Dynamics of Spreading on Heterogeneous Substrates in a Complete

Sep 11, 1999 - (21) D'Ortona, U.; De Coninck, J.; Koplik, J.; Banavar, J. Phys. Rev. E. 1996, 53, 562. (22) Blake, T. D.; Clarke, A.; De Coninck, J.; ...
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Langmuir 1999, 15, 7855-7862

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Dynamics of Spreading on Heterogeneous Substrates in a Complete Wetting Regime M. Voue´,* S. Semal, and J. De Coninck Universite´ de Mons-Hainaut, Mode´ lisation Mole´ culaire, 20, Place du Parc, B-7000 Mons, Belgium Received February 19, 1999. In Final Form: June 28, 1999 The spreading of low molecular weight silicon oils (PDMS) on heterogeneous substrates has been studied by spectroscopic ellipsometry. Using different grafting times, we produced partially OTS-grafted substrates whose surface energy ranged from that of the bare silica surface (about 28 mN/m) to that of the completely OTS-grafted substrate (about 21 mN/m). Conversely to what has been observed in the case of homogeneous substrate, the effective diffusion coefficient of the precursor film decreases monotonically as the grafting becomes more and more effective. Results of Molecular Dynamics simulations of a polymer-like liquid spreading on a heterogeneous substrate are presented and support the experimental data in the case of regular patches of heterogeneities.

I. Introduction The dynamics of spreading is a very active subject of research, not only for academic reasons. To fix the ideas, suppose that we have a drop of a liquid in coexistence with a vapor phase which is put on top of a solid surface. Starting from some initial configuration, the drop will spread to reach its equilibrium shape. The associated dynamics will then be related to the time behavior of the contact area between the drop and the wall or the corresponding base radius due to the cylindrical symmetry of the drop. If the equilibrium shape of the sessile drop remains a drop or becomes a uniform film covering the surface, we will respectively study the associated dynamics of spreading in the so-called partial or complete wetting regimes. Rather recently, interesting progress has been made in the study of the partial wetting regime dynamics on pure substrates. It has been clearly established that the dynamics may be described by a competition between the driving force due to the solid attraction, the friction dissipation on the solid surface, and the viscous dissipation in the liquid.1,2 When heterogeneous substrates made of A and B materials are concerned, the situation may become much more complex. It has been shown by Molecular Dynamics (MD) simulations3 and detailed experiments4 that, to the leading order, the friction dissipation may be simply directly averaged

In a complete wetting regime, since the studies of Cazabat and co-workers,5-7 it is now classical to introduce the experimental diffusion coefficient D of the liquid molecules on the surface defined by

lfilm(t) ) xDt

(2)

where lfilm(t) is the length of the precursor film at time t. The diffusive behavior of the liquid molecules on the solid substrate is analyzed in terms of a competition between ∆W, the affinity of the liquid molecules for the substrate which acts as a driving term, and ζ, their friction process on the substrate, as given by8

D ) ∆W/ζ

(3)

(1)

The aim of this paper will be to reconsider the spreading of a droplet on top of a heterogeneous substrate, in the complete wetting regime. Combining experimental results and MD simulations, we will show hereafter how the effective diffusion coefficient D is affected by the presence of heterogeneities on the substrate and compare these results to what has been observed on homogeneous substrates.9 The organization of this article is the following. Experimental techniques are presented in section II. Results are reported and discussed in section III while related molecular dynamics simulations results are given in section IV. Conclusions are given in section V.

where c and (1 - c) represent the surface fraction of A and B, respectively, ζA and ζB are the friction coefficients of the liquid molecules on the A and B parts, and ζeff is the effective friction coefficient of the liquid on the AB solid, taken as a whole, without distinction between A and B species.

