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Quantitative Spreading Kinetics of a Three Molecular Layer Liquid Patch
Olivier Noel, Jean-Luc Buraud, Laurent Berger, and Dominique Ausserre*
Laboratoire de Physique de l’Etat Condens
e, “Molecular landscapes and biophotonics” group, Universit
e du
Maine, Avenue Olivier Messiaen, 72000 Le Mans, France
Received December 14, 2009. Revised Manuscript Received March 9, 2010
The late stage kinetics of the spreading of a smectic nanodrop on a solid surface was investigated by direct and real
time imaging of a three molecular layer patch using the SEEC microscopy. Experimental data do not conform to the
only available theory, which covers only weakly stratified liquids. A new model is proposed, in remarkable agreement
with experiments, in which the spreading mechanism appears to be a quasi-static process ruled by solid/liquid
interactions, 2D Laplace pressure, and separate edge and surface permeation coefficients.
Finite extended molecular films on solid substrates present
considerable interest in both technological and scientific research.
On the technological side, their control is detrimental in many
applications such as surface patterning, cell adhesion, microfluidics, and biochips. On the fundamental side, they are particularly
interesting as they provide finite 2D or quasi-2D thermodynamic
systems.1-5 They can be obtained by the deposition of a nanodrop
in air, which evolves toward a molecular film by wetting a solid
surface. In practice, the molecules used to make surface patterns
are nonsymmetric and present, at least at the solid-liquid interface, some amphiphilic character. Indeed, because of the ordering
of the first molecular layers induced by the solid surface, every
liquid becomes more or less stratified in the vicinity of a surface,6
and this determines its wetting behavior.7-10 Therefore, understanding the wetting of solid surfaces by stratified liquids is a very
important task, and it has been the object of many studies.11-14
The most extensively studied system is the combination of 4-noctyl-40 -cyanobiphenyl (8CB) liquid crystal with oxidized silicon
wafers. It may be considered as the model system of the field.
Experimental studies on this system were mainly conducted using
ellipsometry,14 atomic force microscopy (AFM),12 or X-ray
reflectivity.13 They have shown that the liquid crystal partially
or totally wets a clean silicon oxide surface, depending on the drop
size.15 This behavior was observed in the whole range extending
from the solid/smectic transition temperature, 21.5 °C, up to
about the nematic/isotrope transition temperature, 40.5 °C.16
Even when the free surface of a macroscopic 8CB drop presents
*Corresponding author. E-mail: Dominique.ausserre@univ-lemans.fr.
(1) De Gennes, P. G.; Cazabat, A. M. C. R. Acad. Sci. 1990, 310, 1601–1606.
(2) Lazar, P.; Schollmeyer, H.; Riegler, H. Phys. Rev. Lett. 2005, 94, 116101-1–
116101-4.
(3) De Jeu, W. H.; Ostrovskii, B. I.; Shalaginov, A. N. Rev. Mod. Phys. 2003, 75,
181–235.
(4) Oshanin, G.; De-Coninck, J.; Cazabat, A.-M.; Moreau., M. Phys. Rev. E
1998, 58, R20–R23.
(5) Wasan, D. T.; Nikolov, A. D. Nature 2003, 423, 156–159.
(6) Israelachvili, J. N.; McGuiggan, P. M. Science 1988, 241, 795–800.
(7) De Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827–863.
(8) Heslot, F.; Fraysse, N.; Cazabat, A. M. Nature 1989, 338, 640–642.
(9) Forcada, M. L.; Mate, C. M. Nature 1993, 363, 527–529.
(10) Bardon, S.; et al. Faraday Discuss. 1996, 104, 307–316.
(11) Lucht, R.; Bahr, C. Phys. Rev. Lett. 2000, 85, 4080–4083.
(12) Xu, L.; Salmeron, M.; Bardon, S. Phys. Rev. Lett. 2000, 84, 1519–1522.
(13) Daillant, J.; Zalczer, G.; Benattar, J. Phys. Rev. A 1992, 46, R6158–R6161.
(14) Bardon, S.; et al. Phys. Rev. E 1999, 59, 6808–6818.
(15) Benichou, O.; et al. Adv. Colloid Interface Sci. 2003, 100-102, 381–398.
(16) Vandenbrouck, F. Thesis Universite-de-Paris VI, Paris France, Paris, 2001.
