1. Colloids and Surfaces A: Physicochem. Eng. Aspects 365 (2010) 95–108
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Colloids and Surfaces A: Physicochemical and
Engineering Aspects
journal homepage: www.elsevier.com/locate/colsurfa
On hydrothermal waves observed during evaporation of sessile droplets
K. Sefiane a , A. Steinchen b,∗ , R. Moffat a
a
School of Engineering, University of Edinburgh, Kings Buildings, Edinburgh, EH9 3JL, United Kingdom
b
Aix, Marseille Université, Université Paul Cézanne, UMR, CNRS 6263 ISM2, Centre St., Jérôme, BP 451, Marseille 13397 Cedex 20, France
a r t i c l e i n f o a b s t r a c t
Article history: Pattern formation driven by the spontaneous evaporation of sessile drops of methanol, ethanol, and FC-
Received 24 October 2009 72 using infrared thermography have been recently observed and, interpreted in terms of hydrothermal
Received in revised form 8 February 2010 waves. New results are presented showing the influence of the temperature and the thermal conductiv-
Accepted 10 February 2010
ity of the substrate on the number of waves observed at the free surface of evaporating pinned droplets.
Available online 18 February 2010
Another correlation shows the influence of the liquid volatility on the number of waves. Additional
experimental data are presented in this paper that corroborates the original findings as well as the inter-
Keywords:
pretation given of the phenomenon. The experimental data seem to reveal the presence of travelling
Drops evaporation
Hydrothermal waves
waves that could be the result of a surface oscillatory instability. To rationalize these results, from the
Marangoni instability theoretical perspective, a three-dimensional stability analysis in spherical coordinates is required that
Pattern formation defines new scaling factors and dimensionless numbers in order to take account of the thermal diffu-
Spherical cap sivity in the substrate as well as the evaporation rate. A formulation of the problem described above is
3D stability analysis attempted 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. This
latter is in turn function of position, time as well as the nature of system studied. A parametric analysis
allowed the deduction of analysis of the form of the solutions giving rise to the experimentally observed
waves. This theoretical analysis represents a first step toward a further more complete description of
experimental observations.
© 2010 Elsevier B.V. All rights reserved.
1. Introduction tigations studying shallow rectangular pools of liquids, Riley and
Neitzel [2] and Schwabe et al. [3]. Key theoretical investigations of
Fluid motion driven by temperature gradients imposed along HTW are summarised in Table 1.
a free liquid–gas interface is a well documented phenomenon For small Pr numbers HTW propagate parallel to the tem-
(termed thermocapillary convection). Smith and Davis [1], studied perature gradient. For large Pr numbers, they travel almost
the stability of an infinite fluid layer with a free surface subjected to perpendicular to the temperature gradient. HTW have also been
a temperature gradient along the interface. This pioneering work investigated in annular geometries, where a temperature gradi-
revealed the oscillatory nature of the instability when in super- ent is imposed between the centre and the outer wall. Garnier [4]
critical conditions. The 3D, time dependent linear stability analysis investigated HTW in thin annular pools of silicon oil, a temperature
of Smith and Davis [1] predicted the occurrence of what is called gradient between a cold centre and a hot outer wall were inves-
hydrothermal waves (HTWs). These regular thermal fluctuations tigated. Liquid depths ranging from 1.2 to 1.9 mm were studied.
are found to depend on the depth of the fluid and also on the tem- The authors reported HTW with spiral–like arms circling the annu-
perature gradient imposed. The waves are found to propagate from lus. For smaller fluid depth (less than the capillary length, c ) the
the cold side to the hot one. The waves are found to have different observed HTW propagate radially from the centre to the outer edge,
underlying forms, with stationary corotative rolls appearing for low see Fig. 1. Many investigations have reported a stability diagram for
Prandtl numbers, and spiral rotating waves for higher Prandtl num- HTW. Depending on the critical value of the T and the fluid depth,
bers. The propagation angle of the spiral rotating waves depends stability regions could be distinguished, see Fig. 2. Garnier [4] pro-
on the Prandtl number value. It is worth noting that in the analysis posed a stability diagram where two different types of HTWs were
of Smith and Davis [1] gravity was neglected. Hydrothermal waves evidenced. Schwabe et al. [3,6] reported temperature fluctuations
have been subsequently confirmed by many experimental inves- associated with HTWs propagating in an annulus, Fig. 3. The tem-
perature oscillation amplitude and frequency is found to depend
on the imposed temperature as well as the fluid depth. From Smith
∗ Corresponding author. Tel.: +33 491288558. and Davis [1] and Schwabe et al. [3] the wavelength was correlated
E-mail address: annie.sanfeld@univ-cezanne.fr (A. Steinchen). to the depth of the fluid as follows; ∝ 2.6 h. It is worth noting
0927-7757/$ – see front matter © 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.colsurfa.2010.02.015

2. 96 K. Sefiane et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 365 (2010) 95–108
Nomenclature
« (r) magnitude of the perturbation of vapour mass frac-
Roman capitals tion
C mass per unit volume of vapour in gas vorticity
D, Dg mass diffusion coefficient of vapour in gas angular velocity of the frame
Ev evaporation number
H humidity Special symbols
K evaporation equilibrium constant universal gas constant
L scaling length ℵ(r) magnitude of the perturbation of evaporation front
Ma Marangoni number velocity
P position vector on the free surface (r) magnitude of the perturbation of temperature
P pressure
Pr Prandtl number
R radius of curvature of the spherical cap
that in the original stability analysis of Smith and Davis [1], as well
Sc Schmidt number
as subsequent studies, heat and mass exchange between the free
T temperature
surface and the atmosphere are neglected.
Tio boiling temperature of the liquid
m A recent infrared thermographic investigation has shown spon-
Y1 Bessel function
taneous hydrothermal waves (HTWs) and patterns in evaporating
Y vapour mass fraction in gas
pinned droplets [5]. The thermal waves are found to exhibit differ-
U(r) magnitude of the perturbation of r-component
ent patterns for different volatile liquids. The experimental results
velocity
presented by Sefiane et al. [5] are explained in terms of self-induced
U velocity scale
hydrothermal waves.
V velocity vector on the free surface
In the present paper, additional experimental results are given.