Substrate Preparation. The substrates were silicon wafers (ACM, France, orientation (100), doping N-Phos, resistivity 5-7 Ω cm, thickness 250-300 µm, polished one side). Their native oxide layer was about 20 Å. These substrates were grafted with an octadecyltrichlorosilane (OTS, Sigma) layer according to the procedure described in ref 10. Heterogeneous substrates were

(1) de Ruijter, M. J. A Microscopic Approach of Partial Wetting: Statics and Dynamics, Ph.D. Thesis, Universite´ de Mons-Hainaut, Mons, Belgium, 1998. (2) Oshanin, G.; de Ruijter, M. J.; De Coninck, J. Langmuir 1999, 15, 2209. (3) Ada˜o, M. H.; de Ruijter, M. J.; Voue´, M.; De Coninck, J. Phys. Rev. E. 1999, 59, 746. (4) Semal, S.; Voue´, M.; de Ruijter, M. J.; Dehuit, J.; De Coninck, J. J. Phys. Chem. B. 1999, 103, 4854.

(5) de Gennes, P. G.; Cazabat, A. M. C. R. Acad. Sci., Ser. II: Mec., Phys., Chim., Sci. Terre Univers. 1990, 310, 1601. (6) Heslot, F.; Fraysse, N.; Cazabat, A. M. Nature 1989, 338, 640. (7) Heslot, F.; Cazabat, A. M.; Levinson, P.; Fraysse, N. Phys. Rev. Lett. 1990, 5, 559. (8) Cazabat, A. M.; Valignat, M. P.; Villette, S.; De Coninck, J.; Louche, F. Langmuir 1997, 13, 4754. (9) Voue´, M.; Valignat, M. P.; Oshanin, G.; Cazabat, A. M.; De Coninck, J. Langmuir 1998, 14, 5951.

ζeff ) cζA + (1 - c)ζB

II. Experimental Setup

10.1021/la990185r CCC: $18.00 © 1999 American Chemical Society Published on Web 09/11/1999

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obtained by varying the grafting time from 10 s to 1 h 30 min at 12 °C and 40% relative humidity.4 Spectroscopic Ellipsometry. Although most of the studies carried out to investigate the dynamics of spreading of microdroplets were done using high spatial resolution single wavelength ellipsometry,6,7 it has been recently shown that spectroscopic ellipsometry (SE), combined with diode array detector technology, is a very interesting alternative to investigate the spreading of simple and complex liquids on solid substrates.11 The spectroscopic ellipsometry measurements were carried out with a Sopra GESP 5 rotating polarizer instrument with an intensified photodiode array (IPDA) detector. The ellipsometric angles ψ and ∆ were measured at 512 energies in the 1.5-4.5 eV range. The polarizer was rotating at 9 Hz. The analyzer angle was set to 20°. The spectroscopic ellipsometer was operating at 75° of incidence, close to the Brewster angle of silicon. Substrate Characterization. As the thickness of the OTS molecule is known, the characterization of the grafting can be done using SE, as proposed in Semal.4 During that substrate characterization, the ellipsometer was operated in a parallel beam configuration (4 mm lateral resolution). The native oxide layer was initially measured before grafting. Keeping the thickness of the grafted layer fixed to the thickness of the OTS molecule (22.0 ( 1.0 Å), the ellipsometric data, measured on the grafted substrate, were fitted with respect to the OTS volume fraction in a mixed layer consisting of void and organic material, according to the effective medium approximation (EMA).12,13 The OTS volume fraction has been shown to be a pertinent variable to characterize the grafted substrates.4 Dynamics of Spreading. To measure the thickness profiles of the microdroplets, due to the use of microspots, the beam originating from a xenon lamp (Hamamatzu, Inc.), initially directed on the entry of a 100 µm optical fiber, was highly focused on the sample surface. The lateral resolution of the experimental setup was 30 µm, and the thickness resolution was 0.2 Å. Typical measurements were done along the droplet diameter every 100 µm. The liquid was a low molecular weight trimethyl-terminated poly(dimethylsiloxane) (PDMS) purchased from ABCR (molecular mass, 1250; viscosity, 0.01 Pa s; surface tension, 20.1 mN/m). The liquid was not further fractionated and will be referred to as “PDMS 10”. The use of this liquid is somewhat restricted because it is slightly volatile. The droplet size is therefore limited to avoid the Marangoni effect, and under these conditions, the way the central part of the droplet acts as a reservoir may be subject to some discussions. The experiments were therefore limited to the early stages of the spreading process. Nevertheless, the use of this liquid was suitable because it wets the partially grafted substrates over the entire range of OTS volume fraction investigated, as well as the pure OTS substrate. A silicon oil of higher molecular weight should not be volatile but should exhibit a wetting transition for OTS volume fractions above a given threshold value. In particular, the PDMS whose viscosity is 0.02 Pa s does not wet the pure OTS surface.