Langmuir 2010, 26(8), 6015–6018
a finite contact angle with the solid surface, the molecular
structure of the edge is made up of a sequence of flat terraces
with a decreasing thickness in the outward direction, lying on a
surface precursor film.12 It is composed itself of a well-defined
bilayer lying on top of a more complex monolayer. In the smectic
range (up to 33.5 °C), the thickness of the former was consistently
reported to be 3.3 nm, which is close to the bulk period 3.2 nm.15
The latter is described as being in a dense state in the region where
it is covered with at least one additional bilayer and as having a
density that smoothly decreases to zero outward into the region
where it is exposed to the vapor phase (see Figure 1). It was
interpreted as a 2D gas phase diffusing along the surface. X-ray
reflectivity showed that whatever its density, its physical thickness
has a constant value of 1.2 nm, slightly smaller than the bulk halfperiod.13 Complementarily, ellipsometry, which is sensitive to the
optical thickness of the monolayer, demonstrated that the surface
density is vanishing in the gas phase.14 When the drop volume is
very small, the entire drop is terraced and complete wetting is
observed.15 At some stage of the spreading kinetics, the structure
reduces to a single bilayer patch lying on the surface monolayer.
Here starts what we may call the late stage spreading, where the
last bilayer shrinks and vanishes while the surface monolayer
spreads all over the solid.
As far as spreading kinetics is concerned, the literature remains
very poor. Especially for late stage kinetics, only few experimental
results were reported, which were not consistent with available
theories.13,15-17 One reason for this is the difficulty to image
molecular layers with a sufficient sensitivity and resolution, in real
time, and over large fields. However, the quantitative study of the
late stage kinetics is particularly relevant in order to understand
the spreading mechanisms at a molecular level. Here we report for
the first time such quantitative spreading kinetics obtained from
real time and high-resolution optical microscopy measurements.
To our knowledge, the only theory dealing with the spreading
kinetics of terraced liquids is the one proposed by de Gennes and
Cazabat in ref 1. We show that this theory cannot reproduce our
results, which is not surprising since it was developed in order to
describe the behavior of weakly stratified liquids. Then we present
a new phenomenological model in perfect agreement with our
experimental data. This model brings new insights into the field.
(17) Betelu, S.; Law, B. M.; Huang, C. C. Phys. Rev. E 1999, 59, 6699–6707.
(18) Ausserre, D.; Abou-Khachfe, R. Langmuir 2007, 23, 8015–8020.
(19) Ausserre, D.; Valignat, M. P. Nano Lett. 2006, 6, 1384–1388.
Published on Web 03/19/2010
DOI: 10.1021/la904704u
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Noel et al.
Figure 2. Bilayer evolution as observed: (a) upper sequence, at
T = 22 °C; (b) lower sequence, at T = 26 °C. Time is progressing
from left to right in each line.
Figure 1. (a) Optical microscopy image of a 8CB nanodrop in the
late stage. (b) Scheme of the internal structure of the drop: a bilayer
on the top of a monolayer; the monolayer is receding outward;
arrows represent the various material flows. (c) 3D view of the
optical thickness profile of the drop.
All the experiments reported below were conducted using
the recent surface-enhanced ellipsometric contrast (SEEC) technique,18,19 which allows direct and real time visualization of
molecular layers on surfaces, with a much higher resolution than
ellipsometry or X-ray reflectivity. This technique commonly
provides height resolution better than 0.1 nm, so that direct and
real time visualization of molecular layers of 8CB becomes
possible. Wetting experiments were conducted using 4-n-octyl40 -cyanobiphenyl (8CB, 99.5%, Merck) on oxidized silicon surfaces, for different temperatures ranging from 21.5 to 33.5 °C. In
this temperature range, 8CB is in a smectic phase. Optical
observations were achieved through a polarization microscope
(Leica DMRX PlanApo Pol Fluotar 20 0.5 objective lens). The
solid support, a thin silicon wafer covered with a 106 nm thick
oxide layer, was cleaned with UV/ozone for 30 s and then was set
on a heating stage with (0.5 °C of accuracy. In order to install
stable and reproducible experimental conditions, the sample was
enclosed in a glovebag in which the relative humidity was kept
constant at 30 ( 5% during the experiment. Pure 8CB was
deposited onto the support using a homemade nanospotter
inserted into the glovebag. Microscope images were recorded
through a tri CCD camera (Optronics DEI 750). Image processing was limited to shadow corrections. The typical size of the
bilayer at the beginning of the late stage is some hundreds of
micrometers, and the typical duration of an experiment is 1 h.
