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Langmuir 1998, 14, 3418-3424
Interplay between Hole Instability and Nanoparticle Array Formation in Ultrathin Liquid Films Pamela C. Ohara† and William M. Gelbart* Department of Chemistry and Biochemistry, University of California, Los Angeles, Los Angeles, California 90095-1569 Received October 21, 1997. In Final Form: February 17, 1998 When dilute solutions of nearly monodisperse (3-5-nm diameter) passivated metal nanoparticles are evaporated on a solid substrate (a transmission electron microscopy grid), the resulting submonolayer structures are annular ringlike arrays. We describe here a theoretical mechanism whereby the growth of holessdry areas of the solid surface which is otherwise wet by solventsis conjectured to be responsible for the formation of these novel close-packed two-dimensional (2D) arrays. Each annular ring is argued to arise from the pinning of the rim of an opening hole by a sufficient number of particles. Hole nucleation in this ultrathin-film system can occur due to two independent driving forces: evaporation (volatile hole nucleation) and disjoining pressure (nonvolatile hole nucleation). The cases are first presented separately, in an appropriate analytical theory, and are subsequently treated jointly in the context of a simulation. We investigate in particular the effect of competing time scales (for hole nucleation, fluid flow, and solvent evaporation) on the average diameter of the resulting ringlike configurations.
I. Introduction Recently, there has been growing interest in the formation of ordered arrays of nanoparticles. In practice, these arrays can be assembled by various means1 and have promising uses in electronic or optical devices.2 In a previous communication,3 we described briefly a new class of two-dimensional (2D) submonolayer arrayss annular ringlike structuresswhich self-assemble from a solution of nanometer-sized metal particles on solid substrates. These micron-sized arrays (0.1-1 µm in diameter) were argued to be formed from holes nucleating in volatile, wetting thin liquid films containing the particles. Figure 1 shows a typical annular ringlike structure observed experimentally. In our earlier work it became apparent that there is a unique interplay between the stability properties of ultrathin wetting films and ultrasmall particles contained in them. This coupling is unique to nanometer-scale colloid science: holes open in films of nanometer thickness and push nanometer-sized particles along the rims of the opening holes. In turn, these particles play an important role in the eventual pinning of the contact line. We previously developed a mechanism for an effectively nonvolatile system, where the rate of evaporation is small compared to that for fluid flow. Now we address the effect of solvent volatility via a Monte Carlo simulation that illustratively interpolates between the various limiting cases of these relative rates. Our goal is to demonstrate the effect of evaporation on ring size. We show in particular that its main effect, if not too fast, is to produce larger rings. †
Present address: IBM Corporation, 5600 Cottle Road, San Jose, CA 95193. * To whom correspondence should be addressed. (1) Herron, N.; Calabrese, J. C.; Farneth, W. E.; Wang, Y. Science 1993, 259, 1426; Vossmeyer, T.; Reck, G.; Katsikas, L.; Haupt, E. T. K.; Schulz, B.; Weller, H. Science 1995, 267, 1476; Eychmuller, A.; Mews, A.; Weller, H. Chem. Phys. Lett. 1993, 208, 59; Murray, C. B.; Kagan, C. R.; Bawendi, M. G. Science 1995, 270, 1335. (2) Takagahara, T. Surf. Sci. 1992, 267, 310; Kagan, C. R.; Murray, C. B.; Nirmal, M.; Bawendi, M. G. Phys. Rev. Lett. 1996, 76, 1517. (3) Ohara, P. C.; Heath, J. R.; Gelbart, W. M. Angew. Chem., Int. Ed. 1997, 36, 1077.
Figure 1. TEM image of a “typical” annular ring observed upon evaporation of hexane from a dilute solution of 3-nm gold particles “coated” by dodecanethiol. See text for discussion. Picture courtesy of Sung-Wook Chung.
Background: Nonvolatile Case. Previously,3 we described the experimental means for obtaining twodimensional (2D) annular ringlike structures. Fairly monodisperse mixtures (2-6-nm diameter metal cores) of organically passivated metal (silver or gold) particles are dissolved in a volatile organic solvent (hexane or toluene). A dilute drop of the solution is placed on a substrate (amorphous-carbon-coated transmission electron microscopy (TEM) grid or organically functionalized Si substrate) which it wets; accordingly, the solvent spreads
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Interplay in Ultrathin Liquid Films
Figure 2. Side-view schematic showing scenario for evaporation of solvent from nanoparticle solution on wetted substrate (top), and its thinning to thickness thole < te (see text), at which point a hole nucleates (middle). The hole then opens, collecting particles in its growing perimeter (bottom), which becomes an annular ring upon pinning of its contact line.
