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Computation of Lifshitz-van der Waals Forces between Alkylthiol Monolayers on Gold Films Thomas Ederth* Physical and Theoretical Chemistry Laboratory, Oxford University, South Parks Road, Oxford OX1 3QZ, United Kingdom Received November 20, 2000. In Final Form: February 28, 2001 The van der Waals interactions between gold films modified with self-assembled monolayers of alkylthiols have been calculated from Lifshitz theory over the range of 1-100 nm, using a detailed multilayer structure to model the surfaces. The Hamaker functions describing the interactions vary with separation in a manner that cannot be approximated with any of the Hamaker constants of the materials constituting the system. The effects of small changes in thickness of various layers have been investigated; the total interaction is very sensitive to the thickness of the adsorbed hydrocarbon layers, and accurate measures of this thickness in real systems is essential for quantitative comparisons with the calculations. For thin gold films on silica supports, the interaction is also very sensitive to the gold thickness, whereas for gold thicknesses greater than 25 nm, the error in assuming the gold layer to be infinite is less than a few percent. Some simplified models have been considered, and in the case of thick gold films, a five-layer model with hydrocarboncoated gold half-spaces in a medium is sufficient to model the system. Comparison of the calculated results with experiments confirms the qualitative features of the calculated Hamaker functions and in some cases also shows good quantitative agreement. However, the quality of available data is not sufficient to enable a detailed quantitative assessment of the conformity. In addition, Hamaker constants for gold and titanium have been calculated using tabulated optical data, and the results are compared with figures based on previously used models for the optical properties.
I. Introduction The adsorption of thiol compounds onto gold is a wellestablished method for modification or control of interfacial properties at solid/vapor or solid/liquid interfaces.1,2 Parameters such as wetting, interfacial energy, surface charge, friction, or the surface density of functional groups can be varied by a change of the thiol compounds, by replacement of functional groups in situ, or by adsorption from mixed thiol solutions. Applications include for example biosensors, catalysis, corrosion protection, photoresists, or microcontact printing, to mention but a few. This method is especially useful for surface force measurements, where interactions between solid bodies across liquids or vapors are studied to gain understanding of phenomena such as colloidal stability, capillary condensation, micro- and nanotribology, rheological properties of thin films, adhesion and adsorption, or the interaction and stability of biomolecules. Widespread methods for direct measurement of forces between solid surfaces (such as the surface force apparatus (SFA)) have been hampered by the restricted number of materials available to form the solid substrate (see e.g. ref 3 for details), and the vast majority of the experiments have been performed with muscovite mica surfaces. The atomic force microscope (AFM) has improved the situation considerably from a materials point of view, and in particular the combination with various methods for chemical surface modification (predominantly silanization and thiol self-assembly onto gold) has produced a subfield known as chemical force * E-mail:
[email protected]. Fax: +44 1865 275410. Tel: +44 1865 275400. (1) Ulman, A. An introduction to ultrathin organic films; Academic Press: Boston, 1991. (2) Xu, J.; Li, H.-L. J. Colloid Interface Sci. 1995, 176, 138. (3) Claesson, P. M.; Ederth, T.; Bergeron, V.; Rutland, M. W. Adv. Colloid Interface Sci. 1996, 67, 119.
microscopy (CFM).4-7 In terms of ease of preparation, stability, and reproducibility, thiolate self-assembled monolayers (SAMs) appear to be superior to most silanization methods, and even though the use of thiols has greatly improved the versatility of direct force measurement techniques, there are some disadvantages, as well. The roughness of the usually polycrystalline metal substrates supporting the SAMs affects the interaction forces and also reduces (and increases scatter in) adhesion. Further, the resulting multilayer structures composed of layers with widely different dielectric properties (hydrocarbons and metals) complicate the analysis of van der Waals interactions, an issue that is the focus of this paper. Many methods have been devised to prepare smooth metal surfaces, particularly for the purpose of scanning probe microscopy,8-12 and one simple and reliable method is to deposit a very thin layer onto a smooth substrate, such as a polished silicon wafer or a flame-polished glass surface. This method has been successfully employed for force measurements using macroscopic thiolate-SAM surfaces,13,14 though these have been limited to comparative data analysis, pending quantitative information on the interaction between such substrates. The very thin metal films (4) Frisbie, C. D.; Rozsnyai, L. F.; Noy, A.; Wrighton, M. S.; Lieber, C. M. Science 1994, 265, 2071. (5) Thomas, R. C.; Tangyunyong, P.; Houston, J. E.; Michalske, T. A.; Crooks, R. M. J. Phys. Chem. 1994, 98, 4493. (6) Ito, T.; Namba, M.; Bu¨hlmann, P.; Umezawa, Y. Langmuir 1997, 13, 4323. (7) van der Vegte, E. W.; Hadziioannou, G. Langmuir 1997, 13, 4357. (8) Buchholz, S.; Fuchs, H.; Rabe, J. P. J. Vac. Sci. Technol., B 1991, 9, 857. (9) Clemmer, C. R.; Beebe, T. P. Scanning Microsc. 1992, 6, 319. (10) Golan, Y.; Margulis, L.; Rubinstein, I. Surf. Sci. 1992, 264, 312. (11) Hegner, M.; Wagner, P.; Semenza, G. Surf. Sci. 1993, 291, 39. (12) Stamou, D.; Gourdon, D.; Liley, M.; Burnham, N. A.; Kulik, A.; Vogel, H.; Duschl, C. Langmuir 1997, 13, 2425. (13) Ederth, T.; Claesson, P.; Liedberg, B. Langmuir 1998, 14, 4782. (14) Ederth, T.; Liedberg, B. Langmuir 2000, 16, 2177.
