Table Of ContentProbing viscoelastic properties of a thin polymer (cid:28)lm sheared between a beads layer
and quartz crystal resonator
J. LØopoldŁs∗ and X.P. Jia†
Laboratoire de Physique des MatØriaux divisØs et Interfaces,
UMR 8108 du CNRS, UniversitØ de Marne la VallØe, CitØe Descartes,
5 Bd Descartes, 77454 Marne la VallØe cedex 2, France
(Dated: January 7, 2009)
Wereportmeasurementsofviscoelasticpropertiesofthinpolymer(cid:28)lmsof10-100nmattheMHz
range. Thesethin(cid:28)lmsarecon(cid:28)nedbetweenaquartzcrystalresonatorandamillimetricbeadlayer,
producinganincreaseofbothresonancefrequencyanddissipationofthequartzresonator. Theshear
modulusanddynamicviscosityofthin(cid:28)lmsextractedfromthesemeasurementsareconsistentwith
the bulk values of the polymer. This modi(cid:28)ed quartz resonator provides an easily realizable and
9 e(cid:27)ective tool for probing the rheological properties of thin (cid:28)lms at ambient environment.
0
0 PACSnumbers:
2
n
From wet sand to the eye cornea, liquid systems con- urationshavebeenproposedtodeterminetheshearmod-
a
J (cid:28)nedintosmallvolumesareubiquitousinnatureandare ulus of thin (cid:28)lms from the frequency shifts of quartz res-
known to alter friction and adhesion at a solid-solid in- onators, including the build-up of composite resonators.
6
terface [1, 2, 3, 4]. Moreover the mechanical properties For example, coating the (cid:28)lm of interest with second
] andstabilityofthin(cid:28)lmsisofparamountimportancefor overlayer (sandwich con(cid:28)guration) allows enhancing the
t
f a number of applications requiring speci(cid:28)c nanometric shear stress and characterizing its rheological properties
o coatings such as optical re(cid:29)ectors or dielectric stacks. It down to nanometric thicknesses [13]. In this Letter, we
s
is then natural to ask weather the properties of con(cid:28)ned describe a new approach to probe the viscoelastic prop-
.
at liquids are similar to their bulk counterpart. Conven- ertiesofthin(cid:28)lmsbyusingaconventionalQCM.Tothis
m tional mechanical testing is not adapted for thin (cid:28)lms end, we deposit gently a layer of spherical bead (glass)
investigation and speci(cid:28)c metrology is needed. Natu- on the top of a (cid:28)lm of thickness h∼10−100 nm coated
-
d rally occurring instabilities such as wrinkling or dewet- on the surface of a quartz crystal. The resulting shifts of
n tingprovidesvaluableinformationaboutrheologicaland resonance frequency and inverse of quality factor can be
o
mechanicalpropertiesofthinpolymer(cid:28)lms [5,6]. Oscil- readilyrelateddirectlytotheelasticmodulusandviscous
c
latory interfacial rheology conducted with a tribometer dissipation of the (cid:28)lm. An entangled polymeric thin (cid:28)lm
[
[7] and the Surface Force Apparatus (SFA) [8, 9] o(cid:27)ers
1 a complementary description of thin (cid:28)lms over several
v
decades of frequency. Local probes analysis performed
0
with Atomic Force Microscopy con(cid:28)rms the result ob-
3
tained with the SFA [10]. Note that the rheological be-
6
0 haviour at high frequency can not be investigated with
. these techniques. However it is important for applica-
1
tions such as hard disk drives.
0
9
Acoustic wave devices have proven to be suitable for
0
the high frequency investigation [11, 12]. Among oth-
:
v ers,quartzcrystal(AT-cut)resonatorsoperatinginshear
i
X mode at ultrasonic frequency from 1 − 100 MHz have
been the most widely used to monitor the viscoelastic
r
a properties of thick (cid:28)lms (h > 1µm) adsorbed on their
surfaces [11]. However for thin (cid:28)lms, i.e. h ≤ 100 nm,
no signi(cid:28)cant shear strain is produced inside it because
the (cid:28)lm is located at an antinode of the standing wave.