In Figure 1a, we reproduce the image of the drop as it is seen
directly from the microscope during the late stage. The bright
patch is the bilayer, with radius R2; the diffuse corona (radius r >
R2) is the monolayer in the gas phase. Figure 1b presents the
schematic view of the internal drop structure that was described
above. The defect line topology appearing at the bilayer edge in
Figure 1b was inferred as a possible one. Another structure is
possible in which the dislocation line is shifted half a period
downward. The important point to notice is that the phenomenological model proposed in the present paper is not conditioned by
the exact structure of the edge defect. The different arrows in the
figure represent the various material flows induced by spreading.
Horizontal arrows demonstrate 2-dimensional intralayer flows.
Because of the radial drop shape, they have a radial symmetry.
The vertical arrows demonstrate the permeation flux, a molecule
per molecule material transfer from the bilayer toward the
6016 DOI: 10.1021/la904704u
monolayer. The curved shape arrow figures how the molecules
of the upper part of the bilayer, with zero radial velocity, pass into
the lower part of the bilayer during bilayer shrinking. Figure 1c
(the “Monument Valley Butte”) shows a topographic view of the
drop right after the late stage structure was attained. The intensity
values of Figure 1a were converted into height values Z by
inferring that image intensity is a regular function of the optical
thickness of the liquid crystal.18 This function was approximated by a second-order polynomial that was determined using
the values known from previous experiments,14,15 i.e., 4.5 nm for
the trilayer thickness, 1.2 nm for the monolayer thickness at the
junction with the bilayer edge, and zero thickness on the bare
substrate. Since the drop shape during spreading remains circular,
it was easy to calculate the average radial thickness profile Z(r) in
the drop from the topographic data. From there, we were able to
probe the time evolution of the total volume of the drop, including
the gas monolayer. Within a 20% error margin, we found this
volume to remain constant for the entire experiment. It confirms
the results already reported by other groups13,15 that in ambient
conditions evaporation is negligible over some days. Once this
point is clarified, it becomes possible to elucidate the kinetic law
that governs material flow from the last bilayer into the surface
monolayer, until the bilayer vanishes completely. Hereafter, we
report our experimental observations and discuss them in
the frame of classical and new theoretical models.
Figure 2 presents two series of top view images of 8CB bilayers
that were arbitrarily extracted from kinetic experiments conducted at two different temperatures: right above the solid/
smectic transition temperature (22 °C) and further away
(26 °C). Both sequences start when spreading of the initial
stratified droplet becomes reduced to the trilayer structure. The
monolayer corona, which corresponds to a smooth intensity
profile extending far beyond the image frame, is not visible.
Shrinking of the bilayer occurs due to material permeation1,20
from the bilayer into the monolayer. Two different behaviors
were observed.
In the upper sequence (a), the bilayer stays dense while
shrinking. At the end of the process, the last molecules to
disappear are located at the center of the initial disk. The domain
evolution is completely deterministic. In the lower sequence (b),
nucleation and growth of holes is observed in the bilayer, in
addition to shrinking.13 Moreover, holes are not uniformly
distributed over the layer but preferentially located in the outer
part of the disk. Ostwald ripening and holes coalescence lead to
the formation of one or several complex bilayer domains before
completely vanishing. These patterns are conditioned by the
random location of the holes that nucleate and, therefore, are
intrinsically unpredictable. In contrast, the deterministic evolution observed at low temperature allows direct comparison with
(20) Helfrich, W. Phys. Rev. Lett. 1969, 23, 372–374.
Langmuir 2010, 26(8), 6015–6018
Noel et al.
Article
theory. Then, we chose to focus on the low temperature regime as
a priority.
The late stage spreading kinetics of stratified liquids was
predicted by de Gennes and Cazabat1 (dGC). Their model was
originally intended to describe the spreading behavior of weakly
layered liquids, for which layering is due to presmectic effects
induced by the solid surface plane. In this model, the driving force
of the spreading is the attractive interaction profile between the
liquid and the solid. This interaction includes a long-range
contribution which is a decreasing function of the distance from
the solid and a short-range contribution affecting only the first
molecular layer. On the other hand, the limiting forces are
twofold: The first one is resistance to molecular permeation
across the interface between adjacent layers, a thermally activated
process. The second one is friction between adjacent layers, which
are slipping against each other (see Figure 1c). Assuming quasiisotropic molecular diffusion in the liquid, dissipation in monolayer slipping becomes so high that the permeation flow is
confined to a thin permeation ribbon near the bilayer edge. The
width of this ribbon, which remains a small portion of the
interlayer interface, was referred to as the permeation length ξ.
