# Modification of the fluctuation dynamics of ultra-thin wetting films

C. Clavaud  CNRS, Sciences et Ingénierie de la Matière Molle, ESPCI Paris, PSL
Research University, Sorbonne Université, 75005 Paris, France  M. Maza-Cuello
CNRS, Sciences et Ingénierie de la Matière Molle, ESPCI Paris, PSL Research
University, Sorbonne Université, 75005 Paris, France  C. Frétigny  CNRS,
Sciences et Ingénierie de la Matière Molle, ESPCI Paris, PSL Research
University, Sorbonne Université, 75005 Paris, France  L. Talini
<EMAIL_ADDRESS>CNRS, Surface du Verre et Interfaces, Saint-Gobain,
93300 Aubervilliers, France  T. Bickel  Univ. Bordeaux, CNRS, Laboratoire
Ondes et Matière d’Aquitaine (UMR 5798), F-33400 Talence, France

###### Abstract

We report on the effect of intermolecular forces on the fluctuations of
supported liquid films. Using an optically-induced thermal gradient, we form
nanometer-thin films of wetting liquids on glass substrates, where van der
Waals forces are balanced by thermocapillary forces. We show that the
fluctuation dynamics of the film interface is strongly modified by
intermolecular forces at lower frequencies. Data spanning three frequency
decades are in excellent agreement with theoretical predictions accounting for
van der Waals forces. Our results emphasize the relevance of intermolecular
forces on thermal fluctuations when fluids are confined at the nanoscale.

When shared with air or another fluid, a liquid interface appears
macroscopically flat but is actually corrugated by thermal motion at smaller
scales. Although of subnanometric amplitude, thermal capillary waves play a
central role in phenomena such as drop formation [1], film break-up [2], jet
destabilisation [3, 4], as well as wetting [5, 6] or dewetting [7] of liquid
films on solid substrates. As first described by Vrij [8] and Sheludko [9],
the rupture of a non-draining liquid film originates from the amplification of
interfacial fluctuations with wavelengths larger than a critical value, which
is set by the balance between surface tension and intermolecular forces. The
lifetime of metastable supported thin liquid films is therefore ruled by the
thermal roughness of their interface. In turn, fluctuations are themselves
modified as the thickness of the liquid layer decreases since confinement
enhances hydrodynamic dissipation [10, 11].

Intermolecular forces are also expected to have a significant effect on the
fluctuations of ultra-thin films. Indeed, dipolar interactions between the
solid substrate, the liquid layer and the gas phase become relevant when the
thickness of the film is smaller than a few tens of nanometers [12]. The
resulting van der Waals forces can be either attractive or repulsive. In the
case of attractive interactions, the fluctuations are amplified by
intermolecular forces and the film is predicted to be unstable [13, 14, 15,
16]. This is confirmed by experimental observations on dewetting polymer films
of nanometric thickness [7]. Still, in this nonequilibrium situation,
measurements are limited to the growth of the more unstable mode and the full
spectrum of thermal capillary waves remains unexplored. In contrast, the
fluctuation amplitude of a liquid layer with repulsive liquid/air and
liquid/solid interactions is expected to decrease under the action of
intermolecular forces. This qualitatively explains the formation and stability
of ultra-thin films of wetting liquids, such as the precursor films involved
in the spreading of a macroscopic droplet. Nevertheless, the effect of van der
Waals forces on interfacial fluctuation has never been evidenced
experimentally. More generally, investigations regarding thermal capillary
waves at the interface of nanoscale liquid films remain almost exclusively
theoretical.

Thermal fluctuations of liquid interfaces were first characterized more than
forty years ago using light scattering techniques [17]. They proved to be an
accurate and non-invasive probe of the macroscopic properties of liquids,
e.g., surface tension or viscoelasticity, that can be deduced from the
capillary wave spectrum for an infinite medium [18]. Owing to experimental
limitations, measurements have long been limited to very thick liquid layers.
The effect of confinement has been addressed more recently thanks to the
development of novel techniques based on either the deflection of a reflected
laser beam [19, 20] or x-ray scattering [21, 22]. Both techniques differ by
the range of the wavevectors that can be probed in the experiments. Small
wavelength x-ray measurements are better suited for highly viscous fluids such
as high molecular weight polymer melts, whereas laser deflection can be used
for simple, low-viscosity liquids. Enhancement of experimental accuracy now
makes it possible to investigate the modification of the fluctuation dynamics
in very thin films [22, 23]. Hydrodynamic boundary conditions at the
liquid/solid interface were also successfully tested [24].

