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ARTICLE IN PRESSModel

OLSUA-16326; No. of Pages 14

Colloids and Surfaces A: Physicochem. Eng. Aspects xxx (2010) xxx–xxx

Contents lists available at ScienceDirect

Colloids and Surfaces A: Physicochemical andEngineering Aspects

journa l homepage: www.e lsev ier .com/ locate /co lsur fa

n hydrothermal waves observed during evaporation of sessile droplets

. Sefianea, A. Steinchenb,∗, R. Moffata

School of Engineering, University of Edinburgh, Kings Buildings, Edinburgh, EH9 3JL, United KingdomAix, Marseille Université, Université Paul Cézanne, UMR, CNRS 6263 ISM2, Centre St., Jérôme, BP 451, Marseille 13397 Cedex 20, France

r t i c l e i n f o

rticle history:eceived 24 October 2009eceived in revised form 8 February 2010ccepted 10 February 2010vailable online xxx

eywords:rops evaporationydrothermal wavesarangoni instability

attern formation

a b s t r a c t

Pattern formation driven by the spontaneous evaporation of sessile drops of methanol, ethanol, and FC-72 using infrared thermography have been recently observed and, interpreted in terms of hydrothermalwaves. New results are presented showing the influence of the temperature and the thermal conductiv-ity of the substrate on the number of waves observed at the free surface of evaporating pinned droplets.Another correlation shows the influence of the liquid volatility on the number of waves. Additionalexperimental data are presented in this paper that corroborates the original findings as well as the inter-pretation given of the phenomenon. The experimental data seem to reveal the presence of travellingwaves that could be the result of a surface oscillatory instability. To rationalize these results, from thetheoretical perspective, a three-dimensional stability analysis in spherical coordinates is required thatdefines new scaling factors and dimensionless numbers in order to take account of the thermal diffu-

pherical capD stability analysis

sivity in the substrate as well as the evaporation rate. A formulation of the problem described above isattempted including the appropriate scaling and relevant dimensionless numbers. The Marangoni num-ber deduced from the scaling of the problem is found to be function of the Evaporation number. Thislatter is in turn function of position, time as well as the nature of system studied. A parametric analysisallowed the deduction of analysis of the form of the solutions giving rise to the experimentally observedwaves. This theoretical analysis represents a first step toward a further more complete description of

s.
experimental observation
. Introduction

Fluid motion driven by temperature gradients imposed alongfree liquid–gas interface is a well documented phenomenon

termed thermocapillary convection). Smith and Davis [1], studiedhe stability of an infinite fluid layer with a free surface subjected totemperature gradient along the interface. This pioneering work

evealed the oscillatory nature of the instability when in super-ritical conditions. The 3D, time dependent linear stability analysisf Smith and Davis [1] predicted the occurrence of what is calledydrothermal waves (HTWs). These regular thermal fluctuationsre found to depend on the depth of the fluid and also on the tem-erature gradient imposed. The waves are found to propagate fromhe cold side to the hot one. The waves are found to have differentnderlying forms, with stationary corotative rolls appearing for lowrandtl numbers, and spiral rotating waves for higher Prandtl num-

Please cite this article in press as: K. Sefiane, et al., On hydrothermal wavPhysicochem. Eng. Aspects (2010), doi:10.1016/j.colsurfa.2010.02.015

ers. The propagation angle of the spiral rotating waves dependsn the Prandtl number value. It is worth noting that in the analysisf Smith and Davis [1] gravity was neglected. Hydrothermal wavesave been subsequently confirmed by many experimental inves-

∗ Corresponding author. Tel.: +33 491288558.E-mail address: [email protected] (A. Steinchen).

927-7757/$ – see front matter © 2010 Elsevier B.V. All rights reserved.oi:10.1016/j.colsurfa.2010.02.015

© 2010 Elsevier B.V. All rights reserved.

tigations studying shallow rectangular pools of liquids, Riley andNeitzel [2] and Schwabe et al. [3]. Key theoretical investigations ofHTW are summarised in Table 1.

For small Pr numbers HTW propagate parallel to the tem-perature gradient. For large Pr numbers, they travel almostperpendicular to the temperature gradient. HTW have also beeninvestigated in annular geometries, where a temperature gradi-ent is imposed between the centre and the outer wall. Garnier [4]investigated HTW in thin annular pools of silicon oil, a temperaturegradient between a cold centre and a hot outer wall were inves-tigated. Liquid depths ranging from 1.2 to 1.9 mm were studied.The authors reported HTW with spiral–like arms circling the annu-lus. For smaller fluid depth (less than the capillary length, �c) theobserved HTW propagate radially from the centre to the outer edge,see Fig. 1. Many investigations have reported a stability diagram forHTW. Depending on the critical value of the �T and the fluid depth,stability regions could be distinguished, see Fig. 2. Garnier [4] pro-posed a stability diagram where two different types of HTWs wereevidenced. Schwabe et al. [3,6] reported temperature fluctuations

es observed during evaporation of sessile droplets, Colloids Surf. A:

associated with HTWs propagating in an annulus, Fig. 3. The tem-perature oscillation amplitude and frequency is found to dependon the imposed temperature as well as the fluid depth. From Smithand Davis [1] and Schwabe et al. [3] the wavelength was correlatedto the depth of the fluid as follows; � ∝ 2.6 h. It is worth noting

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2 K. Sefiane et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects xxx (2010) xxx–xxx

Nomenclature

Roman capitalsC mass per unit volume of vapour in gasD, Dg mass diffusion coefficient of vapour in gasEv evaporation numberH humidityK evaporation equilibrium constantL scaling lengthMa Marangoni number�P position vector on the free surfaceP pressurePr Prandtl numberR radius of curvature of the spherical capSc Schmidt numberT temperatureTo

iboiling temperature of the liquid

Ym1 Bessel function

Y vapour mass fraction in gasU(r) magnitude of the perturbation of r-component

velocityU velocity scale�V velocity vector on the free surface

Roman lower case lettersh drop free surface height above the substrateh�v latent heat of evaporationk heat conductivityr radial coordinaterd triple line radiusrs solid substrate radiust dimensional timetregr; ttot drop regression time�v fluid velocity vector in mobile reference framex, y, z Cartesian coordinates

Superscripts* reference state′ perturbed state◦ initial conditionˆ unit vector∼ dimensionless quantity

Subscriptsg gas phase� liquid phasen normal components solid phasev vapour component index

Greek letters˛ thermal diffusivityı perturbation symbol� longitudinal coordinate�c contact angle�(r) magnitude of the perturbation of pressure� kinematic viscosity� Azimuthal coordinate matter flux density density� surface tension� perturbation evolution time�s heat diffusion time in solid substrate

(r) magnitude of the perturbation of vapour mass frac-tion

� vorticity� angular velocity of the frame

Special symbols� universal gas constantℵ(r) magnitude of the perturbation of evaporation front

velocity(r) magnitude of the perturbation of temperature

that in the original stability analysis of Smith and Davis [1], as wellas subsequent studies, heat and mass exchange between the freesurface and the atmosphere are neglected.