III. Results and Discussion Let us now report the experimental results and let us first consider the pure substrates: the bare silica surface and the completely OTS-grafted surface. The successive thickness profiles of PDMS 10 droplets spreading on bare silicon wafers have been measured at increasing times. They are reported in Figure 1. A 7 Å thick precursor film is growing in front of the macroscopic part of the droplet. To take into account the spreading of the reservoir, the film length lfilm is calculated from these thickness profiles by measuring the drop base radius at (10) Brzoska, J. B.; Shahidzadeh, N.; Rondelez, F. Nature 1992, 360, 719. (11) Voue´, M.; De Coninck, J.; Villette, S.; Valignat, M. P.; Cazabat, A. M. Thin Solid Films 1998, 313-314, 813. (12) Valignat, M. P.; Oshanin, G.; Villette, S.; Cazabat, A. M.; Moreau, M. Phys. Rev. Lett. 1998, 80, 5377. (13) Voue´, M.; Valignat, M. P.; Oshanin, G.; Cazabat, A. M. Langmuir 1999, 15, 1522.

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Figure 1. Ellipsometric thickness profiles of a PDMS 10 droplet on top of a bare silicon wafer. The profiles have been measured 13 min (plain line), 46 min (dashed line), and 168 min (dotted line) after deposition of the droplet. The baseline (about 20 Å) corresponds to the thickness of the native SiO2 layer. A thin wetting film (about 7 Å) is growing at the bottom of the droplet. Inset: Film length versus the square root of the time. The slope of the corresponding straight line provides a value of the diffusion coefficient DSiO2 ) 4.7 × 10-10 m2/s.

mid-height of the growing layer and subtracting the reservoir radius measured 100 Å above the baseline. This film length lfilm grows linearly as a function of t1/2 (Figure 1, inset). From the slope of this growth, we can calculate the diffusion coefficient of PDMS 10 on the bare silica surface, yielding a value of DSiO210 ) 4.7 × 10-10 m2/s. This value is about 1.5 times larger than the one reported in ref 9 for PDMS 20 (DSiO210 ) 3.1 × 10-10 m2/s). In refs 12 and 13, it has been shown that, on substrates on which only a film of molecular thickness appears (i.e., in the low friction regime), D scales as M-1, where M is the mass of the PDMS molecule. As the dependence of the viscosity on M is η ∝ M1.7, the variations of D with the viscosity is D ∝ M-0.59. Given the experimental data obtained in ref 9 and in this study, we obtain DSiO210/DSiO220 ) 1.516 and (ηPDMS10)/ηPDMS20)-0.59 ) 1.505, in close agreement with the given scaling law. A similar experiment can be carried out to study the dynamics of spreading of a PDMS 10 droplet spreading on a completely OTS-grafted substrate. As can be seen in Figure 2 where the time evolution of the thickness profile is reported, a 15 Å thick layer is growing at the bottom of the reservoir. This precursor film can be identified as a bilayer. In this case, the diffusion coefficient is DOTS ) 0.4 × 10-10 m2/s (Figure 2, inset). This value is much lower than the one reported in ref 9 for a comparable experimental system. This discrepancy can be explained by a better quality of the coverage and the high sensitivity of the diffusion coefficient to the surface quality close to the wetting transition. Let us now consider the partially OTS-grafted substrates. Although the thickness profiles measured on either the bare silica surface or the completely grafted surface are rather sharp at the edge of the precursor film, the profiles measured on the partially grafted surfaces appear to be much softer, as indicated by Figure 3 for a 50% grafted substrate. The film which is growing in front of the reservoir has no sharp edge, and its thickness is obviously less than 7 Å, the thickness of the PDMS molecule, having in mind that the thickness determination by ellipsometry gives a thickness averaged over the beam size (30 µm). Whatever the substrate coverage by the OTS grafted layer,

