Probing viscoelastic properties of a thin polymer film sheared between a beads layer and quartz crystal resonator
PProbing viscoelastic properties of a thin polymer (cid:28)lm sheared between a beads layerand 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: October 27, 2018)We report measurements of viscoelastic properties of thin polymer (cid:28)lms of 10-100 nm at the MHzrange. These thin (cid:28)lms are con(cid:28)ned between a quartz crystal resonator and a millimetric bead layer,producing an increase of both resonance frequency and dissipation of the quartz resonator. The shearmodulus and dynamic viscosity of thin (cid:28)lms extracted from these measurements are consistent withthe bulk values of the polymer. This modi(cid:28)ed quartz resonator provides an easily realizable ande(cid:27)ective tool for probing the rheological properties of thin (cid:28)lms at ambient environment.
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From wet sand to the eye cornea, liquid systems con-(cid:28)ned into small volumes are ubiquitous in nature and areknown to alter friction and adhesion at a solid-solid in-terface [1, 2, 3, 4]. Moreover the mechanical propertiesand stability of thin (cid:28)lms is of paramount importance fora number of applications requiring speci(cid:28)c nanometriccoatings such as optical re(cid:29)ectors or dielectric stacks. Itis then natural to ask weather the properties of con(cid:28)nedliquids are similar to their bulk counterpart. Conven-tional mechanical testing is not adapted for thin (cid:28)lmsinvestigation and speci(cid:28)c metrology is needed. Natu-rally occurring instabilities such as wrinkling or dewet-ting provides valuable information about rheological andmechanical properties of thin polymer (cid:28)lms [5, 6]. Oscil-latory interfacial rheology conducted with a tribometer[7] and the Surface Force Apparatus (SFA) [8, 9] o(cid:27)ersa complementary description of thin (cid:28)lms over severaldecades of frequency. Local probes analysis performedwith Atomic Force Microscopy con(cid:28)rms the result ob-tained with the SFA [10]. Note that the rheological be-haviour at high frequency can not be investigated withthese techniques. However it is important for applica-tions such as hard disk drives.Acoustic wave devices have proven to be suitable forthe high frequency investigation [11, 12]. Among oth-ers, quartz crystal (AT-cut) resonators operating in shearmode at ultrasonic frequency from − MHz havebeen the most widely used to monitor the viscoelasticproperties of thick (cid:28)lms ( h > µ m) adsorbed on theirsurfaces [11]. However for thin (cid:28)lms, i.e. h ≤ nm,no signi(cid:28)cant shear strain is produced inside it becausethe (cid:28)lm is located at an antinode of the standing wave.In such a case, the shift of resonance frequency is onlya function of the mass alone and the acoustic propertiesof the (cid:28)lm may be ignored: the quartz crystal behavessimply as microbalance (QCM). Recently, several con(cid:28)g- ∗ Electronic address: [email protected] † Electronic address: [email protected] urations have been proposed to determine the shear mod-ulus of thin (cid:28)lms from the frequency shifts of quartz res-onators, including the build-up of composite resonators.For example, coating the (cid:28)lm of interest with secondoverlayer (sandwich con(cid:28)guration) allows enhancing theshear stress and characterizing its rheological propertiesdown to nanometric thicknesses [13]. In this Letter, wedescribe a new approach to probe the viscoelastic prop-erties of thin (cid:28)lms by using a conventional QCM. To thisend, we deposit gently a layer of spherical bead (glass)on the top of a (cid:28)lm of thickness h ∼ − nm coatedon the surface of a quartz crystal. The resulting shifts ofresonance frequency and inverse of quality factor can bereadily related directly to the elastic modulus and viscousdissipation of the (cid:28)lm. An entangled polymeric thin (cid:28)lm FIG. 1: Sketch of the experiment and modeling of the system (polydimethylsiloxane,