A recent infrared thermographic investigation has shown spon-taneous hydrothermal waves (HTWs) and patterns in evaporatingpinned droplets [5]. The thermal waves are found to exhibit differ-ent patterns for different volatile liquids. The experimental resultspresented by Sefiane et al. [5] are explained in terms of self-inducedhydrothermal waves.

In the present paper, additional experimental results are given.These latter corroborate the analysis proposed by the authors previ-ously. In addition to this, a theoretical formulation of the problem isattempted. The paper is organised as follows, in a first part infraredexperimental results are presented and discussed. In the secondpart the theoretical formulation of the problem is introduced andqualitatively related to experimental observations.

2. Experiments

The four liquids used in this experiment were de-ionised water,methanol, ethanol, and FC-72. The liquids were selected to providea range of volatilities. In addition, all the liquids are readily avail-able and are used in many wide ranging applications. The boilingpoints of each liquid are 100, 78.3, 64.7, and 56 ◦C, respectively,under standard atmospheric conditions. Ethanol and methanolwere purchased from Fisher Scientific (Loughborough, UK). FC-72is a clear, colourless, fully fluorinated liquid. It is thermally andchemically stable, non-flammable, and leaves essentially no residueupon evaporation. FC-72 was purchased from Sigma Chemicals. De-ionised water was supplied from a high purification system calledthe “Barnstead NANOpure Diamond” system. It supplies water witha resistivity of 18.2 m�/cm.

The experimental setup, in this study, consists of a calibrateddroplet delivering syringe, a horizontal substrate mount, tempera-ture controller to control the substrate temperature, and IR analysisequipment.

The drops are delivered at the same temperature as the substrateand the surrounding air is under controlled humidity and at thesame temperature as the solid substrate.

To investigate the effect of substrate material on the test liq-uids, four different substrates were selected. These were chosenso that they represented a large degree of difference in theirrespective thermal conductivities. The four surfaces were PTFE,Macor (ceramic), titanium, and copper, and their respective ther-mal conductivity values were 0.25, 1.46, 21.9, and 401 W m−1 K−1,respectively. Care was taken to ensure homogeneous wettabilitiesand accurate contact angle measurement; to this end, the sub-


strates were coated with a very thin layer of fluoropolymer Cytop.PTFE is a common polymer which is characterised with a low sur-face energy and thermal conductivity. MACOR is a brand name fora glass ceramic material. White in appearance, the material is com-posed of 55% fluorophlogopite mica and 45% borosilicate glass. It




K. Sefiane et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects xxx (2010) xxx–xxx 3

Table 1Summary of theoretical works on HTWs.

Study/theoretical Effects Dimension Pr Results

Smith and Davis (1983a)a Ma 3D 10−3 to 102 HTWSmith and Davis (1983b)b Ma, S 2D 10−3 to 102 Surface wavesLaure and Roux (1989c Ma, Ra 3D 10−3 to 1 HTWGershuni et al (1992)d Ra, Ma 3D 10−2 to 102 HTW, hydrodynamic mode (Pr < 0.1)Parmentier et a. (1993)e Ma, Ra 3D 10−2 to 10 HTWMercier and Normand (1996)f Ma, Ra 3D 7 HTW, stationary rollsPriede and Gerbeth (1997)g Ma, Ra 2D 10−3 to 103 HTW, co-rotating rolls

a M.K. Smith, S.H. Davis, Journal of Fluid Mechanics 132 (1983a) 119–144.b M.K. Smith, S.H. Davis, Journal of Fluid Mechanics 132 (1983b) 145–162.c P. Laure, B. Roux, Journal of Crystal Growth 97 (1989) 226–234.d y Quarterly 2 (3) (1992) 141–151.

ansfer 36 (9) (1993) 2417–2427.

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G.Z.Gershuni, P. Laure, V.M. Myznikov, B. Roux, E.M. Zhukhovitsky, Microgravite M. Parmentier, V.C. Regnier, G. Lebon, International Journal of Heat and Mass Trf J. Mercier, C. Normand, Physics of Fluids 8 (6) (1996) 1433–1445.g J. Priede, G. Gerbeth, Physical Review E 56 (4) (1997) 4187–4199.

as a thermal conductivity that is similar to that of glass. Titaniumnd Copper are common metals that have many wide ranging uses.he thermal conductivity of Copper is higher than that of Titaniumhich is useful in these experiments for giving a comparison of two

onductive materials.The IR camera used in the present investigation is a FLIR Ther-

aCAM SC3000 that has a thermal sensitivity of 20 mK at 30 ◦C, anccuracy of 1% or 1 K of full scale for temperatures up to 150 ◦Cnd 2% or 2 K of full scale for temperatures above 150 ◦C. Theystem provides for automatic atmospheric transmission correc-ion of temperature based on the input distance from the object,tmospheric temperature and relative humidity. The field of viewt minimum focus distance (26 mm) is 10 mm × 7.5 mm and thenstantaneous Field Of View (IFOV) is 1.1 mrad. The system cancquire images in real time or at high speed (up to 750 Hz) withreduction of the picture size so that each frame contains more

han one image. The images acquired are transferred to a dedicatedC with a special built in ThermaCAM research software (by FLIRystem). The spatial resolution of the system depends essentiallyn the IR camera spectral range (8–9 �m for the camera used), theFOV of the camera and the microscope. To obtain images of therop behaviour, the Infrared camera was mounted directly abovehe substrate, facing vertically downwards onto the evaporatingrop. Drops were deposited onto the substrate and then immedi-tely placed under the camera. The investigated drops have contactines that remain pinned to the substrate for the majority of therop lifetime. Hence, basal radius remained essentially constant inime, whilst both the height and drop contact angle decreased inime.

Initial tests were carried out using methanol on titanium sub-trates held at various temperatures. The evaporation of smallrops (controlled equal initial volume for each sample) (triple lineiameter � = 3mm ± 0.1 mm) was recorded using the Thermacamoftware.

Initial tests for the case of water show that the thermal activitys very weak, with the temperature remaining approximately uni-orm spatially. In contrast, the results obtained for the other liquidsested were more interesting. Methanol and ethanol appeared tohow distinct thermal fluctuations that were observed to vary tem-orally. These spoke like wave trains were found to travel aroundhe drop periphery. The number of waves observed and the respec-ive travelling velocities of the waves were found vary dependingn the substrate that was used (Fig. 4(a)). The waves appear toove in a direction that is parallel to the droplet periphery, rotat-

Please cite this article in press as: K. Sefiane, et al., On hydrothermal waves observed during evaporation of sessile droplets, Colloids Surf. A:Physicochem. Eng. Aspects (2010), doi:10.1016/j.colsurfa.2010.02.015

ng around the droplet. It is interesting to notice the similaritiesetween these images and those obtained in thin layer annularhadowgraphy experiments discussed earlier (Fig. 1(a)) [4].

