Langmuir 1993,9, 1906-1909
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A Microscopic Simulation of the Spreading of Layered Droplets J. De Coninck,?N. Fraysse,*M. P. Valignat,t and A. M. Cazabat'tj Coll2ge de France, Physique de la Matiare CondemBe, 11 place M. Berthelot, 75231 Paria Cedex 05, France, and Universit6 de Mom-Hainaut, Facult6 des Sciences, 20 place de Parc, 7000 Mom, Belgium Received February 9,1993. In Final Form: April 28,1993 Spreading dropleta of nonvolatile liquids show striking steplike thickness profiles, revealing a smectic order induced by the solid surface. The dynamics of growth of the successive layera has been studied experimentallyand compared with a numerical study using Monte Carlo simulation. The steplikeprofiles, the diffusive behavior observed at the early stages of spreading,and the slowing-down occurring at long times due to the finite volume of the droplet are satisfactorily accounted for. Introduction Because of ita practical applications in coating, lubrication, adhesion, ..., the old field of wetting phenomena has attracted recently a renewed interest.14 Much work has been devoted to the investigation of the ymesoscopic" scale: where the liquid film is still a continuous medium, controlled by van der Waals interactions and obeying simple, general laws. On the contrary, fiis of molecular thickness do not follow general laws: they are controlled by short range interactions which depend explicitly on the liquid and solid structures and chemical properties. There is increasing evidence that the first molecular liquid layer in contact with a solid surface plays a significant role in technical processes: the strong dissihas pation observed in some forced-flow e~perimenta"~ ita origin at the molecular scale6** and cannot be accounted for by hydrodynamical modekg A way to investigate short range interaction is to follow the spreading dynamics of microscopic droplets of nonvolatile liquids. We already reported4 the ellipsometric observation of the corresponding steplike profiles, where up to four molecular layers were clearly visible. In this paper, we compare with a Monte Carlo simulation a study of the profile and the growth of these layers performed with the same experimental setup.4 Experimental Observations The wetting fluids are PDMS, poly(dimethylsiloxanes), low molecularweight polymerswithout polymericeffects: Mp= 2000, viscosity 7 = 20 cP; Mp = 7300, viscosity t) = 130cP; Mp = 9760, viscosity t) = 210 cP, polydispersity index 1.09 (gel chromatography, courtesy of D. Teyssi6). These liquids are nonvolatile at three dimensions, Le., the volume of the droplets is constant during weeks. The substrates are silicon wafers ([lll]crystallographic plane) covered by 20 A of natural oxide, on which a
* To whom correspondence should be addressed. + Univensit4 de Mons-Hainaut.
* Collage de France.
(1) Derjaguin, B. V.; Churaev, N. V.; Muller, V. M. In Surface Forces; Consultant Bureau: New York, 1987, and referencea therein. (2) de Gennes, P. G. Reu. Mod. Phys. 1985,67, 828. (3) Teletzke, G. F. Ph.D. Thesis, University of Minnesota, 1983. (4) Heslot, F.; Cazabat, A. M.; Levinson, P.; Fraysse, N. Phys.Reo. Lett. 1990,65, 599. (6) Blake, T. D. Wetting kinetics: how do wetting lines move?; Paper la, AIChE International Symposium on the Mechanics of Thin-Film Coating, 6-10 March 1988, New Orlean, LA, 1988 (unpublished). (6) Fermigier, M. Ph.D. Thesis, University of Paria VI, 1989. (7) Calvo, A.; Paterson, I.; Chertcoff, R.; h e n , M.; Hulin, J. P. J. Colloid Interface Sci. 1991, 141, 384. (8) Thomson, P. A. Ph.D. Thesis, Johns Hopkins University, 1990. (9) Cox, R. G. J. Fluid Mech. 1986,168, 169.
0743-7463/93/2409-1906$04.00/0
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Figure 1. Ellipsometricthickness profiles of a microdroplet of poly(dimethylsiloxaue) as a function of time. Profiles are recorded every 24 h. Curve numbers, elapsed time in days Ad = 9760. The substrate is a silicon wafer covered by a g r a f d layer of trimethyl groups.