10.1021/la001607w CCC: $20.00 © 2001 American Chemical Society Published on Web 05/05/2001
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(≈10 nm) complicate the calculation of van der Waals interactions; the thickness of the film might be difficult to control precisely during deposition, and thus the thickness and, consequently, also the interaction will vary from sample to sample. In addition, the practical implementation of this procedure adds to the complexity of the system, because in most situations, where the substrate is for example glass or silica, additional layers are required to promote the adhesion of the gold film to the substrate. These are often thin chromium or titanium layers, whose optical properties might also affect the interaction. Considering that the contribution to the van der Waals interaction is much stronger for the metals than for the other components, it is of interest to determine the effect of variations in the metal layers on the total interaction and to ascertain whether simplified models of the substrates can be used to calculate the interaction with reasonable accuracy. In this paper, van der Waals interactions across air and water between thiolate SAM surfaces have been calculated, using a detailed multilayer model with experimentally relevant parameters and materials. The effects of variations of various parameters are studied, and simplified systems are considered. The intention is to provide a guide to the experimentalist as to how the van der Waals interactions might be calculated and to promote quantitative force studies using thiolate SAM systems. II. Lifshitz Theory The Lifshitz theory of van der Waals forces is a continuum theory including temperature and retardation effects. In principle, the main results can be arrived at by solving Maxwell’s equations with boundary conditions determined by the optical properties of the materials and by assigning the energy of a harmonic oscillator to each allowed electromagnetic mode. The original derivations were quite involved,15,16 and several simplified methods to determine the interactions exist. In the following, only the results of immediate interest to this study are provided, and the reader is referred to for example refs 17 and 18 for further details. The application of Lifshitz theory to interactions between metal surfaces is notoriously difficult, mostly because of variations in available optical data; the optical properties vary between samples and with the method of measurement, and conductors also absorb strongly at low frequencies, where little data is available. However, recent interest in Casimir forces19-24 has shown that Lifshitz theory and available spectroscopic data can be used to describe the interactions between metals accurately. A. Two Half-Spaces. For two parallel plates with dielectric function s interacting across a medium characterized by the permittivity m, the free energy of interaction per area at a separation d is given by15-17 (15) Lifshitz, E. M. Sov. Phys. JETP 1956, 2, 73; Zh. Eksp. Teor. Fiz. 1955, 29, 94. (16) Dzyaloshinskii, I. E.; Lifshitz, E. M.; Pitaevskii, L. P. Adv. Phys. 1961, 10, 165. (17) Mahanty, J.; Ninham, B. W. Dispersion forces; Academic Press: London, 1976. (18) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: London, 1985. (19) Lamoreaux, S. K. Phys. Rev. Lett. 1997, 78, 5. (20) Mohideen, U.; Roy, A. Phys. Rev. Lett. 1998, 81, 4549. (21) Lamoreaux, S. K. Phys. Rev. A 1999, 59, 3149. (22) Lambrecht, A.; Reynaud, S. Eur. J. Phys. D 2000, 8, 309. (23) Bostro¨m, M.; Sernelius, B. E. Phys. Rev. A 2000, 61, 046101. (24) Ederth, T. Phys. Rev. A 2000, 62, 062104.
