Why Drops Bounce on Smooth Surfaces - ACS Publications

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Why Drops Bounce on Smooth Surfaces Rafael Tadmor, Sakshi B. Yadav, Semih Gulec, Aisha Leh, Lan Dang, Hartmann E. N'guessan, Ratul Das, Mireille Turmine, and Maria Tadmor Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b00157 • Publication Date (Web): 06 Mar 2018 Downloaded from http://pubs.acs.org on March 24, 2018

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Why Drops Bounce on Smooth Surfaces Rafael Tadmor1,2*, Sakshi B. Yadav2,#, Semih Gulec2,#, Aisha Leh2, Lan Dang2, Hartmann E. N'guessan2, Ratul Das2, Mireille Turmine3, Maria Tadmor2 * Corresponding Author # These authors contributed equally 1. Department of Mechanical Engineering, Ben Gurion University, Beer Sheva, Israel 2. Dan F. Smith Department of Chemical Engineering, Lamar University, Beaumont, TX 77710. 3. Laboratoire Interfaces et Systèmes Electrochimiques, CNRS, UPR15-LISE; Université Pierre et Marie Curie-Paris 6, Case 133, 4 place Jussieu, 75252 Paris Cedex 05, France

ABSTRACT It is shown that introducing gravity in the energy minimization of drops on surfaces results in different expressions when minimized with respect to volume or with respect to contact angle. This phenomenon correlates with the probability of drops to bounce on smooth surfaces on which they otherwise form a very small contact angle or wet them completely. Theoretically, none of the two minima is stable: the drop should oscillate from one minimum to the other as long as no other force or friction will dissipate the energy. Experimentally, smooth surfaces indeed show drops that bounce on them. In some cases, they bounce after touching the solid surface, and in some cases they bounce from a nanometric air, or vacuum film. The bouncing energy can be stored in the interfaces: liquid-air, liquid-solid and solidair. The lack of a single energy minimum prevents a simple convergence of the drop’s shape on the solid surface, and supports its bouncing back to the air where a single energy minimum exists. Therefore, the lack of a simple minimum described here, supports drop bouncing on hydrophilic surfaces such as that reported by Kolinski et al. Our calculation shows that the smaller the surface tension, the bigger the difference between the contact angles calculated based on the two minima. This agrees with experimental finding where reducing the surface tension, for example by adding surfactants, increases the probability for bouncing of the drops on smooth surfaces.

INTRODUCTION A common way to obtain the Young equation is to use energy minimization under the constraint of constant volume1–3. In this approach, the Laplace pressure term is ignored because the volume is conserved. However, the Laplace equation is derived from arguments concerning volume change of liquid drop, namely volume does not need to be a conserved property. Note that we don’t necessarily consider here the volume change due to evaporation4–6, but mainly due to the density changes that occur as the drop’s molecules fluctuate. These changes give rise to the Laplace equation, which proves that they cannot be neglected. Therefore, while the constant volume approach, provides the correct answer (i.e. the Young equation), ignoring volume minimization as a whole is inconsistent with the Laplace equation and prevents impact of external forces on the equilibrium contact angles3.

1 Environment ACS Paragon Plus

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Here we consider the energy of the drop as a function of both volume (at constant angle) and angle (at constant volume). Both terms (the one of constant angle and the one of constant volume) result in the same Young equation. However, when we introduce gravitational term into the equation, the two minimizations, the one of constant angle and the one of constant volume, lead to different equations, resulting in different, mutually exclusive 'equilibria' contact angles. Lacking a single minimum, the drop theoretically should not settle on the surface suggesting that drop-on-surface systems obtain equilibrium due to other factors such as surface roughness or edge effects7–13. Observations of drops on smooth surfaces by Kolinski et al14 as well as our own, show drops that bounce on smooth surfaces despite having a small Young contact angle. Such drops can spread as they squeeze the air between them and the solid to a nanometric layer before they bounce. The nanometric air layer cannot store elastic bouncing energy since it is viscous. Similarly, the solid cannot store an elastic energy for a liquid drop that hovers nanometers above it since, at these distances, the van der Waals interaction between the drop and the flat surface1,15–19 is always attractive for low refractive index media such as air. On the other hand, the elastic bouncing energy can be stored in the three surface tensions in the problem19–24 since surface tensions are known to store energy almost reversibly22,23. We show here that the surface tensions in the problem do not progress to a single minimum, but rather, to two mutually exclusive local minima. Therefore, to obtain a single interfacial energy minimum the drop needs to be back in the air, namely bounce.

