Friction and Slip at Simple Fluid-Solid Interfaces: The Roles of the

Jun 20, 2005 - (NFLV) is based on total internal reflection-fluorescence recovery after photobleaching ... ing a high surface energy ( s > 72 mJ=m2). Rms rough- .... layers could be the origin of a better momentum transfer and thus of a larger ...
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PRL 94, 244501 (2005)

PHYSICAL REVIEW LETTERS

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Friction and Slip at Simple Fluid-Solid Interfaces: The Roles of the Molecular Shape and the Solid-Liquid Interaction Tatiana Schmatko,* Hubert Hervet, and Liliane Leger† Laboratoire de Physique des Fluides Organise´s, UMR CNRS-Colle`ge de France 7125, 11 Place Marcelin Berthelot, 75231 PARIS Cedex 05, France (Received 9 February 2005; published 20 June 2005) Using total internal reflection-fluorescence recovery after photobleaching, the local velocity, averaged over distances of 50 nm from the solid wall, has been measured for two different simple liquids, squalane and hexadecane, sheared on three smooth surfaces with similar roughness but with gradually decreasing fluid-solid interactions. We show that not only the strength of the fluid-solid interactions, but also the shape of the molecules of the fluid deeply affect the friction and the degree of slip at the wall. DOI: 10.1103/PhysRevLett.94.244501

PACS numbers: 47.27.Nz, 68.08.2p, 81.40.Pq

It is usually assumed that when a liquid is sheared near a solid surface, the no slip boundary condition holds for the flow velocity, meaning that the relative fluid/wall velocity is zero. This hypothesis applies well at macroscopic scales, even if it does not rely on firm physical arguments. The question of its validity at microscopic scales still remains an open one, and an increasing number of experimental evidences of flow with strong slip at the wall for simple liquids [1– 4] have emerged in the past few years. The degree of wall slip was observed to strongly depend on both the wettability and the roughness of the solid surface. These observations were reinforced by molecular dynamic (MD) simulations [5–11]. Available results appear, however, rather scattered and sometimes contradictory. The parameter used to quantify the degree of wall slip is the so-called slip length b or distance from the wall where the velocity profile extrapolates to zero. In a number of experiments, liquids totally wetting the solid show a zero velocity at the wall, which seems intuitively acceptable, as for strong fluid-solid interactions the first molecular layers should adhere to the solid [1,3,4,12]. On the contrary, data seem to agree to support the idea that partial wetting could lead to large slip [1,3,4]. Few experiments have shown, however, that a totally wetting liquid could exhibit noticeable slip, with slip length much larger than the dimensions of the liquid molecules [2,13]. When slip is observed, the measured b values appear quite scattered. For example, for water in glass microcapillaries treated with a self-assembled monolayer of octadecyltrichlorosilane (OTS), Churaev and co-workers [1] have measured a flow rate higher than expected for the no slip boundary condition and concluded that their data were compatible with b  200 nm while using a surface force apparatus, Cottin et al. [11] obtained b  20 nm on apparently similar glass treated surfaces. A possible explanation of such scattered b values comes from the effect of surface roughness, which is expected to decrease the amount of wall slip [14]. In this Letter, we report an experimental investigation, based on direct local measurements of the velocity of 0031-9007=05=94(24)=244501(4)$23.00

the fluid in the immediate vicinity of the solid wall, and conducted to explore in a systematic way the role of the strength of the fluid-solid interactions on the slip length. Comparing the flow behavior of two different fluids, on the same surfaces, at fixed strength of the fluid-solid interactions, we were able for the first time to demonstrate that small changes in the shape of the molecules of the fluid deeply affect the liquid-solid friction and the local velocity at the wall. The technique used, called near field laser velocimetry (NFLV) is based on total internal reflection-fluorescence recovery after photobleaching and has been described before [2,14]. We briefly present here its principle and the improvements in the data analysis we have brought, in order to justify the accuracy in the determination of the slip length values. We shall then compare the slip behavior of two different liquids chosen to have the same surface tension, on a series of chemically modified surfaces, with gradually decreasing strength of solid-fluid interactions but constant roughness. The two liquids are squalane and hexadecane. Their liquid-vapor surface tension are, respectively, 27.4 and 27:6 mJ=m2 at 20  C, as measured using the ring technique [15,16]. Hexadecane (Aldrich 99%) was used as received and squalane (Aldrich 99%) was previously distillated under vacuum. The solid substrate, a bare sapphire -Al2 O3 f0001g, was treated by three different ways: (a) HF treatment (1% 15 min) to remove all inorganic previous grafting residues, followed by a large rinse with tridistilled water (resistivity 18 M) and an oxidative UV=O3 treatment [17,18] yielding a high surface energy (s > 72 mJ=m2 ). Rms roughness (0.4 nm) was measured by x-ray reflectivity (atomic force microscopy (AFM) observations of various 10  10 m zones of the surface lead to mean roughness Ra  0:5 nm. The major part of the surface was very smooth, and a few scratches (due to the difficulty of polishing sapphire) were visible in some zones, with typical depth of 1 to 5 nm, and width of 0:1 m). (b) Grafting in vapor phase of a self-