As the drop evaporated, the number of waves that werebserved decreased, whilst the temperature fluctuations for each

Fig. 1. Hydrothermal waves in an annulus as studied by N. Garnier et al. [4], (a)experimental setup and. various observed patterns depending on experimental con-ditions, (b) sketch of spiral pattern, observed of a temperature gradient �T = 14.25 Kand a pool depth of 1.9 mm, (c) concentric pattern, observed of a temperature gra-dient �T = 5 K and a pool depth of h = 1.9 mm.





Fig. 2. (a) Diagram depicting the influence of fluid depth (h) and temperature gra-do[

wtots

Fig. 3. IR image of hydrothermal waves in an annulus, [6] in microgravity environ-

droplet evaporation, down to 20 at the latter stage of its lifetime.

Fd

ient �T between the centre and the edge of the annulus on the thermal instabilitybserved. (b) Phase diagram with HTW behaviour depicted in regions by Garnier4].

ave increased. It can be noticed that the temperature fluctua-


ions are greater at a later stage in the evaporation; the frequencyf the waves also decreases, indicating less waves. Ethanol washen tested in the same way Fig. 5, and was found to also displayimilar wave like phenomenon. The more volatile methanol drops

ig. 4. (a) Evaporating methanol drops, thermal waves numbered across the drop circumrop on a titanium substrate.

ment. This demonstrates the existence of hydrothermal waves in an annulus usingIR thermography. The figure represents averaged infrared pictures of the tempera-ture distribution at the free surface for a fluid depth h = 8 mm, �T = 5 K, the numberof waves is found to be between 10 and 11.

were found to display much higher numbers of waves than ethanoldroplets, when tested using the same procedure, Fig. 6(a). For a tita-nium substrate, held at 26 ◦C, the number of waves visible in themethanol droplet ranged from around 40 at an early stage in the


In comparison, ethanol displayed a wave range between 14 and 7.These results are displayed in Fig. 6(a), with error bars indicatingthe accuracy limitations in the wave counting due to the imageresolution and the small scale of the fluctuations. Also included is

ference. (b) Evolution of the patterns observed during the evaporation of a FC-72





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ig. 5. Ethanol drops with the same initial pinning radius, at the same initial tembserved at specific intervals during evaporation.

graph showing the change in drop height as a function of time.his is an important factor in this study as HTWs have been foundo depend on the liquid depth. In the case of evaporating droplets,eight is changing constantly and this, in turn, may alter the under-

ying observations. The height was recorded using contact anglenalysis equipment (goniometer).

The decrease in the number of waves appears to follow a lin-ar trend. The effect of substrate temperature was also tested forethanol, with more waves visible at an increased temperature.

ig. 6(c) shows the effect of maintaining the substrate at a tempera-ure of 26 ◦C compared with a temperature of 32 ◦C. It can be clearlyeen that there are more waves appearing in the drop at higherubstrate temperatures. The effect of the base substrate was alsoxamined, with titanium replaced by a ceramic substrate (MACOR).he number of waves was found to be dramatically decreased whenhe relatively non-conductive ceramic substrate was used. Fig. 6(b)hows a comparison of methanol evaporation at 26 ◦C on titaniumnd on the MACOR substrate.

The similarities between the waves in the thin layers in annulitudies and those from our results are very interesting. It is


mportant to note the differences between the two systems, weave studied the evaporatively driven thermal fluctuations fromdroplet surface where as the waves in the two-dimensional thin

ayer studies were induced by thermal gradients. In our work, ther-al waves were observed at ambient temperature driven purely

ure on four substrates are depicted with images showing the thermal behaviour

by the evaporation of the fluid. These waves are visible withoutthe need to implement temperature gradients to induce them. Thephase change as being responsible for the observed behaviour is aninteresting development. It is clear from the recorded images thatas drops evaporate; the behaviour of the waves evolves. In the caseof ethanol, a largely chaotic instability is observed at late stages inthe image lifetimes. This may correspond to a change in the phasefor which the instability occurs. In the case of 2D thin films, the liq-uid depth is an important variable which dictates whether HTWsare observed or not. In the present work, the depth (drop height)is a dynamic quantity which is decreasing in time. In addition it isspace dependent, because of the profile of the drop. A short wave-length (high wavenumber) is known to correspond to a higher heattransport, this is in good agreement with the presented data, whichshow that most waves are observed at the beginning of the dropevaporation process.

In the case of FC-72 a rather different pattern was observed.Cells seem to emerge from the apex region of the drop and thendrift toward the edge. The size of these cells is found to be smallerfor those closer to the edge of the drop, Fig. 4(b). The pattern for-


mation observed in Fig. 4(b) was analysed further by examiningthe dependence of the size of the cells on the local height of thedrop; this dependence is plotted in Fig. 6(d). It is clearly seen thatthe size of the cells increases, approximately linearly, with dropheight; this is in agreement with the previous work of Mancini and






Fig. 6. Graph of number of waves vs. dimensionless time, a linear decrease in the numbeof temperature can be clearly observed, (b) the effect of substrate material can be seen, (local drop height for FC-72 drops evaporating on substrates of varying thermal conductiv

Fig. 7. Moving spherical cone frame for pinned sessile drop geometry; the curva-ture radius grows and the contact angle decreases while the triple line perimeterremains constant. The dashed red lines on (b) limit the angle under which the movingobserver sees the surface of the whole solid substrate.

r of waves can be seen, as well as the difference in waves for each liquid. (a) Effectc) the effect of substrate temperature, and (d) variation of the size of the cells withity.

Maza [7] who studied pattern formation in evaporating layers in theabsence of heating. The cell dimension, however, exhibits a ratherweak dependence on the thermal properties of the substrate.

3. Theoretical model

The purpose of our theoretical analysis is to track the thresholdof the onset of these HTWs by means of a linear hydrodynamic sta-bility analysis. In the present approach, we focus on droplets underatmospheric pressure. Droplets are millimetre sized and hencehave spherical cap geometry as a result of the negligible effect ofgravity. We restrict this analysis to pinned drops, i.e. with a con-stant contact line radius rd. Evaporation takes place as a result of thenon-saturation of the vapour in the semi-open container. At the freesurface, the vapour concentration is assumed to be the saturationconcentration, in the vapour phase a steady diffusive regime sets infrom the saturated layer to the unsaturated region. No natural con-vection is expected for substances heavier than air (methanol andethanol for example). In the reference state, the two phases are ata mechanical equilibrium and isothermal. The perturbation equa-tions have to be written in 3D as azimuthally motion is expected toset in. The key parameters of the problem are the size and volatilityof the drops, the thermal diffusivities of the three phases in con-tact (substrate, liquid and vapour) as well as the vapour diffusioncoefficient and the liquid kinematic viscosity. A very careful scalingprocess allows a pertinent description the specific problem investi-gated. The evaporation process takes place at the free liquid–vapoursurface but due to the relative distance of each surface elementfrom the substrate, the transfer of heat through the liquid dropbecomes a function of the position on the surface. Although themostly accepted models [8,9] assume, by an electrostatic analogy,


that the highest evaporation should occur at the edges, the regionof stronger evaporation is not the region where the temperaturefalls down the most rapidly. The size and thermal conductivity ofthe substrate seem to determine the direction of the Marangoniflow induced by the local cooling of the surface [10].


