Langmuir 2007, 23, 11617-11622
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Slippage of Newtonian Liquids: Influence on the Dynamics of Dewetting Thin Films R. Fetzer† and K. Jacobs* Department of Experimental Physics, Saarland UniVersity, D-66041 Saarbru¨cken, Germany ReceiVed June 13, 2007 Slippage of Newtonian liquids in the presence of a solid substrate is a newly found phenomenon, the origin of which is still under debate. In this article, we present a new analysis method to extract the slip length. Enhancing the slip of liquids is an important issue for microfluidic devices that demand for high throughput at low pumping power. We study the velocity of short-chained liquid polystyrene (PS) films dewetting from nonwettable solid substrates. We show how the dynamics of dewetting is influenced by slippage, and we compare the results of two types of substrates that give rise to different slip lengths. As substrates, Si wafers that have been coated with octadecyltrichlorosilane (OTS) or dodecyltrichlorosilane (DTS) were used. Our results demonstrate that the dewetting velocity for PS films on DTS is significantly larger than on OTS and that this difference originates from the different slip lengths of the liquid on top of the two surfaces. For PS films of thickness between 130 and 230 nm, we find slip lengths between 400 nm and 6 µm, depending on substrate and temperature.
1. Introduction Thin films of liquid polymers play an important role in numerous technological processes ranging from lithography to biological membranes.1 A key to understanding the stability of liquid coatings on solid surfaces was the calculation2-6 and the experimental derivation of the effective interface potential.7-12 In recent years, one major focus of interest has been to understand the dynamics and morphology of polymer films dewetting from nonwettable substrates.13-27 The dewetting rate of the liquid * To whom correspondence should be addressed. E-mail: k.jacobs@ physik.uni-saarland.de. † Present address: Ian Wark Research Institute, University of South Australia, Mawson Lakes SA 5095, Australia. (1) Oron, A.; Davis, S. H.; Bankoff, S. G. ReV. Mod. Phys. 1997, 69, 931. (2) Vrij, A. Discuss. Faraday Soc. 1966, 42, 23. (3) Ruckenstein, E.; Jain, R. K. J. Chem. Soc., Faraday Trans. II 1974, 132. (4) Sharma, A.; Ruckenstein, E. J. Colloid Interface Sci. 1985, 106, 12. (5) Dietrich, S. In Phase Transition and Critical Phenomena; Domb, C., Lebowitz, J. L., Eds.; Academic Press: London 1988; Vol. 12. (6) Schick, M. In Liquids at Interfaces; Charvolin, J., Joanny, J. F., ZinnJustin, J., Eds.; Elsevier Science: Amsterdam 1989. (7) Reiter, G. Phys. ReV. Lett. 1992, 68, 75. (8) Sferrazza, M.; Heppenstall-Butler, M.; Cubitt, R.; Bucknall, D.; Webster, J.; Jones, R. A. L. Phys. ReV. Lett. 1998, 81, 5173. (9) Kim, H. I.; Mate, C. M.; Hannibal, K. A.; Perry, S. S. Phys. ReV. Lett. 1999, 82, 3496. (10) Seemann, R.; Herminghaus, S.; Jacobs, K. Phys. ReV. Lett. 2001, 86, 5534. (11) Seemann, R.; Blossey, R.; Jacobs, K. J. Phys.: Condens. Matter 2001, 13, 4915. (12) Seemann, R.; Herminghaus, S.; Jacobs, K. J. Phys.: Condens. Matter 2001, 13, 4925. (13) Redon, C.; Brochard-Wyart, F.; Rondelez, F. Phys. ReV. Lett. 1991, 66, 715. (14) Redon, C.; Brzoska, J. B.; Brochard-Wyart, F. Macromolecules 1994, 27, 468. (15) Brochard-Wyart, F.; de Gennes, P.-G.; Hervet, H.; Redon, C. Langmuir 1994, 10, 1566. (16) Brochard-Wyart, F.; Debregeas, G.; Fondecave, R.; Martin, P. Macromolecules 1997, 30, 1211. (17) Masson, J.-L.; Green, P. F. Phys. ReV. Lett. 2002, 88, 205504. (18) Damman, P.; Baudelet, N.; Reiter, G. Phys. ReV. Lett. 2003, 91, 216101. (19) Reiter, G. Phys. ReV. Lett. 2001, 87, 186101. (20) Vilmin, T.; Raphae¨l, E.; Damman, P.; Sclavons, S.; Gabriele, S.; Hamieh, M.; Reiter, G. Europhys. Lett. 2006, 73, 906. (21) Jacobs, K.; Seemann, R.; Schatz, G.; Herminghaus, S. Langmuir 1998, 14, 4961. (22) Reiter, G.; Khanna, R. Langmuir 2000, 16, 6351. (23) Seemann, R.; Herminghaus, S.; Jacobs, K. Phys. ReV. Lett. 2001, 87, 196101.