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PHYSICAL REVIEW LETTERS

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assembled monolayer of 1-hydrogeno, 7-chloro octamethyltetrasiloxane with a protocol previously developed for grafting of polydimethylsiloxane brushes on silica [19] and adapted to sapphire (longer reaction times). The thickness (1.25 nm) and the rms roughness (0.4 nm) characterized by x-ray-reflectivity spectra show that the monolayer is close-packed. The advancing contact angle of both hexadecane and squalane on such surface is 20. (c) Grafting of a self-assembled monolayer of OTS in the liquid phase, using the optimized procedure developed for silica [20] and adapted for sapphire (rms roughness of 0.35 nm and thickness of 2.45 nm). The advancing contact angle of hexadecane is 40 with a contact angle hysteresis smaller than 1 , provided optimized conditions of grafting are rigorously followed. In the flow cell, the fluid is confined by capillarity between two parallel discs on a 5 mm wide track [Fig. 1(a)]. One disc is rotated at chosen angular velocity, imposing a permanent planar shear to the fluid. The shear rate is constant within 10% over the width of the ‘‘shear track,’’ and can be varied between 102 and 104 s1 , i.e., well below the critical shear rate _ c  1= r , with r the rotation time of the fluid molecules. The gap is large enough (190 m) to prevent any confinement effect in the liquid. Fluorescent probes (NBD-dihexadecylamine, molecular probes) at low concentration (5 to 50 ppm in weight for hexadecane and 5 to 30 ppm for squalane) act as flow tracers after they had been locally photobleached. Two laser beams are used sequentially to photobleach and read the local fluorescence intensity as a function of time after the photobleaching pulse. They are focused and adjusted to have exactly the same radius in the shear direction (30 or 40 m) at the investigated solid-liquid interface where they intersect. The first beam crosses vertically the fluid cell and photobleaches the probes all through the fluid gap during a short pulse (few ms). The second beam, attenuated by a factor of 100 in order to avoid further photobleaching, impinges the surface in total internal reflection. The associated evanescent wave (penetration depth   50 nm) excites the fluorescence locally at the interface. The fluorescence intensity is recorded using a photomultiplier (Hamamatsu R1104) and large aperture collection optics. Immediately after the photo-

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bleaching pulse, the fluorescence signal is low as the concentration in non bleached probes within the illuminated volume is low. The fluorescence intensity then recovers due to the penetration of nonphotobleached probes in the volume illuminated by the evanescent wave. The characteristic time of fluorescence recovery, , (typically of order of 10 ms for hexadecane and 1 s for squalane) is governed by both flow velocity and diffusion of the photobleached probes. The penetration depth of the evanescent wave, , is the shortest length scale of the experiment, but it does not impose the spatial resolution of the experiment: due to the rapid vertical diffusion of the photobleached probes the flow lines are mixed in the direction normal to the surface (Taylor dispersion). The time fixes the distance z over which this average of the local velocity is effective, through the Stokes-Einstein equation p  z  2D with D the probes diffusion coefficient (D  1:39  1010 m2 =s for hexadecane as calculated by MD [21] and D  4:15  1011 m2 =s for squalane, as measured using fluorescence recovery after photobleaching techniques [22,23]). The experiment is thus sensitive to the average velocity of the probes over the distance z, on the order of a micrometer. With no slip at the wall,   Rz 1 _  _ z z 0 z dz 2 . Assuming that is the time necessary to evacuate the probes from the reading beam, one can also write:   2= , with  the radius of the laser spot at the surface. This leads to  22 =D 1=3 _ 2=3 . A renormalization of the time scale, t  t_ 2=3 , should scale fluorescence recovery curves acquired at different shear rates to a single master curve. With Rzslip at the wall, the average 1 _ b dz  z _ b . velocity over z is:   z 0 z

2=3 The scaling in t  t_ no longer produces a master curve. However, as shown in Fig. 1(b), a ‘‘shear-slip’’ equivalence can be built: the same average velocity can be obtained without slip and with an effective shear rate _ b given by v  2_ b z. This effective shear rate is directly related _ 2b=z [Fig. 1(b)]. A reto the slip length by _ b  1 2=3

normalization in t  t_ b should scale all fluorescence curves to a master curve. The above analysis only holds if the concentration profile in fluorescent probes after the photobleaching pulse remains the same whatever the applied shear rate. As the residence time of a probe in the bleaching beam obviously depends on the shear rate, it is then necessary to adjust the bleaching conditions at each shear rate, so that the initial concentration profile in fluorescent probes does not depend on the shear rate. When such an adjustment is not correctly performed, deviations in the initial shape of the fluorescence recovery curves appear and prevent the formation of a master curve. To extract the slip length from the data one needs to determine which slip length b will give the best master curve. In the previously reported data [2,14], this was achieved visually. To rationalize the procedure, we have developed mathematical criteria based on a minimization of the areas

