Non-linear rheology of a nanoconfined simple fluid
aa r X i v : . [ c ond - m a t . s o f t ] F e b Non-linear rheology of a nanoconfined simple fluid
Lionel Bureau ∗ Institut des Nanosciences de Paris, UMR 7588 CNRS-Universit´e Paris 6, 140 rue de Lourmel, 75015 Paris, France (Dated: October 22, 2018)We probe the rheology of the model liquid octamethylcyclotetrasiloxane (OMCTS) confined intomolecularly thin films, using a unique Surface Forces Apparatus allowing to explore a large rangeof shear rates and confinement. We thus show that OMCTS under increasing confinement exhibitsthe viscosity enhancement and the non-linear flow properties characteristic of a sheared supercooledliquid approaching its glass transition. Besides, we study the drainage of confined OMCTS via thepropagation of “squeeze-out” fronts. The hydrodynamic model proposed by Becker and Mugele[Phys. Rev. Lett. , 166104 (2003)] to describe such front dynamics leads to a conclusion inapparent contradiction with the dynamical slowdown evidenced by rheology measurements, whichsuggests that front propagation is not controlled by large scale flow in the confined films. PACS numbers: 64.70.pm, 83.50.Lh, 83.50-v, 83.60.Rs
Supercooled liquids share qualitative rheological fea-tures upon approaching the glass transition [1]: (i) theirviscosity increases dramatically, and (ii) they exhibitnon-newtonian properties (shear thinning) when the timescale of mechanical forcing becomes shorter than that ofstructural relaxation. The precise origin of such a be-havior is the subject of active theoretical and numericalinvestigations [2]. Recently, an extension to flow situa-tions of the mode-coupling theory (MCT) has been pro-posed, in order to describe this non-linear rheology [3].Now, a stringent test of theoretical predictions againstexperimental results requires measurements, over a largerange of shear rate ( ˙ γ ), of the non-linear properties asjamming is gradually approached. These are extremelychallenging to perform on atomic glass formers, becauseof their elevated glass transition temperature and flowstress. The most comprehensive studies to date have fo-cused on colloidal suspensions of thermosensitive parti-cles, in which the volume fraction, hence the distanceto jamming, can be finely tuned [4]. It has thus beenshown that, in very good agreement with MCT, the flowstress of such suspensions exhibits a rate dependence allthe weaker that the distance to glass transition is small,until a yield stress develops when the suspension getsjammed [4]. Such a behavior can be considered as therheological hallmark of the approach to glass transition.Here, we show that increasing confinement representsan alternative pathway to bring a system close to itsjammed state under well-controlled conditions. SurfaceForces Apparatus (SFA) experiments have shown thatsimple liquids confined between solid walls below thick-nesses of a few molecular diameters exhibit enhancedflow resistance [5]. From SFA experiments probing thelinear response of ultrathin liquid films, Demirel andGranick (DG) concluded to a confinement-induced dy-namical slowdown, akin to what occurs in supercooledliquids [6]. However, this conclusion has been challengedby other groups probing the large strain shear responseof confined fluids [7, 8]. Moreover, experiments by Becker and Mugele (BM) have shown that a confined liquiddrains stepwise by expelling monolayers via the propa-gation of “squeeze-out fronts” [9]. A model of the frontdynamics, extending the work of Persson and Tosatti [10],led them to conclude that the confined fluid retained itsbulk viscosity, dissipation enhancement arising from highfriction on the confining walls.The nature of the mechanisms by which the propertiesof liquids are affected by confinement at the molecularscale therefore remains an open question. Such an issue,which is of interest for the fundamental understanding ofthe jamming transition [11], is also of paramount impor-tance for boundary lubrication [12], and for nanofluidics,where the knowledge of the flow properties of liquids con-fined into nanometer-sized channels or structures is cru-cial [13].In this Letter, we report on the first SFA study inwhich both large strain shear rheology and squeeze-outfronts measurements are performed, in the same exper-imental run, on the nonpolar liquid octamethylcyclote-trasiloxane (OMCTS), which has been used in the afore-mentioned works.(i) We show unambiguously, from flow curves mea-sured over 6 decades of ˙ γ , that OMCTS under increasingconfinement exhibits the viscosity enhancement and non-newtonian features of a supercooled liquid approachingthe glass transition.(ii) We observe squeeze-out front dynamics in quanti-tative agreement with that previously reported [9]. Whenanalyzed within the framework of the BM model, it re-sults in an effective viscosity two orders of magnitudelower than that directly measured in shear. We concludethat such an apparent contradiction arises from an im-proper assumption by BM about the nature of the masstransport mechanism at play during front propagation.Experiments were performed on a home-built SFA [14](Fig. 1). The liquid is confined between two atomicallysmooth backsilvered mica sheets glued onto crossed cylin-drical lenses (radius of curvature ∼ F n is measured by means of a load cell of stiffness9500 N.m − . The “contact” area A , over which the micasheets elastically flatten to form a circular parallel gapin which the liquid is confined, is monitored by videomi-croscopy. The thickness of the film, d , is determined bymultiple beam interferometry [15] and fast spectral cor-relation [14, 16]. Once confined under a given load, overan area A , the liquid is sheared, in a plane Couette ge-ometry, by moving laterally one surface at a velocity V in the range 10 − − µ m.s − , while measuring the re-sulting tangential force F t with a cell of stiffness 5200N.m − . The shear stress sensitivity of the instrument is F t /A ∼