M w ∼ ) is deposited on aquartz resonator of f = ω / π ∼ MHz (Maktex) (Fig-ure 1). The surface of the crystal is polished and gold-coated with a roughness of about nm. This quartz is(cid:28)rst cleaned by snow jet, and then by oxygen plasmaduring 10 minutes. The (cid:28)lm under study is deposited by a r X i v : . [ c ond - m a t . s o f t ] J a n spin coating directly onto the quartz from a heptane so-lution. The (cid:28)lm-coated quartz is placed in a home-madecell and allowed to stabilise at ◦ C during 1 hour. Weuse an impedance analyser (Solartron 1260A) to mea-sure the admittance spectrum of the quartz resonator atdi(cid:27)erent stages of the sample preparation. As shown inFigure 2 (inset), the resonance frequency f is determinedby curve (cid:28)tting with a precision of ∆ f min = ± Hz andthe quality factor is obtained from Q = f / ∆ f ( ∼ )where ∆ f is the width of resonance peak. The depo-sition of the (cid:28)lm lowers the resonance frequency of thecrystal; the measurement of such a frequency shift allowsdetermining the thickness of the (cid:28)lm [11]. Slight (cid:29)uctua-tions of the resonance peak amplitude are observed whenmounting the quartz into the measurement cell from oneexperimental run to another, but the downward shiftsof resonance frequency are very reproducible. In orderto 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 R ∼ µ m (Figure 1a). These beads for abra-sive use (from Centraver) are plasma cleaned and havea surface roughness of about 100 nm. Covering the (cid:28)lmwith glass beads results in a totally di(cid:27)erent behaviourthan 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 amplitudetogether with peak broadening. These results clearly in-dicate elastic enhancement and dissipation increase of thequartz resonator. Moreover both frequency shift ∆ f + and energy dissipation ∆ Q − evolve with time in a simi-lar way roughly according to a power law. This suggests alinear relationship between ∆ Q − and ∆ f + /f as plot-ted in Figure 3 for various thicknesses. Analysis andmodelling: To gain physical insight into our resonancemeasurements, we model the quartz/(cid:28)lm/beads assem-bly as a pair of coupled oscillators (Figure 1 b,c). Asin a previous work [14], the quartz resonator can beviewed as an e(cid:27)ective masse M attached to a spring ofshear sti(cid:27)ness K , determined from the fundamental res-onance frequency K = M ω . The glass bead of mass m is attached to the resonator M via the adsorbed (cid:28)lm.This (cid:28)lm is modelled by a Kelvin-Voigt element withspring of shear sti(cid:27)ness k and damping constant G ” (seediscussions below). No-slip boundary conditions are as-sumed here between both the (cid:28)lm-bead and quartz-(cid:28)lminterfaces, which are ensured by the surface roughness ofbeads and the strong adsorption of polymer (cid:28)lms consid-ered here. Two eigenmodes exist for such coupled oscilla-tors system with a natural frequency either superior ( ω + )or inferior ( ω − ) to the quartz frequency ω . The ω − -mode corresponds to an in-phase motion between m and M giving access to the mass deposition by beads (notdiscussed here), while the ω + - mode corresponds to anout-of-phase motion. The ω + -mode is detected by thequartz crystal (Figure 2a) and to (cid:28)rst-order approxima-tion in k , the associated frequency shift ∆ ω + ( = ω + − ω ) FIG. 2: a. Resonance peak at various stages of the experi-ment. b. Ageing of resonance frequency induced by the layer of beads reads: ∆ ω + ≈ N b k/ √ M K (1)Here N b ∼ is the number of beads e(cid:27)ectively cov-ering the quartz electrode; the oscillation of all beads isassumed to be identical. The determination of ω + en-ables us to characterize the elastic enhancement (sti(cid:27)-ening) and properties of the adsorbed or bonded (cid:28)lms.The minimum shear sti(cid:27)ness k min that can be measuredwith this experiment is determined by ∆ f + min leading