In such a case, the shift of resonance frequency is only
FIG.1: Sketchoftheexperimentandmodelingofthesystem
a function of the mass alone and the acoustic properties
of the (cid:28)lm may be ignored: the quartz crystal behaves
(polydimethylsiloxane, Mw ∼ 90000) is deposited on a
simply as microbalance (QCM). Recently, several con(cid:28)g-
quartz resonator of f =ω /2π ∼5MHz (Maktex) (Fig-
0 0
ure 1). The surface of the crystal is polished and gold-
coated with a roughness of about 2 nm. This quartz is
∗Electronicaddress: [email protected] (cid:28)rst cleaned by snow jet, and then by oxygen plasma
†Electronicaddress: [email protected] during 10 minutes. The (cid:28)lm under study is deposited by
2
spin coating directly onto the quartz from a heptane so-
lution. The (cid:28)lm-coated quartz is placed in a home-made
cell and allowed to stabilise at 40◦C during 1 hour. We
use an impedance analyser (Solartron 1260A) to mea-
sure the admittance spectrum of the quartz resonator at
di(cid:27)erent stages of the sample preparation. As shown in
Figure2(inset),theresonancefrequencyf isdetermined
0
by curve (cid:28)tting with a precision of ∆f = ±1Hz and
min
the quality factor is obtained from Q =f /∆f (∼105)
0 0
where ∆f is the width of resonance peak. The depo-
sition of the (cid:28)lm lowers the resonance frequency of the
crystal;themeasurementofsuchafrequencyshiftallows
determiningthethicknessofthe(cid:28)lm [11]. Slight(cid:29)uctua-
tionsoftheresonancepeakamplitudeareobservedwhen
mounting the quartz into the measurement cell from one
experimental run to another, but the downward shifts
of resonance frequency are very reproducible. In order
to study the viscoelastic properties of a deposited thin
(cid:28)lm beyond the QCM application, we cover the (cid:28)lm-
quartz system with a monolayer of glass beads of diam-
eter 2R ∼ 400µm (Figure 1a). These beads for abra-
sive use (from Centraver) are plasma cleaned and have
a surface roughness of about 100 nm. Covering the (cid:28)lm
with glass beads results in a totally di(cid:27)erent behaviour
than usual mass loading (Figure 2a). We observe an in-
crease of the resonance frequency ∆f+ after the depo-
sition of beads, and a decrease of resonance amplitude
together with peak broadening. These results clearly in- FIG. 2: a. Resonance peak at various stages of the experi-
dicateelasticenhancementanddissipationincreaseofthe ment. b. Ageing of resonance frequency
quartz resonator. Moreover both frequency shift ∆f+
and energy dissipation ∆Q−1 evolve with time in a simi-
larwayroughlyaccordingtoapowerlaw. Thissuggestsa induced by the layer of beads reads:
linear relationship between ∆Q−1 and ∆f+/f as plot- √
0
ted in Figure 3 for various thicknesses. Analysis and ∆ω+ ≈Nbk/2 MK (1)
modelling: To gain physical insight into our resonance
Here N ∼ 300 is the number of beads e(cid:27)ectively cov-
measurements, we model the quartz/(cid:28)lm/beads assem- b
ering the quartz electrode; the oscillation of all beads is
bly as a pair of coupled oscillators (Figure 1 b,c). As
assumed to be identical. The determination of ω+ en-
in a previous work [14], the quartz resonator can be
ables us to characterize the elastic enhancement (sti(cid:27)-
viewed as an e(cid:27)ective masse M attached to a spring of
ening) and properties of the adsorbed or bonded (cid:28)lms.
