Langmuir 2004, 20, 10073-10079
10073
Can Contact-Angle Measurements Determine the Disjoining Pressure in Liquid Nanofilms on Rigid Substrates? Yiao Tee Hsia,‡ Paul Jones,‡ and Lee R. White*,† Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213-3890, and Seagate Technology, 1251 Waterfront Place, Pittsburgh, Pennsylvania 15222-4215 Received February 23, 2004. In Final Form: July 20, 2004 The disjoining pressure of lubricant nanofilms used in the magnetic recording industry controls the equilibrium wetting, the dynamics of film restoration, and the evaporation kinetics of the film. It has been claimed that by measuring the contact angle of nonpolar and polar liquids on lubricant films, the disjoining pressure can be extracted using the method of Girafalco and Good, and such analyses have appeared in the literature. The approximations underlying the method have been discussed before in the literature. In view of the importance of measuring disjoining pressure in nanofilms of lubricants, it is timely to revisit these assumptions to understand the validity of the contact-angle method presently in use. We re-derive the relevant equations using a thermodynamic-interaction-energy approach which is free of the problems inherent in the original derivation and make explicit the assumptions which must be made in the derivation. General interaction energy arguments are then invoked to demonstrate that it does not appear possible to obtain the disjoining pressure in the film from contact-angle measurements in an unambiguous manner.
1. Introduction Perfluoropolyether (PFPE) liquids play an important role in the magnetic recording industry as lubricants. PFPE liquids, when spread on the hard-disk overcoat, help protect the underlying magnetic recording layer and reduce wear on the head. A good lubricant must want to spread on the substrate, as any tendency to dewet and form liquid drops will reduce the protection and interfere with the flying characteristics of the head. The lubricant, although quite viscous, is required to respond quickly to damage to the integrity of the film (such as is occasioned by the contact of the head with the disk surface), restoring the film to its pre-contact state before another incident can occur. The equilibrium stability and dynamic response of a nanofilm of lubricant, L, on a substrate, S, are controlled by the disjoining pressure, ΠSLV, of the substrate/ lubricant system, as discussed below. Before examining the role of disjoining pressure in nanofilm behavior, we should first determine precisely what it is. The surface thermodynamics of thin films have been exhaustively treated in the literature,1-4 and we will therefore only briefly summarize the results relevant to our subsequent discussion. We consider the free energy per unit area, F(h), of a substrate covered with a thin lubricant film of thickness h,
F(h) ) Nµ0L + γSL + γLV + ESLV(h) + ...
(1.1)
We have omitted from eq 1.1 large, bulk energy contributions which are not functions of the film thickness. The first term is the bulk free energy of the film moleculess the free energy these molecules would possess in the bulk * Author to whom correspondence should be addressed. † Carnegie Mellon University. ‡ Seagate Technology. (1) Adamson, A. W. Physical Chemistry of Surfaces, 4th ed.; Wiley: New York, 1982. (2) Derjaguin, B. V. Theory of Stability of Colloids and Thin Films; Plenum: New York, 1989. (3) Starov, V. M. Adv. Colloid Interface Sci. 1992, 39, 147. (4) Sharma, A. Langmuir 1993, 9, 3580; 861.
liquid state. Here, µ0L is the bulk lubricant Gibbs free energy per mole, and N is the number of lubricant moles per unit area of the substrate. The film thickness, h, is defined by
vLN ) h
(1.2)
where vL is the bulk molar volume of the lubricant. Equation 1.2 is the definition of the film thickness determined by IR reflectance spectroscopy or ellipsometry where bulk liquid properties are assumed everywhere across the film in the interpretation of the experimental measurement. The molecules of the film and the interfacial molecules of the substrate are not, however, bulk molecules, and the bulk energy term must be corrected for the presence of the interfaces. The terms γSL and γLV are the excess (over bulk) free energies of the substrate/lubricant interface and the lubricant/vapor interface, respectively. The last term, ESLV(h), is the excess free energy due to the interaction of each interface with the other at close proximity. Clearly, when the film thickness is greater than the range of any surface-generated interaction, the interfaces may be regarded as isolated and
ESLV(h) 9 80 hf∞
(1.3)
When the film is reduced to zero thickness (N ) 0), the free energy, F, must become that of the substrate/vapor interface, viz. γSV, so that
ESLV(0) ) γSV - γSL - γLV
(1.4)
Since the right-hand side of eq 1.4 is the spreading coefficient, S,1 we note that lubricant spreading requires
ESLV(0) > 0
(1.5)
The interaction energy, ESLV(h), is comprised of van der Waals, dipole-dipole, hydrogen bonding, chemical and structural, or steric contributions where appropriate. In a simple lubricant such as the Fomblin Z, with inert
10.1021/la049525l CCC: $27.50 © 2004 American Chemical Society Published on Web 10/02/2004
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Hsia et al.