THEORETICAL METHODS Obtaining the Young equation from energy minimization Consider a liquid drop floating in space. The drop adopts a shape of a perfect sphere with a radius, RS, and its surface tension, , corresponds to the pressure difference, ∆P, between the inside and outside of the drop according to the Laplace relation1,8: ∆P = 2 /RS. Suppose a solid flat surface is brought close to that drop, so that they touch at a point. Now the system has two additional interfacial tensions (interfacial energies): the surface–liquid and the surface-vapor interfacial tensions, , and , respectively. In this case, the drop will begin to adopt a shape of a cap which is a part of a sphere with radius, R, which is bigger than RS. The internal Laplace pressure will reduce because R > RS. As the drop covers more of the solid surface area, the cap shape it adopts corresponds to an ever-increasing sphere size and ever decreasing Laplace pressure. During this spreading process, the flat area between the drop and the surface, ASL, is ever increasing, and the exposed-to-the-air substrate surface flat area, ASV, is ever decreasing, and the change in the cap area of the liquid vapor interface, ALV, decreases for θ > 90° and increases for θ < 90° with the decrease in contact angle, θ. Fig. (1) shows a schematic representation of the interfacial areas - , and and their corresponding surface tensions. At a certain point, the drop will stop advancing, and reach equilibrium because the net-work change, , as a result of this spreading process, becomes zero. The index ng stands for no gravity, since here we consider a system without gravitational force. (a)

(b)


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Figure 1. (a) Two-dimensional representation of a drop on a surface describing interfacial tensions as forces balanced along the x axis which result in Young equation (b) Threedimensional representation of a drop on a surface describing the different parameters in the problem. There are also two terms associated with the three-phase contact line (triple line): one related to the weak energy associated with the 3-body interaction (liquid-solid-vapor)25,26, which we term here as “line tension” and the other is related to the stronger pinning energy that results from molecular re-orientation of the solid molecules at the triple line27,28, which we name here "line retention force". Since, we assume here a perfectly smooth surface, we cannot have triple line retention force (which assumes solid surface deformation at the triple line27,28), yet, it makes little difference to this theoretical analysis as the two parameters have the same mathematical expression. Additionally, for this study, we consider a drop that is barely touching the surface (sometimes there is nanometer gap between the drop and the surface and sometimes it is touching the surface and bounces right back)29 (see section Interfacial Tension as a Function of Film Thickness, in the supplementary material). Hence, it is reasonable to assume small values of line tension for calculation purposes. The differential net-work change ( ) due to a drop spreading can be expressed in terms of the interfacial tensions, the drop volume and the line tension, as described in eq. (1). This expression has a minimum value ( =0) when the drop reaches equilibrium.

= − − − + + −

(1)

where L is the length of the three-phase contact line, k is the energy per triple line length, P and V stand for pressure and volume, respectively, and the indexes in and out relate to the inside and outside of the drop, respectively. Note that, liquids are almost always considered incompressible for all practical purposes; however, Surface Science relies on the compressibility of a liquid in many fundamental equations. This is evident by the mathematical derivation of Laplace pressure difference, inside and outside a drop due to its surface tension. (see the section Derivation of Laplace Pressure using Helmholtz energy function () of the drop and the surrounding media system, in the supplementary material)

Since,

= − and = −

(2)

Eq. (1) can be written as:

= − − − + ∆ −

(3)

Now, the important part to understand is that we need to consider the full differentials (to the best of our knowledge, this is the first study to do that). Thus, the change of each of


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the areas in the problem should be considered by changing the angle at constant volume and by changing the volume at constant angle. Considering the full differentials, we write:

=

(4)

!

'

!

!

+

!

+

"#$%

"

&

#$%

and similarly:

=

(5)

'

"#$%

"

#$%

&

And for the triple line:

= !

(6)

"#$%

+ " !