Dynamics of Spreading on Heterogeneous Substrates

Figure 2. Same experiment as in Figure 1 but on a completely OTS-grafted silicon wafer. Times after deposition of the droplet are 23 min (plain line), 49 min (dashed line), and 218 min (dotted line). The baseline (about 41 Å) corresponds to the thickness of the native SiO2 layer plus that of the OTS layer. A thin wetting film (∼15 Å) is growing in front of the macroscopic part of the droplet. Inset: Film length versus the square root of the time. The diffusion coefficient DOTS ) 0.48 × 10-10 m2/s.

Figure 3. Thickness profiles of a PDMS 10 microdroplet spreading on a 50% OTS-grafted silicon wafer. The profiles have been measured 21 min (plain line), 38 min (dashed line), and 89 min (dotted line) after the deposition of the droplet. In that case, the baseline (about 33 Å) corresponds to the thickness of the native SiO2 layer and the effective thickness of the partially grafted OTS layer. Inset: Time variations of the film length corresponding to a diffusion coefficient D ) 1.23 × 10-10 m2/s.

the thickness profiles are similar to the ones presented in Figure 3. Several recent studies16,17 were devoted to the molecular organization of these self-assembled monolayers (SAMs). Let us briefly review their main results. The surfaces exhibits two types of areas covered with OTS: dense condensed assemblies of molecules and disordered molecules which have not self-assembled to form condensed domains, referred to as LC* phase (liquid condensed) and LE* phase (liquid expanded), respectively, by analogy to (14) Aspnes, D. E.; Theeten, J. B.; Hottier, F. Phys. Rev. B 1979, 20 (4), 3292. (15) Azzam, R. M. A.; Bashara, N. M. Ellipsometry and Polarized Light; North-Holland Publishing Company: Amsterdam, 1977. (16) Davidovits, J. V. De´termination des Conditions d’Obtention de Films Monomole´culaires Organise´s: Application aux Silanes AutoAssemble´s sur Silice. Ph.D. Thesis, Universite´ de Paris VI, France, 1998. (17) Schwartz, D. K.; Steinberg, S.; Israelachvilli, J.; Zasadzinski, J. A. N. Phys. Rev. Lett. 1992, 69, 3354.

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Figure 4. Effective diffusion coefficient D of PDMS 10 microdroplets versus the OTS volume fraction fOTS in the grafted layer. The plain line corresponds to the best fit of the experimental data using eq 6 with Λ* ) 0.22 ( 0.02, DSiO2 ) 4.7 × 10-10 m2/s, and DOTS ) 0.48 × 10-10 m2/s. The dashed lines correspond to the eqs 7 and 8 calculated with the same parameters.