to k min ∼ N/m for K = 3 . N/m and M = 3 . − kg. Moreover, the relative motion between the beadsand the quartz resonator, inherent to the mode, enhancesstrongly the shear strain in the adsorbed (cid:28)lm and inducesan interfacial dissipation which shall be detectable withthe quartz resonator. It has been shown previously thatthe interfacial dissipation between two dry rough solidsurfaces is governed by the interplay of a frictional lossand an interfacial viscoelastic one [4, 15]. However in thepresence of wetting (cid:28)lms as the case here, the viscous lossappears predominant over the other contributions to theenergy dissipation [4, 16]. To characterize such a loss,we calculate the dissipated energy per cycle of oscillationby shearing a thin (cid:28)lm between a sphere and a (cid:29)at sur-face (Figure 1b), ∆ W film ≈ π ( ω + η )( AL / h ) U . Here η is the (cid:28)lm viscosity, U ( ∼ nm) is the vibration ampli-tude of the quartz and A = (16 / lg (2 R/h ) is a geomet- FIG. 3: Elasticity vs dissipation rical constant. In terms of the inverse of quality factor ∆ Q − = (2 π ) − (∆ W film /W q ) where W q = (1 / KU isthe stored energy in the quartz, the additional dissipatedenergy is written as ( ω + ≈ ω ), ∆ Q − = AN b G ” L / hK (2)where G ” = ω η and L is the e(cid:27)ective radius of the con-tact. Measurements of such ∆ Q − ( ∼ − ) allow us todetermine the dynamic viscosity of a thin polymer (cid:28)lmdown to a thickness of 10 nm (Figure 2b).We now focus on the possible mechanisms responsi-ble for the elastic sti(cid:27)ness k observed in our experi-ments. As shown in Figure 2b, the frequency shift forevery measurement is about ∆ f + ∼ Hz, which cor-responds to sti(cid:27)ness of the order of k ∼ N/m for onebead contact. Dybwad [14] previously reported such asti(cid:27)ness magnitude between Au spheres on Au coatedquartz resonator, originating from Van der Waals bonds.This cohesion is not observed in our resonance measure-ments when the glass beads are deposited on the barequartz crystal. This may be related to the roughnessof the beads, which can reduce signi(cid:28)cantly the contactsti(cid:27)ness of Hertz-Mindlin between glass spheres and thequartz resonator, from k HM = 3 . N/m ( >> k min )on smooth interface to k MCI = 1
N/m on rough (multi-contact) interface [15]. This latter value is too low to bedetected with our present apparatus indeed. In the pres-ence of a wetting (cid:28)lm, the capillary force F c increasesthe normal loading on the glass bead (Figure 2b) andcould sti(cid:27)en the above contact sti(cid:27)ness [17]. An estima-tion with the liquid-air surface tension γ yields a forcestrength on rough surface [18], F c ∼ − N, which is lessthan the weight of bead F b ∼ − N; this implies thatno capillary e(cid:27)ect is expected on the contact sti(cid:27)ness inour work. However, the wetting (cid:28)lm may introduce theelastic sti(cid:27)ening via another mechanism, related to thesti(cid:27)ness of larger capillary bridges k c . With a menis-cus formed between a smooth sphere and a (cid:29)at surface,a sti(cid:27)ness k c ∼ πL γ/h of the order of N/m is ex- pected [16]. This mechanism predicts a dependence of k c on thickness that is not evidenced experimentally (Fig-ure 2b). This is possibly due to the e(cid:27)ective coupling andcomplex wetting of mechanisms of rough surfaces, some-how characterized by N b in eqs (1) and (2) (cid:29)uctuatingfrom one experiment to another. Representing the datasuch as ∆ Q − versus ∆ f + /f shall allow overcoming thiscaveat (Figure 3). However, the resulting thickness de-pendence ∆ Q − ∼ /h (∆ f + /f ) is not detected either.We propose here an interfacial mechanism based onthe elastic behaviour of the adsorbed (cid:28)lms. At theMHz range, polydimethylsiloxane is in the Rouse regimeas shown by the characteristic frequencies of a Kuhnmonomer τ − κ = k B T /ζb ∼ Hz and of an entangle-ment strand τ − e = τ − κ N − e ∼ Hz. Here, ζ ∼ − kg/s [19] is the friction coe(cid:30)cient