200 Pa. Our SFA has the unique feature of usingthe normal force signal as the input of a feedback loop,which allows to perform steady-state experiments overlarge shear amplitudes (up to hundreds of microns) un-der constant normal load conditions, whatever the level ofconfinement of the liquid. The mica sheets were preparedas described in [17], glued onto the cylindrical lenses us-ing a UV curing glue (NOA 81, Norland), and cleavedwith adhesive tape immediately before being installed inthe SFA, so as to obtain contaminant-free surfaces [18].OMCTS from Fluka (purum grade ≥ ∼ µ L) of the liquid, fil-tered through a 0.2 µ m membrane, was injected betweenthe surfaces. It was then left at T = 20 ± . ◦ C for 12hin the sealed SFA, containing P O to scavenge residualmoisture, before beginning experiments. FIG. 1: Force vs distance curve during approach of the sur-faces (loading velocity 0.5 nm.s − ). Inset: scheme of thesetup. White light is shone on the confined film, and thetransmitted intensity is sent (i) to a spectrometer for spectralanalysis [15], and (ii) to a CCD camera acquiring images ofthe contact area A at a rate of 55 s − . Fig. 1 shows a force-distance profile measured uponquasi-static approach of the surfaces: it is clearly seenthat below 6 nm[19], the thickness of the confined liquiddecreases by steps of approximately 8 ˚A, which corre-sponds to the minor diameter of the slightly oblate OM-CTS molecule. This reflects the well-documented wall- induced layered structure of the fluid, which gives rise tothe so-called solvation forces [20].We first focus on shear experiments performed on lay-ered OMCTS films with thicknesses ranging from 6 downto 2 monolayers. Over the whole range of confinementand velocity explored, we have observed: (i) a smoothstable shear response (see time trace in the inset of Fig.2b), and (ii) a steady-state value of F t which increaseswith V . On Fig. 2a, we plot the steady-state flow stress σ = F t /A versus shear rate ˙ γ = V /d for the different filmthicknesses. The same data are plotted on Fig. 2b as theeffective viscosity η eff = σ/ ˙ γ versus ˙ γ . FIG. 2: (a) σ ( ˙ γ ) for OMCTS films of ( N ) 6, ( (cid:3) ) 5, ( (cid:4) ) 4, ( ◦ ) 3,and ( • ) 2 monolayers. (b) η eff ( ˙ γ ), symbols as in (a). Insert:time trace of ( • , left scale) σ measured at V = 0 . µ m.s − on a 2nm-thick film, (line, right scale) the forth and backshear motion applied. (c) Master curve showing data from(b) plotted as η eff /η vs ˙ γ/ ˙ γ c . The solid line is a fit of theform η eff /η = 1 / (1 + ˙ γ/ ˙ γ c ) . . Inset: values for η ( • , leftscale) and 1 / ˙ γ c ( ◦ , right scale) used for each n . It can be seen that, as the OMCTS thickness is reducedfrom n =6 to 2 monolayers:(i) The flow stress, and hence the viscosity, steadilyincreases [21].(ii) The dependence of σ on ˙ γ shifts from linear (new-tonian) to sublinear (shear-thinning).(iii) The crossover shear rate, ˙ γ c , above which non-newtonian behavior is observed, shifts to smaller values.(iv) For n ≤ σ ∼ ˙ γ α , α <
1) at low ˙ γ crosses over to a quasi-plateauregime.This set of features is characteristic of the approach tojamming, as reported experimentally [4] and predictedby MCT and numerical simulations [2, 3]. This is fur-ther supported by the fact that η eff ( ˙ γ ) curves measuredfor different n collapse onto a single master curve whenplotted as η eff /η vs ˙ γ/ ˙ γ c , with η the zero shear viscos-ity (Fig. 2c). The inset of Fig. 2c shows that both η and 1 / ˙ γ c ( i.e. the relaxation time of the liquid) sharplyincrease as the film thickness is decreased. The reducedviscosity obeys η eff /η ≃ / (1+ ˙ γ/ ˙ γ c ) . , which is consis-tent with theoretical predictions for sheared supercooledsystems [2]. Finally, the observation of a quasi-plateauregime which does not extend down to the lowest shearrates indicates that, in the present experiments, confinedOMCTS approaches but does not reach jamming. This isconsistent with the fact that (see inset of Fig. 2b), uponcessation of shear, the stress relaxes (i) very slowly, over ∼ F n and d . We observe, as in [9, 17], that a film of thickness n monolayers drains via nucleation/growth of a circularregion of thickness ( n −
