9234
Langmuir 2006, 22, 9234-9237
Density Variation-Induced Sign Change of the Effective Hamaker Constant in Continuum Theory Håkon Kaya MicrograVity Research Centre, SerVice Chimie Physique EP - CP 165/62, UniVersite´ Libre Bruxelles, AVenue F.D. RooseVelt 50, 1050 Brussels, Belgium ReceiVed May 15, 2006. In Final Form: August 16, 2006 The density variation-induced dewetting model is reformulated in terms of the Dzyaloshinskii-Lifshitz-Pitaevskii continuum theory for van der Waals interactions. Though the density dependence is more complicated than in the London-Hamaker microscopic theory, numerical calculations for model polystyrene films show that the two theories yield qualitatively similar results. The physics of the dewetting model is thus not changed. Quantitatively, the differences between the two theories are notable. The calculations also show that a linear approximation suffices to take density variations into account, thus simplifying the density variation-induced dewetting models.
1. Introduction Liquid films deposited on solid substrates are classified as ultrathin when they have thicknesses below 100 nm. A characteristic feature of ultrathin films is the importance of longrange van der Waals (vdW) interactions between molecules in the liquid film and in the surrounding phases, in particular, the solid substrate, and between molecules in the liquid film itself. All intermolecular interactions add up to a net interaction which in the Derjaguin approximation is modeled as that of two planar surfaces separated by a distance equal to the film thickness h.1,2 Treating the vdW forces as nonretarded,1,3 the net interaction per unit area reads
e)-
ASFG 12πh2
(1)
where ASFG is an effective Hamaker constant, giving the magnitude and direction of the interactions between a semi-infinite solid (S) interacting with a gas phase (G) through a film (F).1 ASFG also gives the magnitude of the net interactions acting on the film due to the proximity of the neighboring phases. It is evident from eq 1 that a negative A favors thick films and thus promotes film stability. Conversely, a positive A favors small h, leading the film to thin and rupture. The vdW interactions can also be conceived as a pressure acting on the film surface. This is the conjoining/disjoining pressure4 defined by
Φ)
∂e ASFG ) ∂h 6πh3
(2)
The stability, evolution, and morphology of the film are profoundly affected by Φ.5 When the vdW interactions (1) render the film unstable, Φ is balanced with the capillary pressure created by surface tension forces at the F-G interface to create a pattern of droplets or holes.6 (1) Hunter, R. J. Foundations of Colloid Science; Oxford University Press: New York, 1992; Vol. 1, Chapter 4. (2) Israelchvili, J. N. Intermolecular and Surface Forces, 2nd ed.; World Scientific: New York, 1991. (3) Bergstro¨m, L. AdV. Colloid Interface Sci. 1997, 70, 125. (4) Derjaguin, B. V. Kolloid-Z. 1934, 69, 155. (5) Brochard-Wyart, F.; Daillant, J. Can. J. Phys. 1990, 68, 1084.
The Hamaker constant is usually treated as a material property. Recently, however, Wensink and Je´roˆme showed that density variations in the liquid film itself would induce additional interactions that could possibly change the sign of ASFG.7 Thus, a film expected to be stable could rupture, as was indeed experimentally observed earlier by Demirel and Je´roˆme in ultrathin films of a glass-forming liquid crystal.8 These films were prepared by spin coating solutions of the liquid crystal onto glass substrates, subsequently removing the excess solvent by annealing the films for several hours. The thinnest and the thickest of the prepared films were found to be flat and stable, as expected from the wetting properties of the materials. Films of intermediate thickness, however, broke up into a pattern of regular droplets, similar to those found in spinodal dewetting. After heating, the film morphology changed into a flat film that remained stable. Thus, a transient mechanism seemed to have caused the film to become unstable. With support in experiments, molecular restructuring and density variations were proposed, leading to the theory of density variation-induced dewetting (DVID). The theory of the DVID mechanism was extended and elaborated by Sharma et al., who pointed out that films that were expected to rupture could under the right conditions be stabilized by density variations.9,10 It is in complex fluids such as polymers and liquid crystals that the DVID is expected to play a significant role. Complex fluids have found widespread use as model systems for studies of dewetting phenomena because of their slow dynamics, which permits time-resolved studies. Because of their size and internal degrees of freedom as represented by polymer chain configurations and orientations of liquid crystal molecules, they can exhibit strong deviations from bulk behavior when subjected to confinement in thin-film geometry. Confinement also enhances the effect of the density variations. It restricts free movement, retarding the restructuring and relaxation to equilibrium configurations after deposition. Rapid deposition techniques such as spin coating give complex molecules little time to attain equilibrium conformations, effectively producing films in a nonequilibrium state. When the restructuring to equilibrium time is longer than (6) Donald, A. M., Joanny, J.-F., Mo¨ller, M., Reiter, G., Eds. Focus Point: Unstable Thin Films. Eur. Phys. J. E 2003, 12. (7) Wensink, K. D. F.; Je´roˆme, B. Langmuir 2002, 18, 413. (8) Demirel, A. L.; Je´roˆme, B. Europhys. Lett. 1999, 45, 58. (9) Sharma, A.; Mittal, J. Phys. ReV. Lett. 2002, 89, 186101. (10) Sharma, A.; Mittal, J.; Verma, R. Langmuir 2002, 18, 10213.