These latter corroborate the analysis proposed by the authors previ-
Roman lower case letters
ously. In addition to this, a theoretical formulation of the problem is
h drop free surface height above the substrate
attempted. The paper is organised as follows, in a first part infrared
h v latent heat of evaporation
experimental results are presented and discussed. In the second
k heat conductivity
part the theoretical formulation of the problem is introduced and
r radial coordinate
qualitatively related to experimental observations.
rd triple line radius
rs solid substrate radius
2. Experiments
t dimensional time
tregr ; ttot drop regression time
The four liquids used in this experiment were de-ionised water,
v fluid velocity vector in mobile reference frame
methanol, ethanol, and FC-72. The liquids were selected to provide
x, y, z Cartesian coordinates
a range of volatilities. In addition, all the liquids are readily avail-
Superscripts able and are used in many wide ranging applications. The boiling
* reference state points of each liquid are 100, 78.3, 64.7, and 56 ◦ C, respectively,
perturbed state under standard atmospheric conditions. Ethanol and methanol
◦ initial condition were purchased from Fisher Scientific (Loughborough, UK). FC-72
ˆ unit vector is a clear, colourless, fully fluorinated liquid. It is thermally and
∼ dimensionless quantity chemically stable, non-flammable, and leaves essentially no residue
upon evaporation. FC-72 was purchased from Sigma Chemicals. De-
Subscripts ionised water was supplied from a high purification system called
g gas phase the “Barnstead NANOpure Diamond” system. It supplies water with
liquid phase a resistivity of 18.2 m /cm.
n normal component The experimental setup, in this study, consists of a calibrated
s solid phase droplet delivering syringe, a horizontal substrate mount, tempera-
v vapour component index ture controller to control the substrate temperature, and IR analysis
equipment.
Greek letters The drops are delivered at the same temperature as the substrate
˛ thermal diffusivity and the surrounding air is under controlled humidity and at the
ı perturbation symbol same temperature as the solid substrate.
 longitudinal coordinate To investigate the effect of substrate material on the test liq-
Âc contact angle uids, four different substrates were selected. These were chosen
(r) magnitude of the perturbation of pressure so that they represented a large degree of difference in their
kinematic viscosity respective thermal conductivities. The four surfaces were PTFE,
Azimuthal coordinate Macor (ceramic), titanium, and copper, and their respective ther-
matter flux density mal conductivity values were 0.25, 1.46, 21.9, and 401 W m−1 K−1 ,
density respectively. Care was taken to ensure homogeneous wettabilities
surface tension and accurate contact angle measurement; to this end, the sub-
perturbation evolution time strates were coated with a very thin layer of fluoropolymer Cytop.
s heat diffusion time in solid substrate PTFE is a common polymer which is characterised with a low sur-
face energy and thermal conductivity. MACOR is a brand name for
a glass ceramic material. White in appearance, the material is com-
posed of 55% fluorophlogopite mica and 45% borosilicate glass. It

3. K. Sefiane et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 365 (2010) 95–108 97
Table 1
Summary of theoretical works on HTWs.
Study/theoretical Effects Dimension Pr Results
a −3 2
Smith and Davis (1983a) Ma 3D 10 to 10 HTW
Smith and Davis (1983b)b Ma, S 2D 10−3 to 102 Surface waves
Laure and Roux (1989c Ma, Ra 3D 10−3 to 1 HTW
Gershuni 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 HTW
Mercier and Normand (1996)f Ma, Ra 3D 7 HTW, stationary rolls
Priede 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
G.Z.Gershuni, P. Laure, V.M. Myznikov, B. Roux, E.M. Zhukhovitsky, Microgravity Quarterly 2 (3) (1992) 141–151.
e
M. Parmentier, V.C. Regnier, G. Lebon, International Journal of Heat and Mass Transfer 36 (9) (1993) 2417–2427.
f
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.
has a thermal conductivity that is similar to that of glass. Titanium
and Copper are common metals that have many wide ranging uses.
The thermal conductivity of Copper is higher than that of Titanium
which is useful in these experiments for giving a comparison of two
conductive materials.
The IR camera used in the present investigation is a FLIR Ther-
maCAM SC3000 that has a thermal sensitivity of 20 mK at 30 ◦ C, an
accuracy of 1% or 1 K of full scale for temperatures up to 150 ◦ C
and 2% or 2 K of full scale for temperatures above 150 ◦ C. The
system provides for automatic atmospheric transmission correc-
tion of temperature based on the input distance from the object,
atmospheric temperature and relative humidity. The field of view
at minimum focus distance (26 mm) is 10 mm × 7.5 mm and the
Instantaneous Field Of View (IFOV) is 1.1 mrad. The system can
acquire images in real time or at high speed (up to 750 Hz) with
a reduction of the picture size so that each frame contains more
than one image. The images acquired are transferred to a dedicated
PC with a special built in ThermaCAM research software (by FLIR
System). The spatial resolution of the system depends essentially
on the IR camera spectral range (8–9 m for the camera used), the
IFOV of the camera and the microscope. To obtain images of the
drop behaviour, the Infrared camera was mounted directly above
the substrate, facing vertically downwards onto the evaporating
drop. Drops were deposited onto the substrate and then immedi-
ately placed under the camera. The investigated drops have contact
lines that remain pinned to the substrate for the majority of the
drop lifetime. Hence, basal radius remained essentially constant in
time, whilst both the height and drop contact angle decreased in
time.
Initial tests were carried out using methanol on titanium sub-
strates held at various temperatures. The evaporation of small
drops (controlled equal initial volume for each sample) (triple line
diameter = 3mm ± 0.1 mm) was recorded using the Thermacam
software.
Initial tests for the case of water show that the thermal activity
is very weak, with the temperature remaining approximately uni-
form spatially. In contrast, the results obtained for the other liquids
tested were more interesting. Methanol and ethanol appeared to
show distinct thermal fluctuations that were observed to vary tem-
porally. These spoke like wave trains were found to travel around
the drop periphery. The number of waves observed and the respec-
tive travelling velocities of the waves were found vary depending
on the substrate that was used (Fig. 4(a)). The waves appear to
move in a direction that is parallel to the droplet periphery, rotat-
ing around the droplet. It is interesting to notice the similarities
Fig. 1. Hydrothermal waves in an annulus as studied by N. Garnier et al. [4], (a)
between these images and those obtained in thin layer annular
experimental setup and. various observed patterns depending on experimental con-
shadowgraphy experiments discussed earlier (Fig. 1(a)) [4]. ditions, (b) sketch of spiral pattern, observed of a temperature gradient T = 14.25 K
As the drop evaporated, the number of waves that were and a pool depth of 1.9 mm, (c) concentric pattern, observed of a temperature gra-
observed decreased, whilst the temperature fluctuations for each dient T = 5 K and a pool depth of h = 1.9 mm.

