J . Phys. Chem. 1990, 94, 7581-7585
7581
Spreading at the Microscoplc Scale A. M. Cazabat,* N. Fraysse, F. Heslot, and P. Carles Physique de la MatiPre CondensEe. Collgge de France, I I place Marcelin Berthelot, 75231 Paris Cedex 05, France (Received: December 7, 1989; In Final Form: April 13, 1990)
At the macroscopic scale, the condition for a liquid to spread on a solid surface is that the initial spreading parameter So should be positive. At the microscopic scale, the relevant parameter P(z) for spreading depends on the film thickness z and . molecular thicknesses, P(z) is nothing more than the usual can be expressed as a function of the disjoining pressure ~ ( z )At ) P ( z ) are available in the literature. two-dimensional film pressure. Experimental and theoretical determinations of ~ ( zand These results are used to analyze and predict the thickness profile of films spreading on solid surfaces. It is shown that a relevant case for experiments is the “diffusive” situation, where the film grows from a nearly static macroscopic meniscus. Explicit formulas for the thickness-dependent diffusion coefficient D ( z ) and a discussion of characteristic film profiles are given in this case. The agreement with available experimental results is satisfactory.
Introduction At the present time, the laws of macroscopic spreading are well-known.’ The situation is not so favorable at the microscopic scale. The static properties of thin films, Le., the thickness of which is lower than typically 1 r m , are now satisfactorily described,2” but not the dynamic ones. Only relatively thick films, which can be treated by hydrodynamics, have been investigated,”s and, even in this case, explicit analytic results have been obtained in the particular case of van der Waals liquids only.7 The aim of this paper is to extend the analysis toward the dynamics of molecularly thin films and, more specifically, to establish how the well-known static properties of the films appear in the dynamic equations and in the resulting thickness profiles. At the macroscopic scale, the spontaneous spreading of a liquid on a solid surface is observed if the initial spreading parameter So is positive or zer0.I Thin liquid films on solid surfaces are characterized by the disjoining pressure ~ ( zwhich ) is a function of the film thickness z . ~We consider z to be a continuous variable, namely, a local average over a large number of molecules. For spread-out monolayers, z can thus become smaller than ho, the thickness of a compact layer of molecules. The spreading parameter can be written as
of spreading for a droplet of nonvolatile wetting liquid being a “pancake” of thickness e, with abrupt edges. The critical value e, corresponds exactly to P(e,) = 0. However, this model assumes a truncated form of the disjoining pressure (Figure 1) and therefore ignores any ultrathin film which would in fact destroy the “pancake”, the ultimate stage on a nonlimited surface being a two-dimensional gas. Attempts can be made to calculate ~ ( zfor ) van der Waals fluids at small thicknesses by direct summation of the contributions of the successive molecular layers. The calculation by Beaglehole* leads to a strictly positive, smoothly varying P ( z ) . In this case, no pancake is expected to exist. With regard to the dynamics, it is clear that the thin part of the film, where short-range interactions are important, is not described by available theories. Unfortunately, films of nonvolatile liquids are usually very thin ones?JO For example, an ellipsometric thickness profile of a drop of poly(dimethylsi1oxane) (PDMS) spreading on a bare hydroxylated silica surface is given in Figure 2: the maximum thickness is still in the micrometer range, but a large part of the film is molecular. Also, surface-induced layering, which is a typical molecular effect, can be observed. This asks for a further analysis of this range of thicknesses.
For thin films, the spreading condition is no longer So1 0: it depends on the film thickness z . Actually, it is known that ultrathin films may spread even in nonmacroscopically wetting cases (So< 0). The right parameter is the film pressure, which must be positive for spreading at thickness z to take place. It can be written as3+
Thin Film Properties: Statics Experimental and theoretical information on thin films have been obtained from adsorption isotherms for volatile liquids”J2 and film pressure isotherms for films spreading on liquid surfa~es.~J Adsorption isotherms give the film thickness z as a function of the relative partial pressure PV/PSAT.The disjoining pressure can be written as
P(z) = -zr(z)
+ f r0 ( h )
dh
For thick films, P(z) tends toward S,. For ultrathin molecular films, P ( z ) is the two-dimensional film pressure, widely used for describing monolayers.6 In this respect, the model proposed by de Gennes7 can be given another interpretation. It predicts that the thickness of the film cannot become smaller than a critical value e,., the ultimate stage (1) Cooper, W. A.; Nuttall, W. A. J . Agric. Sci. 1915, 7, 219. (2) Deryaguin, B. V.; Churaev, N. V.; Muller, V. M. Surface Forces; Consultant Bureau: New York, 1987; and references therein.