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K. Sefiane et al. / Colloids and Surfaces A:

.1. Reference state

The drop is set down on the substrate in thermal equilibriumith both the substrate and the ambient air. The driving force for

vaporation is the non-saturation of the ambient air. No gradientf temperature exists in the volumes and on the surfaces. However,vaporation starts due to the non-saturation of the surrounding gas.his endothermic process creates locally a sink of enthalpy that fol-ows the local distribution of the matter flux on the liquid–vapourree surface. In the reference state, a layer of saturated vapour sur-ounds the drop and a steady diffusion sets in within the vapourhase. In this steady state, no convection is expected in both liq-id and vapour phase, however, as the drop volume decreases withime, there is a moving boundary condition for the mass transport.n a reference frame moving with the velocity �V of the curvatureentre of the evaporation front, the governing steady state equationor the diffusion of the vapour reads

� . �∇C + D∇2C = 0 (1)

here D is the diffusion coefficient of the vapour.

.2. Geometry of the system

The choice of the geometry of the moving front here is ambigu-us. For planar fronts, the frame may be located anywhere on thelanar surface and the Euclidian coordinates x, y, and z are, respec-ively, the tangential and normal coordinates. The velocity of thelane is uniform on the plane. In the present problem, the surface isurved and the curvature varies in time. Due to the axial symmetry,2D description in cylindrical coordinates is valid for the reference

tate with the radius of curvature and the contact angle �c functionf time. The spherical cap is pinned on a horizontal substrate andhe velocity of the free surface with regard to the substrate varieslong the surface. At the triple line, the velocity vanishes as longs the cap remains pined and the radius of the triple line remainsonstant. If no deviation to the axial symmetry occurs in the refer-nce state, the frame of reference can be arbitrary placed anywheren the triple line. In a 2D description it could be easier to adopt aotating frame whose angular velocity ˝ = d�c/dt characterizes theelocity of the moving free surface. Such frame is a non-Galileanrame and thus the fluids are rotating relatively to the frame. The

omentum balances have thus to be written with a rotational term.he rotational velocity of the frame is a vector oriented along thezimuthal coordinate. The boundary of the spherical cap, as showny Popov [11], is mapped in toroidal coordinates. In order to solvehe Laplace equation in this coordinates system, the heavy use ofhe special Legendre functions in needed [12]. These coordinatesave been also used in 2D and 3D by Massoud and Felske [13] whoave obtained an exact analytical solution for the stream function

nside the drop.Another way to account for the motion of the reference surface

s to locate a Cartesian frame at the centre of the solid substrate, onhe symmetry axis of the drop. The drop apex is then at a distanceapex = h(t) from the frame and the centre of curvature at a distancecentre = h(t) − R(t).

For a pinned spherical cap droplet, the local normal componentf the evaporation front velocity is sometimes assumed to be uni-orm from the apex to the triple line and equal to the time variationf the radius of curvature; this hypothesis of quasi flat interface isalid if (R/rd) � 1; sin �c ≤ (1/100)


n = R = 12

d

dt

[r2d

h+ h

]= 1

2

[1 − r2

d

h2

]dh

dt(2)

here rd is the constant radius of the triple line.

PRESSochem. Eng. Aspects xxx (2010) xxx–xxx 7

This assumption of a uniform normal surface velocity has fordirect consequence that the normal component of flux of matter isuniform on the cap in opposition with the generally adopted Dee-gan [8] electrostatic analogy. Indeed, the normal mass flux n isgiven by

˚n = Vn(l − v) (3)

However, the circulation of the vector on the surface is the scalarproduct of the flux vector by the unit normal vector to the surfacethat changes its orientation with time. The velocity vector �V on thesurface is indeed the time derivative of the position vector �P of eachpoint on the surface with spherical coordinates (R, �, �) in a movingreference frame fixed at the centre of curvature of the sphericalcap. In this non-Galilean frame, the evaporation front limits a solidangle of radius R(t) and of aperture �c(t) whose angular velocity ˝ =d�c/dt equals the rotational velocity at the triple line. Because of theaxial symmetry of the drop, the unit vector does not rotate at theapex of the sessile drop. Everywhere else on the spherical cap, thenormal unit vector changes its orientation during the evaporationof the pinned drop so that the velocity vector �V of each surfaceelement reads

�V = �P = Rr + R ˙r + R�� + R� ˙� + R� sin �� + R� sin � ˙�[R − R��

− R�� sin2�]r +[2R� − R� sin � cos �

]�

+ [2R� sin � + R�� cos �]� (4)

where the unit vectors time derivatives (see for instance Wikipedia)are

˙r = �� sin � + ��; ˙� = −�r + �� cos �; ˙� = −(�r sin �+� cos �)

(5)

With account of the rotation of the unit vector, for a spherical capwith axial symmetry (� = 0), the normal component of the surfacevelocity reads then

Vn = R − R�� (6)

and the normal component of the local flux of matter on the freesurface is then

˚n = (R − R��)[l − v] (7)

By continuity of the matter flux at the free surface in gas,

˚n = v �∇Y.r (8)

where Y denotes the mass fraction of the vapour in gas phaseAt the apex, �(t) = 0 while at the triple line �(t) = �c(t). The angu-

lar velocity � also vanishes at the apex while it is maximum nearthe triple line (negative value). Because of the negative value of �,the normal matter flux grows from the apex to the triple line, ina similar way as predicted by Deegan et al. who used the Lebedevanalogy of the concentration field in the vapour with the electricfield distribution above the upper surface of a biconvex lens. Noexperimental evidence, however, exists on the local distribution ofthe evaporation flux around the drop neither on the concentrationfield in vapour. Due to the rotational behaviour of the contact angle,by continuity of the angular momentum, vortex motions have todevelop in both neighbouring phases. These rotational motions of


course have to be taken into account for describing the concentra-tion field in the vapour phase. Two dimensionless numbers accountfor these rotational motion: the Rossby and the Eckman numbers.

The free surface of the drop is on top of a spherical cone (seeFig. 7). The aperture of this conical solid angle is 2�c(t). For a pinned


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rop �c(t) is related to the radius of curvature by

rd = R(t) sin �c(t)

�c = arcsinrd

R(t)(9)

or a contact angle smaller than �/2, the radius of curvature ofhe spherical cap is larger than the radius rd of the triple line andhe distance between the equatorial plane of the moving referencerame and the plane of the solid substrate (R − h) ≤ R.

In order to account for the change of shape of the sphericalap during evaporation, we adopt here the moving conical frameith aperture 2�c(t) attached to the centre of curvature of therop. The corresponding time dependent solid angle is equal to� · (1 − cos �c(t)). The symmetry is assumed to be axial aroundhe vertical axis. During the evaporation process, the contact anglehanges with a constant angular velocity ˝ = d�c/dt. The contactngle angular velocity is a pseudo-vector normal to any longitu-inal section of the spherical cap. It corresponds to the rotationalelocity of the mobile reference frame described in Fig. 7. The rota-ion pseudo-vector is normal to the vertical axis of symmetry andescribes the angular velocity of the solid angle of the movingbserver.

