Spreading of Liquid Mixtures at the Microscopic Scale: A Molecular

Modeling of Wetting: A Study of Nanowetting at Rough and Heterogeneous Surfaces. Mathias Lundgren, Neil L. Allan, and Terence Cosgrove. Langmuir 2007 ...
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Spreading of Liquid Mixtures at the Microscopic Scale: A Molecular Dynamics Study of the Surface-Induced Segregation Process M. Voue´,* S. Rovillard, and J. De Coninck Centre de Recherche en Mode´ lisation Mole´ culaire, Universite´ de Mons-Hainaut, 20, Place du Parc, 7000 Mons, Belgium

M. P. Valignat and A. M. Cazabat Physique de la Matie` re Condense´ e, Colle` ge de France, 11, Place Marcelin Berthelot, F-75231 Paris Cedex 05, France Received June 15, 1999. In Final Form: October 6, 1999 The spreading of liquid mixtures on a solid substrate is studied using Molecular Dynamics. The droplets are constituted by 16-atom and 8-atom chains and the solid substrate is described at the atomic level. In all the simulations reported here, the atoms are, for simplicity, equidistributed among the two types of molecules (50%-50%). The influence of the strength of the solid, short molecules interaction on the coating of the solid surface by the liquid first layer is analyzed, for both the “wetting/nonwetting” and “wetting/ wetting” mixtures. A surface-induced segregation process of the mixture is reported in the “wetting/ nonwetting” case and the conditions according to which this segregation of the initial drop occurs are investigated.

1. Introduction In wetting, most of the recent experimental studies,1-4 computer simulations,5-13 and theoretical models14-16 have been devoted to the spreading of pure polymeric liquids. Mixtures of polymers17,18 or asymmetric surface interactions for the ends of the liquid molecules19 have only been addressed in a few studies although the problem of covering a solid substrate by a blend of liquids is of prime importance from a practical point of view because the substrate will usually favor one of the mixture constituents. * To whom correspondence should be addressed. Email: michel. [email protected]. Fax: + 32 65 373881. (1) Beaglehole, D. J. Phys. Chem. 1989, 93, 893. (2) Heslot, F.; Cazabat, A. M.; Levinson, P.; Fraysse, N. Phys. Rev. Lett. 1990, 65, 599. (3) Heslot, F.; Fraysse, N.; Cazabat, A. M. Nature 1989, 338, 640. (4) Heslot, F.; Cazabat, A. M.; Levinson, P. Phys. Rev. Lett. 1989, 62, 1286. (5) De Coninck, J.; Hoorelbeke, S.; Valignat, M. P.; Cazabat, A. M. Phys. Rev. E 1993, 48, 4549. (6) De Coninck, J. Colloids Surf. A 1993, 80, 131. (7) De Coninck, J.; Fraysse, N.; Valignat, M. P.; Cazabat, A. M. Langmuir 1993, 9, 1906. (8) Jang, J. X.; Koplick, J.; Banavar, J. Phys. Rev. Lett. 1991, 67, 3539. (9) Jang, J. X.; Koplick, J.; Banavar, J. Phys. Rev. A. 1992, 46, 7738. (10) Nieminen, J.; Abraham, D.; Karttienen, M.; Kashi, K. Phys. Rev. Lett. 1992, 69, 124. (11) Nieminen, J. A.; Ala-Nissila, T. Phys. Rev. E 1994, 49, 4228. (12) De Coninck, J.; D’Ortona, U.; Koplick, J.; Banavar, J. R. Phys. Rev. Lett. 1995, 74, 928. (13) D’Ortona, U.; De Coninck, J.; Koplick, J.; Banavar, J. R. Phys. Rev. E 1996, 53, 562. (14) de Gennes, P. G.; Cazabat, A. M. C. R. Acad. Sci. 1990, 310, 1601. (15) Abraham, D. B.; De Coninck, J.; Dunlop, F.; Collet, P. Phys. Rev. Lett. 1990, 65, 195. (16) Burlatsky, S. F.; Oshanin, G.; Cazabat, A. M.; Moreau, M. Phys. Rev. Lett. 1996, 76, 86. (17) Steiner, U.; Klein, J.; Eiser, E.; Budkowski, A.; Fetters, L. J. Science 1992, 258, 1126. (18) Steiner, U.; Klein, J. Phys. Rev. Lett. 1996, 77, 2526. (19) Haataja, M.; Nieminen, J. A.; Ala-Nissila, T. Phys. Rev. E 1996, 53, 5111.