hereby strongly depends on the flow velocity at the solid/liquid interface. Hitherto it was common sense that the occurrence of a nonzero flow velocity (“slippage”) at the interface is restricted to non-Newtonian liquids (e.g., high-molecular-weight polymer films at high shear rates).28 Only recently, new experimental techniques have revealed that Newtonian liquids may also exhibit slippage.29-32 The occurrence and the nature of slippage is of great technological interest because a sliding fluid can flow faster through microfluidic devices. Many studies have focused on techniques to measure slippage. These approaches can be classified by direct and indirect methods to determine the fluid velocity profile near the solid/liquid interface. Direct methods use tracer particles in combination with near-field laser velocimetry32,33 or fluorescence recovery after photobleaching techniques.29,30,34 Indirect methods aim at measuring the drainage force of an object moving in a liquid to calculate the amount of slippage. Common techniques use a surface forces apparatus31,35 or an atomic force microscope (AFM) with a colloidal probe.36 For details see the review articles by Lauga et al.37 or Neto et al.38 A common measure of slippage is the slip length, which can be understood as the length below the solid/liquid interface where the velocity extrapolates to zero. For simple fluids, the slip length (24) Becker, J.; Gru¨n, G.; Seemann, R.; Mantz, H.; Jacobs, K.; Mecke, K. R.; Blossey, R. Nat. Mater. 2003, 2, 59. (25) Neto, C.; Jacobs, K.; Seemann, R.; Blossey, R.; Becker, J.; Gru¨n, G. J. Phys.: Condens. Matter 2003, 15, 3355. (26) Fetzer, R.; Jacobs, K.; Mu¨nch, A.; Wagner, B.; Witelski, T. P. Phys. ReV. Lett. 2005, 95, 127801. (27) Vilmin, T.; Raphae¨l, E. Europhys. Lett. 2005, 72, 781. (28) de Gennes, P. G. ReV. Mod. Phys. 1985, 57, 3. (29) Pit, R.; Hervet, H.; Le´ger, L. Phys. ReV. Lett. 2000, 85, 980. (30) Schmatko, T.; Hervet, H.; Leger, L. Phys. ReV. Lett. 2005, 94, 244501. (31) Cottin-Bizonne, C.; Jurine, S.; Baudry, J.; Crassous, J.; Restagno, F.; Charlaix, E. Eur. Phys. J. E 2002, 9, 47. (32) Le´ger, L. J. Phys.: Condens. Matter 2003, 15, S19. (33) Hervet, H.; Le´ger, L. C. R. Phys. 2003, 4, 241. (34) Migler, K. B.; Hervet, H.; Le´ger, L. Phys. ReV. Lett. 1993, 70, 287. (35) Zhu, Y.; Granick, S. Phys. ReV. Lett. 2002, 88, 106102. (36) Craig, V. S. J.; Neto, C.; Williams, D. R. M. Phys. ReV. Lett. 2001, 87, 054504. (37) Lauga, E.; Brenner, M. P.; Stone, H. A. In Handbook of Experimental Fluid Dynamics; Foss, J., Tropea, C., Yarin, A., Eds.; Springer: New York, 2005; Chapter 15; cond-mat/0501557. (38) Neto, C.; Evans, D. R.; Bonaccurso, E.; Butt, H.-J.; Craig, V. S. J. Rep. Prog. Phys. 2005, 68, 2859.