out as a uniform liquid film. At the same time it evaporates due to its volatility, and after completely evaporating, the resulting submonolayer particle arrays are observed by transmission electron microscopy or scanning electron microscopy. For an area coverage of particles ranging from 1 to 10%, these structures are annular rings of particles with diameters of 0.1-1 µm and with annular widths of 10-100 nm: see Figure 1. For large (>2-3-nm diameter) particles the ring diameter is inversely related to the concentration of particlessmore concentrated solutions lead to smaller rings. Also, for any given system (i.e., particular particle size and concentration) and particular solvent, all the rings formed are essentially the same size. We will discuss next how these features are consistent with the formation and pinning of holes in the wetting liquid film. Figure 2 shows a pictorial scenario of our proposed mechanism for ring formation. The solution is placed on the substrate. (a) The liquid wets the substrate as a uniform liquid film. The solvent then evaporates due to its volatility. (b) At a critical thickness thole (to be discussed later), dry holes open up in the liquid film. The shaded circular region indicates the exposure of a dry solid surface as the hole opens. The border between liquid film and solid substrate is referred to here as the “contact line”. (c) The hole continues to open, as indicated by the receding contact line. Most particles are pushed along the liquid rim as the contact line recedes. Eventually, a sufficient number of particles accumulate along the rim of the opening hole so that they cannot be dragged along any further. The contact line then stops receding and is “pinned”. This pinned contact line of particles is the basis for the annular ring. Note that the above mechanism is consistent with the experimental observation that in more concentrated solutions particles accumulate more rapidly along the rim of the opening holes, leading to earlier pinning and hence smaller rings. Particle rings observed in everyday lifesfrom, say, spills of coffee or cooking fluidssare quite different from those described above, and are typically produced from the pinned contact lines of evaporating drops of colloidal solution. These rings are manifestly macroscopic, formed at the pinned contact lines of (nonwetting) drops contain-
Langmuir, Vol. 14, No. 12, 1998 3419
ing macroscopic or micron-sized particles. Deegan et al.,4 under controlled experimental conditions, have investigated the mechanism for the formation of such a ring. They argue that the three-dimensional (3D) buildup of a ring of particles arises from the radially outward convective flux of solvent from the center to the rim of the sessile convex drop, as it struggles to maintain its (spherical portion) shape in the face of evaporative losses. Similar phenomena had been reported earlier by Denkov et al.5 in their discussion of the differences between an ordered array versus ring formation in concave versus convex drops of submicron-particle suspensions. However, these rings are of a different length scale than that shown in Figure 1, being macroscopic as compared to our micron-sized rings. Also, the solvent behavior is different. Our solvent does not form drops on the surface, but rather wets the surface, spreading out to a uniform thin film. Hole nucleation in evaporating wetting films can be caused by two distinct forcessevaporation and disjoining pressure. We previously described the importance of disjoining pressure in effectively nonvolatile (nv) systems.3 There, we assumed that there is a local equilibrium wherein the system equilibrates faster than the rate of disappearance of the fluid. The film spreads across the substrate for energetic reasons; the spreading coefficient S ) γsv - (γ + γsl), which measures the energetic gain per unit area for spreading, is positive. (Here, γsv(l) is the solid-vapor (liquid) interfacial tension and γ is the usual liquid surface tension.) The film continues to spread until it reaches a thickness on the order of nanometers, where the liquid-liquid molecular attractions (as measured by a Hamaker constant AH) are diminished substantially. Upon further thinning, the energy of the system then increases as AH/t2, per unit area, where t is the film thickness. At equilibrium (e), this film will have a “pancake” thickness te ) (3AH/S)1/2 6 which represents a compromise between the desire to spread (as measured by S) and the tendency for the fluid to thicken (measured by AH). In this slow evaporation (nonvolatile) limit, holes can nucleate in thin films with thickness t < te and continue to open in order to reach the equilibrium thickness. Let ∆Ahole,nv denote the free energy associated with the formation of this hole. It is then straightforward to show that this quantity can be written approximately as
∆Ahole,nv ≈ [γt]2πR + S[1 - (te/t)2]πR2
(1)
(Here we have assumed that the hole area πR2 is small compared to the film area per hole As and have directly incorporated the constraint of constant fluid volume. No explicit account has been made of the tension associated with the three-phase contact line.) This free energy has the usual “nucleation form”, balancing surface and bulk contributions, first increasing with R and then going through a maximum at Rmax. Fluctuations corresponding to small holes will grow when the barrier ∆Amax,nv ) πγtRmax is brought down to ≈ kBT, where Rmax ≈ γt/S[(te/ t)2 - 1]. This happens when the film thickness t is sufficiently smaller (by ≈10-15%) than the “pancake” thickness te. Below this thickness, thole, the thin film becomes unstable against the nucleation and growth of holes, in its drive to achieve the optimum thickness te. (4) Deegan, R. D.; Bakajin, O.; Dupont, T. F.; Huber, G.; Nagel, S. R.; Witten, T. A. Nature 1997, 389, 827. (5) Denkov, N. D.; Velev, O. D.; Kralchevsky, P. A.; Ivanov, I. B.; Yoshimura, H.; Nagayama, K. Langmuir 1992, 8, 3183. (6) See, for example, Brochard-Wyart, F.; di Meglio, J. M.; Quere, D.; de Gennes, P. G. Langmuir 1991, 7, 335; de Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827.
