Langmuir 2002, 18, 413-416
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Dewetting Induced by Density Fluctuations K. D. F. Wensink and B. Je´roˆme* Department of Chemical Engineering, University of Amsterdam, Nieuwe Achtergracht 166, 1018 WV Amsterdam, The Netherlands Received October 4, 2001. In Final Form: November 8, 2001 We consider the stability with respect to dewetting of thin films deposited on a substrate they thermodynamically wet. We show that if density fluctuations with a sufficiently large amplitude appear in a film, the resulting gradient of disjoining pressure in the film destibilizes the free surface eventually leading to the breaking up of the film in droplets. This dewetting mechanism is expected to play a role in films undergoing significant restructuring after deposition or for materials close to a critical point.
Introduction Nowadays much research is done on the stability of thin films on substrates, not only because of their numerous technological applications, including coatings, adhesives, lubricants, and dielectric layers but also because of their fundamental interest.1-3 The dewetting of thin liquid films is the process of destabilization of such films leading to the formation of droplets. This dewetting process is generally observed when a liquid has been forced to form a film of uniform thickness on a substrate that it thermodynamically does not wet, i.e., that the energy of the uniform film is larger than the energy of the substrate/ vapor interface. For a homogeneous isotropic liquid on a uniform solid substrate, two main dewetting processes are known: (i) the nucleation of holes at defects or dust particles, and (ii) the amplification of fluctuations of the free surface (e.g., capillary waves) under the destabilizing effect of long-range forcessso-called spinodal dewetting.4-6 These forces comprise in any case van der Waals forces and may include structural forces when these are long range (as in liquid crystals7-9). Although the distinction between the two mentioned dewetting processes is in theory clear, there is still a lot of debate about which of these mechanisms is actually observed experimentally.10-19 Besides these two general * Corresponding author. (1) Frank, C. W.; et al. Science 1996, 912, 912. (2) Le´ger, L.; Joanny, J. F. Rep. Prog. Phys. 1992, 55, 431. (3) de Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827. (4) Vrij, A. Discuss. Faraday Soc. 1966, 42, 23. (5) Scheludko, A. Adv. Colloid Interface Sci. Proc. Int. Conf. 1977, 1, 391. (6) Brochart-Wyart, F.; Daillant, J. Can. J. Phys. 1990, 68, 1084. (7) Vandenbrouck, F.; Valignat, M. P.; Cazabat, A. M. Phys. Rev. Lett. 1999, 82, 2693. (8) Braun, F. N.; Yokoyama, H. Phys. Rev. E 2000, 62, 2974. (9) Ziherl, P.; Podgornik, R.; Zumer, S. Phys. Rev. Lett. 2000, 84, 1228. (10) Reiter, G. Phys. Rev. Lett. 1992, 68, 75. (11) Reiter, G.; Sharma, A.; Casoli, A.; David, M. O.; Khanna, R.; Auroy, P. Langmuir 1999, 15, 2551. (12) Thiele, U.; Velarde, M. G.; Neuffer, K. Phys. Rev. Lett. 2001, 87, 016104. (13) Kim, H. I.; Mate, C. M.; Hannibal, K. A.; Perry, S. C. Phys. Rev. Lett. 1999, 82, 3496. (14) Xie, R.; Karim, A.; Douglas, J. F.; Han, C. C.; Weis, R. A. Phys. Rev. Lett. 1998, 81, 1251. (15) Bischof, J.; Scherer, D.; Herminghaus, S.; Leiderer, P. Phys. Rev. Lett. 1996, 77, 1536. (16) Seemann, R.; Herminghaus, S.; Jakobs, K. Phys. Rev. Lett. 2001, 86, 5534. (17) Herminghaus, S.; Jacobs, K.; Mecke, K.; Bischof, J.; Fery, A.; Ibn-Elhaj, M.; Schlagowski, S.; Seemann, R.; Gau, H.; Monch, W.; Pompe, T. J. Phys. Condens. Mater 1999, 11, A47.