The main features arising from the studies of onecomponent polymeric liquids on top of solid substrates is the “terraced spreading” phenomenon which, under welldefined conditions, leads to the formation of layers of molecular thickness. High-resolution ellipsometric measurements have initially brought experimental evidence of that phenomenon.2 The first attempt to describe its origin is due to de Gennes and Cazabat14 where 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, these authors were able to recover most of the experimental observations. Among these experimental observations, the diffusive nature of the spreading, i.e., the growth of the radius associated to the first layers behaves as t1/2, should be pointed out. Using Molecular Simulations (MD) techniques, De Coninck et al. could recover both the terraced spreading phenomenon and the t1/2 law12-13,20 but these results could only be obtained using a large number of liquid atoms in such a way that the central part of the drop acts as a reservoir, an atomic description of the solid, and an appropriate temperature control of the solid substrate. These authors have considered 25 000 atoms for the fluid and 250 000 atoms for the solid. The microscopic details of the mechanism of spreading and, in particular, the role of the friction at the microscopic level, were recently also investigated using MD simulations.20 The results of this study are in agreement with experimental evidence and support the validity of the de Gennes-Cazabat model based on driving force/friction competition mechanism.14 By use of similar techniques but for different models (monatomic molecules, idealized flat substrate, ...), other power laws were derived in refs 8-11. (20) Cazabat, A. M.; Valignat, M. P.; Villette, S.; De Coninck, J.; Louche, F. Langmuir 1997, 13, 4754.

10.1021/la990770s CCC: $19.00 © 2000 American Chemical Society Published on Web 12/04/1999

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More recently, the one-fluid approach has been extended by Fondecave and co-workers to the study of the spreading of mixtures of low and high molecular weight trimethylterminated poly(dimethylsiloxanes) (PDMS) on octadecyltrichlorisilane (OTS)-grafted silicon wafers.21 This mixture behaves as a polymer solution whose solvent is the low molecular weight PDMS. The mixture does not wet the substrate at high polymer volume fractions φ (φ g 0.63) but, for lower polymer volume fractions, the solution droplet is in equilibrium with a precursor film of microscopic thickness of pure solvent. Wetting phase diagrams for such “wetting/nonwetting” systems have been proposed. In this article, we present computational results concerning the spreading of liquid blends, considering two types of molecules: 16-atom and 8-atom chains. The aim of this work is 2-fold. We first attempt to quantify the segregation phenomenon which may occur when one of the mixture components, satisfying the conditions of partial wetting regime, is in the presence of another component that will perfectly wet the solid substrate. Second, we investigate the influence of the strength of the “solid/fluid” interaction on the rate of spreading of the substrate by the polymer mixture. The article is organized as follows. Section 2 reviews for completeness the MD approach applied to the spreading phenomenon. The results concerning the surface segregation process and the rate of coating of the substrate are presented and discussed in section 3. Concluding remarks are given in section 4. 2. The Molecular Dynamics Model 2.1. Organization of the Simulations. The simulations are organized according to the following three-step scheme: (i) setup of the system by generating and equilibrating the liquid molecules and the substrate independently from each other, (ii) equilibration of the liquid droplet on the substrate with low values of the “solid-fluid” coupling parameters to obtain a configuration representative of the partial wetting regime for all the subsequent simulations, and (iii) increase of the coupling parameters and study of the spreading phenomenon. Details concerning these three steps are given below and in previous articles.12,13,20,22,23 Steps i and ii will be referred to as the equilibration phase, while step iii will be referred to as the production phase. 2.2. Description of the Constituents of the System. The system of which we are investigating the spreading properties is made of a liquid droplet on top of a solid substrate. As in previous studies,12,13 the atoms that we consider here interact with each other via a 6-12 LennardJones potential 6