.3. Hydrodynamic equations

In this moving rotating frame Chandrasekhar [14] (Chapter 3)nd Greenspan [15] (Chapter 3 and 4), the fundamental equationsf hydrodynamics in liquid and vapour phases read,

∂

∂t+ �∇.[(�v − � × �r)] = 0 (10)

∂�v∂t

+ (�v. �∇)�v = − 1

�∇P − 2 � × �v + � �∇2�v (11)

y taking the curl of Eq. (10) and introducing the vorticity

� = �∇ × �v (12)

e get

∂ �ω∂t

+ (�v. �∇)ω = [( �ω + 2 � ). �∇]�v + � �∇2 �ω (13)

oundary conditions have to be written on both solid–liquid andiquid–vapour interfaces. However, if one restricts the domain ofnvestigation to the cone of aperture 2�c(t), the solid–vapour inter-ace is not included in the system. Therefore a larger cone could beaken as moving frame, this last having for aperture a wider anglehan 2�c(t) in order to include the solid domain uncovered by theiquid (red dashed lines in Fig. 7).

The boundary conditions on the free surface couple the energynd matter balances. They have to be written locally on the spher-cal cap and must account for normal deviation of the shape of theurface.

For a pinned droplet, it is assumed that any deviation of shape athe triple line is forbidden. Moreover, we assume that the axial sym-

etry of the drop is preserved in the deformed state. This allowso select even modes in the Legendre polynomials solution of theurface deformation.

.4. Methodology of the stability analysis

In order to analyse the stability of the system, the partial differ-ntial equations and the boundary conditions have to be written


n dimensionless form. The choice of the scaling is then criticalnd very specific to each problem considered. The stability of fluidpheres and spherical shells undergoing heat flux and the influ-nce of rotation have been developed in the famous Chandrasekharook [14], but the special configuration of pinned drops with phase

PRESSochem. Eng. Aspects xxx (2010) xxx–xxx

change and surface tension gradient is not treated. The capillaryoscillations of a constrained inviscid liquid drop pinned on a circleof contact (Rayleigh oscillations) have been recently computed byBostwick and Steen [16]. In this last problem, the volume of the dropis constant so that there is no matter exchange through the free sur-face of the drop. The centre of mass of the system is oscillating whenthe normal modes of vibration are excited but there is no motion ofthe centre of mass associated to a phase change in contrast with theproblem we are concerned. It is then worth noting that in the prob-lem of the pinned evaporating drop, the wavelength as well as theoscillation frequencies and wave amplitudes of the normal modeswill vary during the evaporation process. Some structural analogiesare to be found with crystal growth (see for example Cristini andLowengrub [17]).

In the rotating frame with characteristic angular velocity ˝ anyperturbation of velocity or of temperature generated locally on thefree surface will be driven in a direction normal to the rotationvector � .

3.5. Scaling

The length scale L in the drop is obviously the radius of thetriple line rd. Aspect ratio numbers rs/rd and rv/rd have to be intro-duced to account for the relative dimensions of the solid substrateand the drop contact radius as well as the gas diffusion layer todrop size. As what concerns the time scale, a very careful analysisof the leading phenomenon has to be conducted. At the onset ofevaporation, in isothermal conditions, the driving force is only thenon-saturation of the surrounding gas. As the surface Knudsen pro-cess is in a quasi equilibrium, it is a fast step and only the transportprocess above the saturation layer should define the leading char-acteristic time. This last is often assumed to be purely diffusive, butif one takes into account the motion of the free surface, advectionin the two neighbouring phases has to be considered with a viscousdiffusion. As the process is energy consuming, the heat transportto the evaporation site contributes to the characteristic time. Thelocal cooling induced by evaporation creates on the surface localtemperature gradients in the perturbed state. Heat fluxes from theneighbouring phases carried by diffusion and advection contributeto reduce this local cooling. Classically in liquids the Prandtl num-ber Pr = �/˛ is >1 this is in favour of a dominant role of advectionin heat transfer process, in contrast with the situation prevailingin gas where Pr < 1. In a vapour phase at atmospheric pressure,the Schmidt number Sc = �/D ∼= 1. The matter diffusion and theadvection should then both influence the evaporation kinetics. Therelative heat conductivities of the three contacting phases as wellas the size of the heat-providing phase (here often the solid sub-strate if it is a good heat conductor) also influence the dynamicprocess.

The characteristic time scales associated with the evaporation ofa spherical drop surrounded by a passive gas unsaturated with thevapour of the evaporating component of the droplet are comparedby Ha and Lai [18]. Like many authors concerned with evaporationof drops, either pinned on a substrate or suspended in a gas, theseauthors neglect the regression rate and make a quasi-steady stateapproximation. The fundamental difference between the drop sur-rounded by vapour and the sessile drop lies in the presence of thesubstrate whose physico-chemical properties (thermal conductiv-ity and wettability) have to be taken into account. Our purposehere is indeed to account for the drift of the thermally inducedMarangoni motion under the influence of the change of contact


angle during the evaporation of the sessile drop. Therefore, theangular velocity of the contact angle plays the role of a slow rota-tional motion that carries the smaller wavelength perturbations ofshape of the deformed spherical cap. Similarity can be found withthe geostrophic phenomena although in this case, the axis of rota-


IN PRESSG

C

Physicochem. Eng. Aspects xxx (2010) xxx–xxx 9

tc

tdd

waiwlithgodobs

btttohigtuatmtetc

oe

Tpits

Tc(hi

˚

ARTICLEModel



ion corresponds to the symmetry axis in contrast with the rotationonsidered here whose axis is normal to the symmetry axis.

In the geostrophic flows, the rotation dominates the dynamics ifhe Rossby Ro and Ekman Ek numbers are smaller than 1. These twoimensionless quantities compare the rotation time to the viscousiffusion time and to the advection time.

Ek = �

˝L2= time of rotation (1/˝)

time of viscous diffusion (L2/�)

Ro = U

˝L= time of rotation (1/˝)

time of advection (L/U)

(14)

here U, L and ˝ are, respectively, the scales of velocity, of lengthnd of angular velocity. In our problem, we are seeking for thenfluence of the contact angle velocity on the thermocapillary

aves. These last result from a Marangoni instability induced by theocal cooling during the evaporation process. Two scales of motionnteract, a large toroidal vortex that carries a small wavelength per-urbation at the deformable free surface. A new Marangoni numberas then to be defined including the thermal diffusivities ratios ofas, liquid and solid phases as well as the latent heat of evaporationf the liquid. The stability analysis performed by Ha and Lai for therop surrounded by gas may be adapted for the sessile drop pinnedn a solid substrate but additional boundary conditions have toe satisfied namely the momentum- and thermal-balances at theolid/liquid interface.