4. 98 K. Sefiane et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 365 (2010) 95–108
Fig. 3. IR image of hydrothermal waves in an annulus, [6] in microgravity environ-
ment. This demonstrates the existence of hydrothermal waves in an annulus using
IR 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 number
of waves is found to be between 10 and 11.
Fig. 2. (a) Diagram depicting the influence of fluid depth (h) and temperature gra-
dient T between the centre and the edge of the annulus on the thermal instability
observed. (b) Phase diagram with HTW behaviour depicted in regions by Garnier were found to display much higher numbers of waves than ethanol
[4]. droplets, when tested using the same procedure, Fig. 6(a). For a tita-
nium substrate, held at 26 ◦ C, the number of waves visible in the
methanol droplet ranged from around 40 at an early stage in the
wave increased. It can be noticed that the temperature fluctua- droplet evaporation, down to 20 at the latter stage of its lifetime.
tions are greater at a later stage in the evaporation; the frequency In comparison, ethanol displayed a wave range between 14 and 7.
of the waves also decreases, indicating less waves. Ethanol was These results are displayed in Fig. 6(a), with error bars indicating
then tested in the same way Fig. 5, and was found to also display the accuracy limitations in the wave counting due to the image
similar wave like phenomenon. The more volatile methanol drops resolution and the small scale of the fluctuations. Also included is
Fig. 4. (a) Evaporating methanol drops, thermal waves numbered across the drop circumference. (b) Evolution of the patterns observed during the evaporation of a FC-72
drop on a titanium substrate.

5. K. Sefiane et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 365 (2010) 95–108 99
Fig. 5. Ethanol drops with the same initial pinning radius, at the same initial temperature on four substrates are depicted with images showing the thermal behaviour
observed at specific intervals during evaporation.
a graph showing the change in drop height as a function of time. by the evaporation of the fluid. These waves are visible without
This is an important factor in this study as HTWs have been found the need to implement temperature gradients to induce them. The
to depend on the liquid depth. In the case of evaporating droplets, phase change as being responsible for the observed behaviour is an
height is changing constantly and this, in turn, may alter the under- interesting development. It is clear from the recorded images that
lying observations. The height was recorded using contact angle as drops evaporate; the behaviour of the waves evolves. In the case
analysis equipment (goniometer). of ethanol, a largely chaotic instability is observed at late stages in
The decrease in the number of waves appears to follow a lin- the image lifetimes. This may correspond to a change in the phase
ear trend. The effect of substrate temperature was also tested for for which the instability occurs. In the case of 2D thin films, the liq-
methanol, with more waves visible at an increased temperature. uid depth is an important variable which dictates whether HTWs
Fig. 6(c) shows the effect of maintaining the substrate at a tempera- are observed or not. In the present work, the depth (drop height)
ture of 26 ◦ C compared with a temperature of 32 ◦ C. It can be clearly is a dynamic quantity which is decreasing in time. In addition it is
seen that there are more waves appearing in the drop at higher space dependent, because of the profile of the drop. A short wave-
substrate temperatures. The effect of the base substrate was also length (high wavenumber) is known to correspond to a higher heat
examined, with titanium replaced by a ceramic substrate (MACOR). transport, this is in good agreement with the presented data, which
The number of waves was found to be dramatically decreased when show that most waves are observed at the beginning of the drop
the relatively non-conductive ceramic substrate was used. Fig. 6(b) evaporation process.
shows a comparison of methanol evaporation at 26 ◦ C on titanium In the case of FC-72 a rather different pattern was observed.
and on the MACOR substrate. Cells seem to emerge from the apex region of the drop and then
The similarities between the waves in the thin layers in annuli drift toward the edge. The size of these cells is found to be smaller
studies and those from our results are very interesting. It is for those closer to the edge of the drop, Fig. 4(b). The pattern for-
important to note the differences between the two systems, we mation observed in Fig. 4(b) was analysed further by examining
have studied the evaporatively driven thermal fluctuations from the dependence of the size of the cells on the local height of the
a droplet surface where as the waves in the two-dimensional thin drop; this dependence is plotted in Fig. 6(d). It is clearly seen that
layer studies were induced by thermal gradients. In our work, ther- the size of the cells increases, approximately linearly, with drop
mal waves were observed at ambient temperature driven purely height; this is in agreement with the previous work of Mancini and

6. 100 K. Sefiane et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 365 (2010) 95–108
Fig. 6. Graph of number of waves vs. dimensionless time, a linear decrease in the number of waves can be seen, as well as the difference in waves for each liquid. (a) Effect
of temperature can be clearly observed, (b) the effect of substrate material can be seen, (c) the effect of substrate temperature, and (d) variation of the size of the cells with
local drop height for FC-72 drops evaporating on substrates of varying thermal conductivity.
Maza [7] who studied pattern formation in evaporating layers in the
absence of heating. The cell dimension, however, exhibits a rather
weak dependence on the thermal properties of the substrate.
3. Theoretical model
The purpose of our theoretical analysis is to track the threshold
of the onset of these HTWs by means of a linear hydrodynamic sta-
bility analysis. In the present approach, we focus on droplets under
atmospheric pressure. Droplets are millimetre sized and hence
have spherical cap geometry as a result of the negligible effect of
gravity. We restrict this analysis to pinned drops, i.e. with a con-
stant contact line radius rd . Evaporation takes place as a result of the
non-saturation of the vapour in the semi-open container. At the free
surface, the vapour concentration is assumed to be the saturation
concentration, in the vapour phase a steady diffusive regime sets in
from the saturated layer to the unsaturated region. No natural con-
vection is expected for substances heavier than air (methanol and
ethanol for example). In the reference state, the two phases are at
a mechanical equilibrium and isothermal. The perturbation equa-
tions have to be written in 3D as azimuthally motion is expected to
set in. The key parameters of the problem are the size and volatility
of the drops, the thermal diffusivities of the three phases in con-
tact (substrate, liquid and vapour) as well as the vapour diffusion
coefficient and the liquid kinematic viscosity. A very careful scaling
process allows a pertinent description the specific problem investi-
gated. The evaporation process takes place at the free liquid–vapour
surface but due to the relative distance of each surface element
from the substrate, the transfer of heat through the liquid drop
becomes a function of the position on the surface. Although the
mostly accepted models [8,9] assume, by an electrostatic analogy,
that the highest evaporation should occur at the edges, the region
of stronger evaporation is not the region where the temperature
Fig. 7. Moving spherical cone frame for pinned sessile drop geometry; the curva- falls down the most rapidly. The size and thermal conductivity of
ture radius grows and the contact angle decreases while the triple line perimeter
the substrate seem to determine the direction of the Marangoni
remains constant. The dashed red lines on (b) limit the angle under which the moving
observer sees the surface of the whole solid substrate. flow induced by the local cooling of the surface [10].