(3) Churaev, N. V.; Starov, V. M.; Deryaguin, B. V. J. Colloid Interface Sci. 1982, 89, 16-24. (4) Teletzke, G.F.; Scriven. L. E.;Davis, H. T. J. Chem. Phys. 1982, 77, 5794; 1983, 78, 1431. (5) De Feijter, J. A. In Thin liquidjilms; Ivanov, I. B., Ed.; Surfactant Science Series 29; Marcel Dekker: New York, 1988; p 1. (6) Adamson, A. W . Physical Chemisrry f Surfaces, 4th ed.;Wiley: New . o_ York, 1982. (7) De Gennes, P. G. Rev. Mod. Phys. 1985,57, 828.
0022-3654/90/2094-758 1$02.50/0
* ( z ) = -k T log UO
(5) PSAT
where k is the Boltzmann constant, T the absolute temperature, and uo the molecular volume. Thus, an adsorption isotherm represents z as a function of exp(-uor/k7‘) for positive values of ?r. Note that, if the “true” disjoining pressure has an oscillating behavior (with unstable parts in the curve), these oscillations will not be observed on the isotherms: phase transitions between films (8) Beaglehole, D. Submitted for publication. (9) Leger, L.; Erman, M.; Guinet, A. M.;Ausserre, D.; Strazielle, G.; Benattar, J. J.; Rieutord, F.; Daillant, J.; Bosio, L. Rev. Phys. Appl. 1988, 23, 1047. (IO) Heslot, F.;Cazabat, A. M.; Levinson, P. Phys. Reo. Lett. 1989, 62, 1286. (11) Bassignana, I. C.; Larher, Y. Surf.Sci. 1985, 147, 48. (12) Ball, P. C.; Evans, R. J . Chem. Phys. 1988, 89, 4412. (13) Jarvis, N. L. J. Colloid Interface Sci. 1969, 29, 647-657.
0 1990 American Chemical Society
7582
Cazabat et al.
The Journal of Physical Chemistry, Vol. 94, No. 19, 1990
Figure 1. Schematic r ( z ) variation leading to the “pancake”: van der Waals (-); model isotherm for “pancake” (-); schematic real isotherm (---).
Figure 3. Schematic steplike adsorption isotherms, Le., film thickness z as a function of the relative vapor pressure Pv/PsAT: (a) complete wetting; (b) partial wetting. ho is the thickness of the compact monolayer.
4-1”-
b
Figure 2. Ellipsometric profile of a PDMS drop on an oxide covered silicon wafer after 25 h. The maximum height of the drop is about 1 pm.
of different thicknesses will take place. For relatively thick films ( z 1 50 A) the disjoining pressure ~ ( zis)well-described by the DLVO theory.I4 It is not the range of thicknesses we are interested in. Let us just mention the possibility of phase transition between an ultrathin film and the thick film, as for water films on quartz.15 We are more interested in the low-pressure part of adsorption isotherms, where films have molecular thickness. Smooth or steplike adsorption isotherms can be obtained,I6 according to the relative strength of the thermal energy kT and the ‘fluid-fluid” tff and ”fluid-substrate” tfs interaction energies.”~12Steplike isotherms are an indication for strong interactions. Successive compact liquid monolayers condense on the substrate. The discontinuous jumps in z correspond to phase transitions between films of successive thicknesses (Figure 3). Experimental adsorption isotherms are available only for volatile liquids. However, the behavior for nonvolatile ones is expected to be similar for comparable values of the ratios c f f / k Tand q , / k T . In the range of molecular thicknesses, the static properties of the film may also be inferred from experimental and theoretical film pressure isotherms for monolayers on liquid surface^.^*'^ At equilibrium, only the fluid-fluid interaction energy tff plays a role. Models for this so-called lateral interaction are available.6 For large, positive values of cm/kT,a two-dimensional phase transition may occur in the monolayer between a 2D liquid and a 2D gas (Figure 4a). Moderately positive or negative values of c f f / k T lead to smoother profiles (Figure 4b,c) with in any case a strong increase of P(z) when z approaches the thickness ho of the compact monolayer. In the opposite case of vanishing thickness ( z / h o 0 can be extended to the case of successive layers: the condition for the nth layer to be formed on top of the (n - 1)th one is Pn,,..l 1 0 . Note that m
CPn,,..l = so
n= I
We shall now discuss the role of these static properties in the spreading of wetting films.