mechanisms, other destabilization mechanisms more specific to certain systems have been reported: the confinement of polymer chains20,21 and Marangoni effects in phase-separated systems.22,23 These mechanisms can also be in competition with other processes modifying the dewetting dynamics, such as evaporation24,25 or demixing.23,26,27 Recently the observation of dewetting in ultrathin films of glassy material thermodynamically wetting the substrate28 suggested that a completely different dewetting mechanism is possible. The observed dewetting took place during the first heating of the films after deposition by spin coating producing a glassy uniform film. When the films were heated, the mobility of the molecules increased allowing for restructuring in the film and, in particular, at the interface with the substrate. Simultaneously, fluctuations with a characteristic length scale appeared at the free surface. For thin films with a thickness below 10 nm, these fluctuations lead to the dewetting of the films and the formation of droplets. For thicker films, the fluctuations flattened again as restructuring ceased and the films remained stable at any temperature, even well above the glass transition temperature of the material. The fact that films as thin as 10 nm remain stable even at high temperature shows that they are not just metastable because of the slowing down of fluctuation growth due to spinodal dewetting but that they are rather thermodynamically stable with long-range van der Waals forces favoring wetting. The fact that the appearance of fluctuations coincides with the restructuring of the film suggests that the destabilization of the film is driven by this restructuring. This restructuring has been shown to (18) Jacobs, K.; Herminghaus, S.; Mecke, K. R. Langmuir 1998, 14, 965. (19) Stange, T. G.; Evans, D. F.; Hendrickson, W. A. Langmuir 1997, 13, 4459. (20) Zhao, W.; Rafailovich, M. H.; Sokolov, J.; Fetters, L. J.; Plano, R.; Sanyal, M. K.; Sinha, S. K.; Sauer, B. B. Phys. Rev. Lett. 1993, 70, 1453. (21) Mu¨ller-Buschbaum, P.; Gutmann, J. S.; Stamm, M. Phys. Chem. Chem. Phys. 1999, 1, 3857. (22) Herminghaus, S.; Fery, A.; Schlagowski, S. Science 1998, 282, 916. (23) Yerushalmi-Rozen, R.; Kerle, T.; Klein, J. Science 1999, 285, 1254. (24) Thiele, U.; Mertig, M.; Pompe, W. Phys. Rev. Lett. 1998, 80, 2869. (25) Kargupta, K.; Konnur, R.; Sharma, A. Langmuir 2001, 17, 1294. (26) Mu¨ller-Buschbaum, P.; O’Neill, S. A.; Affrossman, S.; Stamm, M. Macromolecules 1999, 31, 5003. (27) Mu¨ller-Buschbaum, P.; Gutmann, J. S.; Cubitt, R.; Stamm, M. Colloid Polym. Sci. 1999, 277, 1193. (28) Demirel, A. L.; Je´roˆme, B. Europhys. Lett. 1999, 45, 58.
10.1021/la015611z CCC: $22.00 © 2002 American Chemical Society Published on Web 12/27/2001
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A, corresponding to a wetting film. ASL and ALL are related to the density of the solid FS and of the liquid FL by29
Figure 1. Schematic representation of a film with a fluctuating free surface.