7. K. Sefiane et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 365 (2010) 95–108 101
3.1. Reference state This assumption of a uniform normal surface velocity has for
direct consequence that the normal component of flux of matter is
The drop is set down on the substrate in thermal equilibrium uniform on the cap in opposition with the generally adopted Dee-
with both the substrate and the ambient air. The driving force for gan [8] electrostatic analogy. Indeed, the normal mass flux n is
evaporation is the non-saturation of the ambient air. No gradient given by
of temperature exists in the volumes and on the surfaces. However,
evaporation starts due to the non-saturation of the surrounding gas. ˚n = Vn ( l − v) (3)
This endothermic process creates locally a sink of enthalpy that fol-
However, the circulation of the vector on the surface is the scalar
lows the local distribution of the matter flux on the liquid–vapour
product of the flux vector by the unit normal vector to the surface
free surface. In the reference state, a layer of saturated vapour sur-
that changes its orientation with time. The velocity vector V on the
rounds the drop and a steady diffusion sets in within the vapour
surface is indeed the time derivative of the position vector P of each
phase. In this steady state, no convection is expected in both liq-
point on the surface with spherical coordinates (R, Â, ) in a moving
uid and vapour phase, however, as the drop volume decreases with
reference frame fixed at the centre of curvature of the spherical
time, there is a moving boundary condition for the mass transport.
cap. In this non-Galilean frame, the evaporation front limits a solid
In a reference frame moving with the velocity V of the curvature
angle of radius R(t) and of aperture  c (t) whose angular velocity ˝ =
centre of the evaporation front, the governing steady state equation
dÂc /dt equals the rotational velocity at the triple line. Because of the
for the diffusion of the vapour reads
axial symmetry of the drop, the unit vector does not rotate at the
V .∇ C + D∇ 2 C = 0 (1) apex of the sessile drop. Everywhere else on the spherical cap, the
normal unit vector changes its orientation during the evaporation
where D is the diffusion coefficient of the vapour. of the pinned drop so that the velocity vector V of each surface
element reads
3.2. Geometry of the system ˙ ˙ ˙ˆ ˙
ˆ ˙ ˙ ˙
V = P = Rˆ + Rr + R  + R  + R ˙ sin  ˆ + R sin  ˆ [R − R Â
˙r ˆ
The choice of the geometry of the moving front here is ambigu- ˙ ˆ
− R ˙ sin2 Â]ˆ + 2R − R sin  cos  Â
r
ous. For planar fronts, the frame may be located anywhere on the
planar surface and the Euclidian coordinates x, y, and z are, respec- + [2R ˙ sin  + R ˙ cos Â] ˆ (4)
tively, the tangential and normal coordinates. The velocity of the
plane is uniform on the plane. In the present problem, the surface is where the unit vectors time derivatives (see for instance Wikipedia)
curved and the curvature varies in time. Due to the axial symmetry, are
a 2D description in cylindrical coordinates is valid for the reference ˙ ˙
˙ ˙ˆ ˆ ˙r
r = ˙ ˆ sin  +  Â;  = −ˆ + ˙ ˆ cos Â;
ˆ ˆ
ˆ = −( ˙ r sin Â+Â cos Â)
ˆ
state with the radius of curvature and the contact angle  c function
of time. The spherical cap is pinned on a horizontal substrate and (5)
the velocity of the free surface with regard to the substrate varies
along the surface. At the triple line, the velocity vanishes as long With account of the rotation of the unit vector, for a spherical cap
as the cap remains pined and the radius of the triple line remains with axial symmetry ( ˙ = 0), the normal component of the surface
constant. If no deviation to the axial symmetry occurs in the refer- velocity reads then
ence state, the frame of reference can be arbitrary placed anywhere
on the triple line. In a 2D description it could be easier to adopt a ˙ ˙
Vn = R − RÂ Â (6)
rotating frame whose angular velocity ˝ = dÂc /dt characterizes the
velocity of the moving free surface. Such frame is a non-Galilean and the normal component of the local flux of matter on the free
frame and thus the fluids are rotating relatively to the frame. The surface is then
momentum balances have thus to be written with a rotational term. ˙ ˙
˚n = (R − RÂ Â)[ l − v] (7)
The rotational velocity of the frame is a vector oriented along the
Azimuthal coordinate. The boundary of the spherical cap, as shown By continuity of the matter flux at the free surface in gas,
by Popov [11], is mapped in toroidal coordinates. In order to solve
the Laplace equation in this coordinates system, the heavy use of ˚n = v ∇ Y.ˆ
r (8)
the special Legendre functions in needed [12]. These coordinates
have been also used in 2D and 3D by Massoud and Felske [13] who where Y denotes the mass fraction of the vapour in gas phase
have obtained an exact analytical solution for the stream function At the apex, Â(t) = 0 while at the triple line Â(t) = Â c (t). The angu-
inside the drop. ˙
lar velocity  also vanishes at the apex while it is maximum near
Another way to account for the motion of the reference surface the triple line (negative value). Because of the negative value of Â, ˙
is to locate a Cartesian frame at the centre of the solid substrate, on the normal matter flux grows from the apex to the triple line, in
the symmetry axis of the drop. The drop apex is then at a distance a similar way as predicted by Deegan et al. who used the Lebedev
zapex = h(t) from the frame and the centre of curvature at a distance analogy of the concentration field in the vapour with the electric
zcentre = h(t) − R(t). field distribution above the upper surface of a biconvex lens. No
For a pinned spherical cap droplet, the local normal component experimental evidence, however, exists on the local distribution of
of the evaporation front velocity is sometimes assumed to be uni- the evaporation flux around the drop neither on the concentration
form from the apex to the triple line and equal to the time variation field in vapour. Due to the rotational behaviour of the contact angle,
of the radius of curvature; this hypothesis of quasi flat interface is by continuity of the angular momentum, vortex motions have to
valid if (R/rd ) 1; sin Âc ≤ (1/100) develop in both neighbouring phases. These rotational motions of
course have to be taken into account for describing the concentra-
2
rd r2
1 d 1 dh tion field in the vapour phase. Two dimensionless numbers account
˙
Vn = R = +h = 1− d (2)
2 dt h 2 h2 dt for these rotational motion: the Rossby and the Eckman numbers.