10.1021/la061368y CCC: $33.50 © 2006 American Chemical Society Published on Web 09/30/2006
Sign Change of the EffectiVe Hamaker Constant
Langmuir, Vol. 22, No. 22, 2006 9235
the viscous flow time of the overall fluid, additional effects have time to kick in. The study of thin-film rupture has attracted the attention of many research groups working on complex fluid films.6 The DVID has been recognized11 as a particularly interesting model for improving our understanding of rupture and ensuing structure formations observed in complex fluid thin films. In the case of nematic molecules such as those studied in the experiment mentioned above,8 one could also consider the possibility that it is the director orientation that is restricted by confinement, leading to a change in the film-substrate interactions and, consequently, the film stability. This article will exclusively address the physical effects of density variations in a thin film, but it should be mentioned that both density and director orientation can be considered to be order parameters and that the effects of variations and fluctuations in these parameters on the electrostatic interactions across a thin film can generally be treated as pseudo-Casimir effects.13 In the previous publications addressing the DVID,7,9,10,12 the disjoining pressure and the Hamaker constant were given by
Φ)
ASFG 6πh
3
where ASFG ) CLLFF2 - CSLFSFF
(3)
and CLL and CSL are related to the molecular polarizability of the molecules.2 The dipole number densities FF and FS are those of the film and the solid substrate, respectively. The dependence of the Hamaker constant on density as defined in eq 3 is derived from London-Hamaker (LH) theory.2 The net interaction acting on the film is calculated by a pairwise summation over all interacting molecules. The LH treatment produces simple relations (eq 3) between ASFG and the film density FF.1,2 The simple form of eq 3 is a special case of
ASFG ≈
3kBT
∑′[FSRS(ω) - FFRF(ω)][FFRF(ω) - FGRG(ω)] 2 n)0
80
(4)
where the prime indicates that the n ) 0 term is given half weight. FS, FF, and FG are the number densities of dipoles in the respective phases, and RS, RF, and RG are the corresponding molecular polarizabilities.1 Equation 4 is a finite-temperature extension of the original treatment of London, who treated the molecules as fixed and the interactions as pairwise, ignoring many-body effects. This is a valid approximation for a dilute gas at low temperatures but unphysical for the complex-fluid liquid films in question. van der Waals forces result from electrostatic interactions between permanent or induced dipoles. LH theory treats these dipoles as separate particles at fixed positions, thus being a discrete or microscopic approach. Dzyaloshinskii-Lifshitz-Pitaevskii (DLP) theory treats the system as a dielectric continuum at finite temperature.1 The dipoles create an electromagnetic force field interacting with the continuous dielectric medium. DLP theory is thus a continuum or macroscopic theory. The aim of this article is to calculate the effect of density variations on the Hamaker constants using DLP theory, which is superior to the LH approach in obtaining quantitative agreement with experiments.1 DLP theory takes into account many-body effects such as the screening of the intermolecular interactions when calculating the Hamaker constants. Equation 3 is comparable with Lifshitz theory only in the case of a dilute gas, for which the dielectric permittivity (11) Oron, A. Eur. Phys. J. E 2003, 12, 455. (12) Kaya, H.; Je´roˆme, B.; Colinet, P. Europhys. Lett. 2006, 74, 861. (13) Ziherl, P.; Zumer, S. Eur. Phys. J. E 2003, 12, 361.