the Langmuir films. Monte Carlo simulations of the growing process of the dense condensed islands using a two-dimensional diffusion-limited aggregation enforces this process.16 Atomic force microscopy (AFM) measurements show that the coverage consists of domains of different sizes and shapes, including fingering patterns.15 The complete coverage corresponds to a uniform LC* phase. In ref 4, it has been shown that the wetting properties of the LE* phase, as characterized by the parameters of the molecular-kinetic theory,18 are similar to the one measured for the bare silica surfaces. If the PDMS film, spreading on such a heterogeneous surface, wets all the substrate constituents, it will be constituted by different parts of zero, one, or two molecules thick. The first ones will correspond to parts of the substrate not wetted by the liquid, while the others respectively refer to the SiO2 or the OTS constituents, as previously shown (Figures 1 and 2). This combination of three thicknesses will of course be smeared out during the ellipsometric measurement over the surface of the light spot and the resulting signal will lead to a rather soft profile as represented in Figure 3. Nevertheless, the film length, calculated by measuring the drop base radius at 3.5 Å above the baseline and subtracting the reservoir radius, is still representative of the contact area between the liquid and the substrate (Figure 3, inset). Furthermore, it still exhibits the wellknown t1/2 dependence of the diffusive behavior. Using a thickness equal 3.5 Å at above the baseline as a threshold value to determine the drop base radius, although apparently arbitrary, is justified because it avoids determining the intercept of the droplet thickness profile with the substrate baseline when the former has no sharp edges. Using another threshold value (e.g., 2.0 Å) would modify the absolute value of the film length measured at a given time t but would not influence the growth rate of lfilm, i.e., the value of D. We can therefore measure the effective diffusion coefficient D versus the fOTS, the OTS volume fraction in the grafted layer. The values of D monotonically decrease as fOTS increases, i.e., as the critical surface tension of the substrates decreases (Figure 4). (18) Blake, T. D.; Haynes, J. M. J. Colloid. Interface Sci. 1969, 30, 421.

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Figure 5. Snapshots of a droplet spreading on a homogeneous substrate after 100 103 time steps. An equilibration time of 50 × 103 time steps was considered before the droplet being brought to interaction distance of the substrate. The time step value is ∆t ) 0.005 in reduced time units (m/σ2)1/2). 400 chains of 16 atoms are forming the droplet. The substrate is formed by one layer of fcc cells. The coupling parameters for the liquid-solid interaction are either Cfw ) Dfw ) 0.8 or 2.0. The solid atoms are only represented when they interact with the liquid molecules with coupling parameters Cfw ) Dfw ) 2.0. Cfw ) Dfw ) 0.8: (a) top view, (b) front view and Cfw ) Dfw ) 2.0: (c) top view, (d) front view.

This experimental result seems to be characteristic of heterogeneous substrates. It is quite surprising because, on homogeneous ones and over the same range of surface energies, D is not monotonic and exhibits a maximum for substrates of intermediate surface energies, as shown in ref 9. How can we interpret these experimental data? In the case of a pure liquid spreading on an ideal substrate, as described by the stratified droplet model,5 the diffusion coefficient Dk of the liquid molecules on top of the solid substrate k is given by

Dk ) ∆Wk/ζk

(4)

where ∆Wk and ζk are respectively the difference of energies of the liquid molecules between the precursor

film and the central part of the droplet and the friction coefficient of the molecules on the solid k. For the heterogeneous substrates considered in this article, it is easily understood that ∆WOTS/SiO2 ) cOTS∆WOTS + (1 cOTS)∆WSiO2. Using eq 1 to describe the friction process on the heterogeneous substrate, the diffusion coefficient of the liquid molecules is

DOTS/SiO2 )

cOTS∆WOTS + (1 - cOTS)∆WSiO2 cOTSζOTS + (1 - cOTS)ζSiO2

(5)

which is a monotonic function of the heterogeneity concentration cOTS. Replacing the differences of energy ∆Wk (k ) OTS, SiO2) by their expression from eq 4, one obtains

Dynamics of Spreading on Heterogeneous Substrates

DOTS/SiO2 )

cOTSDOTS + (1 - cOTS)DSiO2Λ cOTS + (1 - cOTS)Λ

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(6)

