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Langmuir 1997, 13, 4754-4757
The Mechanism of Spreading: A Microscopic Description A. M. Cazabat,† M. P. Valignat,† S. Villette,† J. De Coninck,*,‡ and F. Louche‡ Physique de la Matie` re Condense´ e, Colle` ge de France, 11, Place M. Berthelot, 75231 Paris Cedex, France, and Centre de Recherche en Mode´ lisation Mole´ culaire, Universite´ de Mons-Hainaut, 20, Place du Parc, 7000 Mons, Belgium Received March 4, 1997. In Final Form: May 20, 1997X By molecular dynamics simulations, we investigate the microscopic details of the mechanism of spreading. Our data are in agreement with experimental evidence and support the validity of the de Gennes-Cazabat model describing the late stages of spreading. The effects specific to chainlike molecules are also accounted for.
The spreading of polymeric liquids on top of solid substrates has been the subject of many experimental and theoretical works. In particular, it has been recognized that some drops which spread on solids may form several layers of molecular thickness1 with a radius growing as the square root of the time in the early stage after deposition. These results have been obtained for several simple or polymeric liquids (trimethyl- or hydroxyl-terminated poly(dimethylsiloxanes) PDMS) on top of well-defined solids (bare or grafted oxidized silicon wafers). The first theoretical attempt to describe the origin of this phenomenon is due to de Gennes and Cazabat.2 In this paper, the drop has a microscopic thickness and a macroscopic radius, thus mimicking the experimental situation. The spreading is described as a competition between the driving force, which is due to the wall attraction, and the friction between layers of liquid and with the solid. Assuming essentially the viscous nature of the friction forces, the authors were able to recover most of the experimental observations. However, friction parameters are introduced at the macroscopic scale and have no microscopic interpretation. Many microscopical approaches have been proposed to support this driving force/friction model: using solid-onsolid or lattice models with an appropriate dynamics (Kawasaki), several groups were able to recapture qualitatively the experimental observations.3-5 However, the only way to justify this choice of dynamics at a microscopic scale is by using molecular dynamics (MD) techniques. Here again, many groups tried to use MD to solve this problem. Some groups6,7 used very small liquid molecules (1 or 2 atoms), and others8 oversimplified the solid substrate considered as purely flat. Recently9,10 for chainlike liquid molecules (with 8 or 16 atoms), atomistic †
Colle`ge de France. Universite de Mons-Hainaut. X Abstract published in Advance ACS Abstracts, July 15, 1997. ‡
(1) Heslot, F.; Cazabat, A. M.; Levinson, P.; Frayss, N. Phys. Rev. Lett. 1990, 65, 599. (2) de Gennes, P. G.; Cazabat, A. M.; C. R. Acad. Sci. 1990, 310, 1601. (3) De Coninck, J.; Hoorelbeke, S.; Valignat, M. P.; Cazabat, A. M. Phys. Rev. E 1993, 48, 4549. (4) De Coninck, J. Colloids Surf. A 1993, 80, 131. (5) De Coninck, J.; Fraysse, N.; Valignat, M. P.; Cazabat, A. M. Langmuir 1993, 9, 1906. (6) Yang, J. X.; Koplik, J.; Banavar, J. R. Phys. Rev. Lett. 1991, 67, 3539. (7) Yang, J. X.; Koplik, J.; Banavar, J. R. Phys. Rev. A 1992, 46, 7738. (8) Nieminen, J.; Abraham, D.; Karttienen, M.; Kaski, K. Phys. Rev. Lett. 1992, 69, 124. (9) De Coninck, J.; D’Ortona, U.; Koplik, J.; Banavar, J. R. Phys. Rev. Lett. 1995, 74, 928. (10) d’Ortona, U.; DeConinck, J.; Koplik, J.; Banavar, J.R. Phys. Rev. E 1996, 53, 562-569.