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Also, because te is on the order of nanometers, it is assumed that there is no difference in rim thickness compared to bulk film thickness, as observed by Elbaum and Lipson7 and incorporated into the theory by Brochard and Daillant8 for thicker thin films. The force acting radially outward on the rim, associated with the drive to thicken, is given by the derivative of the free energy (1) with respect to R; the second term can be shown to dominate for typical values of t, γ, S, and R, leading to8
Fthicken ) 2πRS[(te /t)2 - 1]
(2)
This force increases with hole size R but is also dependent on film thickness, specifically on t compared to te. As each hole opens it sweeps out particles whose number R2φ/r2 is proportional to R2; here φ is the area fraction of particle coverage and r is the particle radius. The thickening force pushes particles along its way until the force per particle fthicken ) 2πr2S[(te/t)2 - 1]/Rφ is balanced by the frictional force per particle ffriction, opposing lateral movement. We associate this frictional force with the downward dispersional attraction between the metal core and the solid substrate. For systems with a very small areal coverage of rings, the solvent thickness would not change much as each hole is opened. For this simple limiting case, it is straightforward to determine the ring radius from the above balance of forces:3
Rpin ) 2πr2S[(t/te)2 - 1]/ffrictionφ
(3)
Here, ffriction is a function only of the particle size and metal-surface interaction strength.9 It then follows that, for a given particle size (r) and concentration (φ), and solvent (S,te), all rings will be characterized by the same radius (Rpin). II. Competing Mechanisms for Hole Growth Evaporation Effects. Hole nucleation in volatile liquid films can also be driven by evaporation,7 analogous to bubble nucleation10 in a bulk superheated liquid, with the difference that holes in thin films are “open” to the vapor above the film. This evaporation can occur both uniformlyschanging the thickness of the filmsor at the rim of the opening hole. Other groups have investigated the possibility of these evaporation rates being different (e.g., where a faster rim evaporation can contribute to the formation of coffee rings).4 Treating the cases separately can also lead to some interesting analytical results. If a thin film evaporates uniformly, its equilibrium thickness can be determined by its volatility. In general, we start with a drop of nanoparticle solution that spreads across the carbon substrate as a thin film. The liquid film is in a large boxsthe laboratory (whose volume is large compared to the volume of liquid)sand evaporates because the equilibrium vapor pressure of the thin-film solvent, P0, greatly exceeds the actual vapor pressure above the (7) Elbaum, M.; Lipson, S. G. Phys. Rev. Lett. 1994, 72, 3562. (8) Brochard-Wyart, F.; Daillant, J. Can. J. Phys. 1990, 58, 1084. (9) We write the lateral frictional force acting on a particle as Kfz,disp, where K is a dimensionless coefficient of order unity and fz,disp is the dispersional attraction (along the vertical [z] direction) between each particle and the substrate. From standard arguments15,17 one obtains: fz,disp ) Am-s (r - δ)3/[3δ2(2r - δ)2/2]. Here Am-s (on the order of 10’s of kBT) is the Hamaker constant appropriate to gold or silver metal (m) interacting through alkanes with the carbon substrate (s). r (2-3 nm) is the radius of the metal particle “core” and δ (1-2 nm) is the thickness of its adsorbed alkylthiol monolayer. (10) DiBenedetti, P. Metastable Liquids; Princeton University Press: Princeton, NJ, 1996.