The quasi-steady state of reference is in fact a process governedy matter diffusion in the non-saturated gas phase; this time scale

scale = L2/Dg . Initially, the system is isothermal. Due to evapora-ion local cooling, a heat transfer process sets in with a time scalescale = L2/˛�. The time of heat transfer through the liquid is tworders of magnitude larger than the time of matter diffusion in gas,owever, the time of heat diffusion in gas and in the solid substrate

s of the same order of magnitude as the matter diffusion time inas. The time of regression of the drop depends on the volume ofhe drop on its volatility and on the thermal properties of the liq-id as well on the solid and vapour and on the wettability (contactngle). This time tregr ∼= �o

c /˝ is long with regard to the diffusionime in gas and with the thermal diffusion in liquid, in the experi-

ents described above, it is taken as the total time of evaporation,tot. Indeed, the drop remains pinned during more than 90% of thevaporation process and the contact angle at the depinning of theriple line falls to a value of difficult access with our device. It islose to zero.

In the rotating frame of reference attached to the moving centref curvature of the free surface, the non-dimensional momentumquation (11) reads

∂�v∂t

+ (�v. �∇)�v = − �∇P − Ek�∇ × �∇ × �v (15)

he dimensionless numbers introduced, adapted for the concernedroblem result from the choice of the scales. The temperature scale

s chosen from the combined mass and thermal boundary condi-ions in the dynamic state at the free surface (accounting for theurface regression).

h�v� ′.r = −k�

�∇T ′�.r

˚′.r = �V ′[� − v].r = (R′ − R��′)[� − v] = vDg�∇Y ′.r

(16)

he length scale for the gradients is the same = rd when the liquid-overed and total surface of the substrate are equal. To make Eq.


16) dimensionless, the obvious temperature scale is then Tscale =�vvDg/k� and the dimensionless thermal boundary condition (16)s then

� ′.r = −Ev �∇T ′

�.r (17)

Fig. 8. Reduced normal flux from apex to triple line for various initial contact anglescalculated from Eq. (16).

The initial condition of the system as a whole being isothermal, theevaporation flux induces a local cooling on the free surface and localgradients of temperature at the liquid/vapour surface. In a quasi-static approach, Eq. (16) is adopted. However, the vapour and solidphase also contribute in the dynamic process of energy transfer.If the solid substrate is a highly conducting material, as soon as aperturbation of temperature reaches the liquid/substrate interface,the solid surface cools down as a whole in a very short time �s ∝r2s /˛s so that we may assume a uniform temperature on the solid

surface.With the scaling used, the dimensionless flux reads

˜ n =(

1 − v

�

)[R

rd˝− R��

rd˝

]

= − 1

sin �0c

(1 − v

�

)[1

tan �0c

+ ��

˝

](18)

In Fig. 8, the reduced normal flux on a droplet with �oc /˝ ≈

120 s and rd = 0.5 mm is calculated in function of �/�c for var-ious values of the initial contact angle �o

c , by assuming a lineardependence of the angular velocity � with �, i.e. � = ˝�/�o

c .The evaporation number Ev = ˚rd/�˛� in the present problem

is locally defined on the free surface and results from the scaling ofEqs. (7) and (8) with a velocity scale vscale = rd˛�/r2

d= ˛�/rd and a

density ratio � = �/v. The thermal Marangoni number is coupledhere with the evaporation number. Indeed, in the reference state,no thermal gradient exists. Only if the process of evaporation is non-uniform on the surface, a local cooling may induce a local gradient ofsurface tension. The gradient of temperature along the free surfaceinduced by non-uniform evaporation is indeed

∂T

∂�= ∂T

∂˚

∂˚

∂�(19)

and the appropriate Marangoni number is then

Ma = −(d�/dT)(∂T/∂˚)(∂˚/∂�) · rd

�˛��

= −(d�/dT)(∂T/∂˚)(∂ ln ˚/∂�) · Ev

��(20)

For evaporating liquid planar films, Zhang [19] showed the influ-


ence of evaporation on the wavelength of the Marangoni cellsduring the system history. He also defines a generalized Marangoninumber that accounts for the rate of evaporation.

From Eq. (18) and from the matter flux profile, the local val-ues of the Marangoni numbers on the surface of drops of water,





Table 2Timescales.

Transport process Time scale Relative time scale Versus heat diffusion in drop

Water Acetone Methanol Ethanol

In vapourMomentum r2

d/�g ˛�/�g 1.1 × 10−2 6.01 × 10−3 6.93 × 10−3 5.95 × 10−3

Matter diffusion r2d/Dg ˛�/Dg 5.86 × 10−3 8.69 × 10−3 7.09 × 10−3 7.59 × 10−3

Heat diffusion r2d/˛g ˛�/˛g 6.1 × 10−3 3.93 × 10−3 4.53 × 10−3 3.89 × 10−3

In liquidMomentum r2

d/�� ˛�/�� 1.6 × 10−1 2.27 × 10−1 1.45 × 10−1 6.5 × 10−2

Heat diffusion r2d/˛� 1 1 1 1 1

In solidHeat diffusion (r2

d/˛s) · (r2

s /r2d) (˛�/˛s) · (r2

s /r2d) 1.7 × 10−1 1.1 × 10−1 1.3 × 10−1 1.1 × 10−1

Surface regression�c/˝ (�c/˝) · (˛�/r2

d) ca 50 ca 35 ca 40 ca 35

Table 3Values used in Marangoni numbers.

H�v (J/mol) Toi

(K) dT/dY (K/% at 298 K) −d�/dT (N/mK)

−4

miit

wg

n

d

3

taetlidatopsdtp

iunder the imposed pressure, Y ′

s(�) the local saturation mass fractionin gas at the surface in the perturbed state, T ′(�) the local perturbedtemperature at the surface, � is the gas constant, the gas phase isassumed to be perfect.

Water 40700 373Acetone 31300 329Ethanol 39330 351Methanol 34400 338

ethanol, ethanol and acetone are obtained with the values shownn Tables 2 and 3. The various profiles for these substances accord-ng to the numerical values at 298 K depend on the initial values ofhe contact angle, respectively.

Ma-water = 472�o

c

∂ ˜

∂�

Ma-ethanol = 2293�o

c

∂ ˜

∂�

Ma-methanol = 13725�o

c

∂ ˜

∂�

Ma-acetone = 33147�o

c

∂ ˜

∂�

(21)

here we have assumed a thickness of the diffusion layer in theas phase ı = 2 �m.

With the profile of matter flux given in Fig. 8, the Marangoniumbers show a linear profile in �/�c, Fig. 9.

The estimated values of the Marangoni number of a waterroplet with rd = 0.5 mm evaporating in dry air are given in Fig. 9.

