6628
Langmuir 2005, 21, 6628-6635
Spreading Dynamics of Chain-like Monolayers: A Molecular Dynamics Study E. Bertrand,* T. D. Blake, and J. De Coninck Centre for Research in Molecular Modelling, University of Mons-Hainaut, Parc Initialis, Av. Copernic, 1, 7000 Mons, Belgium Received November 16, 2004. In Final Form: April 26, 2005 Using large-scale molecular dynamics simulations, we have shown previously that the spreading dynamics of sessile drops on solid surfaces can be described in detail using the molecular-kinetic theory of dynamic wetting. Here we present our first steps in extending this approach to investigate the spreading dynamics of Langmuir-Blodgett monolayers. We make use of a monolayer model originally developed by Karaborni and Toxvaerd, but somewhat simplified to facilitate large-scale simulations. Our preliminary results are in good agreement with recent experimental observations and also support a molecular-kinetic interpretation in which the driving force for spreading is the lateral pressure in the monolayer. Away from equilibrium, initial spreading rates are constant and logarithmically dependent on pressure. However, near equilibrium, spreading is pseudo-diffusive and follows the square root of time. In both regimes the controlling factor is the equilibrium frequency of molecular displacements within the monolayer.
1. Introduction During the spreading of a liquid drop that has a strong affinity for a solid surface and so wets it completely, one typically observes three zones: a macroscopic one controlled by capillarity, a precursor film of molecular thickness which develops in front of the drop, and an intermediate mesoscopic zone in which disjoining pressure strongly influences meniscus shape. The presence of the precursor film can be detected by several techniques, such as ellipsometry or interferometry.1 The properties of the films, which can be millimeters or even centimeters in length, have been established by de Gennes for thicknesses ranging from mesoscopic (several micrometers) to molecular.2,3 Detailed experiments have shown that the dynamic behavior of the film depends on the molecular structure of the liquid in the vicinity of the solid as well as the specific nature of the interactions between the liquid molecules and the solid surface.4 In particular, one observes quasi-universal, pseudo-diffusive behavior, such that the contact radius R of the film as a function of the elapsed time t takes the form
R(t) ) R0 + xDt
(1)
with R0 being the initial contact radius of the reservoir and D being the pseudo-diffusion coefficient of the liquid on the surface. Nevertheless, the precise physical mechanism by which a molecular film and its leading edge migrate across the solid surface is still a matter of speculation. Recently, Baumgart and Offenha¨user have described complementary experiments on the lateral spreading of substrate-supported lipid monolayers that revealed quite different spreading behavior.5 Langmuir-Blodgett (LB) * To whom correspondence should be addressed. E-mail:
[email protected]. (1) Heslot, F.; Fraysse, N.; Cazabat, A. M. Nature 1989, 338, 640. (2) de Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827. (3) de Gennes, P. G.; Cazabat, A. M. C. R. Acad. Sci. II 1990, 310, 1601. (4) Heslot, F.; Cazabat, A. M.; Levinson, P.; Fraysse, N. Phys. Rev. Lett. 1990, 65, 599. (5) Baumgart, T.; Offenha¨user, A. Langmuir 2002, 18, 5899.
Figure 1. Data of Baumgart and Offenha¨user.5 Spreading dynamics at 90% RH of a DMPC monolayer transferred at a pressure of 20 mN/m onto a thin film of chitosan.
monolayers of phospholipids were deposited onto thin, water-swellable polysaccharide films spin-coated onto glass. The influence of lateral deposition pressure and ambient humidity on the spreading dynamics was then investigated. The study showed that the leading edge of the monolayer usually remained sharp and that the initial spreading rates were constant for a given deposition pressure. Furthermore, the rates varied with pressure in a nonlinear way. Overall, spreading rates increased with relative humidity (RH). Figure 1 shows spreading data obtained at 90% RH for a DMPC (1,2-dimyristoyl-sn-glycero-3-phosphatidylcholine) monolayer transferred at a pressure of 20 mN/m onto a thin film of chitosan. Initial spreading rates were measured over distances that were less than 90 µm, that is, much less than the lateral dimensions of the monolayers. Over this distance spreading was linear with respect to time and occurred at a low speed of just 15 nm/s. There was no evidence of the t1/2 dependence found for precursor films and also for lipid bilayers spreading from a reservoir;6 nevertheless, it is still possible that t1/2 dependence might have been found had the experiments been extended to longer times and greater distances. Figure 2 gives the spreading velocities for DMPC on chitosan as a function of deposition pressure at two different humidities: 85% and 90%, respectively. As can (6) He, S.; Ketterson, J. B. Philos. Mag. B 1998, 77, 831.