of a monomer, k B is theBoltzmann constant, T the temperature and b the lenghtof a Kuhn monomer). At such high-frequency range thepolymeric layer would provide a sti(cid:27)ness k e ∼ πL G (cid:48) /h .Comparison between our experiment and this modelleads to G (cid:48) ∼ MPa and G ” ∼ . MPa (eq. 2), whichagree well with those expected for bulk polydimethyl-siloxane [20]. This picture also provides a simple rela-tionship between the polymer loss angle tanδ = G ” /G (cid:48) and the plot of ∆ Q − vs ∆ f + /f . Indeed, combining eqs1 and 2 yields ∆ Q − ≈ Btanδ ∆ f + /f (3)with B ∼ , which is consistent with the scaling be-haviour observed experimentally in Figure 3 and indeedindependent of (cid:28)lm thickness h. We thus conclude thatthe viscoelasticity of the polymer (cid:28)lm appears dominantand responsible for the elastic ( ∆ f + ) and dissipative( ∆ Q − ) responses of our quartz resonator.Our experiment shows no dramatic change of the vis-coelastic response of thin (cid:28)lms. As mentioned earlier,polymer (cid:28)lms may have unusual properties when con-(cid:28)ned into narrow gaps. For example for (cid:28)lm thicknesses h ∼ R g ( R g is the radius of giration) the low frequencydynamic moduli of PDMS show a non monotonic depen-dence on (cid:28)lm thickness and the terminal zone shifts pro-gressively to lower frequencies [21]. No rheological datais however available for the high frequency region. Inthis work the thickness of the (cid:28)lms ranges from − R g ( R g ∼ nm), but no unusual behaviour is detected. Thiscould be expected given the very local probe provided bythe present method. At such high frequency, only Rousemodes are probed (size < nm) and no con(cid:28)nement e(cid:27)ectis expected until the thickness of the (cid:28)lm reaches thecharacteristic size of those modes. The generality of ourobservation has to be con(cid:28)rmed by additional measure-ments down to smaller thicknesses.We now turn to the ageing phenomenon observed bothwith the elastic sti(cid:27)ening ( ∆ f + ) and the energy dissi-pation ( ∆ Q − ) (Figure 2b). In a previous work [3],Bocquet et al. described a thermally activated forma-tion of liquid bridges between rough glass beads, whichis responsible for the logarithmic ageing of capillary co-hesion in a granular medium exposed to water vapour.In our experiments, both ∆ f + and ∆ Q − are seen to in-crease with time following a power law ∼ t . . This couldbe related to a progressive wetting of the glass bead bythe polydimethylsiloxane thin (cid:28)lm. Indeed, we observeby optical microscopy the increase of contact radius Lwith time, revealing an evolution roughly as L t . (notshown here). This result can be understood by the fol-lowing scaling argument. During a wetting process, somepolymer liquids must drain from the thin (cid:28)lm to the freesurface of the beads. If L is the lateral extent of thecontact between the sphere and the quartz, a Poiseuille(cid:29)ow yields dLdt = h ∆ PηL where ∆ P is the Laplace pressure.This leads to L ∼ ( γh L t/η ) / . Assuming a constantaverage thickness of the (cid:28)lm, this prediction agrees rea-sonably well with the observed time evolution of ∆ f + and ∆ Q − . Note that the surface roughness of the beadsis not taken into account in this analysis. This problemis beyond the scope of this work and will be treated inthe future. In summary, we have developed a new ultrasonicmethod for measuring high-frequency shear modulus anddissipation of thin (cid:28)lms down to 10 nm in thickness. Ourresults indicate that the viscoelastic properties of such apolymer (cid:28)lm are not quantitatively di(cid:27)erent from those ofthe bulk. We believe that by using beads of di(cid:27)erent sur-face properties and controlling ambient conditions, thisultrasonic method provides a promising tool for exploit-ing the con(cid:28)nement e(cid:27)ects of nanometric (cid:28)lms, the in-terfacial dynamics and the wetting phenomena at variousboundaries [2, 22]. Acknowledgments