.6. Simplifying assumptions

In the reference state, the temperature is initially uniformhroughout the system and on the free surface so that, if wessume the local evaporation equilibrium to be instantaneouslystablished, the Clausius–Clapeyron equation governs the value ofhe saturation concentration in a thin layer of vapour above theiquid–vapour interface. The far field concentration in the gas phases assumed to be uniform and regulated by the “humidity” controlevice. In a previous analysis of self organised Marangoni motiont evaporating fed drops and menisci Steinchen and Sefiane [20],he scaling used was different in order to deal with a steady statef temperature field and motion sustained by the inflow. In the


resent problem, the regression of the free surface is not compen-ated by a liquid inflow and the contact angle of the pinned dropecreases during the evaporation process while for the fed drop,he contact angle remains constant. The liquid input in the fed droproblem measures the integral evaporation output and maintains

8 1.68 × 1080 1.12 × 10−4

20 1.2 × 10−4

100 7.73 × 10−5

invariant the shape of the evaporation front. In the pinned dropevaporation, however, the front is allowed to move and its radiusof curvature to increase. This change of surface geometry appears inthe new defined Marangoni number Eq. (18) in which the evapora-tion number accounts also for the local variation Eq. (16) of the massflux at the surface that induces the thermal gradient through thelocal equilibrium evaporation constant (Clausius–Clapeyron equa-tion) (Fig. 10).

K(�) = Y ′s(�) = exp

[h�v�To

i

(1 − To

i

T ′(�)

)](22)

where To is the boiling temperature of the evaporating liquid i


Fig. 9. Marangoni numbers on a water droplet from apex to triple line for variousinitial contact angles calculated from Eq. (19).


ARTICLE ING Model


K. Sefiane et al. / Colloids and Surfaces A: Physic

F

ma

shtotrrbbbohfshtbfotcoaiucavtrittvbetamoo

so far cannot be solved analytically. Elaborate and tedious numeri-

ig. 10. Vapour saturation fraction–slopes; acetone, methanol, ethanol and water.

As the process is out of equilibrium, the local mass fraction Y ′s(�)

ay be assumed to be the saturation mass fraction multiplied byn accommodation coefficient ˛ smaller than 1.

In a recent numerical model of the evaporation of a pinnedessile drop on a highly conducting solid, Girard and Antoni [10]ave shown the influence of the size of the solid substrate onhe temperature distribution on the free surface of the drop andn the orientation of the induced Marangoni motion. The rela-ive heat conductivities ks/k� of the solid and liquid as well as theelative radii of the substrate and of the triple line rs/rd seem toule the direction of the induced flow. However, the dynamicalehaviour of the system depends not only on conductivities ratiout also on the thermal diffusivities ratio. As already evidencedy Ha and Lai [18], the heat diffusion time in the vapour is tworders of magnitude larger than in the liquid; in our problem, theeat diffusion in the solid substrate is three orders of magnitude

aster than in the liquid but it is only four times faster than in gaso that the influence of vapour and of solid heat transfers mightave comparable importance in the dynamic process, moreover,he surface of the solid substrate is about ten times the size of thease of the drop. In the present approach, we adopt as a simpli-ying assumption that (ks/k�) � 1, (k�/kv) � 1 and (rs/rd) � 1 butn a good heat conductor as Al, the characteristic times defined byhe ratio of the diffusivities are summarized in Table 2. Numeri-al values are given for droplets with attachment radii of 0.5 mmf various liquids evaporated in an unsaturated gas phase underpressure = 1 atm at 298 K on an Al substrate the radius of which

s 10 times that of the attachment radius. According to the val-es of these characteristic times, the steady state approximationsommonly used for heat, momentum and matter transport in gass well as heat transfer in the substrate are justified. The highalue of heat conductivity of the substrate also allows assume thathe temperature remains constant on the whole solid surface. Aecent 2D numerical model by Barash et al. [21] shows that dur-ng the early stage of evaporation a street of vortices arises nearhe surface of the drop and induces a non-monotonic spatial dis-ribution of the temperature over the drop surface. The number ofortices rapidly decreases with time. This behaviour is confirmedy the thermal visualisation obtained by Sefiane et al. [5]. How-ver, due to the limitations of the 2D model of Barash et al. [21],he circulation of the thermal waves around the symmetry axis


s shown in the movie provided by Sefiane et al. [5] cannot beodelled. This last behaviour could be a consequence of the shift

f the vortices by rotational velocity � of the frame (Eq. (13). Inrder to make asymptotic developments at short times and short

PRESSochem. Eng. Aspects xxx (2010) xxx–xxx 11

wavelengths, it could be better to choose as time scale the regres-sion time. The relative time scales versus the regression time arethen the relative time scales versus the conduction time in liquiddivided by the relative regression time in the liquid conduction timescale.

3.7. Perturbation equations

The velocity–temperature and vapour mol-fraction perturbedfields are expressed in terms of spherical harmonics as well as thevelocity of the free surface (evaporation front).

∂�v′

∂t+ ([�v′ + �V ′] �∇)�v′ = − 1

�∇P ′ − 2 � × �v′ + � �∇2�v′

where

�v′ = �v∗ + ı�v ; �V ′ = �V∗ + ı �V ; P ′ = P∗ + ıP

ı�v = U(r)Ym1 (�, �) exp (ω�); ı �V = ℵ(r)Ym

1 (�, �) exp(ω�);

ıP = �(r)Ym1 (�, �) exp(ω�)

(23)

∂T ′

∂t+ (�v′ + �V ′) · �∇T ′ = ∇2T ′ − (�v′ + �V ′) · �∇T∗

whereT ′ = T∗ + ıTıT = (r)Ym

l(�, �) exp(ω�)

(24)

∂Y ′

∂t+ (�v′ + �V ′) · �∇Y ′ = ∇2Y ′ − (�v′ + �V ′) · �∇Y∗

where∂Y ′ = Y∗ + ıYıY = (r)Ym

l(�, �) exp(ω�)

(25)

The momentum boundary conditions on the free surface have toaccount for the normal deformation of the drop. The evaporationfront curvature has to be perturbed in the same way as done byBlinova et al. [22] for a spherical solidification front.

The reference state is a time dependent state with a character-istic time given by �o

c /˝. The perturbations have a time constant(1/ω) � �/˝. The stability analysis has to be performed for asymp-totic conditions at short time and short wavelength. Exchange ofstability or oscillatory instability might be obtained according tothe values of the real or imaginary parts of ω· The analytical res-olution of the eigen-values problem should lead to knowledge ofthe minimum critical value of the Marangoni number and evapo-ration number at which the most probable mode of deformationof the surface should grow. The short wavelength modes grow-ing on the receding spherical cap may be described by polynomialdevelopments. The most classical ones are the Legendre polynomi-als. With various trial functions to model the perturbations of thedrop surface structures similar to the thermographic images areobtained. In Fig. 11, the patterns obtained on sessile drops withvarious contact angle show that the wavelength increases withdecreasing contact angle as obtained by thermography. Accordingto the trial function used, the pattern obtained on the drop mayshow a decreasing wavelength when approaching the triple line.A deformation in the azimuthal direction may also be predictedwith other trial functions. The normal perturbations of the surfacecorrespond to regions where the vaporization front penetrates incolder or warmer domains removed from the equilibrium condi-tions so that condensation or evaporation may proceed locally onthe crests or in the wells. The surface deformation induces micro-convection in the neighbouring fluids. The problem as formulated