It is related to the various (resistive) interlayer friction coefficients
ζij and to the (accelerating) permeation coefficient C12 between
the bilayer and the first monolayer by the relationship ξ-2=(ζ01 þ
4ζ12)(c12/l ) , where l l1=2l2 is a thickness which is common to
the two layers of the model. From this theory, the kinetic law
obtained for a two layer system is
dR2
¼
dt
W2 -W1
R2
ζ01 R2 ln
R1
ð1Þ
where Wi is the interaction potential of the i index layer with the
solid, ζ01 is the friction coefficient between the solid and the liquid,
and R1 and R2 are respectively the radii of the innermost and
outermost layer from the solid. One might notice that dR2/dt is
independent of ξ, provided that ξ , R2.
However, our experimental results cannot be directly compared with eq 1 for two obvious reasons. The first one is that our
system is composed of one monolayer and one bilayer, the latter
being terminated at the edge by a dislocation line, characteristic of
amphiphilic molecules, while in the dGC model, two identical
layers were considered. The assumption that the monolayer and
the entire bilayer have the same thickness is far from our
experimental reality since we actually have 2l2 . l1. However, it
was shown in ref 17 that taking into account the various layer
thicknesses affects only prefactors in the kinetic law R2(t). The
second one is that in our case the geometry of the liquid layers (see
Figure 1) imposes R1 = R2, which is not compatible with eq 1.
Indeed, molecular diffusion in the gas region of the monolayer
(r > R2) is faster than convective motion in the liquid region (r <
R2). Therefore, the gas region has no impact on the liquid region
kinetics. One could also overcome this difficulty and adapt the
dGC model to the present system, but then we would encounter
three much more serious limitations. The first one is the basic
assumption of the dGC model ξ/R2 , 1. This assumption is true
for isotropic liquids, but it breaks down with highly stratified
liquids such as smectic liquid crystals because in such liquids the
diffusion coefficient is so anisotropic that a layer has enough time
to equilibrate before being significantly affected by permeation.
The second one is that the two-dimensional Laplace pressure in a
layer was ignored, making the model inappropriate for nanodrops. The third one is that it was not considered that permeation
Langmuir 2010, 26(8), 6015–6018
toward the surface monolayer could be easier from the defect line
at the bilayer edge than from rest of the bilayer.
Fully revisiting the dGC theory is beyond the scope of the
present work. However, a phenomenological description that
overcomes these three limitations is proposed hereafter. In this
model, we consider two driving forces for spreading. One is
the molecular interactions, which result in a pressure difference
(W2 - W1)/Σ, Σ denoting the area per molecule. The other one is
line tension, which results in the Laplace pressure difference
(τ2 - τ1)/R2, τi denoting the line tension at the liquid edge of
layer i. Each of them generates material permeation at two levels:
at the plane interface between the bilayer and the monolayer and
at the line interface between the bilayer edge and the monolayer.
These two levels must be distinguished because the energy barrier
that the molecules must overcome in order to jump from the
bilayer to the monolayer is a priori lower at the bilayer edge than
in the rest of the bilayer domain. Resulting fluxes correspond
to the different combinations of driving forces and exchange
channels. Hence, we get the kinetic law for bilayer shrinking:
dS2
τ2 -τ1 W2 -W1
¼ -R
½S2 þ λP2
þ
ð2Þ
dt
R2
Σ
Here S2 and P2 hold for area and perimeter of the bilayer.