12

Cij

[(σr) - (σr) ] ) r

Vij(r) ) 4ij

6

-

Dij r12

(1)

where r denotes the distance between the two atoms i and j. ij is the depth of the potential well and σ the collision diameter of the particles. Cij and Dij refer to the fluid/fluid (ff), fluid/solid (fs), and solid/solid (ss) interactions. We only consider here short-range interactions and, for computational convenience, we cut off the interactions for a distance above rc ) 2.5 σ. (21) Fondecave, R.; Brochard-Wyart, F. Europhys. Lett. 1997, 37, 115. (22) Blake, T. D.; Clarke, A.; De Coninck, J.; de Ruijter, M. J. Langmuir 1997, 13, 2164. (23) De Coninck, J.; Voue´, M. Interface Science 1997, 5, 147.

Table 1. Coupling Constants of the 6-12 Lennard-Jones Potential for the “Fluid-Solid” (fs) Interaction Cfs ) Dfs 8-atom chains

16-atom chains

0.5 1.2 2.0 5.0 1.2

nonwetting wetting wetting wetting wetting

2.2.1. Liquid Molecules. To mimic the dry spreading experiments usually carried out with silicon oils,1-4 neighboring atoms within a given chain molecule interact via a bonding intramolecular potential

Uintra(r) ) Ar6

(2)

The power 6 is here chosen for computational convenience and the constant A equals 1.0. The influence of the value of A has been reported in ref 13 and will not be further discussed here. When two neighboring atoms are considered, the main effect of introducing this intramolecular potential in the force field is to reduce the equilibrium distance from req ) 21/6 in reduced units in the pure LJ case to req ) 1.0 for the dimer and to confine r around this value. To study mixtures of chain molecules, we have considered 16-atom chains and 8-atom chains with the couplings parameters for the “fluid-fluid” interactions set to Cff ) Dff ) 1.0. We are thus dealing with the same type of polymer but with different chain lengths. This ensures that both species are perfectly miscible in the bulk phase. These chain lengths were chosen because their evaporation rate is negligible in the simulation conditions reported here and because they are below the three-dimensional disentanglement threshold. The number of atoms belonging to the short chains (N8) and the long chains (N16) are both equal to 12 800. The effects related to the relative concentration of the mixture constituents will be considered in another publication. 2.2.2. Solid Substrate. The solid substrate on top of which the liquid droplet will spread is composed by two layers of particles (about 300 000 particles). These solid particles interact with each other via a Lennard-Jones potential, as given by eq 1, with parameters Css ) 35.0 and Dss ) 5.0. They are heavier than the liquid atoms (msolid ) 50 mliquid) so that their displacement has a time scale comparable with that of the liquid. These solid particles are initially placed on a (100) fcc (face-centered cubic) lattice, and their motion around these initial positions is restricted with a strong pinning harmonic potential. 2.2.3. “Solid-Fluid” Interaction. In this study, different interactions of the liquid molecules with the substrate have been considered. They are summarized in Table 1. Let us just remind here that for a coupling Cfs ) Dfs g 0.8, the considered liquid wets completely the solid surface,12,13 while coupling constants Cfs ) Dfs ) 0.5 or lower are characteristics of the partial wetting regime.22 That is to say that the 16-atom chains liquid will always completely wet the surface while for the 8-atom chains liquid, we will consider cases relevant of either the partial or the complete wetting regimes. Given the potentials, the motion follows by integrating Newton’s equation of motion using a fifth-order predictorcorrector method. The time step ∆t is measured in units of the dimensionless time variable τ ) (m/σ2)1/2 and its value given by ∆t ) 0.005τ is of the order of 5 × 10-15 s, with  and σ defined as above.

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Figure 1. Initial configuration: spherical cap. No distinction is made between the 8-atom and the 16-atom chains. Substrate has not been represented. There are 25 600 atoms in the liquid droplet and the coupling parameters are Cfs ) Dfs ) 0.4. This configuration is representative of the partial wetting regime. The contact angle calculated from circular fit of density profiles is about 88°.