Thin Film Properties: Dynamics The dynamics of growth of thin films ahead of macroscopic menisci depend on the velocity of the macroscopic liquid front. “Diffusive” films grow from static menisci.I7 The driving force for spreading is proportional to the disjoining pressure gradient along the film. “Adiabatic” films are driven by fronts moving with a velocity large enough to be practically constant over the whole film.’ Although this situation seems to be especially simple to be discussed, it does not correspond to significantly developed films: long films correspond to small driving velocities,’ and the diffusive processes dominate over the main part of their length. In the following, we shall discuss only the diffusive case, which is the one relevant for experimental investigations. Here, the macroscopic meniscus, if any, acts only as a reservoir of liquid. Also, any structural relaxation processes in the liquid film, which would interfere with the spreading, will be ignored. This is certainly a good approximation for simple, compact molecules, but it becomes questionable when polymeric molecules are concerned: disentanglement p r o c e ~ s e s may ~ * ~ play ~ ~ a role in the details of the film profile. (17) Joanny, J. F.;de Gennes, P. G. J . Phys. (Les Ulis, Fr.) 1986,47, 121. (18) Fraysse, N. Unpublished results. (19) Leger, L. Unpublished results.
Spreading at the Microscopic Scale
The Journal of Physical Chemistry, Vol. 94. No. 19. 1990 7583 disjoining pressure gradient, but the thickness dependence of the friction term will be different. A summation of the contribution of the sucassive layers and the introduction of layer-layer friction coefficientsKjkI would be needed. We shall not discuss in more detail this range of thickness. Let us just mention that, if the (n - I)th layer is static (Le., if the liquid spreads over a preexisting film), the diffusion coefficient for the spreading of the nth layer is
Figure 5. Schematic representation of the velocity field in a spreading film.
Let I be the local film thickness, x the longitudinal coordinate, the time, and Uthe average local longitudinal velocity. In the general case the continuity equation can be written as ar =--a ( z w -
f
at
ax
The velocity U results from the balance between the driving term, proportional to the disjoining pressure gradient au/ax, and the friction. Let us first recall briefly the main findings of the hydrodynamic theory, which is relevant for relofiuely f h i c k f i / m ,corresponding to the long-range part of r(z).l’ In this case, the velocity field in the film-is a Poiseuille flow with a no-slip boundary condition at the solid surface (Figure 5 ) . The velocity Ucan be expressed as
The continuity equation becomes
or
”(
at = ax Dh(r)$) where the quantity D,(r) = -(~’/3n)(du/dz) can be interpreted as a thickness-dependent diffusion coefficient in the hydrodynamic regime,” a being the bulk viscosity of the liquid. For fhinnerfilms. the friction term is no longer calculated from a Poiseuille flow with no slip condition. Monolayers, for example, exhibit a slipping behavior, U being the velocity of the individual molecules. Now, the dissipation is due to the friction of the molecules on the surface. Let 01 be the friction coefficient of an isolated molecule on the surface, which accounts for any adsorption-desorption and associated hopping processes. The velocity U can be written as20
Thus, information on a and the K,kI coefficients, which are of obvious importance for layered films, might be obtained experimentally by stopping the lower layers with a nonwettable barrier. Note also that the velocity of the first layer is low in the case of strong adsorption on the solid (a>> K Z d . In this case, the no-slip behavior will be observed even at relatively low thicknesses.