film, P. P0 depends sensitively on the thickness when the film becomes sufficiently thin. In particular, it decreases as t decreases according to P0 ) Pb exp[-2AH/Flt3kBT]; here Pb is the bulk liquid (t f infinity) equilibrium vapor pressure and Fl is the liquid number density. Then, for P , Pb, the film would need to thin, via evaporation, to unphysically small dimensions in order for the liquid to be in equilibrium with its vapor; accordingly, equilibrium is reached only when all of the liquid has evaporated. Assuming that holes open because of rim evaporation exclusively also leads to a simple analytical result for the change in free energy associated with opening a hole. Consider a system with total pressure Ptot ≈ Patm, a given film thickness t, a fixed area of the substrate As, and number of liquid molecules N. The change in the Helmholtz free energy associated with forming a hole of radius R, via the evaporation of Ng ) πR2tFl particles, can be written as
∆Ahole,v ≈ πR2{FltkBT[log(P/P0) - 1] + (S - AH/t2)} + 2πRtγ (4) Here, we have used µl ≈ µl(Patm), µg(P) ) µ0g(Patm) + kBT log(P), and the equilibrium relation between the chemical potentials at P ) P0. Also, Ptot has been neglected compared to FlkBT. Equation 4 is similar to that presented by Elbaum and Lipson,7 with the primary difference that our holes are “dry”, as discussed above in terms of the lack of an equilibrium film thickness for our system. It is also similar to that discussed by Schenning et al.,11 who implicated hole nucleation in the 3D ring formation observed in the evaporation of nonwetting solutions, where film thicknesses are always large enough for disjoining pressure effects to be neglected. If the free energy difference given by (4) is monitored as the hole grows, the barrier associated with hole nucleation can be determined. It is when this barrier ∆Amax,v ) πγtRmax,v is sufficiently small, on the order of kBT, that holes of critical radius Rmax,v ) γt/{tFlkBT‚ [log(P0/P) + 1] - (S - AH/t2)}, will nucleate and grow. (Rmax,v is the value of R at which ∆Ahole,v has its maximum and ∆Amax,v is the free energy barrier at that point. The subscript “v” refers to the solvent being volatile, as opposed to the nonvolatile (nv) case treated earlier.) For our experimental system in which annular rings are observed, typically P , P0, implying that Rmax,v and hence barrier height ∆Amax,v are small even for relatively thick films: holes can nucleate at thicknesses greater than te and can do so at a relatively uniform rate. This is where the importance of competing time scales (which can be varied via controlled evaporation experiments) becomes evident. If the fluid flow is faster than evaporation, these holes will close; they will nucleate and open only when the solvent thickness is less than the “pancake” thickness. Several unique features arise which are associated with the nanometer-scale of our colloidal system. Because our liquid film achieves nanometer thicknesses, the capillary forces12 effective for micron-sized particles are negligible. Here, the decay length for these forces is dominated by its disjoining pressure contribution instead of gravity, and becomes on the order of angstroms rather than millimeters. The particle-solvent interactions are also not sufficiently strong to deform the liquid interface. However, because each particle prefers to be wet by the solvent, its (11) Schenning, A. P. H. J.; Benneker, F. B. G.; Geurts, H. P. M.; Liu, X. Y.; Nolte, R. J. M. J. Am. Chem. Soc. 1996, 118, 8549. (12) Kralchevsky, P. A.; Ivanov, I. B.; Nagayama, K. J. Colloid Interface Sci. 1992, 151, 79 and references therein.
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Langmuir, Vol. 14, No. 12, 1998 3421
tendency to be pulled along in the liquid, compared to being “left in the dry”, is proportional to its wetting energysor particle size. Another special feature of our system involves the dispersional attractions between particles as well as between particle and solvent. Because of the organically passivated surface, attractions between particles are on the order of kBT, sufficiently intermediate in magnitude to allow annealing into close-packed structures. Specifically, the dispersional energies are dominated by the metal core-core attractions. It is because these cores are kept apart by the alkylthiol molecules coating the particle surface that their attractions are on the order of kBT. (And, because of the alkane solvent, attractions between the alkyl chains on different particles make a negligible contribution to their overall interaction energy; for the opposite case, with no solvent, see Luedtke and Landman13). Without the alkylthiol coating, the metal cores could approach each other much more closely, resulting in attractions on the order of 10’s of kBT; in this instance the aggregating particles will not have a chance to anneal, and the resulting structures will be highly ramified (fractal)14 rather than close-packed. The attractions between the particle and the substrate9,15 give rise to frictional forces13,16 opposing lateral movement of the particles, which are size-dependent and play a role in the pinning of contact lines of opening holes. We have described two mechanisms for hole nucleation in volatile, wetting thin liquid films: volatile hole nucleation driven by evaporation and nonvolatile hole nucleation due to disjoining pressure effects. The importance of each