is close to that of free space, so that ≈ 0. Because ultrathin films are considered, interactions are treated as nonretarded,1 thus simplifying the calculations of ASFG and the comparison between the LH and DLP approaches. However, retardation effects are already perceptible in a thin film of 5 nm thickness,2 and the importance of retardation effects has been recognized and accounted for in recent papers.14,15
2. Theory We consider a system consisting of a solid substrate and vacuum (air) separated by a thin liquid film. As in refs 7, 9, 10, and 12, the film is considered to be a single-component liquid, where the density variations are caused by variations in the molecular packing density only. In more complicated cases, one could consider the presence of nanobubbles,16 retained solvent,17 or a second component, as would be the case in, for example, block copolymer films. In DLP theory, ASFG must be calculated numerically from available data on the dielectric response function of the materials in question.1 To this end, we make use of the series expression for the Hamaker constant derived by Hough and White:18
ASLG )
3kBT 2
∑′ ∑
n)0 s)1
(∆SL∆LG)s (5)
s3
The dielectric response difference is given by
∆kj )
ˆ k(iξn) - ˆ j(iξn)
(6)
ˆ k(iξn) + ˆ j(iξn)
where the sampled frequencies are ξn ) (4π2kBT/h)n. The circumflex denotes the dielectric response relative to that of vacuum so that ˆ ) /0. For the dielectric response functions ˆ (iξ), the Ninham-Parsegian representation is used3,19 in which ˆ (iξ) is modeled as a set of Lorentzian oscillators distributed over the electromagnetic spectrum. For organic materials, the response in the UV-vis range of the electromagnetic spectrum dominates, determining the vdW interactions.3 Thus, it suffices to represent ˆ (iξ) by one UV and one IR relaxation
{
ξˆ ) 0
ˆ (0)
ˆ (iξ) ) 1 +
CIR 1 + (ξ/ωIR)
2
+
CUV 1 + (ξ/ωUV)
2
ξˆ > 0
(7)
where CUV and CIR are the absorption strengths of the UV and IR oscillation peaks and ωUV and ωIR are the corresponding peak absorption frequencies, respectively. The values of the physical parameters in eq 7 for a large number of materials can be found in the literature (refs 1 and 3 and references therein). The density variations are taken into account by the ClausiusMossotti (CM) relation,1,20 which is a dielectric constitutive relation connecting ˆ to the number density of the constituent dipoles and their respective polarizabilities. It reads (14) Zhao, H.; Tsui, O. K. C.; Liu, Z. Solid State Commun. 2005, 134, 455. (15) Kargupta, K,; Sharma, A.; Khanna, R. Langmuir 2004, 20 244. (16) Jacobs, K.; Herminghaus, S.; Mecke, K. R. Langmuir 1998, 14, 965. (17) Garcia Turiel, J. Polymer Thin Films: Transport Phenomena and Surface Forces. Ph.D. Thesis, University of Amsterdam, 2005. (18) Hough, D. B.; White, L. R. AdV. Colloid Interface Sci. 1980, 14, 3. (19) Parsegian, V. A.; Ninham, B. W. Nature 1969, 224, 1197. (20) Kayser, R. F. Phys. ReV. B 1986, 34, 3254.
9236 Langmuir, Vol. 22, No. 22, 2006
ˆ (ω) - 1 ˆ (ω) + 2
)
Kaya
1
∑i FiRi(ω)
3
Table 1. Dielectric Parameters for the Materials Discusseda
(8)
where Ri is the molecular polarizability of species i. Another form of the CM relation links the average dielectric response function to the volume fractions of the constituent species21 so that
ˆ (ω) - 1 ˆ (ω) + 2
)
material
ˆ (0)1
CUV1
ωUV [1016 rad/s]
CIR1
ωIR [1014 rad/s]
polystyrene fused quartz ordinary quartz silicon
2.60 3.80 4.29 11.66
1.424 1.098 1.359 10.66
1.432 2.033 2.032 0.608
0.20 1.70 1.93 N/A
5.540 1.880 2.093 N/A
a
The data are collected from refs 1, 3, and 24.
ˆ i(ω) - 1
∑i φiˆ (ω) + 2
(9)
i
where φi is the volume fraction of component i. In the case that the density variations are caused by changes in the molecular packing density (in which case one can have φ > 1), one has (ˆ (ω) - 1)/(ˆ (ω) + 2) ) φ(ˆ L(ω) - 1)/(ˆ L(ω) + 2), where ˆ L is the dielectric response function of the liquid constituting the film. In this one-component case, φ represents a local compression factor rather than a local volume fraction.