shear sti(cid:27)ness K, determined from the fundamental res-
The minimum shear sti(cid:27)ness k that can be measured
onance frequency K = Mω2. The glass bead of mass m min
0 with this experiment is determined by ∆f+ leading to
is attached to the resonator M via the adsorbed (cid:28)lm. min
k ∼ 100N/m for K = 3.1010 N/m and M = 3.10−5
This (cid:28)lm is modelled by a Kelvin-Voigt element with min
kg. Moreover, the relative motion between the beads
spring of shear sti(cid:27)ness k and damping constant G” (see
andthequartzresonator,inherenttothemode,enhances
discussions below). No-slip boundary conditions are as-
stronglytheshearstrainintheadsorbed(cid:28)lmandinduces
sumed here between both the (cid:28)lm-bead and quartz-(cid:28)lm
an interfacial dissipation which shall be detectable with
interfaces, which are ensured by the surface roughness of
the quartz resonator. It has been shown previously that
beadsandthestrongadsorptionofpolymer(cid:28)lmsconsid-
the interfacial dissipation between two dry rough solid
eredhere. Twoeigenmodesexistforsuchcoupledoscilla-
surfaces is governed by the interplay of a frictional loss
torssystemwithanaturalfrequencyeithersuperior(ω+)
andaninterfacialviscoelasticone[4,15]. Howeverinthe
or inferior (ω−) to the quartz frequency ω . The ω− -
0
presenceofwetting(cid:28)lmsasthecasehere,theviscousloss
modecorrespondstoanin-phase motionbetweenmand
appears predominant over the other contributions to the
M giving access to the mass deposition by beads (not
energy dissipation [4, 16]. To characterize such a loss,
discussed here), while the ω+ - mode corresponds to an
wecalculatethedissipatedenergypercycleofoscillation
out-of-phase motion. The ω+ -mode is detected by the
by shearing a thin (cid:28)lm between a sphere and a (cid:29)at sur-
quartz crystal (Figure 2a) and to (cid:28)rst-order approxima-
face (Figure 1b), ∆W ≈2π(ω+η)(AL2/2h)U2. Here
tionink,theassociatedfrequencyshift∆ω+(=ω+−ω ) film
0 η is the (cid:28)lm viscosity, U (∼1nm) is the vibration ampli-
tude of the quartz and A = (16/5lg(2R/h) is a geomet-
3
pected [16]. Thismechanismpredictsadependenceofk
c
on thickness that is not evidenced experimentally (Fig-
ure2b). Thisispossiblyduetothee(cid:27)ectivecouplingand
complex wetting of mechanisms of rough surfaces, some-
how characterized by N in eqs (1) and (2) (cid:29)uctuating
b
from one experiment to another. Representing the data
suchas∆Q−1 versus∆f+/f shallallowovercomingthis
0
caveat (Figure 3). However, the resulting thickness de-
pendence ∆Q−1 ∼1/h(∆f+/f ) is not detected either.
0
We propose here an interfacial mechanism based on
the elastic behaviour of the adsorbed (cid:28)lms. At the
MHz range, polydimethylsiloxane is in the Rouse regime
as shown by the characteristic frequencies of a Kuhn
monomer τ−1 = k T/ζb2 ∼ 108 Hz and of an entangle-
κ B
ment strand τ−1 = τ−1N−2 ∼ 105 Hz. Here, ζ ∼ 10−11
e κ e
kg/s[19]isthefrictioncoe(cid:30)cientofamonomer,k isthe
B
FIG. 3: Elasticity vs dissipation Boltzmannconstant,T thetemperatureandbthelenght
of a Kuhn monomer). At such high-frequency range the
polymeric layer would provide a sti(cid:27)ness k ∼πL2G(cid:48)/h.
e
rical constant. In terms of the inverse of quality factor Comparison between our experiment and this model
∆Q−1 =(2π)−1(∆Wfilm/Wq) where Wq =(1/2)KU2 is leads to G(cid:48) ∼ 5MPa and G” ∼ 0.1MPa (eq. 2), which
thestoredenergyinthequartz,theadditionaldissipated agree well with those expected for bulk polydimethyl-
energy is written as ( ω+ ≈ω0), siloxane [20]. This picture also provides a simple rela-
tionship between the polymer loss angle tanδ = G”/G(cid:48)
∆Q−1 =ANbG”L2/2hK (2) andtheplotof∆Q−1 vs∆f+/f . Indeed,combiningeqs
0
1 and 2 yields
where G”=ω η and L is the e(cid:27)ective radius of the con-
0
tact. Measurements of such ∆Q−1 (∼ 10−5) allow us to ∆Q−1 ≈Btanδ∆f+/f (3)
0
determine the dynamic viscosity of a thin polymer (cid:28)lm
down to a thickness of 10 nm (Figure 2b). with B ∼ 60, which is consistent with the scaling be-
We now focus on the possible mechanisms responsi- haviour observed experimentally in Figure 3 and indeed
ble for the elastic sti(cid:27)ness k observed in our experi- independent of (cid:28)lm thickness h. We thus conclude that
ments. As shown in Figure 2b, the frequency shift for the viscoelasticity of the polymer (cid:28)lm appears dominant
every measurement is about ∆f+ ∼ 100 Hz, which cor- and responsible for the elastic (∆f+) and dissipative
responds to sti(cid:27)ness of the order of k ∼103 N/m for one (∆Q−1) responses of our quartz resonator.