mechanism can be varied by means of controlled evaporation rates (varying the vapor pressure) or by the use of different solvents and substrates. Hole formation and the presence of particles in solutions have been related to mesoscopic annular ring formation. Although both forms of hole nucleation may be simultaneously operative in producing rings, we have presented the two cases separately in order to qualitatively feature their respective properties. The effect of completing fluid flow and evaporation on the resulting annular ringlike configurations must now be addressed, however, and for these results we turn to numerical simulation. Simulation. The focus of this section is to incorporate the effects of fluid flow and evaporation into a single simulation, to illustrate the effects on hole nucleation rates and the resulting ringlike structures. These calculations highlight volatility “corrections” to the nonvolatile (slow evaporation) case of colloidal thin films and provide insight into the coupled system that cannot be obtained analytically, except in specific limits. Recalling the cases discussed in the previous section, we note that a critical thickness for hole nucleation driven by rim evaporation is determined by the vapor pressure of liquid compared to its equilibrium vapor pressure. Because we have a very volatile system (P , P0), hole nucleation can occur at a critical thickness much larger than the “pancake” film thickness te. Once this hole nucleates, it may grow if evaporation is fast enough compared to fluid flow. Alternatively, it may close if fluid flow, driven by the film wanting to spread, is dominant. If evaporation is sufficiently slow, holes will nucleate and
grow only for thicknesses less than te. In the previous example, we mentioned that relative rates are important in determining hole nucleation and growth. We will see the value of the simulation in demonstrating the effect of order of magnitude differences in rates; for example, how fast does fluid flow have to be compared to evaporation? A simulation also affords the luxury of testing the importance of additional mechanisms (not treated here) in ring formation, such as the effects of clustering due to particle-particle attractions,15,17 interactions between two or more growing holes, and both static and sliding frictional forces.16 In the spirit of a mean field approximation, we focus on the evolution of a single hole in a given area of substrate As. We assume that this area As is representative on average of adjacent patches on the substrate, where As is approximately determined by the average number of holes, nholes, on a given total substrate area Atotal: As ) Atotal/ nholes. The number of holes on the substrate as time evolves typically depends on the rate of hole nucleation on the substrate. Focusing on a single hole and making these assumptions is only meant to provide schematic results that illustrate the basic physics involved in a well-defined system. Initial Condition. We return to the same initial system as considered earlier, but now we simultaneously incorporate uniform evaporation, rim evaporation, and fluid flow into a single time-dependent simulation that can test the effect of time scales on hole nucleation and evolution. Before a hole is nucleated we have a uniform thin film of thickness t and area As, sitting under a total pressure Ptot ≈ 760 Torr, but with its solvent vapor pressure P many orders of magnitude smaller than its equilibrium value P0. We write the Helmholtz free energy of the system as (Vl ) Ast):
(13) Luedtke, W. D.; Landman, U. J. Phys. Chem. 1996, 100, 13323. (14) See, for example, Ball, R. C. et al., Phys. Rev. Lett. 1987, 58, 275, and Weitz, D.; Oliveria, M. Phys. Rev. Lett. 1984, 52, 1433. (15) See, for example, Hunter, R. J. Foundations of Colloid Science, Volume 1; Oxford University Press: New York, 1987. (16) See, for example, Krim, J. Langmuir 1996, 12, 4564 and references therein.
The change in free energy ∆A ) Afinal - Ainitial follows directly (assuming that Ptot , FlkBT, which is usually satisfied in systems that we consider):
Ainitial ) Nµl(Ptot) - PtotVl + As(γsl + γ + AH/t2)
(5)
where N ) AstFl is the number of liquid molecules present in the system, with chemical potential µl and at atmospheric pressure Patm ≈ 760 Torr. γ and γsl are as before the surface tension of the liquid and surface-liquid interfacial tensions, respectively, and AH is the effective Hamaker constant15,17 AH ) (Asl - All)/12π. Final Condition. Through a sequence of events (fluid flow, uniform evaporation, and rim evaporation), a hole nucleates. We now wish to determine the free energy of the new system, with a hole of radius R and film thickness t′. Given the number density of liquid hexane Fl, we can easily determine the number of solvent molecules Ng that have evaporated: Ng ) [Ast - (As - πR2)t′] Fl. Now, the total pressure of the system (Ptot) and vapor pressure of hexane (P) change slightly due to the evaporation of hexane. However, if our system is large enough, these pressure changes are effectively negligible. Assuming constant pressures, we can write the final Helmholtz free energy of the system as
Afinal ) (N - Ng)µl(Ptot) + Ngµg(P) - Ptot(As πR2)t′ - NgkBT + (As - πR2)(γsl + γ + AH/t′2) + πR2γsv + 2πRt′γ (6)
(17) See, for example, Safran, S. A. Statistical Thermodynamics of Surfaces, Interfaces, and Membranes; Addison-Wesley: New York, 1994.
