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Physica B 389 (2007) 377–379

www.elsevier.com/locate/physb

Drop formation time from a liquid surface

Heetae Kim�

LCD Development Center, LCD Business, Samsung Electronics Co., Ltd., Asan-City, 336-841, Korea

Received 28 July 2006; received in revised form 9 September 2006; accepted 19 September 2006

Abstract

The surface roughness of the free surface of inviscid liquids excited by thermal energy is proportional to the thermal energy and is

inversely proportional to the surface tension of the liquid at a constant frequency. The surface wave of the liquid becomes unstable above

a critical acceleration and the critical acceleration to make drops out of the liquid surface is predicted using the Rayleigh–Taylor

instability theory. The delayed drop ejection from the surface of the liquid since the applied acceleration, called drop formation time, is

inversely proportional to the square root of the difference between the applied acceleration and the critical acceleration.

r 2006 Elsevier B.V. All rights reserved.

PACS: 47.20.Cq; 47.20.Ma; 47.55.Dz

Keywords: Surface instability; Droplet; Drop formation; Critical acceleration

1. Introduction

Recently, the formation of droplets from a liquid surfacewas intensively investigated by driving a piezo transducerunder the surface of the liquid. A dense fog of heliumdroplets 10–100 mm in diameter was generated using anultrasonic transducer immersed in liquid helium and thesize of helium droplets corresponds to the surfacewavelength [1]. The collision dynamics of water andhydrocarbon droplets was experimentally studied [2] andthe dynamics of water and superfluid fog was alsoinvestigated with diffusing-wave spectroscopy techniques[3]. A dispersion relation was found for capillary waves ofarbitrary amplitude [4] and the surface wave patterns weremeasured with various boundary geometries using trans-mission optics and video image processing [5]. A Ray-leigh–Taylor instability explained the instability of bubbles[6], and the critical acceleration for drop ejection wasderived using the Rayleigh–Taylor instability theory [7]and was measured experimentally for water dropletejection [8].

e front matter r 2006 Elsevier B.V. All rights reserved.

ysb.2006.09.016

275 8245.

ss: heetae7.kim@samsung.com.

In this paper, we show that the surface roughness of aliquid caused by thermal energy at a fixed frequency isproportional to kBT and is inversely proportional to thesurface tension of the liquid. The instability theory ofRayleigh–Taylor is used to predict the critical accelera-tion to generate droplets out of the liquid surface.The drop formation time is calculated in terms of theacceleration and the thermal surface roughness, and isinversely proportional to the square root of thedifference between the applied acceleration and the criticalacceleration.

2. Theory

The free surface of a liquid is a superposition of surfacewave modes. The dispersion relation for the surface wave is

o2 ¼srL

� �k3, (1)

where o is the capillary frequency, k the wave number, sthe surface tension, and rL the density of the liquid. Themeasured surface roughness can be described as

h�2i ¼X

i

hA2i i, (2)

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Fig. 1. Two incompressible fluids with densities rv and rL meet at a

surface. The surface of the liquid is subjected to a vertical oscillation. For

to0, the surface is the plane Z ¼ 0. For times tX0, the liquid surface has a

perturbed shape. The simple case, ZS ¼ ZðtÞcos kx, is illustrated in the

figure.

H. Kim / Physica B 389 (2007) 377–379378

where Ai is the amplitude of the ith mode. From theequipartition theorem in the classical case the amplitudecan be calculated using

1

2ðsk2

i ÞhA2i iL

2 ¼1

2kBT , (3)

where L2 is the surface area.When Bose–Einstein statistics is employed, kBT is

replaced by

_oe_o=kBT � 1

. (4)

Substituting integration for the discreet counting of modes,Eq. (2) can be rewritten as

h�2i ¼

ZZhA2ðkÞi

L2

ð2pÞ2d2k. (5)

With the mode amplitudes calculated using Eq. (3), theexpected surface roughness is expressed as [9]

h�2i ¼

Zjkj

kdk

2p_o

e_o=kBT � 1

� �1

sk2

� �, (6)

where o ¼ o(k) is given by Eq. (1). From Eq. (6) thesurface roughness excited by thermal energy at a fixedfrequency becomes

h�2i ¼kBT

4ps, (7)

except for extremely high frequency, o�kBT=_. Thethermal surface roughness which works for quantum fluidsand classical fluids is proportional to kBT and is inverselyproportional to the surface tension of the liquid.

The surface of the liquid subjected to a verticaloscillation is flat as long as the amplitude of theacceleration remains below a critical threshold accelera-tion. Above the threshold acceleration a surface instabilitysets in, called Faraday instability, and standing wavepatterns appear on the surface. The liquid responds at halfthe frequency of the drive, which is the trademark of theparametric instability.

A second type of instability, the Rayleigh–Taylorinstability, sets in, as the drive level is increased. This isthe instability of an interface between two fluids ofdifferent densities, and occurs when the light fluid pushesthe heavy fluid. As the acceleration increases above acritical acceleration, droplets are ejected from the liquidsurface.