3. Results and Discussion From eqs 5-9, it is evident that the density dependence has a more complicated form than in the simple eq 3. The latter can be derived from DLP theory in the limit of dilute gases for which ˆ f 1. In this limit, eq 8 can be approximated by ˆ - 1 ) FR, whereas eq 9 reduces to ˆ F ) 1 + φ(ˆ L - 1). Inserting these approximations into eq 4, one obtains
ASFG ≈
3kBT [φ2 ′ ˜ L2 - φ ′ ˜ S˜ L] 8 n)0 n)0
∑
∑
Figure 1. Hamaker constants for polystyrene on various substrates, plotted as functions of the density parameter φ. The curves are calculated with DLP and LH theories.
(10)
which is equivalent to eq 4. For the sake of compact notation, a reduced relative dielectric response function ˜ ) ˆ - 1 has been introduced. To compare the two approaches (eqs 5 and 10), the Hamaker constant as a function of density is calculated numerically for polystyrene (PS) films, which have been extensively studied for their dewetting properties.6,22,23 Fused quartz, ordinary quartz, and silicon were considered to be substrates. The relevant dielectric parameters used in eq 7 are listed in Table 1. Calculations with amorphous silica as the substrate were also carried out but gave results very close to those of the system PS/fused quartz. For both eqs 5 and 10, the summation over n is evaluated up to up to n ) 5000, whereas the summation over s in eq 5 is performed up to s ) 100. The Ninham-Parsegian representation (eq 7) for (iξ) is used in both cases. The plots in Figure 1 show the result of the numerical calculations. The effect of the screened interactions yields a notable quantitative difference between the curves calculated in LH theory and those calculated from DLP theory, with the latter having a smaller magnitude in all of the considered systems. All of the curves change sign with an increasing slope. With the exception of PS/silicon, all curves also have a positive slope at the nominal density φ ) 1. Qualitatively, however, the curves in Figure 1 exhibit similar shapes for both DLP and LH theories. This suggests that the conclusions of the papers7,9,10,12 remain valid. Because of this similarity in shape, we attempted to fit the function aφ2 + bφ to the curves calculated from DLP theory. (21) Meli, L.; Pham, Q. J.; Johnston, K. P.; Green, P. F. Phys. ReV. E 2004, 69, 051601. (22) Seemann, R.; Herminghaus, S.; Jacobs, K. Phys. ReV. Lett. 2001, 86, 5534. (23) Xie, R.; Karim, A.; Douglas, J. F.; Han, C. C.; Weiss, R. A. Phys. ReV. Lett. 1998, 81, 1251.
Figure 2. Fits of the function aφ2 + bφ that is valid for the LH theory given by eqs 3 and 10 to the curves for PS/fused quartz and PS/ordinary quartz, both calculated in DLP theory.
The purpose was to check whether eq 10, which is derived from the simpler LH theory, could also describe the DLP curves. The fits, shown in Figure 2, agree quite well, again confirming that the findings in refs 7, 9, 10, and 12 are physically sound. Though the LH and DLP theories agree well qualitatively, the quantitative outcomes of the theories are appreciably different. For the system of PS deposited on fused quartz, DLP theory gives ASFG ) 3.57kBT (14.4 zJ) and a sign change at φ ) 0.98. Thus, the system is thermodynamically unstable but very sensitive to density variations, which would contribute to stabilizing the system. LH theory gives ASFG ) 2.70kBT (10.9 zJ) and a sign change at φ ) 0.92. The stability is the same in both theories, but the sensitivity to density variations turns out to be much greater in DLP theory. For PS on ordinary quartz, DLP theory produces ASFG ) -2.46kBT (-9.96 zJ) and a sign change at φ ) 1.15. LH theory gives ASFG ) -5.35kBT (-21.7 zJ) and a sign change at φ ) 1.15. The system PS/quartz is thermodynamically stable in both cases,
Sign Change of the EffectiVe Hamaker Constant
Langmuir, Vol. 22, No. 22, 2006 9237
Figure 3. Magnification of the curves displayed in Figure 1 around φ ) 1.