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∆A ≈ NgkBT{log(P/P0) - 1} + As(AH/t′2 - AH/t2) + πR2(S - AH/t′2) + 2πRt′γ (7) It is important to note that the above expression reduces to the expected results when taking the appropriate limits. For example, by setting the rim radius R to be zero, the change in free energy reverts to its simple form for uniform evaporation, where the number of molecules that have evaporated is determined by the change in thickness of the film: Ng ) [As(t - t′)]Fl. By setting t ) t′ on the other hand, we eliminate uniform evaporation and (thickening) fluid flow from the system and obtain the change in free energy when rim evaporation is dominant (see eq 4) with Ng ) πR2tFl. Finally, we can make the system nonvolatile by setting the number of evaporated molecules to be zero (Ng ) 0), recovering eq 1 upon using t′ ≈ (1 + πR2/As)t. We now wish to understand the simultaneous effects of fluid flow and evaporation on hole evolution. In doing so, we first note that the time evolution ∆A/dτ of the free energy of the system can be expressed in terms of the rates at which the hole radius R and thickness t change with time, denoted dR/dτ and dt/dτ, respectively. dR/dτ and dt/dτ can be defined with respect to the allowed steps described earlier: rim evaporation, uniform evaporation, and fluid flow. All together, the time evolution of the hole radius R and film thickness t can be expressed as
dR/dτ ) dRnv/dτ + dRevap/dτ dt/dτ ) dtnv/dτ + dtevap/dτ
(8)
where the uniform evaporation rate is written as dtevap/ dτ, the rim evaporation rate as dRevap/dτ, and the nonvolatile fluid flow rate at the rim as dRnv/dτ. Due to conservation-of-volume constraints in the nonvolatile liquid, dRnv/dτ determines the corresponding rate of change in thickness dtnv/dτ (see paragraph below). We make the following assumptions about these rates: both uniform evaporation dtevap/dτ and rim evaporation dRevap/dτ are taken to be constant. The first assumption is based on the fact that the vapor pressure above the film does not change much during the time span over which a hole nucleates and grows. Because evaporation rates depend on the vapor pressure above the liquid, this simplification implies a steady uniform evaporation rate. The second assumptionsof a constant rim evaporation ratesneglects effects of local curvature on evaporation (which is accurate for rings that are large enough) and is also based on the notion of a constant vapor pressure. Finally, we need to formulate phenomenological expressions for the nonvolatile fluid flow rate. By differentiating with respect to time the constraint imposing conservation of volume, we obtain the following relation between dRnv/ dτ and dtnv/dτ: (As - πR2) dtnv/dτ ) 2πRt dRnv/dτ. We then focus on the rim velocity and assume that it is directly proportional to the total outward force exerted on the rim due to film-thickening and frictional effects, i.e., that dRnv/ dτ ) BFtotal. Here, B is a constant (playing the role of an inverse “friction coefficient”) that implicitly includes hydrodynamic factors, and
Ftotal ) Fthicken - (no. of particles)ffriction ) 2πRS[(te/t)2 - 1] - φR2ffriction/r2 (9) Here, we have used eq 2 for Fthicken and the fact that there are φR2/r2 particles in the rim, with ffriction the force per particle needed to overcome static friction due to attraction to the substrate. This assumption is in the spirit of recent
studies of force-velocity relations in dewetting films18 where the force is related to a power of the velocity. It is also based on basic physical considerations: when a rim is pinned, the total force vanishes and so does the velocity, while the larger the force on the rim the faster it advances. Incorporating the expressions for the total force on the rim of advancing holes, we obtain the following phenomenological form for the rim velocity:
dRnv/dτ ) BFtotal,(9) ) bR - cR2
(10)
b ) B2πS(te2/t2 - 1)
(10a)
with
which is associated with the thickening force along the circumference of the hole, and
c ) BRKfz,disp φ/r2
(10b)
where cR2 is the total frictional force (multiplied by B) on the rim exerted by fφR2/r2 particles. Here, each particle exerts a lateral frictional force Kfz,disp, due to downward dispersional attractions between particle and substrate;9 K is of order unity. R is the fraction of particles dragged along the rim, instead of being left behind in the dry, and is typically between 0.5 and 1.0. In performing the simulation, we assume that each step is independent of each other and can be treated separately. Although all steps may in fact be significantly active at any moment in time, each step is carried out assuming that the other steps are not locally active. The algorithm implemented is a Monte Carlo method designed to numerically integrate the equations of motion defined by (8)-(10b) and is similar to kinetic simulations that are typically used for epitaxial growth.19 Before starting, we must choose reasonable values for all of the physical quantities involved (As, P, P0, R, φ, γ, Fl, B, K, fz,disp, r, S, and AH). For the results presented here, As is chosen in the range 107-1012 Å2, P/P0 is set to be ≈10-4, R typically ranges from 0.5 to 1.0 depending on the particle sizeswhich later, r, is taken to be 3-6 nm, K ≈ 0.516, φ ≈ 0.1 or 0.5, and γ ≈ 0.05 kBT/Å2, and fz,disp can be calculated using standard formulas for dispersional attractions.9 The “pancake” thickness te is on the order of nanometers, consistent with a spreading coefficient S of 0.15 kBT/Å2 and an effective Hamaker constant AH of 1 kBT.15,17 B is fixed to be 105 Å2/kBT s to give reasonable fluid flow rates (rim velocities) of ≈1 cm/s. It should be emphasized that only the relative rates are important in the discussion which follows: attempts can be made in the future to specify the absolute rates and hence the actual physical time scales, but accurate experimental estimates for the above quantities should be obtained first. Changing B by a factor of 100, for example, would just require a shifting of choice in the evaporation rates by a factor of 100. Next, we must specify the step sizes Rinc and tinc allowed in each step. The step sizes are chosen to be small enough (fractions of an angstrom) to give an accurate numerical integration over time. For simplicity, we set Rinc ) tinc ) 0.001 Å. (1) We start a hole with radius R0 (R0 ) 0.001 Å) in a film of thickness t. (Note that if t > te, the hole may close.) (2) We calculate the total force on the rim Ftotal, and use this force to find dRnv/dτ ) BFtotal. (3) From (18) Joanny J. F.; Robbins, M. O. J. Chem. Phys. 1990, 92, 3206. (19) See, for example, Fichthorn, K. A.; Weinberg, W. H. J. Chem. Phys. 1991, 95, 1090 and references contained therein.