Dimensional coefficients R and Rλ are proportional to surface
and edge permeation coefficients, respectively. Their ratio is
characterized by the length λ. When writing eq 2, we implicitly
assumed that the hydrodynamic contribution to the 2D pressure
is negligible when compared to other terms. In other words, we
considered that the evolution of the two layers is quasi-static and
that it is regulated by permeation. This is the essence of the
approximation ξ/R2 . 1, which is the exact opposite of the dGC
model approximation. The kinetic law for R2 given by eq 2 has the
following form:
-T
dR2 L1 L2
¼
þ ðL1 þ L2 Þ þ R2
dt
R2
ð3Þ
It is ruled by the three parameters T=(2/R)[Σ/(W2 - W1)], L1 =
[(τ2 - τ1)Σ]/(W2 - W1), and L2=2λ. The first one, T, is a natural
time scale for the experiment. L1 and L2 are two characteristic
lengths to be compared to R2 in order to identify the dominant
driving force and/or dominant permeation mechanism. When
R2 , L1, i.e., when the drop is small enough, the main driving
force for spreading becomes Laplace pressure. On the other hand,
when R2 , L2, permeation is dominated by the edge rather
than by the surface. When R2 , L1 or R2 , L2, the third term
in eq 3 is negligible during the entire kinetic experiment. If R2 , L1
and R2 , L2, the second term becomes also negligible. The
relative values of L1 and L2 condition the overall shape of the
kinetic law R2(t). By looking at eq 3, one readily sees that the
lengths L1 and L2 play exactly symmetric roles. Integral form of
eq 3 is
L1
R2
L2
R2
t -tf
þ
¼
ln 1 þ
ln 1 þ
ð4Þ
L2 -L1
L1
L1 -L2
L2
T
where tf refers to the end of the kinetics and is known from the
experiment. However, this form is valid only if L1 6¼ L2 .When
L1 =L2 L, eq 4 is replaced by
R2
R2
t -tf
þ
¼
-ln 1 þ
L
L þ R2
T
ð5Þ
It is the continuous limit of eq 4. Notice that eq 3 then becomes
DOI: 10.1021/la904704u
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Figure 3. Bilayer radius R2 as a function of time t. Left side:
(black þ) experimental values; (red -) values calculated using
the best parameters in eq 4 or 5; (green -) values calculated using
the best parameters in eq 1 and assuming a dense liquid monolayer.
Right side: (red þ) radius versus time distance Δt between experimental and calculated data.
-T(dR2/dt)=(L2/R2) þ 2L þ R2, exhibiting the two dependent
crossover lengths L and 2L.
Figure 3 shows on the same graph the experimental evolution
of the bilayer radius and the evolution expected from eq 4 after
adjusting T, L1, and L2 to their best values, the two latter
parameters being indiscernible although a priori different. It leads
to T=6840 s, L1=(50 þ ε) μm, and L2=(50 - ε) μm, with ε very
small. Using these parameter values, the agreement between the
model and the experimental data is excellent. Fitting the experimental data with eq 5 gives T=6840 s and L=50 μm with similar
quality. The L value is within the experimental range of R2
(0.5-100 μm), showing that none of the three terms in eq 3 can
be neglected.
6018 DOI: 10.1021/la904704u
Noel et al.
As said previously, eq 1 does not apply to the present geometry.
However, since it is the standard reference, we tried to extend its
use by assuming, as it was done in ref 1, that the monolayer stays a
dense 2-dimensional liquid everywhere on the surface, i.e., by
stating that the sum of the monolayer and bilayer volumes remains
constant. The results obtained in that way are very poor. They are
reported using the green curve in Figure 3 for comparison.
Considering the symmetry of eqs 2 and 3 with respect to L1 and L2
and considering also that the best fitting values of L1 and L2 are
identical, one may wonder if the two parameters should not always
merge into a single one. Were it the case, the entire kinetic problem
would be ruled by a single characteristic length L. This question
cannot be clarified without going deeper into the theory. At this level,
one should consider that the relationship L1=L2 may be accidental.
From our understanding, there are two important lessons in
this work. The first is the role of 2D Laplace pressures in stratified
nanodrop kinetics. We believe that considering these forces may
even help to understand some static properties of these liquids
which are still unclear. For instance, the transition from partial to
complete wetting, depending on drop size, might be understood as
an instability driven by Laplace pressures. The second is that the
spreading kinetics of stratified liquids can be treated in first
approximation as a quasi-static phenomenon, which renders its
model representation much simpler than previously thought. For
instance, a direct consequence is that residual evaporation, as it
could be encountered with materials other than 8CB, would not
qualitatively change the kinetic laws established in the present
work. Finally, it is worth recalling that such questions can be
envisaged only because the SEEC microscopy technique allows
direct imaging of molecular films.
Acknowledgment. This work was entirely supported by ANR
under Project PNANO-07-050. We also thank Stephane Joly
(Nemoptics) for his help with the materials and Marilyn Twell for
her help with the English.
Langmuir 2010, 26(8), 6015–6018