where Λ ) ζSiO2/ζOTS, the ratio of the friction coefficients of the liquid molecules on the homogeneous substrates. As DOTS and DSiO2 can be directly measured from the experiments carried out on the pure substrates, this equation has only one parameter, Λ, whose value can be adjusted over the whole range of concentration by fitting the theoretical curve (Figure 4, plain line) to the experimental data using a standard nonlinear least-squares method.19 Using DOTS ) 0.4 × 10-10 m2 s-1 and DSiO2 ) 4.7 × 10-10 m2 s-1, the experimental values calculated from the time evolution of the thickness profiles represented in Figures 1 and 2, it appears that the best-fit values of D are in quite remarkable agreement with the experimental ones, showing that, at least phenomenologically, eq 6 describes the leading contribution to the effective diffusion coefficient on an heterogeneous substrate. Furthermore, Λ*, the best-fit value of Λ, is 0.22 ( 0.02. Equation 6 can be linearized in a view to determining the asymptotic behavior of D for nearly bare, nearly totally grafted, substrates. To the first order in cOTS, we obtain for the nearly bare substrates (cOTS ≈ 0)

DOTS/SiO2 = DSiO2 +

[

]

DOTS - DSiO2 Λ

cOTS + O(cOTS2) (7)

and for the nearly totally grafted substrates (cOTS ≈ 1)

DOTS/SiO2 = DOTS - (DOTS - DSiO2)Λ(1 - cOTS) + O((1 - cOTS)2) (8) These linear trends are represented in Figure 4 by dashed lines. One can see that although the linear approximation does not hold at low coverage fraction of the substrate, it can be used, within the experimental error bars, for fOTS > 0.80. IV. Molecular Dynamics Simulations To get a better understanding of this experimental result, let us now consider molecular dynamics simulations for which we can measure all the details of the spreading at the atomic scale. Models for the Liquid Molecules and the Substrate. For all the considered atoms, we apply a standard 6-12 Lennard-Jones interaction of the form

Uij(r) ) 4

6

12

Cij

[(σr) - (σr) ] ) r

6

-

Dij r12

(9)

where r denotes the distance between any pairs of atoms i and j, σ is the characteristic diameter of the atoms, and  is the strength of the associated potential. The parameters Cij and Dij are, for simplicity, chosen constant for each species and refer thus to the fluid-fluid (ff), fluidsolid (fs), and solid-solid (ss) interactions. For computational convenience, the tail of the Lennard-Jones potentials are cut off at rc ) 2.5 in reduced units r/σ. This means that we take into account only short-range interactions. (19) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes in Fortran, 2nd ed.; Cambridge University Press: Cambridge, 1992.

Figure 6. Density profiles of the liquid particles as a function of the distance from the droplet symmetry axis. The profiles have been calculated after 67.5 × 103 (open circles), 80.0 × 103 (open triangles), and 92.5 × 103 (open diamonds) time steps. Corresponding sigmoid curves were fitted to these data to guide the eye. (a) Cfw ) Dfw ) 0.8 and (b) Cfw ) Dfw ) 2.0 for all the substrate atoms.

To mimic as much as possible the behavior of the liquid molecules in the experimental conditions, we here consider chainlike molecules instead of single atoms. This choice reduces considerably the evaporation of the liquid and therefore improves the efficiency of the simulation. In practice, we incorporate a confining potential

Uconf(r) ) r6

(10)

for adjoining atoms belonging to a given chain. The power 6 is here chosen for computational convenience. The solid is modeled by one layer of face-centered cubic (fcc) cells, i.e., two layers of atoms. These atoms interact via the Lennard-Jones potential with Cij ) 35 and Dij ) 5. We ascribe to them a heavy mass msolid ) 50mliquid so as to have a time scale comparable with the liquid. Each layer of atoms is initially fixed on a fcc lattice (orientation 100), and these atoms are then allowed to vibrate around their initial positions with a harmonic restoring potential. The temperature of the system is controlled by rescaling the velocities of the atoms. We first equilibrate independently the drop of liquid and the solid. Once we have an equilibrated drop (constant energy), we move it into the vicinity of the solid and then we maintain the temperature of the solid only. This procedure is used to mimic the thermal exchange between the liquid and the thermalized solid, as in a real experiment. The time step ∆t is measured in units of the dimensionless time variable τ ) (m/σ2)1/2 and its typical value given by ∆t ) 0.005. τ is of the order of 5 × 10-15 s, with  and σ defined as before. The