1) layers (see Fig. 3). Nucle-ation is accompanied by elastic relaxation of the confin-ing sheets, which are locally bent in the boundary zone connecting the regions of thickness n − n (Fig. 3inset). This creates a 2D pressure gradient which thendrives the monolayer expulsion [9]. The local curvatureof the mica sheets induces a contrast in the transmit-ted intensity (Fig. 3a-c) which allows us to follow withtime the position of the “squeeze-out” front. We havethus measured, for successive n → n − τ needed to expel one monolayer fromthe contact area A . We have done so before and afterrheology experiments, and did not observe any influenceof shear history on front dynamics. FIG. 3: (a-c): Sequence of images (96 × µ m ) showing thefront propagation during a 3 → µ eff vs n (number of monolayers). ( • ) our results, ( ◦ ) BM results,adapted from [9]. Inset: schematic cross-section of the filmduring squeeze-out. (e): η eff vs n . ( ◦ ) measured in shear, and( • ) deduced from squeeze-out experiments. The horizontalline indicates the bulk viscosity of OMCTS. In the Persson and Tosatti (PT) model [10], thefront velocity is related to the 2D pressure gradient by: ∇ p = − ρ µ eff V , where p ∼ P a and ρ = ρa [9, 10] ( P = F n /A is the applied pressure, ρ the fluiddensity and a the molecular size), V is the front velocityand µ eff a viscous drag coefficient. The latter is deducedfrom the squeeze-out time as [10]: µ eff = 4 πτ P/ ( ρA ). OnFig. 3d we have plotted µ eff as a function of film thicknessfor our experiments, along with the values obtained byBecker and Mugele (BM) [9]. There is quantitative agree-ment between both data sets. BM have extended the PTmodel by assuming that front propagation is controlledby a “layered” Poiseuille flow between the front and theedge of the confinement area (Fig. 3 inset), and thus pro-posed that µ eff should identify with the drag coefficientof a Hele-Shaw flow, i.e. µ eff = 12˜ η eff / ( ρd ), with ˜ η eff ashear viscosity and d the film thickness [9]. We use thisexpression to infer ˜ η eff ( d ) from the front dynamics. OnFig. 3e, we compare [24] it to η eff ( d ) obtained from sheardata. It appears that ˜ η eff , which stays close to the bulkvalue down to 3-layer-thick films, is about two orders ofmagnitude lower than η eff .We propose the following explanation to this apparentparadox. Shear experiments are a straightforward wayto measure η eff , in contrast to squeeze-out experiments,which require modelling of the front dynamics to infera viscosity. Therefore, we consider that the reliable re-sults regarding η eff ( d ) are those from shear rheology. Weare then left with the observation of squeeze-out frontswhich, given the η eff ( d ) obtained in shear, travel muchfaster than expected from the drag mechanism assumedin the BM picture, from which we have also drawn erro-neous conclusions in a recent study [17]. This suggeststhat the front dynamics is not controlled by the coher-ent sliding of adjacent incompressible molecular layersahead of the front. Indeed, another piece of informationemerges from the force-distance profile of Fig. 1: betweentwo steps, the film thickness is observed to decrease byabout 3 ˚A as the force is increased. Such a thicknessvariation is reversible upon load reduction. This showsthat layered OMCTS films are substantially compress-ible, hence contain a non-negligible amount of free vol-ume, which is consistent with the fact that confined filmsdo not reach jamming. This certainly facilitates local re-arrangements, and it is therefore likely that during prop-agation of a squeeze-out front, molecules in the layeredregion ahead of it permeate between layers in order toaccommodate for density variations in the vicinity of thefront. The apparent low resistance to front propagationsuggests that permeation, rather than large scale coher-ent sliding of layers, controls mass transport ahead of thefronts. It implies that, pending further modelling, frontdynamics cannot be used to infer a viscosity.In summary, we have probed the rheology of a simplefluid under molecular confinement, and conclude that itsbehavior is akin to that of a sheared supercooled liquidclose above the glass transition. This shows, as suggestedby recent experiments on colloids [11], that confinementcan be used as an alternative route to finely control theapproach to jamming. Our results now raise two im-portant questions. (i) We observe a liquidlike behaviordown to the thinnest film investigated, which brings upthe issue of how to cross the jamming transition underconfinement. Two routes can be envisaged. It can bedone by varying the chemical corrugation of the walls, asshown in friction experiments [25] or in numerical sim-ulations [26], or, as mentioned above, by changing theorientation between the crystalline lattices of the confin-ing surfaces. (ii) We find that 6 layer-thick films alreadyexhibit a viscosity two orders of magnitude larger thanthe bulk value. This raises the question of the scale below which non-bulk behavior appears, and how it comparesto the range of surface forces.We thank A. N. Morozov for fruitful discussion and C.Caroli for critical reading of the manuscript. ∗ Electronic address: [email protected][1] M.D. Demetriou et al. , Phys. Rev. Lett. , 065502(2006), and references therein. J.H. Simmons et al. , J.Non-Cryst. Solids , 313 (1988).[2] A. Furukawa et al. , Phys. Rev. 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