The free surface of the drop is on top of a spherical cone (see
where rd is the constant radius of the triple line. Fig. 7). The aperture of this conical solid angle is 2Â c (t). For a pinned

8. 102 K. Sefiane et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 365 (2010) 95–108
drop  c (t) is related to the radius of curvature by change and surface tension gradient is not treated. The capillary
oscillations of a constrained inviscid liquid drop pinned on a circle
rd = R(t) sin Âc (t)
r (9) of contact (Rayleigh oscillations) have been recently computed by
Âc = arcsin d Bostwick and Steen [16]. In this last problem, the volume of the drop
R(t)
is constant so that there is no matter exchange through the free sur-
For a contact angle smaller than /2, the radius of curvature of face of the drop. The centre of mass of the system is oscillating when
the spherical cap is larger than the radius rd of the triple line and the normal modes of vibration are excited but there is no motion of
the distance between the equatorial plane of the moving reference the centre of mass associated to a phase change in contrast with the
frame and the plane of the solid substrate (R − h) ≤ R. problem we are concerned. It is then worth noting that in the prob-
In order to account for the change of shape of the spherical lem of the pinned evaporating drop, the wavelength as well as the
cap during evaporation, we adopt here the moving conical frame oscillation frequencies and wave amplitudes of the normal modes
with aperture 2Â c (t) attached to the centre of curvature of the will vary during the evaporation process. Some structural analogies
drop. The corresponding time dependent solid angle is equal to are to be found with crystal growth (see for example Cristini and
2 · (1 − cos Âc (t)). The symmetry is assumed to be axial around Lowengrub [17]).
the vertical axis. During the evaporation process, the contact angle In the rotating frame with characteristic angular velocity ˝ any
changes with a constant angular velocity ˝ = dÂc /dt. The contact perturbation of velocity or of temperature generated locally on the
angle angular velocity is a pseudo-vector normal to any longitu- free surface will be driven in a direction normal to the rotation
dinal section of the spherical cap. It corresponds to the rotational vector ˝.
velocity of the mobile reference frame described in Fig. 7. The rota-
tion pseudo-vector is normal to the vertical axis of symmetry and 3.5. Scaling
describes the angular velocity of the solid angle of the moving
observer. The length scale L in the drop is obviously the radius of the
triple line rd . Aspect ratio numbers rs /rd and rv /rd have to be intro-
3.3. Hydrodynamic equations duced to account for the relative dimensions of the solid substrate
and the drop contact radius as well as the gas diffusion layer to
In this moving rotating frame Chandrasekhar [14] (Chapter 3) drop size. As what concerns the time scale, a very careful analysis
and Greenspan [15] (Chapter 3 and 4), the fundamental equations of the leading phenomenon has to be conducted. At the onset of
of hydrodynamics in liquid and vapour phases read, evaporation, in isothermal conditions, the driving force is only the
∂ non-saturation of the surrounding gas. As the surface Knudsen pro-
+ ∇ .[ (v − ˝ × r)] = 0 (10) cess is in a quasi equilibrium, it is a fast step and only the transport
∂t
process above the saturation layer should define the leading char-
∂v 1 2
+ (v.∇ )v = − ∇ P − 2˝ × v + ∇ v (11) acteristic time. This last is often assumed to be purely diffusive, but
∂t if one takes into account the motion of the free surface, advection
By taking the curl of Eq. (10) and introducing the vorticity in the two neighbouring phases has to be considered with a viscous
diffusion. As the process is energy consuming, the heat transport
ω =∇ ×v (12) to the evaporation site contributes to the characteristic time. The
we get local cooling induced by evaporation creates on the surface local
temperature gradients in the perturbed state. Heat fluxes from the
∂ω 2
+ (v.∇ )ω = [(ω + 2˝).∇ ]v + ∇ ω (13) neighbouring phases carried by diffusion and advection contribute
∂t to reduce this local cooling. Classically in liquids the Prandtl num-
Boundary conditions have to be written on both solid–liquid and ber Pr = /˛ is >1 this is in favour of a dominant role of advection
liquid–vapour interfaces. However, if one restricts the domain of in heat transfer process, in contrast with the situation prevailing
investigation to the cone of aperture 2Â c (t), the solid–vapour inter- in gas where Pr < 1. In a vapour phase at atmospheric pressure,
face is not included in the system. Therefore a larger cone could be the Schmidt number Sc = /D ∼ 1. The matter diffusion and the
=
taken as moving frame, this last having for aperture a wider angle advection should then both influence the evaporation kinetics. The
than 2Â c (t) in order to include the solid domain uncovered by the relative heat conductivities of the three contacting phases as well
liquid (red dashed lines in Fig. 7). as the size of the heat-providing phase (here often the solid sub-
The boundary conditions on the free surface couple the energy strate if it is a good heat conductor) also influence the dynamic
and matter balances. They have to be written locally on the spher- process.
ical cap and must account for normal deviation of the shape of the The characteristic time scales associated with the evaporation of
surface. a spherical drop surrounded by a passive gas unsaturated with the
For a pinned droplet, it is assumed that any deviation of shape at vapour of the evaporating component of the droplet are compared
the triple line is forbidden. Moreover, we assume that the axial sym- by Ha and Lai [18]. Like many authors concerned with evaporation
metry of the drop is preserved in the deformed state. This allows of drops, either pinned on a substrate or suspended in a gas, these
to select even modes in the Legendre polynomials solution of the authors neglect the regression rate and make a quasi-steady state
surface deformation. approximation. The fundamental difference between the drop sur-
rounded by vapour and the sessile drop lies in the presence of the
3.4. Methodology of the stability analysis substrate whose physico-chemical properties (thermal conductiv-
ity and wettability) have to be taken into account. Our purpose
In order to analyse the stability of the system, the partial differ- here is indeed to account for the drift of the thermally induced
ential equations and the boundary conditions have to be written Marangoni motion under the influence of the change of contact
in dimensionless form. The choice of the scaling is then critical angle during the evaporation of the sessile drop. Therefore, the
and very specific to each problem considered. The stability of fluid angular velocity of the contact angle plays the role of a slow rota-
spheres and spherical shells undergoing heat flux and the influ- tional motion that carries the smaller wavelength perturbations of
ence of rotation have been developed in the famous Chandrasekhar shape of the deformed spherical cap. Similarity can be found with
book [14], but the special configuration of pinned drops with phase the geostrophic phenomena although in this case, the axis of rota-

9. K. Sefiane et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 365 (2010) 95–108 103
tion corresponds to the symmetry axis in contrast with the rotation
considered here whose axis is normal to the symmetry axis.