Fig. 1 shows two infinitely extended inviscid fluids thatmeet at a plane interface. The fluids are subjected to aconstant acceleration in a direction perpendicular to theinterface. We write the total acceleration as ~G ¼ ð~a�~gÞ ¼ðaþ gÞ ¼ Gz with g40 and a unit vector z normal to theinterface, pointing into rv. Here ~g ¼ �gz indicates thegravitational acceleration and ~a ¼ az represents a uniformexternal acceleration applied to the system as a whole. IfGo0, then the effective acceleration acts vertically down-ward, the light fluid accelerates the heavy fluid, and the

interface is unstable according to the Rayleigh–Taylortheory.We suppose that the fluid is initially at rest, but that the

interface is perturbed so as to have the form

ZS ¼ ZðtÞ cos kx, (8)

where Z(t) is the time-dependent surface amplitude. Theamplitude of the perturbation is determined by [6]

€ZðtÞ ¼ a2ðkÞZðtÞ, (9)

where

a2ðkÞ ¼ �GrL � rvrL þ rv

� �k �

srL þ rv

� �k3, (10)

with rL being the density of the liquid, rv the density of thevapor, and s the surface tension of the liquid. The interfaceis unstable for a2(k)40. Using Eqs. (1) and (10) we candetermine the critical acceleration [7],

ac ¼rL

rL � rv

� �srL

� �1=3

w4=3. (11)

Because the surface of the liquid becomes unstable whenthe light fluid, the vapor, pushes the heavy fluid, the liquid,the acceleration should be in the negative z direction tomake the surface unstable, thus resulting in the ejection ofa drop from the surface. The drops come out of the liquidsurface when the acceleration is higher than the criticalacceleration. The size of droplets coming out of the liquidsurface corresponds to the surface wavelength and thevelocity of the droplets increases linearly with the velocityof the surface wave [1].In Eq. (11) we can neglect the density of the vapor

compared to the density of the liquid, and using thedispersion relation of Eq. (1), the parameter, a, character-izing the surface acceleration is

a ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiG

rLs

� �1=3o2=3 � o2

r. (12)

ARTICLE IN PRESSH. Kim / Physica B 389 (2007) 377–379 379

From Eqs. (11) and (12), a can be expressed as

a ¼offiffiffiffiffiacp

ffiffiffiffiffiffiffiffiffiffiffiffiffia� acp

, (13)

where a is the applied acceleration and ac is the criticalacceleration. The surface amplitude is then [6]

ZðtÞ ¼ Zð0Þ coshðatÞ, (14)

where Z(0) is the initial surface amplitude and Z(t) is thesurface amplitude at time t. The initial surface amplitu-de,Z(0), corresponds to the thermal surface roughness forthe free surface of the liquid and the surface amplitude attime t, Z(t), corresponds to the increased amplitude of thesurface wave caused by the applied acceleration.

The drop formation time means how long it takes for thefirst drop to appear on the surface of the liquid right afteraccelerating the surface. From Eq. (14) the drop formationtime is

T formation ¼1

aln

ZðtcÞZð0Þþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZðtcÞZð0Þ

� �2

� 1

s24

35, (15)

where a is given by Eq. (13), and Z(tc) ¼ emax ¼ 0.73l sincethe highest amplitude of the capillary wave is 0.73l [4].Since ZðtcÞbZð0Þ, the drop formation time is simplified to

T formation ¼

ffiffiffiffiffiacp

offiffiffiffiffiffiffiffiffiffiffiffiffia� acp ln

2ZðtcÞZð0Þ

� �. (16)

The initial surface amplitude in Eq. (16) can be replacedby the thermal surface roughness, Eq. (7), and Z(tc) can bereplaced by l. Thus, the drop formation time is expressedas

T formation ¼ln 16psl2

kBT

h i2

ffiffiffiffiffiacpffiffiffiffiffiffiffiffiffiffiffiffiffia� acp

� �1

o

� �. (17)

The drop formation time, which is the time differencebetween driving the surface and the first drop ejection, isinversely proportional to the square root of the differencebetween the applied acceleration and the critical accelera-tion, and is also inversely proportional to the capillaryfrequency as we expect. The drops do not come out of thesurface if the applied acceleration is below the critical

acceleration for drop ejection. As the applied accelerationincreases above the critical acceleration, the drop forma-tion time decreases inversely as the square root of thedifference between the applied acceleration and the criticalacceleration. The drop formation time can be applied toinviscid fluids such as water and liquid helium.

3. Conclusions

We have showed the drop formation time from thesurface of a liquid excited by the applied acceleration. Thethermal surface roughness of the free surface of inviscidliquids is proportional to kBT and inversely proportional tothe surface tension of the liquid for a constant frequency.The critical acceleration for drop ejection in the inviscidfluid is predicted using Rayleigh–Taylor instabilitytheory and a drop comes out of the liquid when the heightof the surface wave is above 0.73l. The drop formationtime is inversely proportional to the square root of thedifference between the applied acceleration and the criticalacceleration.

Acknowledgment

The author would like to thank Gary A. Williams forhelpful discussions.

References

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