2.3. Equilibration Phase: Temperature Control and Initial Configuration. The procedure for the temperature control is the one described in refs 12 and 13. The liquid molecules and the solid substrate are generated far away from each other. During step i of the equilibration phase, the temperature of the system is controlled by rescaling the velocities of all the particles. The liquid droplet and the substrate are equilibrated independently from each other. Due the minimization of the free energy, the shape of the liquid droplet becomes spherical. About 50 × 103 time steps later, when the droplet energy reaches a constant value, it is moved into the vicinity of the substrate and starts to spread. From that time, i.e., during steps ii and iii, only the temperature of the substrate is controlled, allowing heat exchange between the liquid and the solid to mimic energy dissipation as it occurs in a real experimental setup. More explicitly, on top of such a square of atoms of the solid, we have first considered a sessile drop made by 12 800 atoms forming 8-atom chains and 12 800 atoms forming 16-atom chains. To consider this initial sessile drop configuration, we equilibrate our system with fixed “solid/fluid” couplings. The corresponding coupling (Cfs ) Dfs ) 0.4) are chosen to have a stable sessile drop as an initial configuration. This equilibration procedure was maintained during a time of the order of 106 time steps. This procedure has been chosen to mimic as much as possible real experimental conditions. A typical snapshot after equilibration is given in Figure 1. Using this procedure, we end up in fact with an initial homogeneous sessile drop which is, as it should be, a piece of a sphere. This configuration is saved and used as the initial configuration of all the subsequent simulations, as described below. 2.4. Production Phase: Investigation of the Spreading Phenomenon. At the end of the equilibration phase, we initiate the change of couplings describing the “solid/ liquid” interaction to reach the appropriate values in 104 time steps. We have of course checked that this particular

Voue´ et al.

Figure 2. Fraction of the total number of atoms at a distance Z of the substrate without distinction between both kinds of atoms (solid lines), for the atoms belonging to the 8-atom chains only (dashed lines) and for the atoms of the 16-atom chains (dotted lines). The global distribution shows evidence for the existence of a precursor film (first layer) but also for a second and a third layer. The atoms belonging to the 8-atom chains appear only in the second and third ones. Time is (120 × 103) ∆t after equilibration time (∆t ) (5 × 10-3) τ). Inset: the 8-atom and 16-atom chains are equally distributed in the Z direction in the initial configuration.

time value has no influence on the net result. That is to say that at each of the first 104 time steps after equilibration, we increase the value of the couplings Cfs ) Dfs by 0.8 × 10-4 for the 16-atom chains and by 0.1 × 10-4, 0.8 × 10-4, 1.6 × 10-4, and 4.6 × 10-4 for the different 8-atom chains cases requiring respectively a final value of 0.5, 1.2, 2.0, and 5.0. Let us denote by teq the time corresponding to the end of this transformation. 3. Results and Discussion For the sake of clarity, we will respectively present our results concerning the segregation process and the coating rate in subsections 3.1 and 3.2. 3.1. Surface Segregation Process. To study the surface segregation process, we have considered a mixture of long and short chains for which the coupling constants defining the “fluid-solid” interactions are respectively Cfs(16) ) Dfs(16) ) 1.2 and Cfs(8) ) Dfs(8) ) 0.5. It is known12,13 that the long chains will wet the substrate, which is not the case for the chains characterized by the couplings Cfs(8) ) Dfs(8) ) 0.5.22 By measuring the number of atoms versus the height z, we observe very clearly in Figure 2 that, starting from an homogeneous drop at t ) teq (Figure 2, inset), we obtain evidence of the existence of a precursor film of molecular thickness at t ) teq + (120 × 103) ∆t. Let us now analyze the dynamics of this precursor film. As a function of time, we have plotted in Figure 3a, N(1)/ Ntot, the fraction of atoms which belong to the precursor film and N16(1)/Ntot, the associated fraction of the 16-atom chains. As can be seen, the latter mainly contribute to the growth of the precursor film. As long as the central part of the droplet acts as a reservoir for the precursor film (i.e., for 125 e t e 1000, in reduced time units), both N(1)/ Ntot and N16(1)/Ntot vary linearly with time. This allows defining the diffusion coefficient D of the liquid molecules on the substrate, as discussed in the next subsection. The accumulation of the long chains in the precursor film naturally leads to the appearance of a depletion zone within the second and third layers as represented in Figure 3b, when we plot the relative accumulation ratio N16/N8