#$%

&

Substituting eq. (4), (5) and (6) in eq. (1) and equating to zero (equilibrium) results in:

= !−

− (7)

"

'

− −

− " ! &

− + ∆ ! + !− "

' "

−

At equilibrium, eq. (7) can only be zero if both first and second terms are zero (because V and θ are independent variables). Specifically, each of the two terms inside the big parenthesis should equal zero. It so happens that the two equations that result from this condition end up being identical, and therefore there is one simple energy minimum for the “no gravity” case. Though each of the terms can only be zero if their terms in the brackets are zero, we start by considering the whole term including the differential to obtain the total energy of the system as a function of volume and contact angle. Let’s define ", and , as: '

− −

− + ∆ !

'

− −

− " ! &

", = − , & = −

"

"

"

(8) (9)

So that:

= ", + , & (10) Solving eq. (8) for ", = 0 results in eq. (11) (see eq. (S4) to (S22) of the supplementary material for detailed calculations): -

− − cos & − = 0 .


(11)

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Similarly, solving eq. (9) for , & = 0 also results in eq. (11) (see eq. (S23) to (S28) of the supplementary material for detailed calculations). Therefore, earlier studies obtain a correct result although only one of the equations was derived. Eq. (11) is the Young equation with an addition of a line tension term and in absence of gravity.

Adding gravity to the total energy expression We now add a gravitational term to the total energy expression. As we shall see, contrary to the previous case which gave the same expressions for the two independent parts of the energy minimizations (the two terms of Eq. (7)), adding gravity results in different equations. It is interesting to note that the previous case considered only parameters which emanated from intermolecular interactions (surface tension, Laplace pressure, and line tension) which are all expressions of the electromagnetic force, while now we add a force of a different origin (gravity). The gravitational term is associated with the height of the drop’s center of mass, hCM, whose differential is:

ℎ01 =

234

!

"#$%

+

234

!

"

#$%

&

(12)

Thus, instead of Eq. (7) we now write:

= !−

−

'

"

− −

− " − 56

234 "

− + ∆ − 56

234

! &

! + !− "

' "

−

(13)

where, ρVg is the weight of the drop which is a constant (ρ is the drop’s density, V is the drop’s volume, and g is the gravitational acceleration). Note that the gravitational term is negative because, as the drop expands, it is doing work on the surrounding (displacing the air around it, away from the center of the drop). Let’s define " and in analogy to eq. (8) and (9), respectively, so that

" = −

'

− −

& = −

'

− −

"

"

− + ∆ − 56

− " − 56

234

234 "

! "

! &

(14) (15)

Simplifying eq. (14) (see eq. (S29) to (S31) of the supplementary material), we obtain (expressed in terms of the radius of the circle the drop makes on the surface – 7, where, 7 = 8 sin &):

;" = − − cos & − - − < B − cos & . =>. ?@A " CDEF? "

;" is work per unit area as follows: Where ;" = "

(17)


(16)

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and at equilibrium:

;" = 0

(18)

When Eq. (16) equals zero, it becomes the modified Young equation considering both line tension and gravitational field when derived from energy minimization with respect to the drop volume. We now proceed to the energy minimization which allows contact angle variation at constant volume ( & = 0). Simplifying eq. (15) (see eq. (S32) to (S37) of supplementary material for detailed calculations), we obtain (expressed in terms of the radius of the circle the drop makes on the surface – 7, where, 7 = 8 sin &): L

?@AK K

; = − − cos & − − I J M . >EF? "DC -

<

√H

K I

(19)

; is work per unit area as follows: Where ; = (20)

"

And at equilibrium,

; = 0 (21)

RESULTS AND DISCUSSION The two equilibria, eq. (16) and eq. (19), are different functions of & . This means, as we show below, that the equilibrium contact angles that result from the two equations are different. Therefore, under a gravitational field, the drop is never in an absolute minimum energy level. The reason for this, is that the minimization of the center of mass w.r.t drop’s volume, and its minimization w.r.t. drop’s contact angle, are different: < I

√H

L

?@AK K

K I

<

B

J>EF? "DCM ≠ =>. ?@A " CDEF? " − cos &

(22)