10.1021/la701746r CCC: $37.00 © 2007 American Chemical Society Published on Web 10/06/2007
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Figure 1. Growth of a hole in a 130-nm-thick PS(13.7k) film on DTS, captured by optical microscopy at 120 °C.
is found to be independent of the shear rate33,39 but is influenced by the wettability of the substrate as well as by the roughness of the wall: Molecular dynamics simulations for a simple liquid with a 140° contact angle on the surface of interest found a slip length on the order of 30 diameters of the fluid molecules but only weak slip for a wetting situation.40 This result is in agreement with experimental data for hexadecane29,30,33 and glycerol.31 The role of surface roughness is ambiguous: simulations revealed that, depending on the pressure, surface roughness can increase or decrease the amount of slippage.41 Experiments demonstrated the importance of the lateral length scale of the roughness.32,42 In our studies, we investigate the dewetting process of thin liquid films, and a typical scenario is depicted in Figure 1. To extract the slip length, we develop a new method for the analysis of the dewetting rates. Dewetting can be described in three different stages,16 starting from the very beginning of hole formation. In the first stage, the dewetted region grows exponentially in time, and in the vicinity of the dry area, a homogeneous thickening of the film can be observed; the hole does not yet exhibit a rim. In the second regime, the “birth” of the rim takes place. In this stage, the rim has an asymmetric shape, and the radius grows linearly in time. In the third stage of dewetting, which starts at radii on the order of the slip length, surface tension rounds the rim, which becomes more symmetric and grows from now on in a self-affine manner. Many experiments confirm this three-stage dewetting scenario.17-20 In the third stage of a “mature” rim, slippage is found to influence the dynamics in a substantial way:14,15 For the case of a no-slip boundary condition or negligible sliding friction at the solid/liquid interface, viscous dissipation (which occurs dominantly in the vicinity of the three phase contact line) as well as contact line friction results in the linear growth of holes. However, if dissipation at the solid/liquid interface dominates the dewetting process, an R ∝ t2/3 power law is expected. For the latter case of large slippage, the third stage might be followed by a forth regime, when the rim height is large compared to the slip length.15 In this case, sliding dissipation again can be neglected, and a linear growth R ∝ t is expected. In our studies, we are interested in the stage of a mature rim that grows in a self-affine manner in width and height. In the following text, we will demonstrate how we can deduce the slip length from the dynamics of the hole growth. We hereby restrict ourselves to liquids of Newtonian behavior and therefore investigate only the dewetting dynamics of short-chained polymer melts below the entanglement length at low shear rates. A short glance at Figure 3 reveals that the growth law for the hole radius is clearly nonlinear. We therefore suppose that slippage cannot be neglected. However, for short-chained polymer melts, the slip length is predicted to be in the nanometer range.28 Hence, slippage is not expected to dominate the growth of holes at any (39) Priezjev, N. V.; Troian, S. M. Phys. ReV. Lett. 2004, 92, 018302. (40) Barrat, J.-L.; Bocquet, L. Phys. ReV. Lett. 1999, 82, 4671. (41) Cottin-Bizonne, C.; Barrat, J.-L.; Bocquet, L.; Charlaix, E. Nat. Mater. 2003, 2, 237. (42) Joseph, P.; Cottin-Bizonne, C.; Benoit, J.-M.; Ybert, C.; Journet, C.; Tabeling, P.; Bocquet, L. Phys. ReV. Lett. 2006, 97, 156104.
Figure 2. Snapshots of the rim profile of a dewetting PS(13.7k) film on OTS, measured in situ with tapping mode AFM. The radii of the respective holes range from 2 to 10 µm. The dynamic contact angle stays constant during the growth of the hole. Here, the threephase contact line is shifted to the origin.