bead contact. Dybwad [14] previously reported such a Our experiment shows no dramatic change of the vis-
sti(cid:27)ness magnitude between Au spheres on Au coated coelastic response of thin (cid:28)lms. As mentioned earlier,
quartzresonator, originatingfromVanderWaalsbonds. polymer (cid:28)lms may have unusual properties when con-
This cohesion is not observed in our resonance measure- (cid:28)ned into narrow gaps. For example for (cid:28)lm thicknesses
ments when the glass beads are deposited on the bare h∼10R (R istheradiusofgiration)thelowfrequency
g g
quartz crystal. This may be related to the roughness dynamic moduli of PDMS show a non monotonic depen-
of the beads, which can reduce signi(cid:28)cantly the contact dence on (cid:28)lm thickness and the terminal zone shifts pro-
sti(cid:27)ness of Hertz-Mindlin between glass spheres and the gressively to lower frequencies [21]. No rheological data
quartz resonator, from k = 3.104 N/m (>> k ) is however available for the high frequency region. In
HM min
on smooth interface to k =1 N/m on rough (multi- thisworkthethicknessofthe(cid:28)lmsrangesfrom1−10R
MCI g
contact)interface [15]. Thislattervalueistoolowtobe (R ∼7nm),butnounusualbehaviourisdetected. This
g
detectedwithourpresentapparatusindeed. Inthepres- couldbeexpectedgiventheverylocalprobeprovidedby
ence of a wetting (cid:28)lm, the capillary force F increases the present method. At such high frequency, only Rouse
c
the normal loading on the glass bead (Figure 2b) and modes are probed (size < nm) and no con(cid:28)nement e(cid:27)ect
couldsti(cid:27)entheabovecontactsti(cid:27)ness [17]. Anestima- is expected until the thickness of the (cid:28)lm reaches the
tion with the liquid-air surface tension γ yields a force characteristic size of those modes. The generality of our
strengthonroughsurface [18],F ∼10−7N,whichisless observation has to be con(cid:28)rmed by additional measure-
c
than the weight of bead F ∼ 10−6N; this implies that ments down to smaller thicknesses.
b
no capillary e(cid:27)ect is expected on the contact sti(cid:27)ness in Wenowturntotheageingphenomenonobservedboth
our work. However, the wetting (cid:28)lm may introduce the with the elastic sti(cid:27)ening (∆f+) and the energy dissi-
elastic sti(cid:27)ening via another mechanism, related to the pation (∆Q−1) (Figure 2b). In a previous work [3],
sti(cid:27)ness of larger capillary bridges k . With a menis- Bocquet et al. described a thermally activated forma-
c
cus formed between a smooth sphere and a (cid:29)at surface, tion of liquid bridges between rough glass beads, which
a sti(cid:27)ness k ∼ 2πL2γ/h2 of the order of 103N/m is ex- is responsible for the logarithmic ageing of capillary co-
c
4
hesion in a granular medium exposed to water vapour. In summary, we have developed a new ultrasonic
In our experiments, both ∆f+ and ∆Q−1 are seen to in- methodformeasuringhigh-frequencyshearmodulusand
creasewithtimefollowingapowerlaw∼t0.3. Thiscould dissipationofthin(cid:28)lmsdownto10nminthickness. Our
be related to a progressive wetting of the glass bead by results indicate that the viscoelastic properties of such a
the polydimethylsiloxane thin (cid:28)lm. Indeed, we observe polymer(cid:28)lmarenotquantitativelydi(cid:27)erentfromthoseof
by optical microscopy the increase of contact radius L thebulk. Webelievethatbyusingbeadsofdi(cid:27)erentsur-
with time, revealing an evolution roughly as L t0.16 (not face properties and controlling ambient conditions, this
shown here). This result can be understood by the fol- ultrasonic method provides a promising tool for exploit-
lowingscalingargument. Duringawettingprocess,some ing the con(cid:28)nement e(cid:27)ects of nanometric (cid:28)lms, the in-
polymer liquids must drain from the thin (cid:28)lm to the free terfacialdynamicsandthewettingphenomenaatvarious
surface of the beads. If L is the lateral extent of the boundaries [2, 22].
contact between the sphere and the quartz, a Poiseuille
(cid:29)owyields dL = h2∆P where∆P istheLaplacepressure.
dt ηL
This leads to L ∼ (γh4L2t/η)1/7. Assuming a constant
Acknowledgments
average thickness of the (cid:28)lm, this prediction agrees rea-
sonably well with the observed time evolution of ∆f+
and∆Q−1. Notethatthesurfaceroughnessofthebeads We wish to thank J. Laurent for his assistance to
is not taken into account in this analysis. This problem impedance measurements and the design of the cell, D.
is beyond the scope of this work and will be treated in HautemayouandH.Sizunforthecellrealization,andY.
the future. Leprince for useful discussions.
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