However, the contribution of intermolecular forces on the capillary spectrum
has never been evidenced yet. The experimental challenges are multiple since
stable films of controlled nanometric thickness must be produced in the first
place. Then, interfacial deformations have to be probed at the molecular
scales and over long times. To achieve these goals, we make use of the
thermocapillary effect to create ultra-thin films of wetting liquids. We then
measure the fluctuation dynamics of the interface in a large frequency range
using the previously developed Surface Fluctuation Specular Reflection
technique [19]. We further compare the experimental data with theoretical
predictions based on the lubrication approximation. Our analysis therefore
shows that the thickness of the liquid layer results from a balance between
thermocapillary and intermolecular forces, and that the fluctuation spectra
are dominated by van der Waals forces at low frequencies.

Figure 1: Schematized experimental set-up (not to scale). The laser is focused
on the liquid film spread on a colored glass substrate. Light absorption by
the substrate creates a thermal gradient in the liquid layer over the
lengthscale $L$, which also define the extension of the film thinning.

Silicone oils (Rhodorsil) of two different viscosities ($9.3$ and $19$
mPa$\cdot$s at room temperature) are spin-coated on colored glass substrates
(Hoya Longpass optical filters, diameter $50$ mm, thickness 2.5 mm) to form
films of uniform initial thickness, between 250 and 1150 nm. The orange color
of the glass substrate (optical index $n=1.526$) is chosen for its poor but
non-zero absorption of the He-Ne laser light (wavelength 632.8 nm). The spin-
coated films are placed in the set-up described in Fig. 1. The laser is
strongly focused on the film with a $\times 50$ microscope objective. The
other optical elements are used to avoid back laser light and to collect the
light reflected by the interface on a two-quadrant photodiode. Special care is
taken to suppress parasitic light on the photodiode and to place the liquid
film at the laser beam waist within the Rayleigh length ($\approx 20~{}\mu$m).
The reflected beam is centered on the two quadrants and the variations of its
position are detected by computing the Power Spectrum Density (PSD) of the
difference between the voltages delivered by the quadrants. Owing to either
low or high-frequency parasitic noise, fluctuation spectra are obtained for
frequencies ranging from 1 Hz to 1 kHz.

As previously described [24], the laser is used for three different purposes.
First, it induces a thermocapillary flow in the liquid film, so that the
thickness decreases locally. Second, the laser reflections at the air/liquid
and liquid/solid interface interfere and are collected on the photodiode,
allowing accurate measurements of the film thickness by interferometry.
Lastly, the laser is used to measure the fluctuations of the slope of the
liquid layer interface, which result in fluctuations of the position of the
reflected beam.

We first focus on the thinning of the oil film due to the thermocapillary
effect. The colored glass substrate is slightly heated by the laser beam. The
temperature of the film therefore increases by thermal diffusion within a few
seconds. Convection is negligible in the liquid phase since the thermal Péclet
number is much smaller than unity. The resulting surface tension gradient
creates a liquid flow directed away from the laser beam, which induces a local
thinning of the film (see Fig. 1). Since the absorption length of the glass
substrate is very large compared to its thickness ($=2.5$ mm), it is the
latter millimetric length that sets the lateral scale of the thermal gradient,
and hence of film thinning. The extension of the thin flat region is therefore
three orders of magnitude larger than the actual beam size ($R=1.5~{}\mu$m).

To describe the thinning of the film, we denote $\theta(r)=T(r)-T_{0}$ the
temperature rise, with $T_{0}$ the room temperature and $r$ the radial
coordinate with origin at the beam center. The temperature rise being small
($<1$ K, see below), we neglect the variations of the physical properties of
the liquid that are at most 1% of their equilibrium values. The temperature
dependance of the surface tension is only accounted for in the thermocapillary
stress ($\approx 0.1$ Pa) resulting from the thermal gradient ($\approx
10^{3}$ K$\cdot$m-1). The film thickness $h(r,t)$ then follows the thin-film
equation [25, 26]