Therefore, the drop will spread to accommodate the contact angle that corresponds to one of the minima, but then shrink back as it tries to accommodate the other minimum. As it moves away from one of the minima, it stores surface energy. If, in addition, the drop is given kinetic energy, the inability to settle in one minimum will be accompanied by the drop bouncing. The further away the two minima are, the higher is capacity for storing an elastic response, and the higher is the probability of the drop to bounce. To appreciate the differences between the two minima, we plot in fig. 2 the derivative ;" = " and w.r.t the angle ; = " . of the energy w.r.t. the volume

R

R

Considering a common case where, = 0.072 , = 0.04 , S


S

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R S - ;" and ; = 0.03 S, 6 = 9.8 %X , = 2 ∗ 10[\ ]B , = 10[^_ ` ,and 5 = 1000 SI. o o cross the x axis (i.e. the energy functions reach minima) at θ0 = 83.14 and θ0 = 82.79 , respectively.

100

100 82.79

50

50 83.14

0

0 177

179 -50

-50

-100

0

60

120

180

-100

o Contact Angle ( )

;" ) and w.r.t contact angle ( ; ), based on Figure 2. Drop energy derivative w.r.t. volume ( R R R eq.s (16) and (19), for water ( = 0.072 ), assuming = 0.04 , = 0.03 , = 10

[^_

` , 6 = 9.8

S

%X

S

, = 2 ∗ 10 ] and 5 = 1000 [\

B

-

SI

).

S

S

Fig. 2 shows how the drop energy derivative (considering eq. (16) and (19)) changes with respect to changes in drop contact angle. At & = 83.14 and & = 82.79 , the two energy derivatives reach zero and hence the drop energy function reaches minimum value. The different equilibria shown in fig. 2 are rather small but they correspond to just one set of parameters. Fig. 2 represents all the energy derivatives that can exists, mathematically, for a particular liquid-solid system. With the change in the free surface and interfacial energies, the energy derivative functions will also change (as shown in fig. 3 and 4 below). This can be better understood by an analogy of a spring with a weight attached to its free end. When the spring with the weight is released, it oscillates around the equilibrium state, from one energy state to another till it reaches equilibrium. Similarly, a drop bounces from one energy states to another till it reaches a minimum energy state.


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4

100

80 Eq 18 Eq 21 θ (Eq 21) - θ (Eq 18)

60

3

2 40

∆θ (o)

o

Equillibrium Contact Angle ( )

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1 20

0

30

60

0 120

90

γLV (N/m) Figure 3. The of equilibrium contact angles obtained from eq. (18) and (21) as a function of R R S γLV when = 0.04 , = 0.03 , = 10[^_ `, 6 = 9.8 X , = 2 ∗ 10[\ ]B and 5 = 1000

-

. SI

S

S

%

Fig. 3, shows how the contact angle at equilibrium changes with the change in , for ;" = 0 and ; = 0 for a given system. In fig. 3, we changed only the liquid’s surface tension, but often when one changes the liquid’s surface tension, the other interfacial tensions also change. For example, adding a surfactant to the liquid will reduce at least two of the surface tensions, and if the substrate is a liquid or covered with a liquid (such as mica that has a nanometric water layer on it) all three interfacial tensions in the problem can change. To describe such a condition, we define, a , a and a as the dimensionless interfacial surface tensions and ̅%$cde as a factor with surface tension dimensions such that:

− − cos & = ̅%$cde a − a − a cos &

(23)

We rewrite eqs. (18) and (21) using the new definition in eq. (23) to obtain equilibrium contact angle. Fig. 4 shows a plot that corresponds to equilibrium contact angles obtained from eq. (18) and (21), by varying ̅%$cde :


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85 Eq 18 Eq 21

θ(Eq 21) - θ(Eq 18)

3

84 2

∆θ (ο)

Equillibrium Contact Angle (o)

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83 1

82

0

5

0 15

10

(N/m)

Figure 4. Comparison of equilibrium contact angles obtained from eq. (18) and (21) as a R R R function of ̅%$cde (obtained from eq. (23)) (a = 0.072 , a = 0.04 , a = 0.03 , S

S

-

= 10[^_ `, 6 = 9.8 X , = 2 ∗ 10[\ ]B and 5 = 1000 I ). % S

S

S

Figure 4 shows that smaller the surface tension, the higher is the difference between the two equilibrium contact angles. If this is indeed the reason for bouncing of drops in hydrophilic surfaces, then we expect higher bouncing probability for lower surface tensions. Preliminary experiments that we did in our lab, shows exactly this trend.

