Festoon instabilities of slightly volatile liquids during
spreading
C. Redon, F. Brochard-Wyart, F. Rondelez
To cite this version:
C. Redon, F. Brochard-Wyart, F. Rondelez. Festoon instabilities of slightly volatile liquids during spreading. Journal de Physique II, EDP Sciences, 1992, 2 (9), pp.1671-1676.
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Phys.
J.
France
II
(1992)
2
1671-1676
1992,
SBPTBMBBR
1671
PAGE
Classification
Physics
Abstracts
46.30
66.90
Communication
Short
instabilities
Festoon
Redon,
C.
Brochard-Wya~
F.
Curie,
Institut
05,
le
Nous
(degrd
contour
t°"( it)
ondule
la
en
dessous
R
de
goutte
We
the
same
these
h~,
a
rim
time,
observed
generated
at
induit
the
formed
the
two
bord
au
the
at
line
contact
spreading speeds
of Marangoni
terms
drop edge by liquid
and
tend
to
induce
deformations
modulations
a
liquid
and
the
impose
larger
velocities
(*)
Laboratoire
associd
of the
rim
contour
to
au
first
the
CNRS,
contact
URA
time
I)
:
R
as
increases
instabilities
induced
R
as
by
spectacular
t
the
of
silicone
first, the drop
t°"( it) below a
at
develops
contour
radius
fronts
line.
The
of
1379,
the
et
h
to
extemal
studied
during
observed
been
Fingers
parts
n°
submitted
intensively
have
line
undulates.
thicker
with
the
le
"~.
We
surface
oils
contour
critical
festoons.
At
interpreted
tension
gradient
have
evaporation.
contact
the
near
and
par
large, flat, droplets
wafers
silicon
et
Marangoni produit par
l'Evaporation du liquide.
l'effet
goutte
of
s'accdmre
l'dtalement
temps,
par
spreading
the
up
driven
Fingering instabilities of advancing liquid
gravity, thermal gradient
have been
Periodic
la
de
in the
stages
mdme
En
ces
force,
8].
1992)
rdsultats
20) deposited on bare
<
drop radius R increases
N
the
in
results
festons.
en
interpr6tons
superficielle,
is
I6July
N
instabilit6
Nous
(degree of polymerization
is perfectly
and
circular
thickness
est
une
have
final form
deux
d'une
~t~'~.
tension
AbstracL
in
polymdrisation
de
comme
gradient
Chimie,
Laboratoire
de
Physico-Chimie,
Laboratoire
de
Pierre
Marie
75231Paris
(*), 11 rue
Cude,
et
et
d'huiles
r6gimes
d'6taIement
pour de grandes gouttes plates
d'abord,
pastilles
de
silicium
I)
20)
ddposdes
des
nues
:
sur
<
parfaitement
r6gulier et le rayon R croft en temps
comme
qui
bourrelet
forrne h la ligne de
6paisseur seuil h~, un
contact
se
observ6
avons
de
ddveloppe
et
croft
R
Rondelez
Interfaces
et
accepted
1992,
9June
Rksumk.
R
spreading
France
(Received
silicones
Surfaces
des
liquids during
volatile
F.
and
Physique
de
Section
Physico-Chimie
Cedex
slightly
of
rim is
are
whenever
usually
then
unstable
formed
centrifugal
forces
the
when
last
few
the
extemal
rim.
l'Universit6
Pierre
et
years
[1-
driving forces
towards
peristaltic
the
Marie
Cude.
forces
1672
JOURNAL
study
We
here
instabilities
spontaneous
the
PHYSIQUE
DE
slightly
N° 9
II
observed
with
gradient
extema1thermal
no
during the spreading
spreading were performed by Bascom et al. [9], monitoring the spreading of squalane on steel
impurities increase
observed
that
volatile
drastically the rate of spreading and
plate. They
lead to the
formation
of a ridge at the advancing edge. Later
Williams
observed
digitation
11 0]
of
for
droplets
of
slightly
volatile
silicone
oils. He
liquid
undulation
the
small
contour
or
contribution
exist
for
the
volatile
liquids.
However,
noticed
that a Marangoni
he
must
most
line,
independent
of
suggested altematively an inherent instability of the advancing
contact
this
perform
systematic
gradient,
which
is
probably
ruled
In
spreading
out.
paper,
we
any
experiments for a series of slightly volatile silicone oils (molecular weights ranging from 400 to
in the late
1400
Daltons).
The
is
unstable
stages of spreading (for large radii, with
contour
weights (index of polymerization
drops flattened by gravity). For relatively high molecular
circular (and this rules out
Williams's
20), the contour
remains
suggestion). In section 1, we
section
2,
interpret
in
of a Maragnoni
describe
experiments.