$\partial_{t}h=\frac{1}{3\eta
r}\partial_{r}rh^{3}\left\\{\partial_{r}\left[-\gamma_{0}\nabla^{2}h+\phi^{\prime}(h)\right]+\frac{3\gamma_{\theta}}{2h}\frac{\text{d}\theta}{\text{d}r}\right\\}\
,$ (1)

with $\eta$ the liquid viscosity, $\gamma_{0}$ the equilibrium surface
tension, and $\gamma_{\theta}=|\partial\gamma/\partial\theta|$ the derivative
of surface tension with respect to temperature. Since we focus on sub-micron
scales, gravity can safely be neglected. The first term in parentheses
corresponds to the Laplace pressure. The second term $\phi^{\prime}(h)$ is the
derivative of the van der Waals interaction potential (per unit area); it is
the opposite of the so-called disjoining pressure. Consistently with the
literature on oil films [27], we assume $\phi^{\prime}(h)=A/(6\pi h^{3})$ with
a negative Hamaker constant $A<0$. Both intermolecular and Laplace forces
oppose the thinning of the film, which is driven by the thermocapillary stress
— the third term in parentheses.

Figure 2: Thickness of film of a silicone oil (viscosity $9.3$ mPa$\cdot$s)
under the laser beam as a function of time. Insert shows details of the
plateau corresponding to $h_{0}=10.0\pm 0.3$ nm.

A comprehensive study of the film thinning kinetics will be detailed in a
future work. We focus here on the stationary version of Eq. (1). Neglecting
the Laplace pressure term, a double integration of Eq. (1) yields the
stationary thickness under the beam $h_{0}=h(r=0)$ [26]

$h_{0}=\left(\frac{|A|}{6\pi\gamma_{\theta}\theta(0)}\right)^{1/2}\ ,$ (2)

where $\theta(0)$ is the maximum excess temperature. The thickness is
therefore set by $\theta(0)$ but is independent of both the initial film
thickness, the viscosity and the details of the temperature profile. With
$\gamma_{\theta}=6.5\times 10^{-5}$ N$\cdot$m${}^{-1}\cdot$K-1 [28] and taking
for instance $\theta(0)=1$ K, we predict a stationary thickness $h_{0}\approx
9$ nm for $|A|=10^{-19}$ J.

We show in Fig. 2 an exemple of the measured variation of an oil film
thickness. After an initial decrease, the stationary state is reached within a
few minutes. The thickness value at the plateau $h_{0}=10.0\pm 0.3$ nm remains
remarkably stable for several hours. Let us emphasize that the liquid plateau
is very flat since it actually extends over the millimetric size of the
thermal gradient. The original Eq. (2) expresses the balance between
intermolecular and thermocapillary forces. From this perspective, our
experiment is a novel analogous to the Thin Film Pressure Balance, in which
the disjoining pressure of a liquid film is measured at equilibrium with an
imposed pressure [29]. Here, the film thickness is simply controlled by
adjusting the intensity of the laser beam. Using a variable neutral density,
we can check that thickness varies as predicted with laser intensity (see Fig.
S1 in the Supplemental Material [26]). Stationary thicknesses ranging from 8
to 30 nm are obtained by tuning the laser intensity.

In the following, we take advantage of the long-lasting stability of the thin
films to measure fluctuation spectra over a wide frequency range. We
especially focus on low frequencies that are most sensitive to intermolecular
forces. As already mentioned, the same laser is used to measure interfacial
fluctuations. More precisely, it is the fluctuating slope of the liquid
surface that results in fluctuations in the position of the specularly
reflected beam off the surface. In contrast to scattering experiments, in
which a single surface mode is selected, the contributions of all the surface
modes are measured but with different weights. Actually, the finite radius $R$
of the beam sets the lenghtscale of the measurement and the main contribution
to the signal comes from the fluctuating modes with wavevectors $q\sim 1/R$
[19]. The power spectral density (PSD) can then be quantitatively expressed as
$S(\omega)\propto\int_{0}^{\infty}P(q,\omega)\Phi(q)\text{d}q$ — see Ref. [24]
for a comprehensive discussion of $S(\omega)$ and all the parameters on which
it depends. The weight function $\Phi(q)$ only depends on the beam
characteristics (size and divergence), and is maximum for $q=2\pi/R$. The mode
density $P(q,\omega)$ represents the PSD of the liquid film:
$P(q,\omega)=\langle|\tilde{h}_{\mathbf{q},\omega}|^{2}\rangle$, with
$\tilde{h}_{\mathbf{q},\omega}$ the Fourier transform of the height
$h(\mathbf{r},t)$.