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Figure 2. Late stage of spreading for a 2D-nonvolatilemicrodroplet on the same substrate (Mp= 7300): f i t profile (dotted), 26 h after drop deposition;second profile (fullline), 170h; third profile (dotted), 262 h.
hydrophobic layer of trimethyl groups has been grafted. The critical surface tension of this surface is ye = 22.2 dyn/cm (for the homologous series of alkanes). It is completelywetted by the PDMS (y- 20.6-21 dyrdcm)and behavesas a low energy~urface.~ A typical series of drop profiles is shown in Figure 1for the PDMS with Mp = 9760. The steplike profile has steep edges, indicating that the liquid is also nonvolatile at two dimensions. The late stage of spreading is shown in Figure 2 for the sample with Mp = 7300: A compact, monomolecular layer of molecules stays for weeks, at least if the surface is smooth and chemically homogeneous. This is no longer the case for the lightest oil (Mp= 2000), the monolayer of which is known to spread due to the 2D diffusion 0 1993 American Chemical Society
Spreading of Layered Droplets
Langmuir, Vol. 9, No. 7, 1993 1907 recovery of all the experimentalfads, at least qualitatively. To study in more detail these diffusive behaviors, we performed a Monte Carlo simulation on a 3-Dlattice for a slice of a drop in the presence of an impenetrable wall. Each site of the lattice may be occupied by one molecule (nj= 1)or empty (ni = 0). The corresponding Hamiltonian describing the interactions between the molecules is of the following form16
2 51
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i
Y
where Ui describes the attraction of the wall which we chooeeof the van der Waals type in this qualitativeanalyeis, the short range interactions are accountedfor by a specific interaction A for the first layer
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and vij represents the attraction between two molecules at sites i and j (vij < 0). Here, for simplicity, we consider a nearest neighbor interaction -
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Figure 3. Radius of the fmt layer as a function of the square root of the time for the lighter PDMS on the same substrate.Mp = 2OOO. First diffusion coefficient D = 2.4 X W0mas-l. Late mz s-l. diffueion coefficient D 6 X of the molecules on the
This PDMS is nonvolatile at 3D, but slightly volatile at 2D. The 2D diffusion is slow compared tothe growthof the layers,which are still well-defined, althoughthe edges are smoother than in the previous ~8888.The length of the fmt layer is plotted versus the square root of the time dt in Figure 3. Two domains of spreadingare clearlyvisible. At short times, the central part of the drop acta as a reservoir. The successive layers grow as dt. At long times, the maximum height of the drop is only a few monolayers. The emptying of each layer into the next lower one causes the upper layers to slow down and ultimately recede and dieappear. Finally, all the molecules are in the first layer, which goes on spreading for 2Dvolatile liquids. In this final sfage a diffusion-like law is again o k e d . For 2D-nonvolatile liquids on homogeneous surfaces, this second diffusion law does not take place.
Microscopic Simulation: Model Numerical attempts for describing the spreading of microscopic droplets are under way. They go to the rescue of theoretical modelsl1J2 which either disagree with experiments" (the first layer is predicted to grow linearly with time) or are not easily used in a predictive way.12 However, a recent molecular dynamicssimulation13found the length of the first monolayer to grow like dlog(t) instead of dt. This result might be due to the small size of the drop (4OOO molecules for all the drop), which is in the crossover between the two diffusive laws, and not to disagree in fact with experiments (the dlog(t) law is actually o b s e ~ e din these simulations when the drop is completely layered13). To understand the meaning of all the microscopic parameters which appear into the problem, it is of great interest to fiid a microscopic process which enables the (10) H d o t , F.; Cambat, A. M.; Levineon, P. Phys.Rev. Lett. 1989,62, 1286. (11) Abraham, D. B.; Collet, P.; De Coninclr, J.; Dunlop,F. Phys.Rev. Lett. 1990,66, 196. (12) de ( 3 % ~ eP. ~ ,G.; Cazabat, A. M. C. R. Acad. Sci. 1990,310,1601. (13) Yang, J.; Koplik, J.;Banavar,J. R.Phys.Reu.Lett. 1991,67,3639.