10.1021/la047185y CCC: $30.25 © 2005 American Chemical Society Published on Web 06/01/2005
Spreading Dynamics of Chain-like Monolayers
Langmuir, Vol. 21, No. 14, 2005 6629
Figure 3. Data of Baumgart and Offenha¨user.5 Surface pressure-area isotherm for DMPC at 25 °C. Figure 2. Data of Baumgart and Offenha¨user.5 Spreading velocities of DMPC monolayers at 85% and 90% RH as a function of deposition pressure. The curves were obtained by fitting eq 3 simultaneously to both sets of data.
be seen, the slope of each graph increases monotonically. Such nonlinear behavior is in contrast to that observed for the spreading of lipids on mesoscopically thin films of water,6 where it is supposed that the force opposing spreading is simply the hydrodynamic (viscous) dissipation within the underlying water film. Baumgart and Offenha¨user modeled the spreading behavior on the polysaccharide films by balancing viscous friction at the monolayer/substrate interface by the gain in surface free energy, effectively balancing frictional dissipation by a local Marangoni surface-tension force. For spreading over small distances, this gave a constant spreading velocity, as observed experimentally. The influence of humidity was accounted for through its effect on the friction. An explanation advanced for the nonlinear dependence of spreading velocity on deposition pressure was that the friction varied with monolayer density. But, as Baumgart and Offenha¨user also pointed out, the selfdiffusion coefficient for the monolayer is expected to increase with decreasing monolayer density (i.e., with decreasing film pressure), which is inconsistent with Figure 2. In a recent note,7 we proposed an alternative explanation of monolayer spreading based on the molecular-kinetic theory of wetting. This theory, which derives from the Frenkel-Eyring model of liquid transport as a stressmodified molecular rate process,8 has proved useful in describing the dynamics of wetting for a wide range of experimental systems.9 However, it is only in the past few years that molecular dynamics (MD) simulations have shown the theory to have some fundamental validity.10 The basic concept is as follows: we picture the wetting line at the molecular level and suppose its motion to be determined by the statistical dynamics of the molecules within the three-phase zone where the liquid interface meets the solid surface. The key parameters are κ0, the equilibrium frequency of the random molecular displacements occurring at the wetting front, and λ, the average distance of each displacement. In the simplest case, λ is supposed to be the distance between adsorption sites on the solid surface. The driving force for the wetting line to move is taken to be the change in the solid-liquid interfacial tensions (uncompensated Young force) that arises when mechanical equilibrium is disturbed. For a (7) Blake, T. D.; De Coninck, J. Langmuir 2004, 20, 2977. (8) Blake, T. D.; Haynes, J. M. J. Colloid Interface Sci. 1969, 30, 421. (9) Blake, T. D. In Wettability; Berg, J. C., Ed.; Marcel Dekker: New York, 1993; p 251. (10) de Ruijter, M. J.; Blake, T. D.; De Coninck, J. Langmuir 1999, 15, 7836.