Diffusive Film Profiles: z(x) In the general case, solving a diffusionlike equation with a thickness-dependent diffusion coefficient can only be done numerically2’ for given boundary conditions. Only in the case of simple laws for u (nonretarded van der Waals for example) can a self-similar solution hold for the whole film profile r ( x ) . This situation has been studied by de Gennes and Joanny. The implicit equation is” x2 = 2 D ( z ) f . For nonretarded van der Waals interactions, Dh = it so that z(x) = x - ~ . Usually. and this is the case in the thin oart of the film we are at preseni/nvtitigating, a self-similarsolutibn d m not hold. Only a aualitauve. but not without interest.discussion of the film Drofile can be made: the film profile depends both on the general’shape of the U ( Z ) curve, which is an equilibrium property, and on the friction coefficients. Large D values are associated with extended, relatively flat parts of the profile (of typical length = (Df)Il2),while low D values are associated with steep parts of the height profile z(x). Every (horizontal) transition in u(r)or P ( r ) corresponds to a nearly vertical part of the profile: thus, steplike isotherms (Figure 3) are associated with steplike profiles. The profile is not strictly vertical because of the higher order terms Q(dr/dx)’ which have k e n ignored in our simple model where the film slope is assumed to be small. Such terms become significant close to a phase transition and will produce a slight smoothing out of the profile.” However, the spatial extent of the perturbed zone is expected to be of the order of the healing Le., much less than 1 pm for molecular films. This correction is negligible since film lengths approach a few centimeters and since the spatial resolution of the experiments is typically a few tens of micrometers. At the very edge of the film, the velocity U(t) is practically the same everywhere, as suggested by de Gennes.’s This allows one to predict the film profile in this range. In the reference frame of the tip, one gets rU = -D&)
VO au
(I=--
a ax
This can be rewritten as dr _
The corresponding monolayer diffusion coefficient DMis
dx At vanishing thicknesses ( L K1,*. be the same for the three profiles.
10
-
4a). The corresponding profile will exhibit a well-developed, rapidly growing monomolecular tongue ( z h,) terminated by a sharp edge (Figure 6a). For moderately attractive or repulsive interactions, the profile is smoother, with a shorter molecular tongue (Figure 6b). The same trends are observed in the few next layers, but now the friction coefficients can change the profile significantly. For example, if a >> K2.1,the second layer will spread on top of the first one which is practically static: the profile is terminated by a sharp edge a t z L ho (Figure 6c). Finally, one has to keep in mind that all structural relaxations have been ignored in this model. In particular, reptation and disentanglement processes are expected to play a role for high polymers.
Comparison with Experiments Let us compare these predictions with experimental findings. Ellipsometric measurements of film profiles have been performed on various low-volatility liquids: light poly(dimethylsi1oxane) (PDMS),9J0*2629tetrakis(2-ethy1hexoxy)silane (TK27b928),and squalane.l0 The substrates were hydrophilic oxide covered silicon wafers,”10*29 as well as hydrophobic, but still wettable by PDMS, Langmuir-Blodgett layer covered silica surfaces.29 The main part of the film profiles corresponds to thicknesses below, say, 200 A. The crossover toward the macroscopic meniscus is always extremely short (-0.1 mm) and could be studied only by specific setups, as the microscopic imaging ellipsometer developed by B e a g l e h ~ l e . ~ JFigure ~ 7 (adapted from ref 27b) gives an example of an ellipsometric profile recorded on TK after 300 h in the capillary rise geometry. The surface is an oxide covered silicon wafer. Experimentally, in the “thick” part of the films, e.g., between 20 and 200 A typically, the plot 1 / z versus x is approximately linear (see Figure 7 for example). For a diffusive film of a van der Waals fluid in the hydrodynamic range, the predicted profile scales as z iz: x - ~ , ~whereas ’ a z 2: x-l law is expected for adiabatic films only.7 It had then been concluded that some ‘remnant motion” of the macroscopic meniscus might be responsible for the observed profile^.