In the geostrophic flows, the rotation dominates the dynamics if
the Rossby Ro and Ekman Ek numbers are smaller than 1. These two
dimensionless quantities compare the rotation time to the viscous
diffusion time and to the advection time.
time of rotation (1/˝)
Ek = =
˝L2 time of viscous diffusion (L2 / )
(14)
U time of rotation (1/˝)
Ro = =
˝L time of advection (L/U)
where U, L and ˝ are, respectively, the scales of velocity, of length
and of angular velocity. In our problem, we are seeking for the
influence of the contact angle velocity on the thermocapillary
waves. These last result from a Marangoni instability induced by the
local cooling during the evaporation process. Two scales of motion
interact, a large toroidal vortex that carries a small wavelength per-
turbation at the deformable free surface. A new Marangoni number Fig. 8. Reduced normal flux from apex to triple line for various initial contact angles
has then to be defined including the thermal diffusivities ratios of calculated from Eq. (16).
gas, liquid and solid phases as well as the latent heat of evaporation
of the liquid. The stability analysis performed by Ha and Lai for the
The initial condition of the system as a whole being isothermal, the
drop surrounded by gas may be adapted for the sessile drop pinned
evaporation flux induces a local cooling on the free surface and local
on a solid substrate but additional boundary conditions have to
gradients of temperature at the liquid/vapour surface. In a quasi-
be satisfied namely the momentum- and thermal-balances at the
static approach, Eq. (16) is adopted. However, the vapour and solid
solid/liquid interface.
phase also contribute in the dynamic process of energy transfer.
The quasi-steady state of reference is in fact a process governed
If the solid substrate is a highly conducting material, as soon as a
by matter diffusion in the non-saturated gas phase; this time scale
perturbation of temperature reaches the liquid/substrate interface,
tscale = L2 /Dg . Initially, the system is isothermal. Due to evapora-
the solid surface cools down as a whole in a very short time s ∝
tion local cooling, a heat transfer process sets in with a time scale 2
rs /˛s so that we may assume a uniform temperature on the solid
tscale = L2 /˛ . The time of heat transfer through the liquid is two
surface.With the scaling used, the dimensionless flux reads
orders of magnitude larger than the time of matter diffusion in gas,
however, the time of heat diffusion in gas and in the solid substrate v ˙
R RÂ Â˙
˜
˚n = 1− −
is of the same order of magnitude as the matter diffusion time in rd ˝ rd ˝
gas. The time of regression of the drop depends on the volume of
the drop on its volatility and on the thermal properties of the liq- 1 v 1 ˙
ÂÂ
=− 1− + (18)
uid as well on the solid and vapour and on the wettability (contact 0
sin Âc 0
tan Âc ˝
angle). This time tregr ∼ Âc /˝ is long with regard to the diffusion
= o
In Fig. 8, the reduced normal flux on a droplet with Âc /˝ ≈ o
time in gas and with the thermal diffusion in liquid, in the experi-
ments described above, it is taken as the total time of evaporation, 120 s and rd = 0.5 mm is calculated in function of Â/Â c for var-
o
ious values of the initial contact angle Âc , by assuming a linear
ttot . Indeed, the drop remains pinned during more than 90% of the
evaporation process and the contact angle at the depinning of the ˙ ˙
dependence of the angular velocity  with Â, i.e.  = ˝Â/Âc .o
triple line falls to a value of difficult access with our device. It is The evaporation number Ev = ˚rd / ˛ in the present problem
close to zero. is locally defined on the free surface and results from the scaling of
2
Eqs. (7) and (8) with a velocity scale vscale = rd ˛ /rd = ˛ /rd and a
In the rotating frame of reference attached to the moving centre
of curvature of the free surface, the non-dimensional momentum density ratio = / v . The thermal Marangoni number is coupled
equation (11) reads here with the evaporation number. Indeed, in the reference state,
no thermal gradient exists. Only if the process of evaporation is non-
˜
∂v uniform on the surface, a local cooling may induce a local gradient of
˜ ˜ ˜ ˜˜ ˜ ˜
+ (v.∇ )v = −∇ P − Ek ∇ × ∇ × v
˜ (15)
∂t surface tension. The gradient of temperature along the free surface
induced by non-uniform evaporation is indeed
The dimensionless numbers introduced, adapted for the concerned
problem result from the choice of the scales. The temperature scale ∂T ∂T ∂˚
= (19)
is chosen from the combined mass and thermal boundary condi- ∂ ∂˚ ∂Â
tions in the dynamic state at the free surface (accounting for the
and the appropriate Marangoni number is then
surface regression).
−(d /dT )(∂T /∂˚)(∂˚/∂Â) · rd
Ma =
h v ˚ .ˆ = −k ∇ T .ˆ
r r ˛
(16)
˙ ˙
˚ .ˆ = V [ − v ].ˆ = (R − RÂ Â )[
r r − v ] = v Dg ∇ Y .ˆ
r
−(d /dT )(∂T /∂˚)(∂ ln ˚/∂Â) · Ev
= (20)
The length scale for the gradients is the same = rd when the liquid-
covered and total surface of the substrate are equal. To make Eq. For evaporating liquid planar films, Zhang [19] showed the influ-
(16) dimensionless, the obvious temperature scale is then Tscale = ence of evaporation on the wavelength of the Marangoni cells
h v v Dg /k and the dimensionless thermal boundary condition (16) during the system history. He also defines a generalized Marangoni
is then number that accounts for the rate of evaporation.
˜ ˜˜ r From Eq. (18) and from the matter flux profile, the local val-
˚ .ˆ = −Ev ∇ T .ˆ
r (17) ues of the Marangoni numbers on the surface of drops of water,

10. 104 K. Sefiane et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 365 (2010) 95–108
Table 2
Timescales.