Spreading of Liquid Mixtures

Figure 3. “Wetting-nonwetting” mixture. (a) Fraction of the total number of atoms in the precursor film as a function of the time (plain line) and the contribution of the atoms belonging to the 16-atom chains (dotted line). The precursor film contains mainly long wetting chains. As 50% of the atoms belong to the long chains and 50% to the short ones, it can be seen that almost all the 16-atom chains are in the first layer at the end of the simulation. (b) Dynamics of the accumulation ratio N16/N8 for the first three liquid layers. A important segregation of the blend is induced by the large asymmetry between the “liquidsolid” coupling constants for both types of chains. The coupling parameters are Cfs ) Dfs ) 0.5 for the 8-atom chains and Cfs ) Dfs ) 1.2 for the 16-atom chains. The segregation process occurs during the first 300 reduced time units of the simulation, as shown by the important increase of the accumulation ratio in the first layer.

of atoms belonging to the 8-atom and 16-atom molecules for the three first liquid layers versus time. The dynamics of the accumulation ratio N16/N8 obviously is evidence of the fast segregation process between the wetting and the nonwetting species (Figure 3b). Rapidly larger than 10, it reaches for the first layer a plateau value around 30 and then decreases slowly, as the spreading of the droplet proceeds. For the second and the third layers, this process is inverted: these are 8-atom chain-rich layers. To study in more detail the mechanisms corresponding to that spreading, we have computed the fluxes of the different atoms between the first and the second layers of our drop. The atoms going from the first to the second layer, i.e., contributing to the formation of the precursor film, are counted positively, while the atoms leaving the first layer are counted negatively. The data are averaged over 104 time steps at time t ) (15 × 103) ∆t, (115 × 103) ∆t, and (215 × 103) ∆t. The results are presented in Figure 5a-c as a function of R, where R denotes the radial distance measured from the axis of symmetry of the drop.

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Figure 4. Fluxes between the first and the second layer as a function of the distance from the drop axis (plain lines, 16atom chains; dashed lines, 8-atom molecules). Transfer of atoms is counted positively when occurring from the first layer to the second one and negatively in the opposite case. The data are averaged over (5 × 103) ∆t. (a) At t ) (15 × 103) ∆t, the long chains are transferred from the second to the first layer mainly at the border of the drop (b) idem at t ) (115 × 103) ∆t, (c) idem at t ) (215 × 103) ∆t.

During the initial stages of the spreading (Figure 4a), two coupled processes occur: the transfer of the 16-atom chains in the immediate vicinity of the substrate and the displacement of the 8-atom chains from the first to the second layer. The fluxes mainly occur in the outer part of the drop. The segregation process characterized by the transfer of the 8-atom chains from the first to the second layer is very fast and only occurs during the early stages of the spreading. After these stages, the fluxes of the short chains damp out (Figure 4b,c) to a value which is not significantly different from 0 while the 16-atom chains continue to contribute to the growth of the precursor film. As the segregation process is clearly evidenced, it may be analyzed in terms of the dynamics of adsorption24 of (24) In this context, “adsorption” has to be understood in the widest sense of the term because, as will be highlighted in the next paragraphs, although accumulating at the solid surface, the long chains still have a surface mobility and, except in some peculiar cases (see subsection 3.2), they are not immobilized on the substrate.