S0743-7463(97)00242-4 CCC: $14.00
representation of the solid, and sufficiently large systems to mimic the existence of a reservoir (up to 250 000 atoms), it has been recognized that MD techniques lead to the detailed recovery of the experimental observations. Attempts to recover the features of wetting behavior were proposed in refs 10 and 11, stressing the importance of friction against the substrate in this phenomenon. Here we complement this analysis at a microscopic level with new experimental and numerical MD results, for chainlike liquid molecules, studying for the same system the dynamics of spreading and the friction against the substrate. This approach thus leads to a detailed analysis of the de Gennes-Cazabat model. To begin, let us first for completeness recall here the basic features of the stratified model due to de Gennes and Cazabat. General Features of the Models Stratified Model.2 In this model, the liquid is supposed to be incompressible. The drop is formed of superimposed compact monolayers of (macroscopic) radii R1 (on the surface), R2, R3, ..., Rn. The spreading proceeds in two ways: (i) a flow inside the layers, i.e., parallel to the substrate, where a viscous friction between adjacent layers is assumed and characterized by a friction coefficient ζn,n-1 ) ζ for n > 1 and ζ1 for n ) 1; (ii) a vertical permeation flow which depends on the difference in the chemical potentials in layers n, and n - 1 and is not associated to friction. The energy at the edge of the nth layer is Wn. Using hydrodynamical considerations, the authors were able to establish a direct relationship between the dynamics of spreading measured in R1, R2, ..., and the friction against the substrate
dR12 R1 W2 - W1 ln ) 2 dt R2 ξ1
(1)
It is our aim in this paper to study in detail the validity of that prediction relating spreading properties to viscous friction. Let us now introduce our molecular dynamics model. Molecular Dynamics Simulations. For all the atoms, we have considered a standard interaction of the Lennard-Jones type
Vij(r) )
Cij r
12
Dij r
6
{(σr) - (σr) }
) 4
12
6
(2)
where r denotes the distance between the two atoms i and j. Cij and Dij refer to the fluid/fluid (ff), fluid/solid (fs), and (11) Cieplak, M.; Smith, E.; Robbins, M. Terraced spreading and the frictional force on monolayers. Preprint, 1994; Science 1994, 265, 1209.
© 1997 American Chemical Society
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solid/solid (ss) interactions as a function of the nature of i and j. For computational convenience, the tails of the potentials are cut off at rc ) 2.5 in reduced units or, in other words, in terms of the classical hard-core radius σ associated with the Lennard-Jones potential. Moreover, for adjoining atoms belonging to a given chain, we have incorporated a confining potential
Vconf(r) ) Ar6
(3)
This additional potential considerably reduces the evaporation of the molecules and allows for a better description of dry spreading. The typical chains considered here are made by 8 or 16 atoms, and the power 6 is just chosen for computational convenience. The substrate is made by one layer of fcc lattice with the (100) surface exposed to the fluid, at each site of which is an atom. We chose a heavy mass for the atoms of the solid so as to have comparable time steps for the solid and the liquid. The solid atoms execute thermal oscillations around their lattice sites, being attached to these last ones by a deep harmonic potential. For the new simulations which are presented here, we have chosen the parameters Cff ) Dff ) 1, Css ) 36.45, Dss ) 5.0, and A ) 1.0 with Cfs ) Dfs ) 1.2 and 2.0. These values are not supposed to describe any realistic potentials but are instead selected for numerical reasons. At that stage, we are indeed more interested in recovering a qualitative agreement with the observed experimental properties. Given the potentials, the motion follows from integrating classical Newton equations. The temperature of the solid alone is kept fixed during the simulation by rescaling the velocities of the atoms so as to mimic the dissipation of energy of the substrate. For this model and these parameters, it is known that a precursor film of molecular thickness will form during the spreading of the drop.9,10 Let us first analyze the friction process. Investigation of the Friction Process. On top of a square of atoms of the solid, let us consider a monolayer of atoms of the liquid at a given density. These atoms are randomly associated in chainlike molecules of given length, thus allowing the chains to cross each other in the initial state. With appropriate periodic boundary conditions in the x and y directions, we let the film of chainlike molecules equilibrate itself during a time of the order of 10 000 time steps with a time step ∆t ) 0.005τ, where τ is the natural time unit, x/(mσ2) ) 10-12 s, where m denotes the unitary mass for the considered fluid atoms, a process which will be discussed further in the following. The equilibration process leads to a monolayer of segregated chains, in good agreement with the expected behavior of 2D polymer melts.12,13 Then we naturally apply a constant force in the x > 0 direction on each atom.14 Let VCM be the computed velocity of the center of mass. In a viscous regime, one has F ) ζV. Typical results are reproduced in Figure 1, where VCM is plotted vs time. As a function of the amplitude of the applied force, three regimes are observed: a first regime where VCM is compatible with zero for very small applied force F; a second regime where VCM exhibits a plateau as a function of the time, which usually appears 100.000 steps after equilibration; and a third regime where a plateau was not observable within our simulations. (12) de Gennes, P. G. Scaling concepts in polymer physics; Cornell University: Ithaca, NY, 1985. (13) Carmesin, I.; Kremer, K. J. Phys. Fr. 1990, 51 915. (14) Persson, B. N. J. Phys. Rev. Lett. 1993, 71, 1212.