becoming unstable only when the density is significantly increased (by 15% in both cases). The calculations for the system PS/silicon showed that the system is strongly stable, with a nominal Hamaker constant of ASFG ) -17.15kBT (-69.44 zJ) as calculated from DLP theory. The LH calculations gave ASFG ) -124.65kBT (-504.68 zJ) (not shown in Figure 1). Clearly, the oxide layer on the Si layers have played a significant role in the destabilization observed in refs 22 and 23. The system PS/silicon will not be considered further. In Figure 3b, ASFG(φ) has been plotted in the vicinity of nominal density φ ) 1, around which most of the density variations are expected to take place.7-10 All of the curves exhibit a linear trend in this region, suggesting that the effect of the density variations on the Hamaker constant can be approximated by a linear expansion around φ ) 1:
∂ASFG | ∂φ φ)1
ASFG ≈ ASFG(φ ) 1) + (φ - 1)
(11)
Normalizing with the nominal value of ASFG, one obtains the dimensionless form
∂φASFG|φ)1 A ˜ SFG ) 1 + (φ - 1) ASFG(φ)1)
(12)
which can be compared with that used in refs 7, 9, 10, and 12
A ˜ SFG ) φ2 -
ASL φ ALL
(13)
where ALL ) CLLFL2(φ ) 1) and ASL ) CSLFSFL(φ ) 1) are the liquid-liquid and solid-liquid Hamaker constants, respectively, evaluated at the nominal density. Equation 12 is linear in φ and therefore provides a simpler model for the effect of density variations on ASFG than eq 13, which is quadratic in φ. It should be stressed that the numerical calculations have shown that the model (eq 13) still gives valid results. In fact, by linearizing eq 13 around φ ) 1 and normalizing, one obtains
A ˜ SFG ) 1 +
2ALL - ASL (φ - 1) ALL - ASL
(14)
Comparing with the definition of ASFG in eq 3, one sees that eq 12 becomes eq 14 when the density dependence of ASFG is given by LH theory. Hence, eq 12 also represents a more general
representation the DVID mechanism. The dimensionless parameter describing the density variations is in either case given by the ratio ∂φASFG/ASFG, which in the case of the LH approach takes the form in eq 14 that is known from the preceding works on DVID. The hydrodynamic modeling of thin films has not been a topic of this article, but given the important role that ASFG plays in this context, some discussion of this point is warranted. In the case of ultrathin films of complex fluids, one could argue that we are no longer in the continuum limit when the film thickness is comparable with molecular size. The application of continuum models derived from hydrodynamics to ultrathin films is frequently, almost routinely, questioned. Still, mesoscopic models show excellent agreement with experiments.25 Wensink and Je´roˆme focused on the effect of the sign change of ASFG on the film stability,7 but as pointed out by Sharma et al., the density dependence of ASFG creates additional terms in the spinodal parameter ∂hΦ that have even stronger effects on the stability and the appearance of the dewetting pattern.9,10 By treating φ as a function of the film thickness h only, it was shown that the characteristic distance between droplets forming on an unstable film scaled as hn, where n was between 1.5 and 2. This phenomenon has been observed experimentally22,23 in PS films and was attributed to the presence of an additional term ∂hA/6πh3 in the spinodal parameter by Sharma et al.9,10
4. Conclusions The density variation-induced dewetting scenario proposed in refs 7, 9, and 10 has been revisited and reformulated in terms of the macroscopic continuum theory for van der Waals interactions, where it was previously treated in the framework of microscopic theory. Though the continuum theory has a more complicated density dependence through eqs 5-9, numerical calculations showed that the theories yielded qualitatively the same dependence on density for the Hamaker constants. Quantitatively, however, appreciable differences were found. For the important and intensively studied polystyrene films on fused quartz or amorphous silica, the calculations in the continuum theory showed a notably higher sensitivity to density variations than in the corresponding microscopic theory. Because the continuum theory is held to be quantitatively more exact, this is of importance when considering the effect of density variations in thin-film experiments. The numerical calculations for the considered one-component films also showed that the density dependence of the Hamaker constant can be sufficiently described by a linear approximation around the nominal density. This will simplify the models for the density variation-induced dewetting mechanism and also presents the notation in a more general and physically more exact form. Instead of the liquid-liquid and solid-liquid Hamaker constants ALL and ASL, respectively, which are concepts originating from the London-Hamaker approach treating all interactions as pairwise, one should discuss the dewetting mechanism in terms of the nominal Hamaker constant ASFG(φ ) 1) and its derivative with respect to the density, ∂φASF. Acknowledgment. This work was supported by the ARCHIMEDES project of the Communaute´ Franc¸ aise de Belgique (Actions de Recherche Concertes). LA061368Y (24) Sabisky, E. S.; Anderson, C. H. Phys. ReV. A 1973, 7, 790. (25) Becker, J.; Gru¨n, G.; Seemann R.; Mantz H.; Jacobs, K.; Mecke, K. R.; Blossey, R. Nat. Mater. 2003, 2, 59.