Table 1. Parameters Obtained by Fitting Equation 3 to the Data of Figure 2 % RH
κ0 (s-1)
λ (cm)
% RH
κ0 (s-1)
λ (cm)
85
3
7.8 × 10-8
90
14.9
7.8 × 10-8
partially wetting liquid this is γLV(cos θ0 - cos θ), where γLV is the surface tension of the liquid and θ0 and θ are, respectively, the equilibrium and instantaneous dynamic contact angles of the liquid on the solid. The resulting equation for the wetting line velocity is then
dx ) 2κ0λ sinh[γLV(cosθ0 - cosθ)λ2/2kBT] dt
(2)
where kB is the Boltzmann constant and T is the temperature. If the liquid completely wets the solid surface (θ0 ) 0) then the driving force is augmented by the surface pressure of the liquid at the solid/vapor interface, πLV.11 For a LB monolayer, the natural driving force for spreading would appear to be simply the lateral surface pressure at which the layer was deposited πLB. Given that the leading edge of the layer remains sharp, we can assume two-dimensional liquidlike behavior. Hence, if such a layer were allowed to spread from confinement at this pressure, we might anticipate that the initial dynamics of the leading edge would be described by eq 2 in the form
dx ) 2κ0λ sinh(πLBλ2/2kBT) dt
(3)
This simple expression predicts a nonlinear relationship as required, becoming logarithmic for sufficiently large πLB and linear when πLB is sufficiently low. The curves in Figure 2 were obtained by fitting eq 3 simultaneously to the spreading data for both relative humidities. Evidently the equation is very effective at describing monolayer spreading from an initially confined state. The values of κ0 and λ obtained from the fit are shown in Table 1. The value of λ is reasonable, being of molecular size and equivalent to a site area of about 60 Å2, which is close to the mean cross-sectional area of the DMPC molecule over the range of pressures investigated. Figure 3 shows the surface pressure-area isotherm for DMPC reported by Baumgart and Offenha¨user. However, we also expect λ to be strongly influenced by the molecular structure of the substrate. This was the motive for holding λ constant during the fitting procedure, rather than using values obtained directly from the isotherm. The values of κ0 obtained for the monolayer are low compared with those reported in the literature for the dynamic wetting of solids by simple liquids9 but not necessarily unreasonable given the fact that the mono(11) Blake, T. D.; De Coninck, J. Adv. Colloid Interface Sci. 2002, 96, 21.
6630
Langmuir, Vol. 21, No. 14, 2005
Bertrand et al.
layers are strongly adsorbed on the substrate and spread only very slowly ( 1 to avoid evaporation of the monolayer, while still allowing some spreading on the solid surface. Too small a value allows strong evaporation while too large a value restricts all the molecules to their initial positions. Furthermore, by adjusting B, we are able to simulate the effects of humidity on monolayer mobility as observed by Baumgart and Offenha¨user.5 Three different values of B were used, respectively 2.3, 4.6, and 6.9, in a total of eight simulations. The theoretical range of the Lennard-Jones 12-6 interactions extends to infinity. Strictly, one should, therefore, evaluate the interactions between all possible pairs in the system. Fortunately, the potentials decrease rapidly as the distance becomes large. We, therefore, apply a spherical cutoff at 20 Å, that is, at about 5σ. This makes the computation manageable, while retaining all the essential physics. To be consistent, the dipole-dipole interactions are then cut at a distance that yields a repulsive potential equal to the Lennard-Jones attraction at the 20-Å boundary. This distance is approximately half the width of the monolayer within the periodic boundary conditions. To summarize, we consider a simple monolayer of chainlike, 10-atom, amphiphilic molecules on top of a surface made of a planar solid lattice. Although the model is very simple, we will show that it contains all the essential ingredients to reproduce realistic results within practical computational time scales. A perspective view of the system at the start of a simulation is shown in Figure 4. 3. Results Monolayer at Equilibrium. The first step was to establish the lateral pressure isotherm of our simulated monolayer at equilibrium. For a square monolayer comprising 25 × 25 ) 625 molecular chains with 10 atoms per chain, we computed the lateral pressure using periodic boundary conditions in both the x and y directions. Simulations were launched for a range of molecular areas
Figure 5. Isotherm of the lateral pressure versus molecular area for B ) 2.3. The curve through the data is a fit to the two-dimensional van der Waals equation.