involve a densification of the interfacial layer at the interface with the substrate.28 This requires the motion of molecules toward the interface, which creates regions of reduced density in the film. These density fluctuations imply fluctuations of the van der Waals interaction potential between the two surfaces of the film via the Hamaker constant,29 which depends on the film density. Therefore a gradient of disjoining pressure will appear in the film leading to a modulation of film thickness. In this article, we show that density fluctuations can indeed lead to the destabilization of a thermodynamically stable film. For this we generalize in the next section the theoretical description of spinodal dewetting, corresponding to the case of destabilizing van der Waals forces,6 to the case of stabilizing van der Waals forces with a fluctuating Hamaker constant. The calculations show that this case also leads to the destabilization of the fluctuations of the free surface. Subsequently we discuss under which conditions this destabilization takes place and which experimental systems are likely to present this type of dewetting mechanism. Theory Following the notations and the general scheme of the theoretical description of spinodal dewetting,6 we consider a film of average thickness e (in the z direction) on an infinite rigid substrate (Figure 1). We allow the thickness of the film to fluctuate along the surface (in the x direction) by writing it in the form
ζ ) e + uqeiqxe-t/τ
(1)
where q is the wave vector of the fluctuations and τ their decay time. These modulations give rise to fluctuations of the pressure applied onto the film
p ) p0 - γ
∂2ζ + Π(ζ) ∂x2
(2)
where p0 is the external pressure, the second term is the Laplace pressure (with γ the surface tension), and Π(ζ) is the disjoining pressure arising from the van der Waals forces between the two interfaces of the film
Π(ζ) ) -dP(ζ)/dζ
(3)
ASL ) CSLFSFL
(4)
ALL ) CLL FL2
(5)
where CSL and CLL are two constants related to the corresponding interatomic van der Waals pair potential. We assume that the density of the liquid fluctuates along the surface (we neglect for simplicity variations in the z direction)
FL ) F0 + rpeipxe-t/δ
(6)
To first order in rp, the expression of the parameter A becomes
A ) CSLFSF0 + CLLF02 + (CSLFS + 2CLLF0)rpeipxe-t/δ ) A0 + apeipxe-t/δ
(7)
The fluctuations of A resulting from the fluctuations in density give rise to additional fluctuations of the pressure p. We consider here the case A0 > 0 corresponding to a thermodynamically stable film. The pressure gradient in the film gives rise to a flow v(x,z) in the film obeying the hydrodynamic equations of motion.30 As we deal with a slow viscous flow in a thin film, the inertial terms are negligible. Moreover, we can use the lubrication approximation, stating that there is a Poiseuille flow in the film with slow variations along the surface
vz ) 0
(8)
∂2vx/∂x2 , ∂2vx/∂z2
(9)
We also neglect the dependence of the viscosity η of the liquid on the density FL and therefore treat η as a constant. The equation of motion takes then the following simple form:
-
∂2vx ∂p +η 2 )0 ∂x ∂z
(10)
By using the boundary conditions at the interface with the substrate (no slip) and at the free surface (zero shear stress), one gets
vx(z) )
(
)
1 ∂p z2 - ζz η ∂x 2
(11)
It is convenient to introduce the total flow over the film
F)
∫0ζ vx(z) dz
(12)
with Using eqs 1, 2, and 11 yields 2
P(ζ) ) A/12πζ
F)
[
]
ζ3 ∂3ζ ∂Π γ 3η ∂x3 ∂x
(13)
A is the difference between the solid-liquid and liquidliquid Hamaker constants: A ) ASL - ALL. A is negative for a nonwetting film, which corresponds to the case of spinodal dewetting. We consider here the case of a positive
Note that this result is independent of whether the density in the film is constant or not.
(29) Israelachvili, J. N. Intermolecular and surface forces, 2nd ed.; World Scientific: New York, 1991.
(30) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport phenomena; Wiley International Edition: New York, 1960.