In
results
terms
we
our
our
tension
gradients are generated by evaporation near the contact
line.
We
effect : the
surface
instability « festoons ». The festoons are very
different
from the fingering
observed
call this
for a drop placed in an
gradient [5, 6].
extemal
of
volatile
oils.
silicone
The
observations
first
of
anomalous
~
Experimental
1.
We
studied
have
System)
the
thickness
e
500
than
CDD
system
: a
T.V.
monitor
spreading
for
minutes,
a
1000
PDMS
silicone
as
the
drop
molecular
different
flattens.
weights,
in
have
fashion.
by
Periodic
:
I) the
power x 20) and a
different
of the
stages
the
contact
line
of
spreading
variation
R
oscillations
(t) of the
remains
silicone
of
radius
oils
with
it) the onset of the festoon at a critical
radius R~. Since the
radius is always at
larger than the amplitude of the contact line oscillations, R (t ) is a well defined
variation
A typical
of R (t ) for a PDMS droplet (volume 5 ~l,
molecular
weight 770)
in figure 2 in
logarithmic
coordinates
four
decades.
One
clearly
two
over
sees
regimes : I) a slow regime with R (t ) t°.~~ * °.°( below a critical radius R~, previously
t°.~~ * °'°~. The
with high
molecular
weights [12] and it) a fast regime, where R (t)
radius R~
between
the
regimes is the critical
radius
of the
festoon
instability.
two
studied R~ as a
function
of the
molecular
weight, M~ and volume, fl, of the sessile
and
times
For
each
weight,
molecular
f1°.~~*°.~~
R~
For
we
monitored
have
droplets
flat
the
the
here,
considered
variation
appears
at
critical
a
thickness
h~=
h~,
of R~
arR~ ho.
fl
=
instability
video
a
(magnifying
dynamics
the
radius,
by
monitored
was
and
mm
Then
sinusoidal
studied
101.6
droplet
The
weight 410 Daltons). During the
liquid rim appears at the
a
modulations
of the moving front
molecular
fringes.
these
characterized
mm,
describe
now
circular
of
l.5
telephotolens
a
is Z
wafer.
the
~
We
interference
diameter
from
=
~
=
(PDMS,
Petrarch
Siltronix,
are
oils
purchased
wafer
~
«~
a
number
We
The
(volume 2 ~l,
oil
drop spreads
the
silicone
completely
wets
resolution.
of
wafers,
The
state.
»
oil
CE) with
XC-77
lines
wedge, together with colored
begin to form (Fig. I). The
constant
silicone
(Sony
camera
clean
«
droplets
of
wafers.
capillary length
the
with
first
two
silicon
A
~m.
=
larger
always
on
spreading
of
horizontal
(oxided-covered)
bare
their
in
dynamics
the
deposited
used
R,
observations.
~~,
fl
with
least
independent
of
ten
quantity.
is plotted
spreading
observed
cross-over
We
have
droplets.
found
and
conclude
We
of
time
the
that
the
volume
arR~
and
fl
h~
14
=
depending
±
2 ~m
for
only
on
M~
770
=
weight.
molecular
and
h~
10
=
±
3
found
We
~m
for
M~
h~
40±5
=
250.
=
~m
for
M~
410,
=
9
N°
FESTOON
m"
Fig.
1.
can
see
Snapshot
line
is
OF
SLIGHTLY
LIQUIDS
VOLATILE
1673
t
spreading
of the
interferences
colored
contact
INSTABILITIES
on
periodically
for
PDMS
a
drop
the
silicone
surface.
0-5
11
#
O
£l
°
O
y~
droplet (n
rim
is
=
formed
2 ~cl,
at
M~
the
410 Daltons) :
drop periphery and
we
=
the
modulated.
~
)
oil
liquid
A
11
~Ll
2 ~cl
5 ~Li
=
"
O~
~~
~
I'
~~
O"
%*
~
O
~a
o°°
~o
°
~OO
~
~
'-
°°
°
°°
,:"
,"
--
°
-'
Q
_~'
C£
O-
~-
t
i
io°
ioJ
io3
1o2
tbne
Fig.
2.
Log-log plot
symbols
(o) : £l
5
different
The
2 ~Ll,
£l
=
2.
=
Interpretation
2.I
onset
SLow
of
dynamics
forces
are
of
SPRBADING
the
of
of the
variation
to
the
versus
three
ios
time
different
for
a
droplet of
droplets : (+)
PDMS
volume
molecular
£l
=
weight
0.5 ~cl
770.
(Q)
~cl.
spreading
the
(h ~h~).
instability is
wetting and
dominant).
radius
correspond
io4
(s)
We
regimes.