Koplik, J. Privata communication. (14) Healot, F.; fiayeee, N.; Cazabat, A. M. Nature 1989,338,640.
if i j are n-n 0 otherwise (3) The model is equivalent to a 3-DIsing model with local fields uij = Y
where
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1
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Within this formalism, we consider two phases [+]and [-I in equilibrium with a particular initial configuration
corresponding to a cube of occupied sites (thisc o n f i i ration mimics in fact the first layers of molecules at the beginning of the experiment, cf. Figure 1). We then let evolve the system according to a Monte Carlo dynamics which preserves the total number of [+I spins and [-I spins: the volume of the drop is indeed kept f i e d during the experiment. This is the Kawasaki double spin exchange dynamics.1s-18 The system we consider is of 3000 occupied sites for this slice of the drop. At each step, the spins ( g i , uj) may interchange positions (uj, ci) according to the Kawasaki algorithm with a probability of transition defined with the Boltzmann factor corresponding to the Hamiltonian (l),
all the other spins of the system being kept fiied, k is the Boltzmann constant and T is the temperature.
Microscopic Simulation: Results In our numerical simulations, we have considered a lattice of 1500 X 20 X 100sites and appropriate boundary conditions (periodic in the y direction, the width of the slice, and free everywhere else on top of the substrate) (16) Pandit, R.; et al. Phys.Rev. B 1982,26,6112. (16) Kawasaki, K. In P h e Transitions and Critical Phenomena; Domb, C., Green, M. S., Eds.;Academic Prese: London, 1972; Vol. 2. (17) Kawasaki, K. Phys.Rev. 1966,146,224. (18) Binder, K., Ed.Monte-Carlo methods in Statistical Mechanics: An Introduction; Springer: Berlin, 1988.
1908 Langmuir, Vol. 9, No. 7, 1993
De Coninck et 01.
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-m '? ?ll vl 90 Bn ry G Figure 4. Thickness of the simulated profile in unita of sites. The profiles have been reproduced after 1 (a), 28 OOO (b), and 94 OOO (c) Monte Carlo steps per occupied site. . 47
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Figure 5. (a) Radii of the four fmt layers as a function of the square root of MC steps per site with a total number of occupied sites (volumeof the drop) = 2940 and the parameters: @A= 100; @B= 33.3; @J = 0.33. The results reported in this figure are statistical averages over 10 independent realizations. The total number of MC steps per site considered in this simulation is of the order of 108. Once all the moleculesare in the first layer, they still diffuse but more slowly. The dotted straight lines are obtained by best linear fit. (b, insert) Number of occupied sites in the four first layers as function of the square root of the number of MC steps per site for the Same value of the parameters: BA = 100; BB = 33.3; @J = 0.33. Once the number of occupied sites in the second layer becomes zero, all the occupied sites are in the first layer, then the number of occupied sites in the first layer remains constant. The results reported here correspond to statisticalaveragesover 10independent realizations. The dotted straight line is obtained by best linear fit.