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Figure 7. Snapshots of a droplet spreading on heterogeneous substrates (top views). Same conditions of simulation as in Figure 5 except that the substrate is formed by a matrix of atoms (Cfw ) Dfw ) 0.8) in which are embedded square patches of atoms interacting with the liquid with Cfw ) Dfw ) 2.0 (dots). The dimension of the patches is such that the fraction of the latter atoms is (a) 8.2%, (b) 22.7%, (c) 44.4%, and (d) 73.4%.

trajectories of the atoms are then computed solving the associated Newton’s equations with Cff ) Dff ) 1.0. The system size we consider in this paper is 6400 atoms for the liquid (400 16-atom chains) and 45000 atoms to represent the wall. The interaction between the liquid and the solid itself is modulated by the constants Cfs and Dfs. To mimic the existence of two completely wettable species in the substrate, we have considered Cfs ) Dfs ) 0.8 for some solid atoms and Cfs ) Dfs ) 2.0 for the others. That is to say that the substrate is constituted by one species, A, which interacts strongly with the liquid (Cfs ) Dfs ) 0.8) and another species, B, which interacts even more strongly (Cfs ) Dfs ) 2.0). Previous studies have shown that the interactions of 1.0 and 1.2 lead to a complete wetting regime characterized by a single monolayer covering the surface,8,20,21 while interactions of 0.5 and (20) De Coninck, J.; D’Ortona, U.; Koplik, J.; Banavar, J. Phys. Rev. Lett. 1995, 74, 928.

lower are characteristics of the partial wetting regime.22 A system with liquid molecules interacting with the substrate with a coupling Cfs ) Dfs ) 0.8 is close to the wetting/nonwetting transition but remains relevant of the complete wetting regime. Simulation Results. The relative concentrations of A and B are denoted as c and (1 - c), respectively. Intuitively, we may expect that the larger the concentration of B is, the faster the spreading should be. To compute the associated base radius, we proceed as follows. We first locate the first layer of the liquid droplet in contact with the solid. We then locate the extremity of the layer as the distance where the density falls below a cutoff value of 0.4 times the liquid density. To check the consistency of the (21) D’Ortona, U.; De Coninck, J.; Koplik, J.; Banavar, J. Phys. Rev. E. 1996, 53, 562. (22) Blake, T. D.; Clarke, A.; De Coninck, J.; de Ruijter, M. J. Langmuir 1997, 13, 21.

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Figure 9. Time evolution of the contact area between the liquid droplet and the substrate. Each individual set of data can be fitted by a straight line. From the slope of this line, the value of the effective diffusion coefficient D can be calculated.

Figure 8. Density profiles of the liquid particles spreading on heterogeneous substrates. The profiles were calculated in the same conditions as in Figure 6. The fraction of heterogeneities is (a) 8.2% and (b) 73.4%.

method, different cutoff values were considered and gave almost identical results. Typical snapshots obtained for droplets spreading on the pure substrates (i.e., for c ) 0.00 and 1.00) are given in Figure 5. The density profiles and their fit using a sigmoid function23 at time t equal to 337.5, 400.0, and 462.5 (in τ units) are reproduced in Figure 6. These profiles show that the edge of the first layer in contact with the pure substrate becomes more and more soft as the spreading proceeds. Obviously, the spreading is faster on the substrate for which c ) 1.00. From these extremities of the layers, we can then measure the base radius R of the drop versus time and therefore calculate the diffusion coefficient of the liquid molecules on the substrate. Intuitively, we expect that, contrarily to what has been measured for equivalent substrates in the partial wetting regime,3 the details of the spreading phenomenon will depend on the geometry of the substrate because the liquid will try to wet “at best” the substrate atoms which are the most wettable. To mimic in a first approximation the occurrence of the growing LC* islands, regularly distributed patches of heterogeneities will be considered in this study. The shape has been chosen rectangular, for simplicity. Starting from a pure substrate characterized by coupling constants Cfs ) Dfs ) 0.8, the heterogeneitys are build up as squares of atoms with a coupling Cfs ) Dfs ) 1.2. By varying the size of these more wettable patches, we investigated heterogeneity concentrations equal to 0.0, 8.2, 44.4, 73.4, and 100.0%. Typical snapshots and typical density profiles are given in Figures 7 and 8, respectively. From Figure 7, it can be (23) Rowlinson, J. S. Widom, B. Molecular Theory of Capillarity; Clarendon Press: Oxford, U.K., 1984.