The
spreading of the large drop (R » « ~~) before the
by gravity. In previous works [11, 12], we studied the
the shape of the
so-called « heavy » droplets (because
gravitational
showed
the
existence
of a new
regime of spreading for droplets
controlled
:
JOURNAL
DE
3
~
1674
radius
between
R~ (~)
and
«~
ln
«
=
N° 9
PHYSIQUE
II
cm),
where
(
-v
a
is
molecular
a
size.
In
this
a
dissipation in the wedge of the drop,
drop relaxes faster, and the drop
the
part
contact
near
flat
disk
of
thickness
pictured
ho, terminated by a liquid
quasi-static
It
be
has a
shape.
as a
can
«~~.
between
relationship
size
The
ho and
0~(~l),
and
of
angle
wedge
contact
flattened by gravity [14]) given by :
is (as for a static droplet
«
regime, (the
important
most
line, is
practice),
in
one
dominant.
ho
This
has
~
velocity U~
spreading
The
experimentally
checked
been
in
of
of
(I)
[12].
reference
the
the
9d
"
"
line
contact
dt
=
viscous
the
central
The
v*
d~
~~~S~W~~
~
V*
where
is
typical
a
=
velocity
wetting
and
'1
divergence
of
dissipation
the
in
Tanner
the
law
[13]
~
:
~~~
In
k
wedge (In
a
given by
is
logarithmic
a
factor
describing
the
~10).
ln
a
The
conservation
volume
for
gravity
flat
a
fl
=
equations (1, 2)
and
lead
to
written
is
as
:
arR~ ho
velocity
spreading
the
pancake
(3)
:
n3
~~
~
~~
~~~S~W~~~°~~W~fi
~
and
R
to
(t) by
integration
[I1]
U~
The
under
threshold
gravity,
the
wedge
~#
~3
In
~r3
fl3t.
comparing
by
obtained
is
R~
ignoring
velocity
of
the
contact
~
The
quasi-static regime
shows
up
I)
velocities:
:
given by
line,
(5)
two
~j
~@~ ~~~~
it) the
~~~
:
R~=
(~)
~
~~
if U~
the
dR
v*
dt
~91n
<
3R'
law
Tanner
(~ho)~
~~
U~, I.e. R
:
~~
<
~~
91n
R~
3
=
'
w
ln
~ -1
the
spreading
velocity
N°
9
FESTOON
2.2
ONSET
the
THE
oF
surface,
and
Marangoni
a
(h
INSTABILITY
by the liquid
line
contact
INSTABILITIES
=
[5, 6, 9]
gravity
the
~~
1675
is
stress
a
«~~
=
induced
near
dx
~~
liquid free
the
at
dx
:
~~
V*
ho.
=
~~
here,
considered
case
gradient
tension
induced
which
LIQUIDS
VOLATILE
surface
A
evaporation,
flow
SLIGHTLY
h~).
UM
In
OF
Ay,
«
=
dx
(6)
Ay
where
the
is
total
surface
tension
variation
the wedge.
silicone
oils, the evaporation is weak and UM is small.
For
over
our
However,
when UM becomes
just behind the contact
larger than U~, the liquid is
accumulated
line [9]. The resulting liquid rim is
unstable
with
periodic
deformations
[5, 8, 9]
respect to
(Rayleigh type instability) [15]. The
critical
condition
U~=U~
gives the
thickness
h~ for
the
of
onset
instability
the
:
h)
~~
4.5
(7)
ln
=
Pg
We
Ay
evaluate
can
from
the
experimental
values
~~
of h~
2
10~
x
~
for
M~
410).
As
=
Y
the
molecular
If
observed.
weight of PDMS
increases,
decreases
and the
Ay
drops of different
volumes
deposited,
instability
the
are
instability
will
show
is
no
up
for
longer
critical
i
R~ related
radii
data.
dependence
Ay
The
n
h~ by equation (3) (R~
to
via
2
The
"hc
=
equation (7)
is
tested
not
fl
~' ~
yet,
dependence
Ay
because
with
agrees
is
our
directly
not
measurable.
2.3
FAST
dx
evolution
by
(driven
~~
gradient
h~).
gravity)
(h
SPREADING
spreading
R
quasi-static
«
=
Ay,
which
ho
R(t)
For
<
and
remains
during
spreading time,
the
spreading droplet in the Marangoni regime :
assumption (2) for the droplet shape (Eq. (I)), we find
(t) of the
1~~
$
leads
faster.
grows
constant
dR
which
Marangoni flow
overcomes
Assuming again a
h~, the
<
to
dy
YdX
I
2
we
spontaneous
surface
tension
deduce
can
equation (6)
from
and
n
~~~
arR~
:
2
Festoon
instabilities
(2) The
of
the
y
(9)
ar
remarks.
evaporation.
part
the
:
R3=~V*«~~~t.
