the two right-hand sides and omitting thenumerical factors, |∇2

‖pl| is much smaller than |∇2p| if

µgµl� (kh)3∆Φ

P . (78)

The viscosity ratio on the left-hand side is µg/µl ≈ 1.6×10−3. Even though this is small, the right-hand side iseven smaller. Indeed, the values of ∆Φ

P for p0 → ∞,p0 = 1 atm and p0 = 0.1 atm are (using the respectivevalues of hc for the film width) 1, 32 and 280; the right-hand side values are then approximately 8×10−6, 3×10−5

and 1 × 10−4. Thus, condition (78) is satisfied; in fact,given a small prefactor (1/12) in Eq. (68), this conditionis probably even more stringent than necessary. For thedrop-drop collision, the left-hand side of Eq. (78) is ≈8.6 × 10−3 and the right-hand side is ≈ 1.2 × 10−3 and7.7 × 10−4 for the two cases we have considered; again,Eq. (78) is satisfied. It is worth noting, however, that theratios of the left- and right-hand sides are not very largein some of the cases; thus, under different (but still nottoo extreme) conditions, particularly, for more viscousliquids, the system will be in a different regime where adifferent set of assumptions will need to be made.

Third, we have assumed that the speed of the liquidnear the interface in the direction parallel to it is sosmall that the Couette/plug component of the air flowis negligible compared to the Poiseuille component. ForPoiseuille flow, shear stress at the boundary is

τP ∼ h∇p (79)

(the prefactor is always between 1/2 and 1 for a diffuselyscattering boundary, regardless of the properties of theother boundary). This drives the outer fluid at speed

vl ∼h∇pµlk

, (80)

which results in the plug/Couette mass flow rate ∼ρgvlS ∼ (ρgSh∇p)/(µlk). This is much smaller thanthe Poiseuille mass flow rate ∼ (ρgSh

2∇p∆ΦP )/µg when

µgµl� kh∆Φ

P . (81)

For drop impact and p0 → ∞, when ∆ΦP = 1, the left-

hand side is ≈ 1.6 × 10−3 and the right-hand side is ≈

17

2.1×10−2, so the condition is satisfied. In all other cases,for both impact and collisions, ∆Φ

P is considerably larger,so the ratio of the left-hand side and the right-hand sideis even smaller.

It is also necessary to check that inertial effects aresmall in both the air film and the liquid, as we haveassumed from the outset. Since we are considering lin-ear growth, the nonlinear convective derivative term isassumed negligible and the relevant comparison is be-tween the time derivative term and the viscous term inthe NS equation. The dimensionless parameters that arerequired to be small are then

Il =ρlσmax

µlk2max

(82)

in the liquid and

Ig =ρgσmax

µg/h2c

(83)

in the film. In fact, for the first of these Il . 1 is suffi-cient, since we only need to confirm that the outer liquidflow is negligible, as we have assumed. For drop im-pacts, using the values of kmax and hc for 1 atm, we getIl ' 3 × 10−3 and Ig ' 2 × 10−7; for drop-drop colli-sions, using the values for 2.4 atm gives Il ' 9 × 10−3,Ig ' 2 × 10−6. Other sets of parameters give the valuesof the same order of magnitude. Thus, neglecting inertiais justified.

There are two potentially important effects that wehave neglected in our considerations. First, contact oc-curs while the drop(s) is (are) spreading radially; thismotion and the associated flows in both the drop(s) andthe air film are neglected. Crudely, one can switch tothe frame of reference moving with the drop(s) near thelocation of minimum film thickness where the instabilityarises; however, this is problematic particularly in thecase of drop impact, as the solid surface will no longerbe immobile. Good agreement of our theoretical resultswith numerical simulations appears to indicate that thisis not a significant problem; however, the issue deservesfurther study. Second, we have assumed the air film tobe infinite and its thickness to be the same everywhere,but this is, obviously, not the case in our problems. Inpractice, the infinite flat film approximation is likely tobe adequate when the thickness changes little over thelength ' 2π/kmax. This is more-or-less true for the filmmode of contact, but not necessarily for the kink mode,as the kink is only a few µm in width, comparable to thewavelength of the instability. This can potentially havetwo effects. First, it imposes a restriction on possible val-ues of the wave vector k, which is particularly importantwhen only axisymmetric solutions are allowed. Second, itis possible that features of the interface with a particularlength scale, in effect, “pre-select” the modes with wave-lengths similar to that scale, making them “easier” togrow than for flat surfaces. Nevertheless, our estimates

of hc are surprisingly accurate even in the kink mode, forboth drop impacts and drop-drop collisions.VII. ADDITIONAL DETAILS OF SIMULATION

RESULTS

A. “Dimple collapse”

The computational colormap in Fig. 2 in the Letterhas intentionally been extended to longer times com-pared to the experimental one, to reveal a small speckon the drop axis at t ≈ 6.3 ms corresponding to thefilm thickness suddenly dropping below the experimen-tally imposed 500 nm threshold right before the dropstarts to lift off. We term this phenomenon “dimple col-lapse”, as it corresponds to a very sudden disappearanceof the “dimple” on the axis that exists for the whole du-ration of the collision, as the free surface goes from beingconcave to convex at the axis of symmetry. The cor-responding rapid dip in the minimum film thickness isseen in Fig. 3 in the Letter and Figs. 12–14 in this SM.It has also been obtained theoretically and computation-ally for collisions with liquid surfaces [40] and drop-dropcollisions (see Refs. [4, 6] and Fig. 8).

B. Simulations for higher impact speeds in theKn → 0 limit

To achieve contact with the solid surface without GKE(or in the Kn → 0 limit), much higher impact speedsare required than in other cases considered in the paper,which presents some numerical difficulties when trying todetermine the exact wetting threshold. These are asso-ciated with topological changes occurring at the uppersurface of the drop. Specifically, at 1.1 m/s the simula-tion runs without contact until the film thickness beginsto increase, suggesting a bouncing regime, but the simu-lation cannot capture the full rebound due to an unsta-ble jet that forms from the drop’s apex, whilst for speedsbetween 1.3 and 1.8 m/s the upper surface of the droptouches itself at the axis, forming a bubble, as shown inFig. 15 and previously observed experimentally for im-pact on a hydrophobic surface [41]. Topological changessuch as these cannot be handled by our computationalapproach. At 2 m/s contact does occur before bubbleformation, therefore the threshold is likely to lie between1.1 and 2 m/s. Based on the fact that by the time thebubble forms the separation between the drop and thesurface already starts to increase without obvious signsof a vdW instability, it is highly likely that the drop willbounce off in the range of speeds in which the bubble isobserved and therefore the threshold should be close to1.9 m/s. However, destabilization of the drop due to thebubble leading to contact cannot be ruled out completelyand further studies using an approach allowing topolog-ical changes of the drop surface are required alongsidenew experimental evidence.

18

FIG. 15. The profile of the drop for impact speed 1.3 m/s andno GKE immediately before the creation of a trapped bubble.

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