Figure 3. Radius of the dewetted region as a function of time, measured for 130-nm-thick PS(13.7k) films on OTS as well as on DTS substrates at two different temperatures.
stage. In the Theoretical Section, we will describe a model assuming that neither viscous dissipation at the contact line nor sliding friction at the solid/liquid interface entirely dominates the dynamics. We rather propose that the two different dissipation mechanisms are balanced and both have to be taken into account simultaneously.21 2. Experimental Section For the experiments, atactic polystyrene (PS, by Polymer Standard Service, Mainz, Germany) with a molecular weight of 13.7 kg/mol and a polydispersity of Mw/Mn ) 1.03 was used. The samples were prepared by spin casting a toluene solution of PS onto mica, floating the films on Millipore water, and then transferring them to hydrophobic substrates. For the hydrophobization, we used two different self-assembled monolayer coatings on top of Si wafers (2.1 nm native oxide layer), octadecyltrichlorosilane (OTS) and the shorter dodecyltrichlorosilane (DTS) prepared by standard techniques.43 By ellipsometry (EP3 by Nanofilm, Go¨ttingen, Germany), the thicknesses of the SAMs were found to be dOTS ) 2.4 nm and (43) Wasserman, S. R.; Tao, Y.-T.; Whitesides, G. M. Langmuir 1989, 5, 1074.
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Langmuir, Vol. 23, No. 23, 2007 11619 on OTS, we expect a significantly larger slip length for PS(13.7k) on the DTS coating.
3. Theoretical Section: Energy Dissipation Models
Figure 4. Data shown in Figure 3 on a logarithmic scale. For different dewetting temperatures and on the different substrates, the radii grow algebraically in time with an exponent of R ) 0.82(5), when the analysis is restricted to the stage of the mature rim. Fitting the data points in an earlier stage would reveal larger exponents. dDTS ) 1.5 nm. These values were corroborated by X-ray and neutron scattering, which also revealed that the molecules stand upright and at maximum area density.44 Surface characterization by atomic force microscopy (AFM, Multimode by Veeco, Santa Barbara, CA) revealed rms roughnesses of 0.09(1) nm (OTS) and 0.13(2) nm (DTS) at a 1 µm2 scan size and a static contact angle of polystyrene droplets of θY ) 67(3)° on both coatings. Optically measured advancing contact angles of different liquids are given in the table below and show that the two surfaces exhibit slightly different surface energies when probed with nonpolar liquids (γOTS ) 23.9 mN/m and γDTS ) 26.4 mN/m). Water shows a low contact angle hysteresis of 6° on OTS and 5° on DTS, which indicates the small roughness and good quality of the respective coatings. The surface tension of polystyrene is γ ) 30.8 mN/m.
Millipore water bicyclohexane 1-bromonaphthaline
θadv on OTS (deg)
θadv on DTS (deg)
116(1) 45(3) 62(4)
114(1) 38(4) 57(5)
The polystyrene films in this study are either 130(5) or 230(5) nm thick. To induce dewetting, the films were heated above the glass-transition temperature of PS (93 °C) to three different temperatures (110, 120, 130 °C). After a few seconds, circular holes are born and instantly start to grow. An example of a typical temporal series captured by optical microscopy is shown in Figure 1. Because of mass conservation of the liquid, rims develop that surround each hole. Cross sections of such a growing rim are shown in Figure 2, as measured by in situ AFM. The dynamic contact angle of the liquid is θdyn ) 56(2)° and stays constant during dewetting. In Figure 3, the optically measured radii of the emerging holes in the PS(13.7k) films as a function of time are depicted for two temperatures and two types of silane substrates. First, for each series, we find a nonlinear growth law. To emphasize this, the data are additionally drawn in Figure 4 in a logarithmic diagram, revealing the algebraic growth of the radius in time with an exponent of R ) 0.82(5). Second, dewetting clearly progresses faster on DTS- than on OTS-coated substrates. We qualitatively interpret this observation as follows. At the same temperature, the polymer films on both samples have identical properties: the viscosity η and the surface tension γ do not depend on the substrate underneath. Additionally, the static contact angle θY of polystyrene on both surfaces is constant within the experimental error. Therefore, the spreading coefficient S ) γ(cos θY - 1) ) -0.0188 N/m, which is the driving force of the dewetting process, is identical on OTS and DTS substrates. The only parameter that could be different is the boundary condition at the solid/liquid interface of the two substrates. Because dewetting on DTS is much faster than (44) Magerl, A. Private communication.