Figure 1. Speed of the center of mass of the 16-chain monolayer for an applied force F ) 0.08 in reduced units.
Figure 2. Speed of the center of mass as a function of the amplitude of the applied force for 2 different chain lengths, 8 (diamonds) and 16 (crosses), with Cfs ) Dfs ) 1.2. The simulations were realized for 1024 atoms in the monolayer and 6400 atoms in the solid. Larger systems have been checked and were consistent with the reported results.
The second regime is the one relevant for the present analysis. We give in Figure 2 the observed velocity VCM of the plateau as a function of the amplitude of the applied force in reduced units and that for two different chain lengths. The error bars correspond to the statistical dispersion over the duration of the plateau, typically of the order of 100 000 time steps. The F vs VCM dependence is approximately linear. At least for Cfs ) 1.2, the existence of this linear regime already reported in ref 11 thus guarantees the validity of the viscous nature of the friction force as assumed in the model.2 Let us observe here that the associated friction coefficient is independent of the chain length, which is to be expected in 2D motion of segregated polymer chains.9,12 In Figure 3 we plot VCM vs F for 16-chain molecules with 2 different interactions with the substrate; we choose the parameters Cfs ) Dfs ) 1.2 and 2.0. Clearly, the friction increases with increasing substrate interaction. De Gennes’ Model Revisited. To study the associated driving force, we have reconsidered the behavior of a new
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Figure 3. Speed of the center of mass as a function of the amplitude of the applied force for 16 atoms per chain, with 2 different amplitude of interactions Cfs ) Dfs ) 1.2 (diamonds) and 2.0 (crosses) in reduced units. The associated linear fits lead to σ1-1 ) (5.48 ( 1.64) × 10-3 for Csf ) 1.2 and σ1-1 ) (2.73 ( 1.17) × 10-3 for Csf ) 2.0.
Figure 5. From top to bottom, the radii of the first R1 and the second R2 layers vs time in τ units for the spreading drop with 16 atoms per chain, Cfs ) Dfs ) 1.2, 25 600 atoms for the fluid, and 160 000 for the solid in dots. We have represented in dotted lines fourth order polynomial fits.
where R1 are constant parameters independent of time and volume, related to the friction coefficients between the first layer of fluid and the solid σ1 and between the first and the second layer of fluid σ
R1 )
2 (W - W1) σ1 2
(
2 σ R2 ) (W3 - W2) 1 + 4 σ σ1
Figure 4. Side view of a drop of 1600 16-atom molecules during spreading.
spreading drop of chainlike molecules using the same model as described before. Within this simulation, it is clearly observable that there appears several layers during spreading, as represented in the snapshot given in Figure 4. The radii R can be measured by determining when the associated density of atoms fall below 0.5. More details about the techniques of simulations can be found in refs 9 and 10. According to the stratified droplet model,2 the two first layers should obey the following hydrodynamical equations:
dR12 R1 ) R1 ln dt R2
(4)
dR12 dR22 R2 ln ) R2 + dt dt R3
(5)
(
)
(6)
)
(7)
Wn denotes the potential energy in the nth layer according to eq 2. Studying the radii R1 and R2 of the spreading drop as a function of time, we can compute R1, which can be compared to the friction σ1 measured previously for monolayers. Using a polynomial fit for R1(t) and R2(t), as presented in Figure 5, we get R1(t). The results for R1(t) are presented in Figure 6. The horizontal dotted line has to be associated with the computed value 2(W2 - W1)/σ1. As can be easily seen, the agreement is good, with no adjustable parameters, and, moreover, again qualitatively compatible with experimental determinations.15 Comparison with Experiment In this part, we shall reexamine the previous results in reference to the experimental observations. Two-Dimensional Disentanglement Process. We shall first discuss the equilibration process allowing chainlike molecules with N monomers in a monolayer to disentangle and segregate. The experiments give nice evidence of this process. We study the behavior of light PDMS (molecular mass Mp ) 1250, N ) 17snonvolatile at 3D but volatile at 2D14) spreading on a homogeneous surface. A convenient substrate is a grafted oxidized silicon wafer. For this specific range of chain length (say, N between 10 and 30), the monolayer has a very striking shape, illustrated on Figure 7. The profile cannot be fitted by a spherical shape cap and Gaussian feet, as should be (15) Fraysse, N.; Valignat, M. P.; Cazabat, A. M.; Heslot, F.; Levinson, P. J. Colloid Int. Sci. 1993, 158, 27.