Figure 10. Diffusion coefficient D versus cA, the relative concentration of the less wettable species (Cfs ) Dfs ) 0.8). The dashed line corresponds to the best fit of the simulation data using eq 6 with DA ) 0.102 and DB ) 0.047. The fitted value of the ratio of the friction coefficients is Λ* ) 0.24 ( 0.04.

seen that on such patterned substrates, three regions can be identified in the liquid drop: (a) the reservoir, (b) an intermediate compact liquid layer, and (c) an outer region whose structure is highly dependent on the substrate structure. In that region, the liquid molecules cover only the atoms of the substrate characterized by Cfs ) Dfs ) 2.0, although, on the basis of what is seen on the pure substrates (Figure 5a), they completely wet both types of substrate atoms. This effect is notsor lesssvisible in the intermediate region because the liquid molecules have more neighbors and that the liquid-liquid interactions compensate the trends for these molecules to partially dewet the substrate. A comparison between the density profiles corresponding to the pure substrates (Figure 6) and to the heterogeneous ones (Figure 8) shows the spreading occurs at intermediate speeds on the latter ones. To characterize it more accurately, the time variations of the contact area between the liquid and the substrate for different concentrations of A and B are given in Figure 9. For a given concentration, the contact area linearly varies with times with a slope characteristics of the associated effective diffusion constant D. The values of D presented in Figure 10 are in good agreement with the data calculated on the basis of the eq 6, with the values obtained from the simulations carried

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out for the pure substrates (DA ) 0.102; DB ) 0.047) and with the fitted value of the ratio of the friction coefficients Λ* ) 0.24 ( 0.04. As already evidenced for the experimental system, the linear approximation holds better at low CA values that at high ones. The snapshots (Figures 5 and 7) also evidence fluctuations of the three-phase contact line, suggesting that the geometry of the heterogeneity patches (and not only their concentration) influences the dynamics of spreading. That an important question is still to be answered and the details of this study will be presented in a forthcoming article. So, both experimental and MD studies give the same trends concerning the variation of the effective diffusion of a polymeric liquid on a heterogeneous substrate, i.e., a monotonic decrease of D as the lyophilicity of the substrate increases. This feature seems to be in contradiction with the results of our previous study concerning the spreading of a similar liquid (PDMS 20) on pure substrates of increasing energies.9 Indeed, we observe that if we increase the substrate surface energy from the one of the OTS substrate to the one of the SiO2 surface either by considering intermediate pure substrates, as in ref 9, or by considering intermediate heterogeneous substrates, as in this study, the dependence of D on the surface energy

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variations is fundamentally different: a monotonic increase in the former case and a non-monotonic behavior in the latter one. V. Conclusions The present studies bring experimental evidence that, in the complete spreading regime, the diffusion coefficient of low molecular weight PDMS on partially OTS-grafted substrates is a monotonically increasing function of their surface energy, as characterized by the fraction of the substrate covered by the OTS layer. Furthermore, this evidence is supported by the results of Molecular Dynamics simulations carried out for chainlike molecules spreading on a substrate characterized by regularly distributed rectangular patches of heterogeneity. To date, this behavior has to be considered as a generalization of the competition process between driving and friction terms. An approximate expression of D is given as a function of the diffusion coefficients of the liquid molecules on the pure substrates and of the ratio of the friction coefficients. Acknowledgment. This research has been partially supported by the Ministe`re de la Re´gion Wallonne. Thanks are due to M. J. de Ruijter for fruitful discussions. LA990185R