between 21 and 55 Å2. The unit of time was fixed to 1 fs. The values chosen for these parameters do not qualitatively affect the spreading behavior but are helpful in comparing the observed dynamics with physical experiments. For each simulation, we first let the system equilibrate for a period of time, typically about 50 000 time steps, keeping the temperature constant at 250 K. To check that equilibrium had been attained, we measured the lateral pressure π given by
1 π ) (Pxx + Pyy)H h 2
(11)
where Pxx and Pyy are computed according to20 N N 1 N PRβ ) ( piRpiβ/mi + rijRfijβ) V i)1 i)1 j>i
∑
∑∑
(12)
with H h being the average height and V being the volume of the part of the monolayer considered containing N atoms. Here, R, β are the x, y, or z coordinates, piR is the Rth component of the ith particle momentum, mi is the ith atom mass, rijR is the distance between atoms i and j along Rth axis, and fijβ is the force between atoms i and j along the β axis. A typical isotherm of the lateral pressure versus the area per molecule is given in Figure 5 for B ) 2.3. The pressure is rather large at small areas, but otherwise the general behavior is consistent with experimental isotherms. As can be seen, the data are well described by the two-dimensional van der Waals equation of state:
π)
b a - a0
(13)
where a is the molecular area. Using a LevenbergMarquard algorithm, the best fit gives b ) 1.835 ( 0.249
6632
Langmuir, Vol. 21, No. 14, 2005
Bertrand et al.
Figure 6. Typical snapshots of the monolayer (a) in its initial state, (b) after equilibration at 50 000 time steps, and during spreading after (c) 80 000, (d) 110 000, (e) 140 000, (f) 170 000, (g) 200 000, and (h) 230 000 time steps. Table 3. Two-Dimensional van der Waals Coefficients simulation
B
b (10-20 J)
a0 (Å2)
R2a
1-5 6, 7 8
2.3 4.6 6.9
1.835 ( 0.249 2.533 ( 0.406 2.772 ( 0.477
18.90 ( 0.45 17.73 ( 0.63 17.30 ( 0.74
0.982 0.970 0.967
a
R2 is a measure of the quality of the fit.
× 10-20 J and a0 ) 18.90 ( 0.45 Å2. The parameter a0 corresponds to the minimum area of each molecule and is consistent with the amphiphile cross section. Because of tail bending and the dipole-dipole repulsion, the value is slightly larger than the cross-sectional area of the head atoms, 14 Å2. The values of a0 and b obtained with the three B coefficients used in the simulations are listed in Table 3. Dynamics of Spreading. For the spreading simulations the monolayer comprised 425 molecular chains (17 in the x direction and 25 in the y direction). The solid was a single layer of atoms extending 85 atoms in the x direction and 31-35 in the y direction, to allow us to investigate several initial areas per molecule. After equilibrating the monolayer for 50 000 time steps, we removed the periodic boundary conditions in the x direction, thereby allowing the molecules to spread in this direction only. A series of simulations were run, varying the S-H interaction parameter B and the initial area per molecule. Figure 6 shows typical snapshots of the film in its initial state, on equilibration, and during successive stages of spreading. An example of the evolution of the overall lateral pressure within the monolayer during the equilibration period is shown in Figure 7, where we plot the pressure versus time for simulation 1 with B ) 2.3 and a ) 29.8 Å2. We also show the running average of the pressure over 20 000 successive time steps. Because of the moderate size of the simulation, the raw trace is somewhat noisy; however, the monolayer clearly reaches a steady state within 50 000 time steps, in this case at an equilibrium pressure of 0.18 N m-1. To understand how the lateral pressure inside the monolayer evolves in more detail, we studied the pressure at five different locations. The pressures were sampled (20) Allen, M. P.; Tildesley, D. J. Computer Simulations of Liquids; Oxford University Press: New York, 1991.
Figure 7. Overall lateral pressure within the monolayer versus time for simulation 1 with B ) 2.3 and a ) 29.8 Å2. The smooth line corresponds to the running average over 20 000 consecutive time steps.