with 106 MC steps per occupied site (MCS/S). Similar results were obtained for different widths and different number of molecules. It is not our aim to present in this paper all the different regimes we have observed as a function of the microscopicparameters, but we would like
:m WI m .w m Figure 6. Number of occupied sites in the fmt layer as function of the square root of the number of MC steps per site: For different amplitudes of particlewall interactiom Ui. The AIB ratio is kept constant. AIB = 3: (a) PA = 100, BB * 33.3, and @J=0.33;(b)BA =0.6,BB=0.2,andBJ=0.33. Inthislastawe, the spreading is slower. For different volumee of drops with fixed interactions (PA = 100, BB = 33.3) (a) 2940 occupied sitae and (c) 6300 occupied sites, the first regime of spading is more extended. n
im
to stresa the adequacy of the microacopic dynamica which is here considered with the experimental observations. This is the reason why we limit ourselves in this paper to present data corresponding to one triplet of parameters &.I = 0.33, @B = 33.3, and PA = 100 (@ = l/kT). We have reproduced in Figure 4 the shape of the drop for successive times, i.e. the number of MCS/S. It is clearly seen in this figure that this kind of dynamica allows ua to recover the sharp experimental profiles. However, one must be aware of the huge difference in the lateral d e a of the two casea: In the simulation,t h i c k " and lateral scalesare comparable,while in the experimentsthe lateral scale is macroscopic. In Figure 5a, we have presented the length of the f i i t four layers of molecules as a function of the square root of the time, i.e. the square root of the number of MC stepa per site. These simulations have been performed with the same set of parameters &.I = 0.33, @B= 33.3, and @A = 100. The main features of the experimentaldata (Figure 3), i.e. the two limiting diffusion laws, the second one occurring when the second layer disappears, are nicely accounted for. This property is also clearly reproduced in Figure 5b where we have plotted the number of molecules in the first four layers as function of the square root of the number of MC steps. Increasingthe drop volume shifts the cromover to larger times as illustrated in Figure 6. The late process reflects the 2D evaporation of a 2D liquid since the values of the couplings@Jensuresthat we are working below the 3D critical point but above the 2D criticalpoint. Let us also point out here that this property can only be studied on top of a 2D substrate since, for the 1D system, there cannot be phase transitions for short range interactions. From the experimental point of view, one must be very careful about the exact nature of the observed 2D-late spreading proceaa. As a matter of fact, 2D critical
Spreading of Layered Droplets temperatures for PDMS are not known. If no late diffusion is observed, the liquid is certainly below ita 2D critical point. If on the contrary late diffusion is observed, surface heterogeneity can also play an important role in the dynamics. Further experimental investigationswith high spatial resolution techniques like AFM are needed to determine the exact part of 2D evaporation and heterogeneity in the measured profiles. Finally, let us point out here that microscopic dynamics allows us to interpret very nicely the notion of “rubber of permeation” already predicted by de Gennes and Cazabat.12 Indeed in our model, the molecules are considered as hard cores. The motion of the molecules can only appear through holes which have been created by thermal fluctuations. Since the molecules are preserved during the time and since a molecule can only jump from one site to a neighboring site, there indeed appears an annulus of permeation near the surface of the drop. Let us also point out here that our model allows to take into account the growing of the layers which is due to molecules coming through the atmosphere by evaporation of the drop. As recalled above, our model considers the molecules as hard cores. This is plausible for compact molecules like tetraki~(2ethylhexoxy)silane~~ but not for chainlike molecules like PDMS. However, the drop shapes are very similar in both cases, at least for short PDMS chains, well below the entanglementthreshold. The stepped-pyramid shape of the profiles supporta the assumptionthat at least in the mobile part of the drop the PDMS molecules are
Langmuir, Vol. 9, No. 7, 1993 1909
in a flat configuration and belong to a given layer. However, a quantitative description of the permeation process at the molecule scale would require a more elaborated model with several lattice sites per molecule. Attampta are presently under way. The final aim is to account for the shape and dynamics of the whole profile, which is known to depend on the surface energy of the substrate and on the liquid itself as illustrated in Figure 6 where we have reproduced the length of the f i t layer for two different amplitudes of interaction, @A= 100,@B= 33.3, and @J= 0.33 and @A= 0.60, @B= 0.20, and @J= 0.33. We are now studying liquid-liquid and liquid-wall interactions in a systematic way: the interaction parameters can be varied, and more elaborated liquid-wall interactions can be introduced. For the experimental part, various liquids and surfaces are being presently investigated.
Acknowledgment. Fruitful discussions with P. G. de Gennes are gratefully acknowledged. We thank J. Koplik and J. R. Banavar for helpful comments. The ellipsometric setup has been assembled with F. Heslot and P. Levinson. The experimental investigations have been supported by a French DRET grant. This text presenta results of the Belgian programme in Interuniversity Poles of attraction initiated by the Belgian state, Prime Minister’s office, SciencePolicy Programming. The scientificresponsibility is assumed by ita authors.