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Figure 5. Time evolution of the distribution of the number of atoms adsorbed per 16-atom molecule (a, c, e) or per 8-atom molecule (b, d, f). (a and b) t ) (21 × 103) ∆t, (c and d) t ) (121 × 103) ∆t, and (e and f) t ) (321 × 103) ∆t. The adsorption process is evidenced as a function of the time by an increasing number of long molecules with a greater number of adsorbed atoms, going from an uniform distribution to a distribution where all the 16-atom molecules are almost completely in contact with the solid substrate. No significant differences are observed for the short molecules as a function of the time.

molecules on the solid substrate. It can practically also be studied by determining the statistical distribution of the number of atoms per molecule in contact with the solid as a function of the time for the 16-atom chains. It will be referred to as the number of atoms adsorbed per molecule. These results are presented in Figure 5a-e for

t ) (21 × 103) ∆t (a and b), t ) (121 × 103) ∆t (c and d), and t ) (321 × 103) ∆t (e and f). In the early stage of the spreading (Figure 5a), most of the 16-atom chains are not adsorbed on the solid surface. The numbers of atoms adsorbed per molecule are distributed between 1 and 16 with an equal probability. Most of the short chains have

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Figure 6. Time evolution of the distribution of the number of atoms adsorbed per molecule (mean value ( standard deviation) for the 16-atom (open circles) and 8-atom chains (open triangles). The arrows indicate times at which the distributions represented in Figure 6a-c have been evaluated. The adsorption process of the long chains occurs from t ) 0 to t ) 1250 (in reduced time units). It is characterized by a wide distribution of the number of adsorbed atoms per molecule and is followed by a plateau region (1250 e 2000) although the distribution of the 8-atom chains remains unchanged throughout the whole simulation.

no atom in contact with the substrate, and the probability of having a 8-atom chain with a nonzero number of adsorbed atoms vanishes as this number tends to the chain length (Figure 5b). Later on, while the distribution calculated for the 8-atom chains is not significantly different from the one obtained previously (Figure 5d), the distribution calculated for the 16-atom chains clearly shows that these molecules tend to adsorb strongly on the substrate (Figure 5c). In the late stages of the spreading process (Figure 5e), the longer chains are mainly in the precursor film and totally adsorbed on the solid substrate. The distribution is sharp. Some atoms of the shortest chains are now in contact with the solid (Figure 5f). At this time, the blend is almost completely surface segregated. The time evolution of the average number of atoms adsorbed per molecule is given in Figure 6 (mean ( standard deviation). Two regimes clearly appear in the time evolution of the number of atoms adsorbed per 16atom molecules. The first regime takes place during the most important part of the simulation. It is mainly characterized by a rapid increase of the average number of atoms adsorbed by molecule, going from 2 to 13, and by a large distribution of the data. The second regime takes place in the late stages of the spreading process when most of the 16-atom chains have reached the substrate and lay flat on it. The adsorption rate slows down, and the distribution of the average number of atoms adsorbed by molecule is much sharper than that in the first regime. The adsorption of the atoms belonging to the short chains does not take place, as expected. The averaged number of adsorbed atoms drops in the early stages of the spreading to increase slowly after a while. These results are similar to those presented in Figure 3a, indicating that the atoms of the chain reach the substrate one after each other as already evidenced in the case of pure liquids.23 3.2. Spreading Rate of the Blend. We will now focus on the influence that the “fluid/solid” interaction exerts on the spreading rate of the blend. To that purpose, we have computed the diffusion coefficient D of the chain molecules on the substrate.

Figure 7. Dynamics of the accumulation ratio N16/N8 for the “wetting-wetting” mixtures. (a) The coupling parameters are Cfs ) Dfs ) 1.2 for the 8-atom chains and Cfs ) Dfs ) 1.2 for the 16-atom chains. No significant influence of the chain length is observed. The mixture spreads as an homogeneous fluid. (b) The coupling parameters are Cfs ) Dfs ) 2.0 for the 8-atom chains and Cfs ) Dfs ) 1.2 for the 16-atom chains. The blend segregates and the short chains slightly accumulate at the solid substrate.