Figure 8. Five transverse strips across the monolayer. Each strip contains 360 atoms at the beginning of the simulation.
within transverse strips across the monolayer as indicated in Figure 8. At the beginning of the simulation, each strip contained 360 atoms, that is, about 8.5% of the total. During the course of the simulation, the length and width of each strip were held constant. The positions of strips 2-4 were also held constant, but strips 0 and 1 were allowed to track with the leading edge of the monolayer. In Figure 9 we show the evolution of the pressure within each strip with time as the monolayer equilibrates. Here, the curves are running averages over 7000 time steps.
Spreading Dynamics of Chain-like Monolayers
Figure 9. Evolution of the lateral pressure during simulation 1 within each of the five strips defined in Figure 8. The curves are the running average over 7000 consecutive time steps.
Figure 10. Evolution of the lateral pressure during simulation 1 within each of the five strips of the monolayer as it spreads. The curves are running averages over 20 000 time steps.
Although there is considerable scatter, the pressures within each strip evidently converge to the same, moreor-less constant value (0.18 N m-1) after about 35 000 time steps. In other words, once equilibrium is achieved, the lateral pressure is uniform across the monolayer, as in a real system. After equilibration, we removed the periodic boundary conditions in the x direction and followed the continued evolution of the pressure within each strip as the monolayer spread. The results are given in Figure 10, in this case as a running average over 20 000 time steps. The most striking feature is that the lateral pressure of the edge of the monolayer (strips 0 and 1) falls off very rapidly and approaches some low equilibrium value. However, the pressure within the monolayer decreases very much more slowly, changing by less than 20% over the same period. To quantify the dynamics of spreading, it was first necessary to locate the edge of the monolayer. To do this we proceeded as usual by computing the density of atoms in the x direction. We located the edge at the point where the density of the monolayer fell to half the density at the center. We also studied the maximum positions of the atoms belonging to the monolayer in the x direction. This gave essentially the same results. The position of the edge of the monolayer as a function of time is plotted in Figure 11.
Langmuir, Vol. 21, No. 14, 2005 6633
Figure 11. Position of the edge of the monolayer versus time for simulation 1. The straight line through the steep part of the plot was obtained by fitting the data up to about 100 000 time steps using eq 15. The shallow line through the upper part of the plot was obtained by fitting the data after about 100 000 time steps using eq 20.
The plot reveals two different regimes: an initial steep regime in which spreading is evidently linear in time (Figure 11) and a subsequent shallow one, beginning after about 100 000 time steps, in which, as we show below, spreading appears to become asymptotically proportional to t1/2, that is, pseudo-diffusive. Exactly the same result is obtained if we track the positions of the headgroups alone rather than the edge of the monolayer as a whole. Because the sample is much smaller, the results are statistically noisier, but the data remain strongly suggestive of an initial linear regime followed by the transition to a slower diffusive one. Comparison with Figure 10 suggests that the second regime commences when the lateral pressure inside the spreading monolayer begins to reduce significantly and approach equilibrium. Although Baumgart and Offenha¨user5 did not observe 1/2 t behavior in their system, spreading was followed for only a relatively short period in the time frame of the phenomena. In addition to this, we know that pseudodiffusive behavior is commonly observed in the spreading of thin liquid films, both in experiment and in simulation.3,21 Furthermore, it has come to our attention recently that, during Monte Carlo simulations to model the spreading of liquid monolayers, an initial brief linear regime has been detected.22 The existence of both regimes in real systems, therefore, seems very plausible. 4. Discussion As we have seen, one of the principal results of our simulations of monolayer spreading on a solid substrate is that the pressure within the monolayer does not fall off uniformly. Evidently, the edges, having more freedom, spread first, gradually reducing the confinement of the interior, which only then relaxes toward equilibrium. Thus, the information flow regarding the release of confinement appears to percolate backward from the advancing front toward the interior as the monolayer spreads. The second important result is that there are two regimes of monolayer spreading: an initial regime, in which the monolayer spreads linearly with time, and a subsequent one in which spreading appears to become (21) De Coninck, J.; D’Ortona, U.; Koplik, J.; Banavar, J. Phys. Rev. E 1996, 53, 562. (22) Dietrich, S.; Popescu, M. N. Personal communication, 2004.
6634
Langmuir, Vol. 21, No. 14, 2005
Bertrand et al.