An intuitive example for a system with no single equilibrium One can ask for an intuitive example for a system that lacks a single minimum in the energy. Such an example is the case of a hypothetical moon that orbits around two planets making the shape “∞” as shown in fig. 5. As it is around one of the planets it aims to fall into that planet, but then it enters the region between the planets where a mountain at the other planet captures it to the other planet’s gravitational field. Each of the two planets have one mountain, and they alternate capturing the moon to their gravitational field.


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Figure 5. A hypothetical schematic of a moon that orbits around two planets: planet L and planet R. In its current position, the moon aims to fall into the planet R since it is trapped in the planet R’s gravitational field. However, when it will reach the region between the two planets, a mountain in the planet L will cause a higher attraction towards the planet L and distort the moon’s orbit towards the gravitational field of the planet L. The moon will remain trapped in the gravitational field of the planet L for only one cycle because planet R also has a mountain that will be closest to the moon, when the moon reaches the region between the two planets in the next round. When this happens, the moon will again be trapped in the gravitational field of the planet R from which it will reach back to the currently drawn configuration. For this example, the two planets are kept at the same distance from each other. Fig. 5 represents an analogy to better understand the nature of a bouncing drop. The analogy is a hypothetical schematic of two planets and a moon system, wherein the moon is trapped in the gravitational field of two close planets and orbits them in a “∞” orbit path. This “∞” orbit path of the moon around the two planets can persists as long as no perturbations exist. However, any slight deviation in the gravitational pull from one planet can cause the moon to collapse into the other planet. Similarly, in an ideal situation, a drop can oscillate from one energy minimum to another in the absence of external disturbances. This lack of convergence to a single minimum could result in oscillation between the two minima, but for some reason, possibly due to addition of kinetic energy, the drop sweeps across the two minima and bounces back to air. Any uneven perturbation in the moon’s orbit will cause the moon to collapse into one of the planets. Similarly, any uneven perturbation will cause the drop to stop bouncing and spread on the surface. This can be characterized in terms of the probability to bounce, or the average number of bounces for a given system. Finally, preliminary experiments that we have done, by dripping water drops with and without surfactant on mica surfaces, show that the bouncing probability is clearly higher for drops with surfactants. An exact evaluation of this probability study requires normalization with respect to the height, but the effect is so clear that we already note it here. This agrees with fig. 3 and 4 which show bigger difference between the two minima as the surface energies go down (as happens when surfactant is added). Another aspect that should be noted is the relation of dynamic contact angle values and the equilibrium contact angles noted here. The two are related because the bouncing mechanisms on hydrophobic and hydrophilic surfaces are different. Yet the exact relation between dynamic and equilibrium contact angles is beyond the scope of this study, which only aims to show that two minima exist and that their difference correlate with the


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probability of drops to bounce. This aspect should be considered for a detailed analysis of the bouncing drops phenomenon.

CONCLUSION In this paper, we show that two, mutually exclusive, energy minima states co-exist for a drop on a surface when considering gravity in the energy minimization. We use this to explain the reason for drops to bounce on flat hydrophilic surfaces that they otherwise would wet, or form a finite contact angle on. Our results suggest that the energetic difference between the two minima increases when all the interfacial tensions in the problem decrease. Indeed, experiments show that the probability for drops to bounce, as well as the number of bounces, is higher for systems in which all surface tensions are lower. In the absence of external disturbances, a drop will continue to bounce from one energy minima to another. However, other forces and perturbation cause the drop to eventually settle in one of the minima. Preliminary experimental results show that reducing the surface tension results in a higher probability for drops to bounce, which is in agreement with our analysis as shown in fig. 3 and 4.

ASSOCIATED CONTENT Supporting Information Derivation of modified Young equation, drop energy minimization, Laplace pressure difference using Helmholtz energy function.

AUTHOR INFORMATION Corresponding Author * Email: [email protected]. Author Contributions The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Notes The authors declare no competing financial interest. ACKNOWLEDGMENT This study was supported by NSF grants CMMI-1405109 and CBET-1428398 and CBET0960229.


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Why Drops Bounce on Smooth Surfaces - ACS Publications

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