For the solid-liquid interactions Cfs ) Dfs g 0.8, it is expected that the liquid will wet the surface and form a monomolecular precursor film.12,13,23 Let us then study the characteristics of this film as a function of the wettability of the surface with respect to the 8-atom molecules. For Cfs ) Dfs ) 1.2 or 2.0, we have indeed observed the appearance of a precursor film. We present in Figure 7 the accumulation ratio N16/N8 for the first three layers versus the reduced time in units of τ. We do not observe a significant effect related to the length of the molecule whenever the attraction to the solid is the same (Figure 7a). However, when the 8-atom chains are more strongly attracted to the solid than the 16-atom molecules, we obtain as expected an accumulation of 8-atom chains in the first layer and a depletion for the second and third layers (Figure 7b). The radius of the precursor film R(1)(t) is measured from the density profiles by determining the point at which F, the density of the first layer in contact with the substrate, falls below 0.4 Fbulk, with Fbulk the liquid density in the bulk phase. The factor 0.4 acts as a cutoff value. Several cutoffs were tested, and the choice does not significantly influence the final result. Below the two-dimensional liquid-gas transition temperature, the density of the precursor film F(r) does not vary with the radial distance r. The number of particles belonging to the precursor film

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coating and for each species, as a function of the Csf ) Dsf coupling constants of the 8-atom chains (Csf ) Dsf), i.e., as a function of the strength of the “solid-liquid” interaction. The total spreading rate is obviously not a monotonic function of the “solid/liquid” interactions. It reaches a maximum for substrates characterized by intermediate coupling constants, i.e., by an intermediate surface energy. On low-energy substrates (i.e., for coupling constants Csf ) Dsf ) 0.5 of the 8-atom chains), the spreading of these short chains occurs at a very low rate. Although a droplet composed only by 8-atom chains does not wet the substrate in these conditions, the spreading of the short chains occurs on the solid surface (i.e., mainly as a second layer) because they are in contact with a more or less homogeneous precursor film made up by the 16-atom chains. The 8-atom chains are spreading on a liquid precursor film and not on the solid substrate itself. Let us now propose a simple interpretation of this phenomenon, given in terms of the competition between the driving term ∆W and the friction coefficient ζ of the liquid layers on each other or on the substrate. As a consequence of our simulation results, we have investigated by ellipsometry how the surface energy of the substrate modified the spreading rate of low molecular weight poly(dimethylsiloxane)s. The results of this study have been reported elsewere.25 For this experimental system, the diffusion coefficient D calculated from the precursor film length reaches a maximum value for surfaces of intermediate energies, confirming thus our simulations. In that article, it is also shown that the diffusion coefficient D, defined by the ratio ∆W/ζ scales as Figure 8. Influence of the “fluid/solid” interactions of the short chains on the total spreading rate of the mixture. (a) Time evolution of N(1)(t), the fraction of atoms in the precursor film for different “fluid/solid” interactions. (b) Individual contributions of the long and short chains have also been evaluated. The spreading rate D is obviously not a monotonic function of the coupling parameters and it shows an optimum when the “fluid-solid” interactions of the long chains and the short chains are equal.

can therefore be used to determine the diffusion coefficient D of the liquid molecules on the substrate, as explained below. The contact area A(t) of the droplet with the solid substrate measured at time t is given by A(t) ) πR(1)2(t), where R(1)(t) is the radius of the precursor film. It is proportional to N(1)(t) F, with N(1)(t) ) N8(1)(t) + N16(1)(t) being the number of atoms in the film. As shown in Figure 3a for the “wetting/nonwetting” case, N(1)(t)/Ntot is a linear function of t over some time period. We therefore obtain the following equation

N8(1)(t) + N16(1)(t) = R(1)2(t) = Dt Ntot

(3)

which defines the spreading rate D of the liquid molecules on the substrate and links it to the number of atoms in the precursor film. Figure 8a represents the time variations of N(1)(t)/Ntot for different interactions between the short molecules and the solid substrate. These variations are linear and, for each set of Cfs and Dfs parameters, the spreading rate D of the blend can be computed. Furthermore, it appears that not only N(1)(t)/Ntot but also N16(1)(t)/Ntot and N8(1)(t)/ Ntot are linear functions of t, allowing us to define the spreading rates of each of the constituents of the blend. Their variations are presented in Figure 8b for the global

(

D = C1(fs - *) exp -

)