To gain more insight into the mechanisms involved in the dewetting process, a detailed analysis of the velocity is required. For this purpose, we first recall the different dewetting models introduced by Brochard-Wyart et al.15 The velocity of the moving front V ) R˙ is given by the force balance between the driving force |S| and the power P dissipated in the dewetting process, |S|V ) P. Assuming no slip at the solid/liquid interface, energy is dissipated by viscous friction only within the liquid volume and at the three-phase contact line. Because the highest shear rates arise in the vicinity of the contact line, viscous dissipation Pv takes place predominantly at this line and is therefore independent of the size of the rim. However, Pv is influenced by the flow geometry near the contact line (i.e., by the dynamic contact angle θdyn). Thus, the dewetting velocity V ) Vv is given by
|S| η
Vv ) Cv(θdyn)
(1)
with η being the viscosity of the liquid and Cv(θdyn) being a function of the dynamic contact angle that counts for the flow geometry.15 As shown in Figure 2, the dynamic contact angle is temporally constant in our experiments. Thus, viscous dissipation depends neither on hole size nor on time. However, the driving force |S| does not vary during the dewetting process. Hence, in the model of the no-slip boundary condition, the dewetting velocity V ) Vv is constant, and the radius R grows linearly in time, R ∝ t, which holds for Newtonian liquids as shown in ref 13. Because the data in Figure 3 clearly do not show linear growth of the radius R but rather a decreasing velocity, our experiments do not allow the assumption of a no-slip boundary condition. Another model, the so-called “plug flow” model, introduced by Brochard-Wyart et al.15 to explain the dewetting rate of entangled polymer melts assumes large slip lengths b at the solid/ liquid interface as compared to the film thickness h (i.e., h/b , 1). Here, energy dissipation predominantly occurs at this interface over a distance of about the width of the rim w. Hence, the power dissipated in the dewetting process is proportional to w, and force balance results in the velocity V ) Vv
Vs )
1 |S| b 3 η w
(2)
with the slip length b. Self-similarity of the growing mature rim (which means a constant dynamic contact angle, cf. Figure 2) and mass conservation of the liquid yield w ∝ xR. Hence, in this model, the velocity decreases with increasing hole size, Vs ∝ 1/xR, and the hole radius increases algebraically in time with an exponent R ) 2/3, R ∝ t2/3. For simplicity, we can introduce the constant K and write Vs ) K/xR. Although the data in Figure 4 show an algebraic behavior R ∝ tR, the exponents fitted to the data achieve values clearly above 2/3; the mean value of the exponents of the four curves shown in Figure 4 is R ) 0.82(5). This is the first indication that the plug flow model is also not adequate to describe our experimental results in the stage of the mature rim. Additionally, from rim shape analysis of the same experimental system, we know that the lowest value for the slip length b is in the range of the film thickness h.45 Thus, the precondition for plug flow, h/b , 1, is not satisfied. To evaluate the role of slippage further,
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we will analyze the dewetting velocity in the following in more detail. In that way, we also avoid the determination of exponents, which often is subject to errors. Another model combining energy dissipation at the threephase contact line and sliding dissipation at the solid/liquid interface is required. In the literature, there are some suggestions of a transition within the dewetting process, meaning that at an early stage viscous dissipation at the contact line dominates the dynamics of dewetting, R ∝ t, whereas in a later stage the holes grow mostly as a result of slip effects, R ∝ t2/3.16-18 In our experiments, for some small holes, we can see such a transition from an early stage of almost constant dewetting velocity to a later stage of decreasing velocity. Nevertheless, the model of slip-dominated dissipation is not appropriate for the experimental data, not even in the later stage. In our earlier study,21 the addition of both dissipation mechanisms is suggested, |S|V ) Pv + Ps. In this case, the film is allowed to exhibit viscous flow and, at the same time, sliding motion along the interface. Thus, the overall velocity V is given by the sum of both contributions
V ) Vv + Vs
Figure 5. Data of Figures 3 and 4 plotted in a different manner. Except for very small holes, the data of the dewetting velocity V ) R˙ are in line when plotted versus 1/xR. Extrapolation for infinitely large holes yields a finite velocity V0.