Table 4. Parameters for the Linear Spreading Regime simulation
B
πinitial (N m-1)
c (m s-1)
κ0 (ns-1)
1 2 3 4 5 6 7 8
2.3 2.3 2.3 2.3 2.3 4.6 4.6 6.9
0.180 0.149 0.125 0.105 0.090 0.205 0.165 0.215
15.00 ( 0.02 8.98 ( 0.02 6.64 ( 0.01 5.42 ( 0.01 2.84 ( 0.02 6.73 ( 0.01 4.13 ( 0.02 7.20 ( 0.02
0.218 0.309 0.449 0.649 0.519 0.0482 0.0915 0.0393
proportional to t1/2 at long times. To understand the governing mechanism, we need to examine these two regimes in more detail. First Regime: x ∝ t. Within this regime, the edge of the monolayer advances at a constant speed, as found experimentally by Baumgart and Offenha¨user:5
x(t) ) ct + x0
(14)
where x is the distance spread from some initial position x0 in time t. Such behavior is consistent with the molecularkinetic theory, eq 3. Because the lateral pressure inside the monolayer changes only slowly during the initial stages of spreading, the right-hand side of eq 3 is essentially constant; hence, the associated spreading rate is also constant. Setting πLB ) πinitial, where πinitial is the lateral pressure at the start of spreading, that is, the equilibrium pressure,
dx ) c ) 2κ0λ sinh[πinitialλ2/2kBT] dt
(15)
All the parameters in this expression are independently quantifiable except the molecular displacement frequency κ0. This can, therefore, be estimated from a linear fit to the data in the first regime, as in Figure 11. Here, we have assigned data by inspection and set λ ) 4.4 Å, the equilibrium distance between two solid atoms. In Table 4, we give the result for each of the simulations, listing κ0 along with the other associated parameters. Although there is a fair amount of scatter, κ0 shows the expected reduction in mobility as the H-S interaction parameter B is increased. Taking mean values where we have more than one simulation for a given B, κ0 decreases more than 10-fold, from 0.43, through 0.070, to 0.039 ns-1 for B ) 2.3, 4.6, and 6.9, respectively. Using the data for B ) 2.3, where we have five simulations for a range of initial lateral pressures, we can also check that the initial spreading velocities are a nonlinear function of πinitial, as found by Baumgart and Offenha¨user and predicted by eq 15. Thus, in Figure 12 we have plotted the initial spreading velocity versus πinitial, as well as the curve given by eq 15 with λ ) 4.4 Å and κ0 ) 0.429 ns-1 (i.e., the mean value). Clearly the relationship is nonlinear as found experimentally. Given that the curve is obtained directly from these few results and not by some fitting procedure, the overall agreement with the molecular-kinetic theory is very good. Furthermore, the variation in the H-S interaction parameter B enables us to mimic the effects of humidity observed by Baumgart and Offenha¨user. In Figure 12 we have, therefore, included the predicted curves and data for B ) 4.6 and 6.9. Although we do not have sufficient data to test the relationships properly, the curves show the expected reduction in mobility with increasing interaction, which is qualitatively very similar to the effect of reducing humidity (Figure 2). Second Regime: x ∝ t1/2. In the vicinity of equilibrium, both the lateral pressure and the spreading rate are small;
Figure 12. Plot of initial spreading velocity versus πinitial using data from the simulations (Table 4): O, B ) 2.3; 4, B ) 4.6; 0, B ) 6.9. The theoretical lines through the data were calculated using eq 15 with λ ) 4.4 Å and κ0 ) 0.429, 0.07, and 0.039 ns-1, respectively.