cal techniques are necessary to give a full solution to the problem.Instead of going down that route, we chose to adopt a differentapproach, which aims to reveal the nature and form of the per-turbations which could lead to surface waves similar to the ones





ation o

owersi

Tafb

Fig. 11. Surface wavy perturb

bserved experimentally. Using as an initial steady state a dropith a spherical cap shape, a parametric study is undertaken to

valuate the form of the solutions to perturbations which may giveise to waves similar to the ones observe experimentally. In a firsttage (trial function 1) we have introduced a perturbation of thenterface, following the functions given below,

x + ıx = cos ϕ ·(

cos[

�

2− �

]+ 1

64cos 64

[�

2− �

])

y + ıy = sin ϕ ·(

cos[

�

2− �

]+ 1

64cos 64

[�

2− �

])

z + ız =(

sin[

�

2− �

]+ 1

64sin64

[�

2− �

])(26)

he resulting waves are of similar wavelength travelling from thepex to the contact line region. In a second stage the perturbationunctions are altered (trial function 2) according to the format givenelow,

x + ıx = cos ϕ ·(

cos[

�

2− �

]+ 1

64cos 64

[�

2− �

]2)

( [ ] [ ]2)


y + ıy = cos ϕ · cos�

2− � + 1

64cos 64

�

2− �

z + ız =(

sin[

�

2− �

]+ 1

64sin 64

[�

2− �

]2)

(27)

btained with trial function 2.

The resulting waves are still travelling from the apex to the con-tact line, with the wavelength decreasing as we move from theapex to the contact line, Fig. 11. In a last stage, and in additionto the perturbations on the longitudinal angle, a perturbation onthe Azimuthal angle is introduced. The perturbations functions areselected in such a way that the contact line is not deformed (sincewe are dealing with pinned drops). This last form of perturbationfunctions seems to yield waves that are very similar to the onesobserved in the experiments. Circling waves, with a wavelengththat depends on position are obtained when trying the followingfunctions (trial function 3),

x + ıx ={

cos(ϕ −[

�

2− �

]) + 1

6cos 6ϕ · sin(� − �c)

}

·(

cos[

�

2− �

]+ 1

64cos 64

[�

2− �

]2)

y + ıy ={

sin(ϕ −[

�

2− �

]) + 1

6sin 6ϕ · sin(� − �c)

}

·(

cos[

�

2− �

]+ 1

64cos 64

[�

2− �

]2)

( [�

]1

[�

]2)

(28)


z + ız = sin2

− � +64

cos 642

− �

In addition to circling waves on the drop surface whichreplicates the observed experimental trend, the above pertur-bation functions, and in agreement with experiments, indicate





obta

tF

npicftt[

4

ssmaa

bioc

Fig. 12. Azimuthal motion

hat the wave number decreases as the drop evaporates,ig. 12.

The approach adopted above may be qualitative in nature,onetheless it unravels very useful information on the form theerturbations may have in order to give rise to waves sim-

lar to the ones observed experimentally. This approach alsoan guide modellers in selecting the right form of perturbationunctions, when attempting a more complete theoretical resolu-ion of the problem. This approach can also be used to unravelhe influence of the substrates thermal properties as shown in23].

. Conclusions

In this paper the complexity of the evaporative process of ses-ile drops has been investigated by means of IR thermography. Thepontaneously occurring thermal instabilities shown by ethanol,ethanol and FC-72 are clearly distinct from each other, and can

lso be manipulated by altering the evaporation parameters suchs substrate material and substrate temperature.


Hydrothermal waves on evaporating sessile drops are evidencedy infrared thermography. A theoretical method for the rational-

zation of the results is proposed on the basis of a stability analysisf the normal deformation of the drop surface resulting from aombined Marangoni instability carried by the time dependent

[[

ined with trial function 3.

regression motion due to evaporation. Trial functions are used toreproduce the patterns observed experimentally.

Acknowledgements

ESA MAP FASES contract has partly sponsored one of us, theFrench Space agency CNES as well as the CNRS GDR FAM pro-gramme gave a financial support of the present contribution.

The authors acknowledge the support of the British Engineeringand Physical Sciences Research Council (EPSRC), for their financialsupport through DTA grant.

References

[1] M.K. Smith, S.H. Davis, J. Fluid Mech. 132 (1983) 119–144.[2] R.J. Riley, G.P. Neitzel, J. Fluid Mech. 359 (1998) 143–164.[3] D. Schwabe, U. Möller, J. Schneider, A. Scharmann, Phys. Fluids (1992)

2368–2381.[4] Garnier, N. PhD Paris 7 (2000).[5] K. Sefiane, J.R. Moffat, O.K. Matar, R.V. Craster, Appl. Phys. Lett. 93 (2008)

074103.[6] D. Schwabe, A. Zebib, B.C. Sim, J. Fluid Mech. 491 (2003) 239–258.


[7] H. Mancini, D. Maza, Europhys. Lett. 66 (2004) 812–818.[8] G.J. Dunn, S.K. Wilson, B.R. Duffy, S. David, K. Sefiane, Coll. Surf. A: Physicochem.

ng. Aspects 323 (2008) 50–55.[9] H. Hu, R. Larson, Langmuir 21 (2005) 3963.10] F. Girard, M. Antoni, Langmuir 24 (2008) 11342.11] Y.O. Popov, Phys. Rev. E 71 (2005) 36313.


ING

C

1 Physic

[

[[

[

[[[

[

[

ARTICLEModel


4 K. Sefiane et al. / Colloids and Surfaces A:

12] N.N. Lebedev, Special Functions and Their Applications, Prentice-Hall, Engle-wood Cliffs, 1965.

13] H. Massoud, J.D. Felske, Phys. Fluids 21 (2009) 042102.14] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Clarendon


Press, Oxford, 1961.15] H.D. Greenspan, The Theory of Rotating Fluids, Breukelen Press, Brookline,

1990.16] J.B. Bostwick, P.H. Steen, Phys. Fluids 21 (2009) 032108.17] V. Cristini, J. Lowengrub, J. Cryst. Growth 240 (2002) 267.18] V.M. Ha, C.L. Lai, Int. J. Heat Mass Trans. 45 (2002) 5143.

[

[

[

PRESSochem. Eng. Aspects xxx (2010) xxx–xxx

19] N. Zhang, Surface tension-driven convection flow in evaporating liquid lay-ers, in: Surface Tension Driven Flows and Applications, Vol. 37/661, Signpost,Trivandrum, Kerala, India, 2006, p. 147.

20] A. Steinchen, K. Sefiane, J. Non-Equilib. Thermodyn. 30 (2005) 39.


21] L.Y. Barash, T.P. Bigioni, V.M. Vinokur, L.N. Shchur, Phys. Rev. E (Stat. NonlinearSoft Matter Phys.) 79 (2009) 046301.

22] I.V. Blinova, V.V. Gusarov, Y.O. Popov, Z. Angew. Math. Phys. 60 (2009)178.

23] G.J. Dunn, S.K. Wilson, B.R. Duffy, S. David, K. Sefiane, J. Fluid Mech. 623 (2009)329–351.


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