(3)
Because viscous flow and sliding motion are driven by the same spreading coefficient S and the power dissipated in each of the processes depends only on the respective velocity contribution Vv or Vs, we assume that the two contributions Vv and Vs are given by the above expressions for solely viscous flow (eq 1) and plug flow (eq 2), respectively. Using Vs ) K/xR in the “combined model” (eq 3), separation of the variables yields the function
t - t0 )
(
(
))
xR 1 R - 2γ xR + 2γ2 ln 1 + , Vv γ
γ)
K (4) Vv
Figure 6. For the evaluation of the combined model, four representative data sets of dewetting velocities of PS(13.7k) films at 120 °C are plotted versus 1/xR.
Unfortunately, this implicit equation for the radius is rather cumbersome, and the fit parameters K and Vv are interdependent. Therefore, this function is not suited to critically test the combined model and its assumptions.
4. Testing the Combined Model An alternative and more comprehensive way to check the combined model (eq 3) is to plot the velocity versus 1/xR. If the combined model is valid, then the data points of the mature rim regime should appear on a straight line with positive intercept Vv and slope K. Note that even the special cases Vv ) 0 and K ) 0 are included in the model, indicating that viscous flow is negligible compared to sliding motion and vice versa, respectively. As shown in Figure 5, our experimental data indeed fall on such a straight line with a finite velocity V0 extrapolating 1/xR to zero, i.e., for infinitely large holes. This gives the first hint that the simple combined model V ) Vv + Vs is an adequate description of the dewetting rate of short-chain polymer films. Just the data of small holes on DTS with radii below 4 µm deviate from this line. Because the rims of these holes are not yet in the mature regime, we can neglect these data points for further analysis. How can we further test that the combined model is indeed an appropriate description? If the intercept V0 represents the dewetting velocity of viscous flow Vv, then it should fulfill three conditions: (i) the temperature dependence of V0 should be dominated by the inverse viscosity (45) Fetzer, R.; Rauscher, M.; Mu¨nch, A.; Wagner, B. A.; Jacobs, K. Europhys. Lett. 2006, 75, 638.
Figure 7. Finite velocity V0 extrapolated for infinite large hole radii increases with increasing dewetting temperature. This behavior qualitatively compares well to the temperature dependence of the inverse melt viscosity. Additionally, V0 depends neither on the initial film thickness nor on the substrate coating.
of the polymer melt, (ii) V0 should be independent of the initial film thickness h0, which is a very strong condition, and (iii) the intercept V0 should depend only on the contact angle, not on the slip length. To test these three conditions, we repeated our dewetting experiments at three different temperatures and with films of two different thicknesses on both coatings DTS and OTS (cf. Figure 6 for the experiments at 120 °C). As shown in Figure 7, the extrapolated velocity V0 increases for increasing dewetting temperature; moreover, the qualitative run compares well to the inverse viscosity of PS(13.7k) as measured by a rheometer. Note
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Figure 8. Width of the rim as a function of the hole radius, shown for a 130-nm-thick PS(13.7k) film dewetting from OTS at 110 °C. By fitting eq 5 to the data, we get Cs ) 1.96.
that viscous dissipation in the vicinity of the three-phase contact line is not necessarily expected to follow the 1/η behavior exactly because there is an additional contribution to the shear flow that is derived directly from the contact line dynamics (i.e., thermally activated jumps over an energy barrier, pinning, etc.).46 Thus, the first condition for V0 concerning the temperature dependence is satisfied sufficiently. Note that, within the temperature range of the performed experiments, the driving force |S| is not affected by temperature changes. Concentrating on the influence of the initial film thickness on the dewetting velocity, we learn from Figures 6 and 7 that the differences in velocity can be attributed to different slopes K rather than to different intercepts V0. In Figure 7, within the error bars, no systematic impact of the film thickness h0 on the extrapolated velocity V0 can be observed.47 With this, the second condition is fulfilled. Furthermore, V0 is not influenced by the substrate, OTS- or DTS-covered silicon wafers, on which the PS films are supposed to show different slip lengths. This satisfies the third condition, the independence of slippage of V0. Note that in general V0 depends on the substrate because viscous dissipation is determined by the dynamic contact angle θdyn. In our experiments, however, we found the same θdyn on both substrates and, consequently, the same values for V0. With the described tests, we found all three conditions satisfied and therefore can assume that the intercept V0 indeed compares to Vv, which allows further analysis of the data.