The mode density can be obtained in the lubrication regime $qh_{0}\ll 1$, with
$h_{0}$ a few tens of nanometers. Note that the lubrication theory cannot
always be used to describe thin film dynamics: for instance, it has been shown
recently to fail to describe the very last moments before film rupture [30].
Here however, the weight function $\Phi(q)$ is maximum for $q\sim 1/R$, with
$R=1.5~{}\mu\text{m}\gg h_{0}$. The lubrication approximation is therefore
fully legitimate (see SM for the derivation of the fluctuation spectra beyond
the lubrication approximation [26]). Our derivation of the stochastic version
of the thin film Eq. (1) then follows Ref. [13]. Assuming small deviations
with respect to the stationary height $h_{0}$, we can write
$h(\mathbf{r},t)=h_{0}+\delta h(\mathbf{r},t)$ with $|\delta h|\ll h_{0}$. The
thin film equation can then be linearized and one obtains at lowest order [26]

$\partial_{t}\delta
h=\frac{h_{0}^{3}}{3\eta}\left[\phi^{\prime\prime}(h_{0})\nabla^{2}\delta
h-\gamma_{0}\nabla^{4}\delta h\right]+\kappa\bm{\nabla}\cdot\mathbf{f}\ ,$ (3)

where the stochastic term $\mathbf{f}=(f_{x},f_{y})$ follows a Gaussian
distribution with ensemble average $\langle f_{i}(\mathbf{r},t)\rangle=0$ and
$\langle
f_{i}(\mathbf{r},t)f_{j}(\mathbf{r}^{\prime},t^{\prime})\rangle=\delta_{ij}\delta(\mathbf{r}-\mathbf{r}^{\prime})\delta(t-t^{\prime})$.
The noise amplitude is $\kappa=\sqrt{2k_{B}T_{0}h_{0}^{3}/(3\eta})$. It is
important to note that the thermocapillary stress happens to be irrelevant as
far as fluctuations are concerned. Indeed, although it sets the stationary
thickness of the film, its contribution to the fluctuation spectrum is several
orders of magnitude smaller than the capillary stress and the disjoining
pressure [26]. It is then straightforward to show that the mode density
$P(q,\omega)$ is a Lorentzian function of the frequency

$P(q,\omega)=\frac{k_{B}T_{0}}{\pi}\frac{\Gamma(q,h_{0})}{\omega^{2}+\omega_{q}^{2}}\
.$ (4)

The relaxation rate is given by
$\omega_{q}=\frac{\gamma_{0}}{\lambda^{2}}\Gamma(q,h_{0})(1+q^{2}\lambda^{2})$,
with $\Gamma(q,h_{0})=q^{2}h_{0}^{3}/(3\eta)$ the dissipation kernel. Since
gravity is negligible, the capillary length is replaced by $\lambda=\sqrt{2\pi
h_{0}^{4}\gamma_{0}/|A|}$. This lengthscale lies in the micrometer range for
typical values $|A|\approx 10^{-19}$ J, $\gamma_{0}\approx 10^{-2}$
N$\cdot$m-1, and $h_{0}\approx 10^{-8}$ m. As a consequence, the contributions
from both capillary and intermolecular forces are equally significant in the
PSD. Interestingly, $\lambda$ also corresponds to the critical wavelength of
the unstable modes leading to the rupture of metastable films [8, 13]. Eq. (4)
moreover reveals that the main contribution of van der Waals forces is to
shift the low cut-off frequency of the Lorentzian function. Indeed, no
modification of the capillary spectrum is expected at high frequencies, where
$P(q,\omega)\sim\omega^{-2}$. In contrast, one expects a signature of van der
Waals forces on the fluctuation spectrum at low frequencies, below a few tens
of Hz taking the same numerical values as above.