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The conservation of mass implies that
〈uq2〉 )
∂(FLF ) ∂(FLζ) )0 ∂x ∂t
[
(14)
ap 2 ipx -t/δ e3 γq4uqeiqxe-t/τ pe e + 3η 6πe3 A0 erp ipx -t/δ uq iqx -t/τ e e e e u q2eiqxe-t/τ - F0 )0 4 q δ τ 2πe (15)
]
This equation can only be fulfilled for all values of the variables x and t if p ) q and δ ) τ. Moreover ap/A0 ) rp/F0. Therefore
1 1 ) τ rp u q + F0 e
[(
)
]
)
(17)
This is only possible if
uq rp >3 F0 e
(18)
The raising time |τ| of the instability passes through a minimum for q ) qM
qM )
1 e2
[(
) ]
rp e A0 -3 F0 uq 12πγ
1/2
(19)
for which the raising time is given by
(
)
2u γe3 rp e 1 q 1 ) -3 |τM| η F0 uq 3e rp uq + F0 e
e2 (A0/2πγ)1/2
(22)
l is of the order of a molecular dimension, L is a macroscopic length, and avdw is of the order of a few 100 nm to 1 µm for film thicknesses of a few 100 Å (taking γ ≈ 0.1 N m-1 and A0 ≈ 10-20 J). Since l , avdw , L, the expression of 〈uq2〉 can be written as
avdw e4 kT kT ln 2 ) ln 2 4πγ 4πγ l (A /2πγ) l
(23)
0
The fluctuations of the free surface will increase if 1/τ < 0, which implies that
A0 1 rp uq 3 F0 e 2πγe3u q
avdw )
2
(16)
(
where l is the intrinsic width of the interface (without fluctuations), L is the size of the system (in the direction parallel to the surface), and avdw is a characteristic length related to the strength of the van der Waals interactions
〈uq2〉 )
uq 1 rp A0 2 γe3 uq 4 q + q e 3 F0 6πηe 3η e
q2
3(uq/e) implies that the density fluctuations in the film should be of a few percent to destabilize the film, which is achievable in a real system. The threshold in rp/F0 above which destabilization occurs decreases as the film thickness e increases, making in principle the destabilization easier to occur. However the raising time |τM| of the fluctuations increases as e increases (eq 20), making thick films kinetically metastable. We have considered here the case when the fluctuations of the effective Hamaker constant A are due to density fluctuations in the film. In principle, the same dewetting mechanism can occur if A fluctuates for other reasons (e.g., fluctuations in dielectric properties). The expression of the time constant τ (eq 16) is then somewhat different (because FL is constant in eq 14), but the condition for destabilization to take place is the same: ap/A0 > 3(uq/e). Let us finally discuss which experimental systems are likely to present such a dewetting process driven by density fluctuations. The first candidate is a film of a material (single compound or mixture) close to a critical point. The density or concentration fluctuations intrinsically present in such a system could lead to dewetting by the present mechanism. There are two other systems in which the occurrence of density fluctuations is possible but needs to be proven experimentally. The first one is the system made of a low-molecularweight glass former on glass for which dewetting has been observed in thin films when they are first heated after deposition.28 The occurrence of this dewetting was found to coincide with the restructuring taking place in the films upon heating, suggesting that density fluctuations induced by this restructuring could indeed be at the origin of the dewetting. We should at this point emphasize that the present calculation (performed in the case of small surface fluctuations) shows that density fluctuations can indeed destabilize the free surface. However the time evolution of the fluctuations might not be correctly predicted. In the first place, as the fluctuations of the free surface increase in amplitude, the lubrication approximation used in our calculation breaks down. Second, as restructuring proceeds and the system reaches structural equilibrium, the driving
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force behind the appearance of density fluctuations disappears, modifying the nature of the system. Another system in which a similar dewetting process could occur is a thin film of a binary A-B mixture quenched in a glassy state after deposition by spin coating from a solution. If the two compounds are totally miscible, the film formed by spin coating will have a uniform composition, since the solvent evaporation is too fast to allow for any restructuring. If one component (say A) of the mixture has a more favorable interaction with the substrate than
Wensink and Je´ roˆ me
the other, heating the sample will lead to some diffusion in the film to create a gradient of concentration across the film, with a higher concentration of A close to the substrate. Since this diffusion cannot take place precisely in the same way over the whole film, this is likely to create fluctuations in the mixture composition along the surface, and dewetting through the present mechanism could occur. LA015611Z