5. Results: The Slip Length In the second step, the velocity component Vs is analyzed, and the slip length b is extracted. In the case of large slippage at the solid/liquid interface (plug flow), the velocity is given by eq 2, where the width of the rim reads
w ) Csxh0R
(5)
Hence, the slope K shown in Figure 5 is expected to be
K)
1 |S| b 3 η C h sx 0
(6)
First, the constant Cs has to be determined by plotting the width of the rim, w, versus the hole radius R (cf. Figure 8). Here, w is taken as the lateral distance between the three-phase contact line (x ) 0) and the position where the rim height is dropped (46) de Ruijter, M. J.; De Coninck, J.; Oshanin, G. Langmuir 1999, 15, 2209. (47) Note that in order to strictly exclude any influence of the film thickness on the intercept V0, experiments with larger variations in film thickness would be required.
Figure 9. (Left) Slope K of the straight lines in Figure 5 as a function of the melt temperature. The parameter K depends on the initial film thickness as well as on the type of coating. (Right) Slip length for polystyrene on top of both DTS- and OTS-covered Si wafers. The values are determined from K.
to 110% of the prepared film thickness h0 (i.e., h(w) ) 1.1h0). Fitting eq 5 to the data of 130- and 230-nm-thick PS films, we obtain Cs ) 1.96(5) on OTS-covered wafers and Cs ) 2.1(1) on DTS-covered wafers. Knowing Cs, the spreading coefficient |S| ) 0.0118 N/m, the initial film thickness h0, and the viscosity η of the investigated films (cf. Figure 7), we can now use eq 6 to directly determine the slip length b from the slope K. Although the values for K are affected by the film thickness (cf. Figures 6 and 9a), this dependence is canceled out very well for the slip length, as shown in Figure 9b. This again indicates that the combined model (eq 2) can be used to analyze the data of our experimental system. We find b decreasing for increasing dewetting temperature. Additionally, the slip length of PS(13.7k) is clearly larger on DTS than on OTS, but the difference decreases with increasing temperature, a fact that corroborates the results for the slip lengths determined by rim shape analysis.45
6. Conclusions We have developed a new analysis method to extract slip lengths from dewetting rates. The experiments give evidence that a model has to be put forward that combines frictional energy dissipation at the interface with (viscous) dissipation at the threephase contact line. We show that the simple addition of two different velocity components is an adequate description for highly viscous Newtonian liquids dewetting from smooth surfaces. Moreover, we were able to demonstrate that this description also captures a variable friction coefficient at the solid/liquid interface. Components Vv and Vs are given by the well-known expressions derived by Brochard-Wyart et al.15 for the limiting cases of no slip and plug flow, respectively. From the slope of the velocity data plotted versus 1/xR, the slip length can be determined. For short-chained polystyrene melts on OTS- and DTS-covered wafers, we found slip lengths between 400 nm and 6 µm. Slippage in this system decreases for increasing dewetting temperature. On the DTS coating, the slip lengths are up to 1 order of magnitude larger than on OTS. So far, we have hypotheses only for the molecular mechanisms at the solid/liquid interface that give rise to the different slip lengths on OTS- and DTS-covered wafers. Commonly investigated system parameters such as interaction forces between solid and liquid (i.e., contact angle and long-ranged dispersion forces) or substrate roughness are identical on both types of
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substrates. Thus, we have no straightforward explanation. The molecular origin of slippage in our system has to be postponed for future research. The simple analysis method of dewetting rates, however, is a powerful tool that allows for extensive studies of various systems to get a comprehensive picture of slippage of simple and complex liquids.
Fetzer and Jacobs
Acknowledgment. We acknowledge financial support by the European Graduate School GRK 532 and by grant JA 905/3 within DFG priority program 1164 and the generous contribution of Si wafers by Siltronic AG, Burghausen, Germany. LA701746R