^' From the present analysis, it appears that another possible explanation is the Occurrence of partial slipping in the film: with D(z) proportional to z2 du/dz, a x-l profile would be observed as well. Thus, keeping in mind the narrow range over which the fit is done, we could also interpret the observed profile (26) (a) Beaglehole, D.J . Phys. Chem. 1989,93,893and previous (1984) unpublished results. (b) Beaglehole. D.; Heslot, F.; Cazabat, A. M. In Proceedings of the E.P.S. meeting, Hydrodynamics of Dispersed Media; North-Holland: Amsterdam, 1990. (27) (a) Heslot, F.; Cazabat, A. M.; Fraysse, N. J . fhys.: Condens. Matfer 1989,I , 5793. (b) Cazabat, A. M.;Fraysse, N.; Heslot, F.; Carles, P.; Levinson, P. To be published in frog. Colloid folym. Sci. (28) Heslot, F.; Fraysse, N.; Cazabat, A. M. Nufure 1989,338, 640. (29) Heslot, F.; Cazabat, A. M.; Levinson, P.; Fraysse, N. To be published in Phys. Rev. Lett. (30) Marmur, A.; Lelah, M.D. J . Colloid Interface Sci. 1980,78,262. (31) Beaglehole, D.Rev. Sci. Insfrum. 1988, 59, 2557.
5
0
Figure 7. Ellipsometric profile of a film of TK in the capillary rise geometry: measurement after 300 h. Insert: plot of the inverse thickness z-’ versus x coordinate (16 first points of the curve). The origin of the x axis has been arbitrarily taken as the intersection of the dotted straight line with the x axis.
as resulting from a smooth crossover between no-slip and complete slipping behaviors. In the thinner part of the films ( z 5 20 A), steplike profiles corresponding to successive superimposed monolayers have been actually observed for PDMS and T K (Figures 2 and 7). The layering is especially striking for TK, which is an indication for strong lateral interactions. As expected (cf. Figure 6a), the first monolayer forms a well-developed tongue, several centimeters long after 10 days,27bwhile the rest of the film (from tongue to macroscopic meniscus) extends over less than 5 mm. On the contrary, a smooth profile-with no particular behavior of the first monolayer (Figure 6b)-is obtained for squalane:I0 this is an indication for moderate lateral interactions. The analysis is not so straightforward for PDMS. Although a light one, it is a polymeric molecule, the conformation of which changes significantly in the vicinity of a wall. Thus, the cff interaction will in fact depend on the nature of the solid surface. On clean hydroxylated silica, one observes a gentle profile terminated by a well-defined edge (Figure 2). On LangmuirBlodgett layer covered silica, the thicker part of the profile is steep and practically static; only one molecular layer with sharp edge spreads rapidly over the ~urface.2~ In light of the preceding model, we propose the following interpretation. On high-energy surfaces, the surface-molecule interaction cy, is much larger than the molecule-molecule interaction cfP All the monomers lie on the surface, strongly adsorbed and with restricted mobility, which forbids any further reorganization of the layer. P ( z ) is not expected to be especially steep (Figures 4b and 6c). On low-energy surfaces, tff and tfs are comparable, and the mobility is larger (less adsorption). A more compact conformation of the molecules and thus a steeper P ( z ) are expected. From the thickness of the monomolecular tongue (7 A), it appears that the molecules lie still flat on the surface (Figures 4a and 6a). There is a last point which deserves discussion: when comparing theoretical predictions and experimental results,32the influence of surface heterogeneities must not be ignored. Residual roughness or chemical heterogeneities will cause a scattering in the values of D(z) and a smoothing of the profile. This is especially critical in the case of a well-developed tongue: the width of the tip ( z (32)Scheutjens, J. Private communication.
J . Phys. Chem. 1990, 94, 7585-7588
< h,) may be partly due to the scattering in the values of D(h0). As a result, a diffusion coefficient estimated from the width of the tip might rather be an indication for the surface heterogeneity: Dexp(z