Interplay in Ultrathin Liquid Films
the given rates, we determine the frequencies (rates divided by the increments Rinc and tinc); based on these relative frequencies, one of the three events is chosen randomly. (4) The event is then carried out by its appropriate incremental change in R and/or t, and time τ. The number of gas molecules and the change in free energy are calculated (Ng ) [Ast - (As - πR2)t′]Fl and ∆A ) ∆Aeq7, respectively). Finally, we check to see if the rim is pinned (i.e., if Ftotal is no longer positive). If pinning has not yet occurred, the process is repeated from step 2. The above simulation is useful in demonstrating the effect of order-of-magnitude differences in evaporation and fluid flow rates on determining whether a single process or several processes are dominant, and their effect on ringlike structures (when holes are pinned). This is an interesting problem in hole nucleation theory, where a phenomenological expression for the nucleation rate is typically assumed.10 In a nonvolatile system, holes will not nucleate until the film thickness is small enough compared to te (typically 85-90%te), such that the barrier ∆Amax is on the order of kBT. Thus, hole nucleation occurs at distinct thicknesses and is effectively separated from hole growth. This should lead to fairly monodisperse ringlike structures, analogous to the classic mechanism where nucleation of particles is separated from particle growth,20 resulting in monodisperse particles. Experimentally, fairly monodisperse rings are typically observed. In volatile systems (with evaporation rates fast compared to fluid flow), however, hole nucleation occurs over a wide range of thicknesses (>te) due to the high equilibrium vapor pressure of solvent P0 . P. There nucleation is not separated from growth and is expected to lead to more polydisperse sizes of rings. The specific reason our experimental system probably lies in an intermediate regime is due to the time scales for opening and pinning a hole, compared with the time required for the solvent to completely evaporate. Holes will not be pinned before the solvent disappears if evaporation is too fast. Conversely, evaporation may take an unphysically long period of time if evaporation is too slow (a necessary condition in order for fluid flow to be dominant). In the intermediate regime it is difficult to obtain analytical expressions for hole nucleation barriers (and hence rates) because of the coupled nature of the deterministic rate equations. A numerical integration, such as via the Monte Carlo simulation presented here, is necessary. Preliminary estimates for evaporation rates21 and fluid flow rates22 can be made for bulk liquids and are approximately of the same order of magnitude. However, these would necessarily be overestimates for the thin liquid film present here.23 More accurate estimates for these quantities appropriate to our system must be made. In the next section, we will discuss briefly some general results from these simulations, where the uniform evapo(20) Reiss, H. J. Chem. Phys. 1951, 19, 482. (21) Using the Langmuir equation for the net evaporation rate from a volatile bulk liquid film (see discussion, say, in Xia, T. K.; Landman, U. J. Chem. Phys. 1994, 101, 2498) leads to an estimate of about 1 ns for the time required for a R ) 1 nm hole to open. However, one must be careful about using bulk viscosities and evaporation rates from films as thin as those in our system. (22) A simple dimensional analysis of the thin-film Navier-Stokes equation, balancing the viscous force density by the pressure gradient associated with Laplace curvature, leads to the estimate U ) (γ/η)(t/ R)3, where η is the fluid viscosity. Equating this to the ratio of the hole size R divided by the time τ for filling this hole, and taking η ) 0.1 poise and t ) R ) 1 nm, we find τ on the order of a nanosecond, and hence comparable to the evaporation time estimated in ref 19. (23) Elbaum and Lipson,7 for example, conclude that evaporation rates in their thin films are up to 100 times smaller than the corresponding bulk values.