Concluding
the
An
are
early
clearly the
manifestation
experiment suggesting this
quasi-static shape
drop.
is
valid
now
as
long
as
U~
of
a
was
<
U~,
Marangoni flow
the following :
to
assume
a
fast
induced
if
by
the
one
illuminates
relaxation
of the
liquid
the
central
1676
JOURNAL
PHYSIQUE
DE
II
N° 9
increases.
The physical origin
generated
by the liquid
gradient
to a
can
for the
concentration
effects : the polymer melt is polydisperse
evaporation, or to direct
chains.
This
surface
tension
is larger for the
shorter
oligomers
interested
here, the
was
observed a long time ago by Gaines et al. [16], and interpreted recently by de Gennes [17]. The
depleted by evaporation in rite wedge and rite surface tension is lower
smaller
chains
are
more
decide
the two
line. We are not able to
the
between
in the thick regions than
contact
near
instabilities
references
[5, 9] on
mechanisms.
of the
thermal
shift following
A rough
estimate
wavelength) or equation (7) for by (h~ ) leads
gradients (from ho and the oscillation
in extemal
would
directly.
be
AT
0.01
°C,
which
not
to
measure
easy to
sample
with
of the
strong
a
light,
of
source
gradient
tension
surface
the
be
thickness
threshold
due
h~
thermal
either
~
The
festoons
«
gradient.
in
Our
~,
start
the
a
driven
is
of
the
fingers
«
rim
observed
»
which
breaks
at
constant
out
in
extemal
an
microdroplets.
velocity and leaves
thermal
But,
into
a
in
an
mark
wet
a
geometry
regime,
x
we
microdroplet
each
from
with
finger.
of large,
flat droplets
leads
the gradient is
localised
because
in
capillary length. In the central part
form
the
cases,
gradient,
extemal
different
very
are
»
both
In
simple
wedge
to
the
flat,
is
which
spreading laws
the droplet,
of
surface
in
in
tension
Marangoni
region of size
the
a
is
uniform.
Acknowledgements.
We
thank
P. G.
de
for
Gennes
reading
critical
a
of
the
manuscript.
References
[Ii HUPPERT H. E., Nature
(London) 300 (1982) 427.
[2] SILVI N. and DussAN E. B., Phys. Fluids 28 (1985) 5.
[3] MELD F., JOANNY J. F. and FAUVE S., Phys. Rev. Lett. 63 (1989) 1958.
[4] TROIAN S. M., Wu X. L. and SAmAN S. A., Phys. Rev. Lett. 62 (1989) 1496.
[5] CAzABAT A. M., HESLOT F., TROIAN S. and CARLES P., Nature
(London) 346 (1990)
[6] HOCKING L. M., J. Fluid Mech. 211 (1990) 372.
[7] TROIAN S. M.,
HERBOLzHEIMER
E., SAmAN S. A. and JOANNY J. F., Europhys. Lett.
824.
10
(1989)
25.
[8]
[9]
[10]
II Ii
[12]
[13]
[14]
J. B.,
BRzOSKA
BAscom
W.
BROCHARD-WYART
D.,
Adhesion,
WILLIAMS R.,
COTTINGTOM
F.
Nature
F.,
BROCHARD-WYART
(1991)
REDON C.,
(1992)
GENNES
DE
ADAMSON
Lord
Fowekes
and
and
(American
Ed.
F.,
RONDELEz
Europhys.
C. R.,
SINGLETERRY
Chemical
19 (1992) 97.
Angle, Wettability
Washington DC) p. 381.
and
Lett.
Contact
in
Society
(London)
266
(1977)
153.
HERVET
H.,
REDON
C.
and
RONDELEz
F., J.
Colloid
Inte~fiace Sci.
142
HERVET
H.
and
RONDELEz
F.,
Colloid
Inte~fiace
149
Phys. 57 (1985)
Chemistry Series 43
827.
518.
BROCHARD-WYART
F.,
J.
Sci.
580.
G.,
Rev.
Mod.
A. W.,
Adv.
in
P.
Washington
[15]
[16]
[17]
M.
F.
R. L.
DC,
RAYLEIGH,
LE
GRAND
DE
GENNES P. G.,
D.
and
Proc.
1964) pp.
London
GAINES
C-R-
G.,
Acad.
Contact
Angle,
Wettability
57-73.
Math
J.
Sci.
Sac.
10
(1879) 4.
ColloidInte~fiace
(Paris)
3o7 II
Sci.
(1988)
31(1969)
1841.
162.
and
Adhesion
(ACS