fluorine terminations, the interaction energy is most likely a combination of van der Waals and (perhaps) structural contributions, whereas the hydroxyl-terminated Fomblin Zdol can also contribute hydrogen bonding or chemical interactions with substrate moieties. In general, these contributions are not additive. For example, the hydroxyl interaction with the substrate in Zdol films will grossly perturb the bulk conformation of the polymer, yielding structural or entropic contributions to the interaction energy. The disjoining pressure of this thin-film system is defined by
ΠSLV(h) ) -
∂ESLV ∂h
(1.6)
and plays the role of an excess film pressure. The various contributions to the interaction energy listed above will produce their separate contributions to the disjoining pressure and these will be nonadditive in general. Note that nonadditivity in this context does not prevent the decomposition of interaction energy or disjoining pressure into components labeled van der Waals, structural, or chemical but asserts that these components are not independentsthe magnitude of each contribution being a functional of the others. For the simple Fomblin Z, the disjoining pressure is expected to reflect the van der Waals contribution and decay monotonically with film thickness. For Fomblin Zdol, the hydroxyl interaction with the substrate produces a first layer with significantly altered structure and mobility compared to those of subsequent layers. The molecularity of the polymer liquid is thus manifested and can result in a nonmonotonic disjoining pressure on length scales typical of the molecular size. Films will spontaneously thicken (i.e. spread), given a source of new lubricant, if
ΠSLV(h) > 0
(1.7)
since an increase of h will lower the free energy, F(h). Thus, thermodynamically, a liquid will spread as a film over a substrate when the disjoining pressure is positive. Negative disjoining pressure at a film thickness, h, will give rise to the spontaneous breakup of the uniform film to create a film of smaller thickness (still with negative disjoining pressure), with the excess liquid appearing as bulk liquid sessile drops since, again, this process lowers the system free energy. Uniform film stability also requires2
∂ΠSLV ( dAB (A.7)
where dAB is a normal distance of closest approach of entities A and B across the z ) 0 interface, as shown in Figure 2. The neglect of correlations across the interface (23) Fowkes, F. M. J. Phys. Chemistry 1968, 72, 3700.
B
AB
and that
) 0 z1 - z2 < dAB
(A.10)
AA
Φ)
bulk z < (>) 0 FA(B)(r) ) FA(B)
(A.9)
Thus, we have
A
required to produce eq 2.7. We have therefore neglected any structural rearrangements from the bulk configuration that might occur when the S and l phases are contacted. To proceed with our pairwise approximation, we make the assumption that bulk densities hold up to the interface from each side
(A.8)
AB
νA ) νB and dAB ) dAA ) dBB
(A.14)
Alternative mixing rules can be derived. For example, let us assume dAB ) dAA ) dBB and eliminate the frequencies νA and νB from eq A.11 using eq A.10 and its equivalent (24) Hunter, R. J. Foundations of Colloid Science; Clarendon: Oxford, 1987; Vol. 1.
Disjoining Pressure in Liquid Nanofilms
Langmuir, Vol. 20, No. 23, 2004 10079
for B. We obtain
EdAVB(0) )
φpAB ) -
2EdAVA(0)EdBVB(0) Fbulk Fbulk B RB(0) A RA(0) + EdBVB(0) bulk EdAVA(0) bulk FA RA(0) FB RB(0) (A.15)
bulk and, if we make the assumption Fbulk A RA(0) ) FB RB(0), we have a new mixing rule
EdAVB(0) )
2EdAVA(0)EdBVB(0) EdAVA(0)
+
EdBVB(0)
(A.16)
the possibility of which was realized by Wu19 and shown to yield better results for the calculation of interfacial tensions of polymers than the geometric mean. We do not advocate the replacement of one mixing rule with another since they are all based on essentially the same ad hoc assumptions. The fact that small variations in the assumptions can lead to completely different mixing rules should lead one to question the nature of the pairwise approach to interaction energies and should certainly lead to serious doubt as to the general validity of eq 2.7 or eq 2.11. The use of a geometric mixing term for the polar contribution (eq 2.10) follows in a similar way to the above derivation. The thermally averaged permanent dipolepermanent dipole interaction between polar submolecular entities is the Keesom energy24
2(µAµB)2 3kT|rA - rB|6
(A.17)
The major additional assumption here is that the probability of a given orientation of two interacting dipoles is determined by the permanent dipole interaction energy only and that no other interactions that might restrict dipolar orientation, e.g., chemical binding to the substrate, are present. Identical arguments and approximations as those above for the dispersion contribution can now be made for the pairwise addition of these polar interaction energies across the AB interface to obtain
EpAVB(0) ) -
lAAlBB p (EAVA(0)EpBVB(0))1/2 l2AB
(A.18)
where lAB is the normal distance of closest approach of the polar entities across the AB interface. Since, in the argument presented here, we have only one polarizable entity per material, lAB can be taken as equal to the dAB of the dispersive interaction. In a more general argument, this would not be so. In any event, we make the further approximation
lAB2 )1 lAAlBB
(A.19)
on these cutoff distances to obtain eq 2.10. Acknowledgment. L.R.W. gratefully acknowledges the financial support of Seagate Technology administered through the Data Storage Systems Consortium at Carnegie Mellon. LA049525L