Langmuir, Vol. 14, No. 12, 1998 3423
Figure 3. Top-view schematic of situation which arises when the rims of holes are not pinned. Here, four holes “percolate” to form a “drop” at their interstices (top). Each drop continues to evaporate, pushing particles together (middle), resulting in close-packed, size-segregated structures (bottom) like those described in earlier work.24
ration rate is set equal to the rim evaporation rate. Even allowing for faster rim evaporation, these quantities should still not differ by much. The relevant consideration is not so much how they compare to each other but rather to the rate of (nonvolatile) fluid flow. The two limiting cases are again considered: the nonvolatile limit and the fast limit. III. Results and Discussion We have previously made estimates for the evaporation rate21 and fluid flow rates.22 Both quantities are on the order of 107-108 Å/s for bulk liquids, but should decrease significantly due to the viscosity of the thin films that we treat here. In the absence of better estimates for these quantities, we take 108 Å/s as a strong upper limit for both evaporation and fluid flow rates. In the nonvolatile limit, where the evaporation rate is negligible with respect to fluid flow, the results are straightforward. If small holes are artificially opened at
3424 Langmuir, Vol. 14, No. 12, 1998
various initial thicknesses, the resulting pinned radii are strongly dependent on thickness. For sufficiently large systems (As . πRpin2), where the film thickness remains relatively unchanged as a hole opens up and gets pinned, these ring sizes are given approximately by Rpin ) 2πr2S[(t/ te)2 - 1]/ffrictionφ. However, we must also consider the rate at which these holes are nucleated. This rate of hole nucleation is strongly dependent on film thickness because the barrier to hole nucleation can be formidably high until the thickness drops around 10-15% below te. This nucleation rate increases by orders of magnitude with decreasing thickness, so that eventually it will be fast enough to be sufficiently active in the system. In the limit where evaporation is dominant, typically where the evaporation rate is greater than 104 Å/s, the ring sizes can be as much as an order of magnitude larger. For example, in the particular case for which tinital ) 0.9te, Rpin increases by more than a factor of 2 (from 0.2 to 0.4 µm) upon increasing the evaporation rates from 102 to 103 Å/s. Furthermore, when evaporation is still more dominant, the solvent may completely disappear from the system before the hole rim can be pinned. The cause for larger ring sizes lies in the t-dependence of the thickening force given in eq 2: Fthicken ) 2πRS[(te/t)2 - 1]. This thickening force is proportional to the rim velocity dRnv/ dτ and is faster in the volatile case because of the decreasing film thickness (compared to te) due to evaporation. In this system there is also a unique interplay between particles and solvent. The film thickness must reach nanometer scales so that the film thickness can drop below the particle size. Because the particles protrude from the solvent film, these particles can feel the frictional forces thatsdue to attraction to the substratescause pinning. Furthermore, the film thickness must drop below the “pancake” thickness te in order for there to be radially outward hole-rim forces. Finally, we note that if the evaporation is sufficiently fast and/or the particle concentration is low enough, the growing holes will not be pinned until they have run into one another. In this case the ordered arrays will comprise compact domains rather than annular ringsssee schematic picture in Figure 3sas the particles collect in the interstices of the percolated
Ohara and Gelbart
holes. Indeed, this corresponds to the situation studied earlier for the same system but with smaller particles.24 In summary, we have presented a theoretical model for hole nucleation in volatile, wetting thin liquid films. This theory incorporates two distinctly different driving forcessevaporation and equilibrium forces (disjoining pressure effects) into a single Monte Carlo simulation that interpolates between limiting cases where uniform evaporation, rim evaporation, or fluid flow are dominant. The limits are tested against analytical results for hole nucleation barriers and corresponding critical radii. The effects of time scales on ring formation are also investigated, with the following result: faster evaporation (both uniform and at the hole rims) results in larger rings. These simulations constitute a zero-order attempt for predicting experimental results under controlled evaporation situations, using different solvents and substrates, or a wide range of other possibilities. Eventually, a more detailed treatment may be developed which incorporates the effects of particle clustering, hydrodynamics (including Marangoni flow), nonuniform film thickness and evaporation, and particle-solvent interactions. Although our model is preliminary and phenomenological, we argue that it captures basic physics of the system. The effects of particle size and concentration have been addressed and are currently being pursued experimentally. Acknowledgment. We are pleased to thank Jim Heath for his experimental discovery and pursuit of the ring structures discussed here and for the many very helpful conversations about their properties. Sung-Wook Chung is gratefully acknowledged for providing the image shown in Figure 1, from his transmission electron microscopy studies of these metal nanoparticle systems. This work was supported by the National Science Foundation (Grant No. CHE-95-20808), the American Chemical Society (Grant No. PRF31797-AC9), and a UCLA Dissertation Year Fellowship to P.C.O. LA971147F (24) Ohara, P. C.; Leff, D. V.; Heath, J. R.; Gelbart, W. M. Phys. Rev. Lett. 1995, 75, 3466.