Transport process Time scale Relative time scale Versus heat diffusion in drop
Water Acetone Methanol Ethanol
In vapour
Momentum 2
rd / g ˛ / g 1.1 × 10−2 6.01 × 10−3 6.93 × 10−3 5.95 × 10−3
Matter diffusion 2
rd /Dg ˛ /Dg 5.86 × 10−3 8.69 × 10−3 7.09 × 10−3 7.59 × 10−3
Heat diffusion 2
rd /˛g ˛ /˛g 6.1 × 10−3 3.93 × 10−3 4.53 × 10−3 3.89 × 10−3
In liquid
Momentum 2
rd / ˛ / 1.6 × 10−1 2.27 × 10−1 1.45 × 10−1 6.5 × 10−2
2
Heat diffusion rd /˛ 1 1 1 1 1
In solid
Heat diffusion (rd /˛s ) · (rs /rd )
2 2 2
(˛ /˛s ) · (rs /rd )
2 2
1.7 × 10−1 1.1 × 10−1 1.3 × 10−1 1.1 × 10−1
Surface regression
Âc /˝ (Âc /˝) · (˛ /rd )
2
ca 50 ca 35 ca 40 ca 35
Table 3
Values used in Marangoni numbers.
H v (J/mol) Tio (K) dT/dY (K/% at 298 K) −d /dT (N/mK)
Water 40700 373 8 1.68 × 10−4
Acetone 31300 329 80 1.12 × 10−4
Ethanol 39330 351 20 1.2 × 10−4
Methanol 34400 338 100 7.73 × 10−5
methanol, ethanol and acetone are obtained with the values shown invariant the shape of the evaporation front. In the pinned drop
in Tables 2 and 3. The various profiles for these substances accord- evaporation, however, the front is allowed to move and its radius
ing to the numerical values at 298 K depend on the initial values of of curvature to increase. This change of surface geometry appears in
the contact angle, respectively. the new defined Marangoni number Eq. (18) in which the evapora-
tion number accounts also for the local variation Eq. (16) of the mass
˜
472 ∂˚
Ma-water = o
flux at the surface that induces the thermal gradient through the
˜
Âc ∂ local equilibrium evaporation constant (Clausius–Clapeyron equa-
˜
2293 ∂˚ tion) (Fig. 10).
Ma-ethanol = o
Âc ∂˜
(21) h v Tio
13725 ∂˚ ˜ K(Â) = Ys (Â) = exp 1− (22)
Ma-methanol = o
Tio T (Â)
Âc ∂˜
33147 ∂˚˜ where Tio is the boiling temperature of the evaporating liquid i
Ma-acetone = o
Âc ˜
∂ under the imposed pressure, Ys (Â) the local saturation mass fraction
in gas at the surface in the perturbed state, T (Â) the local perturbed
where we have assumed a thickness of the diffusion layer in the temperature at the surface, is the gas constant, the gas phase is
gas phase ı = 2 m. assumed to be perfect.
With the profile of matter flux given in Fig. 8, the Marangoni
numbers show a linear profile in Â/Â c , Fig. 9.
The estimated values of the Marangoni number of a water
droplet with rd = 0.5 mm evaporating in dry air are given in Fig. 9.
3.6. Simplifying assumptions
In the reference state, the temperature is initially uniform
throughout the system and on the free surface so that, if we
assume the local evaporation equilibrium to be instantaneously
established, the Clausius–Clapeyron equation governs the value of
the saturation concentration in a thin layer of vapour above the
liquid–vapour interface. The far field concentration in the gas phase
is assumed to be uniform and regulated by the “humidity” control
device. In a previous analysis of self organised Marangoni motion
at evaporating fed drops and menisci Steinchen and Sefiane [20],
the scaling used was different in order to deal with a steady state
of temperature field and motion sustained by the inflow. In the
present problem, the regression of the free surface is not compen-
sated by a liquid inflow and the contact angle of the pinned drop
decreases during the evaporation process while for the fed drop,
the contact angle remains constant. The liquid input in the fed drop Fig. 9. Marangoni numbers on a water droplet from apex to triple line for various
problem measures the integral evaporation output and maintains initial contact angles calculated from Eq. (19).

11. K. Sefiane et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 365 (2010) 95–108 105
wavelengths, it could be better to choose as time scale the regres-
sion time. The relative time scales versus the regression time are
then the relative time scales versus the conduction time in liquid
divided by the relative regression time in the liquid conduction time
scale.
3.7. Perturbation equations
The velocity–temperature and vapour mol-fraction perturbed
fields are expressed in terms of spherical harmonics as well as the
velocity of the free surface (evaporation front).
∂v 1 2
+ ([v + V ]∇ )v = − ∇ P − 2˝ × v + ∇ v
∂t
where
∗ (23)
v = v + ıv ; V = V ∗ + ıV ; P = P ∗ + ıP
Fig. 10. Vapour saturation fraction–slopes; acetone, methanol, ethanol and water.
m
ıv = U(r)Y1 (Â, ) exp (ω ); ıV = ℵ(r)Y1 (Â, ) exp(ω );
m
ıP = m
(r)Y1 (Â, ) exp(ω )
∂T
As the process is out of equilibrium, the local mass fraction Ys (Â) + (v + V ) · ∇ T = ∇ 2 T − (v + V ) · ∇ T ∗
∂t
may be assumed to be the saturation mass fraction multiplied by where (24)
an accommodation coefficient ˛ smaller than 1. T = T ∗ + ıT
In a recent numerical model of the evaporation of a pinned ıT = (r)Ylm (Â, ) exp(ω )
sessile drop on a highly conducting solid, Girard and Antoni [10]
∂Y
have shown the influence of the size of the solid substrate on + (v + V ) · ∇ Y = ∇ 2 Y − (v + V ) · ∇ Y ∗
the temperature distribution on the free surface of the drop and ∂t
where (25)
on the orientation of the induced Marangoni motion. The rela- ∂Y = Y ∗ + ıY
tive heat conductivities ks /k of the solid and liquid as well as the ıY = « (r)Ylm (Â, ) exp(ω )
relative radii of the substrate and of the triple line rs /rd seem to
rule the direction of the induced flow. However, the dynamical The momentum boundary conditions on the free surface have to
behaviour of the system depends not only on conductivities ratio account for the normal deformation of the drop. The evaporation
but also on the thermal diffusivities ratio. As already evidenced front curvature has to be perturbed in the same way as done by
by Ha and Lai [18], the heat diffusion time in the vapour is two Blinova et al. [22] for a spherical solidification front.