C2fs kT

(4)

where C1 and C2 are numerical prefactors and fs is the standard energy parameter of the Lennard-Jones interaction. The linear factor corresponds to the driving term ∆W which is proportional to the strength of the fluidsolid interactions, * being a threshold value depending on the strength of the liquid-liquid interactions and influencing the wetting/nonwetting transition. The exponential factor indicates that the molecular friction follows an Arrhenius-type law and increases as the fluidsolid interactions increase. As shown below, the behavior of the 8-atom chains can be explained using eq 4. As, in the second layer, the short chains undergo liquidliquid interactions with coupling constants Cff ) Dff ) 1.0, their mean coupling constants Csf ) Dsf are greater than 0.5. This results in an increase of the effective driving term ∆W. As, for values of Cfs ) Dfs e 1.2, the contribution to the diffusion process of the linear factor (fs - *) is more important than the one of the exponential term, the spreading occurs at a faster rate. This process is a consequence of the onset of the precursor film and can occur only if two conditions are fulfilled: the segregation process between the long wetting chains and the short nonwetting ones has to occur rapidly and there must be an important reservoir of 16-atom molecules (as in the case of this 50%-50% mixture). When the affinity of the short chains for the substrate increases, their individual spreading rate will increase as a consequence of the increase of the ∆W factor. But, as the molecules segregate less easily because the process is not driven anymore by a large difference between the “solid(25) Voue´, M.; Valignat, M. P.; Oshanin, G.; Cazabat, A. M.; De Coninck, J. Langmuir 1998, 14, 5951.

Spreading of Liquid Mixtures

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the solid yields an increase of the driving term but also a larger increase of the friction term: the molecules are more attracted by the solid surface but exhibit a strong tendency to stay in the deep potential wells of the substrate and, finally, to adsorb on it. This results in a nonmonotonic spreading rate of the mixture. Such nonmonotonic variations of the spreading rate have already been experimentally observed by Davis and co-workers for watersurfactant solutions spreading on gold-coated quartz substrates grafted by mixed organosulfur monolayers.26-29

Figure 9. Mixture spreading on a solid substrate (top view). The coupling parameters are Cfs ) Dfs ) 5.0 for the 8-atom chains and Cfs ) Dfs ) 1.2 for the 16-atom ones. The short chains (small dots) are stuck on the substrate and become obstacles for the wetting 16-atom chains (open circles). This results in an asymmetric spreading of the droplet.

liquid” interactions, the spreading rate of the long wetting chains is slowed. On high-energy surfaces, for which the leading factor in eq 4 is the exponential one, the short chains remain trapped in the potential wells of the substrate and act as obstacles for the 16-atom chains. They diffuse very slowly on the surface. The droplet loses its circular shape and, in some cases (Figure 9), this leads to an asymmetric spreading. Recalling that the spreading process is controlled by the relative influence of the “solid-liquid” interaction, the driving term, and the friction term at the microscopic scale,20 the increase of the affinity of the short chains for

4. Conclusions We have shown that a strong asymmetry in the solidliquid interactions acts as a driving force inducing a segregation process of the binary mixture in the vicinity of the substrate. We have also observed an obvious nonmonotonic dependence for the spreading rate showing that, for each substrate, there should be an optimal composition of the polymer blend for the coating. The influence of the mixture composition (i.e., relative concentrations), the geometry of the molecules during spreading, and the effects of the chain length will be investigated in forthcoming papers. Acknowledgment. This research is partially supported by the European Community (Grant CHRX-CT940448 and COST Grant D5/0003/95) and by the Ministe`re de la Re´gion Wallonne. The authors gratefully thank G. Oshanin, M.J. de Ruijter, and T.D. Blake for fruitful discussions. LA990770S (26) Stoebe, T.; Lin, Z.; Hill, R. M.; Ward, M. D.; Davis, H. T. Langmuir 1996, 12, 337. (27) Stoebe, T.; Lin, Z.; Hill, R. M.; Ward, M. D.; Davis, H. T. Langmuir 1997, 13, 7270. (28) Stoebe, T.; Hill, R. M.; Ward, M. D.; Davis, H. T. Langmuir 1997, 13, 7276. (29) Stoebe, T.; Lin, Z.; Hill, R. M.; Ward, M. D.; Davis, H. T. Langmuir 1997, 13, 7282.