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Langmuir, Vol. 13, No. 17, 1997 4757
This experimental observation has a numerical counterpart: the time needed for the monolayer to reach equilibrium (i.e., the situation where the chains are segregated) was found to be unexpectedly large and increasing steeply with the chain length. In fact, chains with 16 monomers were able to reach equilibrium within normal computing times, while this equilibrium could never be obtained with 32-monomer chains. We believe that in confined geometry, a 2D disentanglement threshold could be defined, the value of which might be around 15 monomers in the present case, i.e., M ≈ 1000. Friction Coefficient ζ1
Figure 6. R1 as a function of the number of time steps in reduced units compared to (W2 - W1)/σ1 in dots computed with the observed distance between the substrate and the first (respectively second) layer equal to 0.85 (respectively 1.75) in reduced units.
Figure 7. Ellipsometric profile of a monolayer of PDMS Mp ) 1250, η ) 10 cP, and Ip = 1.25. The absence of lighter chains has been checked by GPC. Another very sensitive test is the absence of Marangoni effect in capillary rise geometry. Upper profile 3 h, lower profile 5 days after drop deposition. The thickness of the compact monolayer is 8 Å. The baseline (21 Å) is the silica layer.
the case for polymer melts:16 the central part of the monolayer is almost static while the edges spread normally.16 For shorter chains, the profiles are satisfactorily fitted by polymer melt models,16 and it is still the case at low temperatures, when the bulk viscosity of the polymer is large: the relevant parameter is clearly the chain length. For longer chains, the monolayer does not spread any more.17 Our interpretation is the difficulty for the chains in the center of the monolayer to “disentangle” in this confined geometry.13 (16) Albrecht, Otto, Leiderer, Phys. Rev. Lett. 1992, 68, 3192. (17) Cazabat, A. M.; De Coninck, J.; Hoorelbeke, S.; Valignat, M. P.; Villette, S. Phys. Rev. E 1994, 49, 4149.
Let us now compare the calculated values of the friction coefficient σ1 and the experimental data and discuss first the influence of solid/liquid interactions. The relevant experimental parameter is the diffusion coefficient D1 of the first layer at the early stages of spreading, when the central part of the drop can be considered as a reservoir. D1 is the ratio of a driving term to the friction coefficient σ1 and is found to decrease when the interaction increases.1 As Csf ) Dsf, in the present case, D1 is proportional to Csf/σ1, a quantity which actually decreases with increasing Csf (Csf/σ1 = 6.6 × 10-3 for Csf ) 1.2 and = 5.4 × 10-3 for Csf ) 2). Thus, a qualitative agreement with experiment is obtained. The influence of the chain length is more complex. As both the driving term and the friction term are proportional to N, the experimental value of D1 should be independent of N, like in the numerical prediction. In reality, experimental D1 values scale like η-1, i.e., approximately N-1.14 We propose the following interpretation: the viscosity here plays in the process of molecules extracting themselves from the 3D melt to reach the 2D first monolayer. The probability for a molecule to reach the first monolayer depends on the bulk viscosity, just like in the analysis of Cherry and Holmes.18 As a matter of fact, a probabilistic approach might provide clues for a complete description of spreading.19 In conclusion, this research based on microscopical considerations supports very well the description of spreading based on the driving force/friction competition mechanism. Specific effects in confined geometry appear to play a significant role for chainlike molecules and deserve further analysis. Moreover, the chain-length dependence of the experimental diffusion coefficients is not accounted for at this moment. Acknowledgment. This research is supported by the Belgian Program on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister’s Office, Science Policy Programming, by the European Community with Grant CHRX-CT94-0448, and COST Grant D5/0003/ 95. A.M.C. gratefully thanks J. Israelachvili and S. Karaborni for fruitful discussions and preprints. LA970242L (18) Cherry, B. W.; Holmes, C. M. J. Colloid Interface Sci. 1969, 29, 174. (19) Burlatsky, S. F.; Oshanin, G.; Cazabat, A. M.; Moreau, M. Phys. Rev. Lett. 1995, 76, 86.