the right-hand side of eq 3, therefore, reduces to its linear form:
dx λ3κ0 ) π dt kBT
(16)
Furthermore, close to equilibrium the lateral pressure is essentially uniform across the monolayer, and so π will again be given by the two-dimensional van der Waals equation of state, eq 13. Thus,
dx λ3κ0 b ) dt kBT a - a0
(17)
If the monolayer contains N molecules and has length 2x and width w, then the area per molecule is
a)
2xw N
(18)
whence,
dx λ3κ0 bN ) dt kBT 2xw - a0N
(19)
On integration, this gives the expression for the position of the edge of the monolayer as a function of time as
x(t) )
[(
) ]
λ3κ0 bN t+C kBT w
1/2
+
a0N 2w
(20)
For a given set of conditions, C is a constant that depends on system parameters and the position of the crossover between the first and the second regime. Thus, under these near-equilibrium conditions, x(t) becomes asymptotically proportional to t1/2 at long times, as proposed. All the parameters in eq 20 are quantifiable except κ0 and C. For simulation 1, evaluation yields
x(t) ) x1410κ0t + C + 29.42
(21)
where x is expressed in angstroms. Moreover, by fitting the data for the second regime using eq 20, as in Figure 11, we can explicitly determine both κ0 and C. The results for all the simulations are listed in Table 5, along with the values of κ0 found for the linear regime.
Spreading Dynamics of Chain-like Monolayers
Langmuir, Vol. 21, No. 14, 2005 6635
Table 5. Parameters for the Second Spreading Regime and Comparison of K0 Values from Both Regimes
simulation
B
πinitial (N m-1)
1 2 3 4 5 6 7 8
2.3 2.3 2.3 2.3 2.3 4.6 4.6 6.9
0.180 0.149 0.125 0.105 0.090 0.205 0.165 0.215
C (Å2) 602.0 ( 1.5 660.8 ( 1.8 579.6 ( 6.5 683.8 ( 1.8 753.7( 2.4 557.5 ( 1.2 494.3 ( 0.7 448.7 ( 1.0
κ0 κ0 (t-1/2 regime) (linear regime) (ns-1) (ns-1) 0.202 0.211 0.534 0.547 0.516 0.069 0.107 0.033
0.218 0.309 0.449 0.649 0.519 0.0482 0.0915 0.0393
The data show that C becomes smaller as B increases, suggesting that the crossover to the pseudo-diffusive regime occurs at an earlier stage in the evolution of the monolayer as its mobility decreases. However, the striking thing to notice is how closely the values of κ0 from the two regimes agree. This is a strong indication that the underlying mechanism is the same in both cases and is well-described by the molecular-kinetic theory. 5. Conclusions Pseudo-diffusive spreading with t1/2 dependence has been widely observed for precursor films of completely wetting liquids,4 lipid films on water,6 and metals on metal surfaces.23 Theoretical explanations have been advanced,3 and the behavior has been successfully modeled using MD.21 On the other hand, it is well-known that the dynamics of partial wetting does not follow t1/2, but exhibits a constant rate of progression for a given driving force. Here, we have used moderately large-scale MD to model (23) Popescu, M. N.; Dietrich, S. Phys. Rev. E 2004, 69, 061602.
the spreading of a monolayer on a solid substrate. Our results are in good agreement with recent experiments5 and suggest that an initial linear regime is followed by one that is pseudo-diffusive at long times. Significantly, we have demonstrated that the molecular-kinetic theory, already successful in describing partial wetting dynamics,9 is also capable of accurately modeling both the initial linear spreading behavior and the later t1/2 dependence. It would seem that this offers the possibility of realizing a single molecular model of wetting that describes the dynamics of both partial and complete wetting in a consistent way. As this was our first step in exploring monolayer spreading using MD, we have restricted the size of our simulations. Nevertheless, we believe our results are already sufficiently stable to be convincing. In the present study, the total expansion is about 16 Å, which represents some 18% of the initial length of the monolayer. In the future, we plan to carry out simulations using significantly larger numbers of molecules for longer times. This will both reduce the statistical noise and enable us to explore the pseudo-diffusive regime more completely. We will also refine the fitting procedures to obtain κ0 from the two regimes. However, the results obtained so far are more than adequate to demonstrate the existence of the two regimes and to show that they are two aspects of the same underlying molecular mechanism. In this context, the agreement observed between the values of κ0 from the two regimes is particularly telling. Acknowledgment. The authors thank the Fonds National de la Recherche Scientifique for partial financial support. LA047185Y