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between the different scaled curves. For a given applied shear rate _ i , after scaling with a try _ b and integrating the intensities between 0 and a fixed t

, the standard deviation between the area Ai and Amean , the mean area of all shear rates, is calculated. The average of standard deviations i Ai  Amean 2 =i is then plotted as a function of tried slip length. The minimum gives the slip length corresponding to the best master curve for the liquidsurface investigated. Fluorescence recovery curves obtained for hexadecane on OTS modified sapphire and applied shear rates between 100 and 5000 s1 are shown in Fig. 2(a). In Fig. 2(b) the best master curve, obtained by trial and error varying the slip length and _ b , is presented. The standard deviation of the area between curves is reported in Fig. 2(c). The minimum in Fig. 2(c) defines the optimum slip length, b  350 nm 50 nm. For hexadecane on the totally wetting bare sapphire, the best fit gives a slip length b  110 nm 50 nm. The fluorescence recovery is clearly faster than expected for the no slip boundary condition but also clearly slower than for the same liquid on the OTS layer. Comparing these results to the previously published one [2], the rationalized procedure leads to slightly smaller slip length on comparable surfaces, and to increased accuracy. The different results for the two liquids and the three investigated surfaces are summarized in Fig. 3, in term of slip length as a function of the advancing contact angle with hexadecane, chosen as an indicator of the strength of the fluid-surface interactions. Two important results appear clearly. First, for the two liquids, the slip length increases smoothly when the strength of the fluid-surface interactions is decreased, provided the roughness of the surface is kept constant. Second, squalane always slips less (slip lengths smaller by a factor of 3) than hexadecane, on corresponding surfaces. This is not the result of a variation of the fluid-surface interactions, as the two liquids have essentially the same surface tension. We suspect that the shape of the molecules, which can affect their organization near the wall, could emphasize or prevent slip, as previously observed for alcohol molecules [24]. Such a hypothesis is supported by molecular simulations which compare linear and branched alkanes confined between two walls [25–28]. Confinement indeed tends to induce layering order in the fluid, as it has indeed been observed in AFM force versus distance measurements [29,30]. The simulations show a layering particularly pronounced for hexadecane molecules whose aspect ratio allows alignment parallel to the wall, while for squalane and other branched molecules whose shape is closer to a ball, the ordering appears less pronounced with weaker density oscillations. The resulting interdigitation of the methyl branch between layers could be the origin of a better momentum transfer and thus of a larger friction between layers, leading to a lower slip at the wall. Recent experiments by Cho et al. [31] which correlate slip at the wall to orientation of

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 t_ 2=3 b . (c) Illustration of the procedure used to determine the optimum effective shear rate: The standard deviation between the curves obtained for different effective shear rates is plotted as a function of corresponding slip length b. The minimum defines the optimum effective shear rate and gives the slip length b  350 nm 50 nm.

dipolar moments rather than to wettability go along the same lines. The two liquids used in the present study are nonpolar, they are not highly confined between two plates, but the smooth surface may orient the fluid molecules parallel to the wall in the first layers. The resulting density oscillations (that we are trying to visualize through neutron reflectivity experiments at present) could lead to a preferential plane of slip within a few molecular distances from the wall, even for wetting surfaces. The NFLV experiment, which is far from having a resolution in distance from the

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hexadecane squalane 500 400 300 200 100 0 -100

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FIG. 3 (color online). Slip lengths obtained for hexadecane on three different sapphire modified surfaces and squalane on two different surfaces, characterized by the advancing contact angle of hexadecane.

wall comparable to a molecular size, cannot help in localizing such a slip plane. Finally, our results do not show any dependence of the slip length versus the shear rate (all data for one liquid on one surface are compatible with the same slip length whatever the applied shear rate in the range 102 to 104 s1 ), as expected by MD simulations [32] and in agreement with others’ experiments [12,24,33]. The difference in the viscosities between the two liquids had no incidence on the results. To conclude, we have measured the slip length with two different Newtonian liquids, on three different surfaces having the same roughness but very different wettabilities for the two fluids (chosen to develop similar interactions on each of those surfaces). The strength of the solid-liquid interactions appears definitely to be an important parameter that governs slip at the wall and friction at simple fluidsolid interfaces. Contrary to what is usually admitted, we confirm that a totally wetting surface can lead to noticeable slip at the wall. More surprisingly, we have shown that the shape of the fluid molecules is an important factor, which can facilitate wall slip in case of linear elongated cigars or, on the contrary, decrease the aptitude to slip in case of branched molecules. This is an important effect, as branches as small as methyl groups can lead to a decrease in slip length by a factor of 3 compared to the linear alkane. Theses observations can be qualitatively rationalized in terms of layering of the fluid in the immediate vicinity of the solid wall.

*Present address: FOM Institute for atomic and Molecular Physics, Kruislaam 407. 1009 DB Amsterdam, The Netherlands † Corresponding author. [1] N. V. Churaev, V. D. Sobolev, and A. N. Somov, J. Colloid Interface Sci. 97, 574 (1984).

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