Figure 3: Fluctuation spectra of oil films of thickness $30.3$ nm (viscosity
$19$ mPa$\cdot$s, blue, left axis) and $12.9$ nm (viscosity $9.3$ mPa$\cdot$s,
red, right axis). Spurious peaks result from parasitic noise either of
mechanical or electromagnetical origin. For each spectrum, the full lines
represent the theoretical spectra without (grey lines) and with (black lines)
intermolecular forces effects. Dash-dotted lines correspond to spectra
computed with Hamaker constants differing by $\pm 50\%$.

We thus anticipate a significant contribution of intermolecular forces for
thicknesses smaller than a few tens of nanometers and at low frequencies. Fig.
3 displays the experimental spectra of two different oil films with two
different thicknesses: $12.9$ nm (viscosity $9.3$ mPa$\cdot$s) and $30.3$ nm
(viscosity $19$ mPa$\cdot$s). The data are obtained in the stationary regime
and averaged over at least 20 spectra (the acquisition time of each spectrum
being 50 s). The experimental spectra are in excellent agreement with the
theoretical prediction obtained by numerical integration of the mode density
Eq. (4). Tabulated values of viscosity, surface tension, and refractive
indexes were used, together with measured values of film height and beam
radius. The only adjusted parameter is therefore the Hamaker constant: we get
$|A|=1.2\times 10^{-19}$ J and $|A|=2\times 10^{-19}$ J for the less and more
viscous oil, respectively. For comparison, the theoretical spectra without the
contribution of intermolecular forces are also shown. Clearly, it is the low-
frequency part of spectra which is most strongly modified by van der Waals
forces. One can notice that the Hamaker constants for silicon oils on glass
are one order of magnitude larger than for alkane films on mica, but they are
very close to alkane films on steel [27]. Silicone oils being known to be
particularly difficult to remove from solid surfaces, this finding
consistently indicates a strong repulsion between glass/oil and oil/air
surfaces.

Figure 4: Fluctuation spectra (in logarithmic scale) of oil films (viscosity
$19$ mPa$\cdot$s) at $4$ Hz and $8$0 Hz, as a function of film thickness. The
full lines correspond to theoretical predictions without (lighter lines) and
with (darker lines) intermolecular forces, with $|A|=10^{-19}$ J and
$|A|=2\times 10^{-19}$ J. Inset: Fluctuation spectra (in linear scale) at $4$
Hz of the two oil films with viscosity $19$ mPa$\cdot$s (blue circles, same
data as main figure) and $9.3$ mPa$\cdot$s (red circles). Vertical error bars
correspond to the statistical uncertainty. Horizontal error bars are too small
to appear.

Additional experiments are performed where the film thickness is varied by
adjusting the laser intensity. In each experiment, the fluctuation spectra of
a stationary thin film are measured at two different frequencies, respectively
4 Hz and 80 Hz. The corresponding values of the PSD, respectively averaged
over a $0.8$ Hz-wide and $16$ Hz-wide frequency range, are reported in Fig. 4
as a function of the film thickness. Note that for the thicker films,
measurements are conducted before reaching the plateau as the film is thinning
down. Only three spectra could then be recorded and the large error bars
result from poor averaging. All data points are in excellent agreement with
our theoretical predictions. Fig. 4 confirms that the effect of intermolecular
forces is most significant for the lower frequencies and for smaller
thicknesses. In contrast, intermolecular forces do not modify the spectra at
the larger frequencies, even for film thicknesses below 10 nm. Similar data
are obtained for the two different oils. Note that the Hamaker constants are
expected to be very close since optical indexes of the two silicon oils only
differ by 0.1%.

To summarize, we have developed a novel method to form oil films of tunable
nanometric thicknesses using a laser-induced thermocapillary effect. The films
are stable for several hours, allowing for the characterization of the
fluctuation dynamics of their interface. We thus present the first evidence of
the contribution of intermolecular forces on the fluctuation spectra of
supported liquid films, over a large frequency range. The experimental data
are fully described by our theoretical predictions accounting for van der
Waals forces, the Hamaker constant being the only adjusted parameter. A
significant decrease of the amplitude of the spectra is observed at low
frequencies, showing the relevance of intermolecular forces in nanometric
liquid layers. Our results therefore provides new insight on the dynamics of
ultra-thin liquid films.

###### Acknowledgements.

This project has received funding from the European Union’s Horizon 2020
research and innovation program under the Marie Sklodowska-Curie grant
agreement No 754387.

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