orders of magnitude larger than in the liquid; in our problem, the The reference state is a time dependent state with a character-
heat diffusion in the solid substrate is three orders of magnitude o
istic time given by Âc /˝. The perturbations have a time constant
faster than in the liquid but it is only four times faster than in gas (1/ω) Â/˝. The stability analysis has to be performed for asymp-
so that the influence of vapour and of solid heat transfers might totic conditions at short time and short wavelength. Exchange of
have comparable importance in the dynamic process, moreover, stability or oscillatory instability might be obtained according to
the surface of the solid substrate is about ten times the size of the the values of the real or imaginary parts of ω· The analytical res-
base of the drop. In the present approach, we adopt as a simpli- olution of the eigen-values problem should lead to knowledge of
fying assumption that (ks /k ) 1, (k /kv ) 1 and (rs /rd ) 1 but the minimum critical value of the Marangoni number and evapo-
on a good heat conductor as Al, the characteristic times defined by ration number at which the most probable mode of deformation
the ratio of the diffusivities are summarized in Table 2. Numeri- of the surface should grow. The short wavelength modes grow-
cal values are given for droplets with attachment radii of 0.5 mm ing on the receding spherical cap may be described by polynomial
of various liquids evaporated in an unsaturated gas phase under developments. The most classical ones are the Legendre polynomi-
a pressure = 1 atm at 298 K on an Al substrate the radius of which als. With various trial functions to model the perturbations of the
is 10 times that of the attachment radius. According to the val- drop surface structures similar to the thermographic images are
ues of these characteristic times, the steady state approximations obtained. In Fig. 11, the patterns obtained on sessile drops with
commonly used for heat, momentum and matter transport in gas various contact angle show that the wavelength increases with
as well as heat transfer in the substrate are justified. The high decreasing contact angle as obtained by thermography. According
value of heat conductivity of the substrate also allows assume that to the trial function used, the pattern obtained on the drop may
the temperature remains constant on the whole solid surface. A show a decreasing wavelength when approaching the triple line.
recent 2D numerical model by Barash et al. [21] shows that dur- A deformation in the azimuthal direction may also be predicted
ing the early stage of evaporation a street of vortices arises near with other trial functions. The normal perturbations of the surface
the surface of the drop and induces a non-monotonic spatial dis- correspond to regions where the vaporization front penetrates in
tribution of the temperature over the drop surface. The number of colder or warmer domains removed from the equilibrium condi-
vortices rapidly decreases with time. This behaviour is confirmed tions so that condensation or evaporation may proceed locally on
by the thermal visualisation obtained by Sefiane et al. [5]. How- the crests or in the wells. The surface deformation induces micro-
ever, due to the limitations of the 2D model of Barash et al. [21], convection in the neighbouring fluids. The problem as formulated
the circulation of the thermal waves around the symmetry axis so far cannot be solved analytically. Elaborate and tedious numeri-
as shown in the movie provided by Sefiane et al. [5] cannot be cal techniques are necessary to give a full solution to the problem.
modelled. This last behaviour could be a consequence of the shift Instead of going down that route, we chose to adopt a different
of the vortices by rotational velocity ˝ of the frame (Eq. (13). In approach, which aims to reveal the nature and form of the per-
order to make asymptotic developments at short times and short turbations which could lead to surface waves similar to the ones

12. 106 K. Sefiane et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 365 (2010) 95–108
Fig. 11. Surface wavy perturbation obtained with trial function 2.
observed experimentally. Using as an initial steady state a drop The resulting waves are still travelling from the apex to the con-
with a spherical cap shape, a parametric study is undertaken to tact line, with the wavelength decreasing as we move from the
evaluate the form of the solutions to perturbations which may give apex to the contact line, Fig. 11. In a last stage, and in addition
rise to waves similar to the ones observe experimentally. In a first to the perturbations on the longitudinal angle, a perturbation on
stage (trial function 1) we have introduced a perturbation of the the Azimuthal angle is introduced. The perturbations functions are
interface, following the functions given below, selected in such a way that the contact line is not deformed (since
we are dealing with pinned drops). This last form of perturbation
1
x + ıx = cos ϕ · cos −Â +
cos 64 −Â functions seems to yield waves that are very similar to the ones
264 2 observed in the experiments. Circling waves, with a wavelength
1 that depends on position are obtained when trying the following
y + ıy = sin ϕ · cos −Â + cos 64 −Â (26)
2 64 2 functions (trial function 3),
1
z + ız = sin −Â + sin64 −Â 1
2 64 2 x + ıx = cos(ϕ −− Â ) + cos 6ϕ · sin(Â − Âc )
2 6
2
The resulting waves are of similar wavelength travelling from the 1
· cos −Â + cos 64 −Â
apex to the contact line region. In a second stage the perturbation 2 64 2
functions are altered (trial function 2) according to the format given 1
below, y + ıy = sin(ϕ −− Â ) + sin 6ϕ · sin(Â − Âc ) (28)
2 6
2
2 1
1 · cos −Â + cos 64 −Â
x + ıx = cos ϕ · cos −Â + cos 64 −Â 2 64 2
2 64 2
2
2 1
1 z + ız = sin −Â + cos 64 −Â
y + ıy = cos ϕ · cos −Â + cos 64 −Â (27) 2 64 2
2 64 2
2 In addition to circling waves on the drop surface which
1
z + ız = sin −Â + sin 64 −Â replicates the observed experimental trend, the above pertur-
2 64 2
bation functions, and in agreement with experiments, indicate

13. K. Sefiane et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 365 (2010) 95–108 107
Fig. 12. Azimuthal motion obtained with trial function 3.
that the wave number decreases as the drop evaporates, regression motion due to evaporation. Trial functions are used to
Fig. 12. reproduce the patterns observed experimentally.
The approach adopted above may be qualitative in nature,
nonetheless it unravels very useful information on the form the Acknowledgements
perturbations may have in order to give rise to waves sim-
ilar to the ones observed experimentally. This approach also ESA MAP FASES contract has partly sponsored one of us, the
can guide modellers in selecting the right form of perturbation French Space agency CNES as well as the CNRS GDR FAM pro-
functions, when attempting a more complete theoretical resolu- gramme gave a financial support of the present contribution.
tion of the problem. This approach can also be used to unravel The authors acknowledge the support of the British Engineering
the influence of the substrates thermal properties as shown in and Physical Sciences Research Council (EPSRC), for their financial
[23]. support through DTA grant.
4. Conclusions References
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