Ederth
kT
F(d, T) )
∞
∑ ′I(ξn, d) 2 n)0
(1)
8πd
with the convention that F f 0 as d f ∞. The prime on the summation means that the term n ) 0 should be halved. Further, ξn ) 2πnkT/p, and
I(ξn, d) )
(
)
2ξndxm c
∫1∞
{(
2
(
)) ))}
2pξndxm + c 2pξndxm ln 1 - ∆ms2 exp p dp (2) c
ln 1 - ∆ h ms2 exp -
(
(
where the reflection coefficients for the TM and TE polarizations at the boundary are, respectively,
∆ h ms )
ssm - ps ssm + ps
∆ms )
ss - p ss + p
and
ss ) xp2 - 1 + s/m
j ) j(iξ)
The n ) 0 case needs special attention because the ξn2 factor in (2) vanishes, while the integral over p is divergent. This is circumvented by changing to the variable x ) 2pξnd xm/c, whereupon (2) is transformed to
I(xn, d) )
∫x∞ln(1 - ∆h ms2e-x) + ln(1 - ∆ms2e-x)x dx n
(3)
and xn ) 2ξndxm/c. If the plates are dielectric, then as n is set to 0
∆ h ms f
m - s m + s
∆ms f 0
(4)
and the integral is reduced to
I(0, d) )
( (
) )
m - s 2 -x e x dx m + s
∫0∞ln 1 -
(5)
For later use, we note that this integral can be performed analytically using the series ∞
ln(1 + t) )
tn (-1)n-1 n n)1
∑
(6)
and termwise integration, yielding ∞
I(0, d) ) -
( )
1 m - s
∑ 3 t)1 t
m
+ s
2t
(7)
On the other hand, if the plates are metallic, for which the permittivity tends to infinity as ξ f 0, it is important that the limit s f ∞ is taken before n is set to 0, to ensure that the correct boundary conditions are imposed.25 In this case, both ∆ h ms2 and ∆ms2 f 1 as s f ∞, and (3) reduces to (25) Schwinger, J.; DeRaad, L. L.; Milton, K. A. Ann. Phys. (NY) 1978, 115, 1.
Computation of Lifshitz-van der Waals Forces
I(0, d) ) 2
∫x∞ln(1 - e-x)x dx ) -2ζ(3) 0
Langmuir, Vol. 17, No. 11, 2001 3331
(8)
where ζ(3) ≈ 1.202 is the Riemann ζ function. B. Hamaker Constants. Using Lifshitz theory, it is possible to derive Hamaker constants for the interaction in the nonretarded regime, where the functional form of the interaction is the same as in the pairwise additivity (“Hamaker”) approach. The Hamaker result gives the free energy of interaction between the surfaces as26
A F(d) ) 12πd2
(9)
where the Hamaker constant A is a function of the number density of atoms in the interacting materials and the magnitude of the atom-atom pair potentials. The additivity approach does not work well for interactions in a medium and at separations where retardation effects are significant. However, Lifshitz theory can be used to calculate a corresponding “Hamaker function”, thus giving a more appropriate description of the interaction across, say, liquids and/or at large separations. We can define the Hamaker function A(d, T) through the relation
F(d, T) ) -
A(d, T)
(10)
12πd2
where by comparison with eq 1 we can identify A(d, T) as
A(d, T) ) -
3kT 2
∞
∑ ′I(ξn, d)
(11)
n)0
Using the form of I(ξn, d) in (3) and further noting that with x ) 2pξndxm/c, the limit d f 0 is equivalent to the limit ξn f 0, and we can enter the result from (7) into (11) to obtain
A(0, T) )
3kT 2
∞
∞
∑∑ ′
( )
1 m - s
3 n)0 t)1t
2t
m + s
(12)
The 1/t3 factor causes the terms in the second sum to decrease rapidly, and only very few terms are required to obtain good precision. Again, for metals, the result in (8) must be used for the static term. C. Multilayers and Thin Films. The surfaces which are the subject of this study are composed of many different layers (see Figure 1), and the result for semi-infinite halfspaces cannot be applied directly. The Lifshitz result (eqs 1 and 2) can be generalized to multilayer systems, according to a procedure described by Parsegian and Ninham17,27 where, for two semi-infinite media covered h ms and ∆ms in eq by N layers of thickness li, i ) 1 to N, ∆ 2 are substituted by
∆ h ms )
( (
h Ns exp ∆ h mN + ∆
) x )
2ξnlNsNxm c
2ξnlNsN m 1+∆ h mN∆ h Ns exp c
(13)
(similarly for ∆ms) which can be calculated after repeated use of relations of the type (26) Hamaker, H. C. Physica 1937, 4, 1058. (27) Parsegian, V. A.; Ninham, B. J. Theor. Biol. 1973, 38, 101.
Figure 1. The surface model used in the calculations. The model is shown to scale (top) for a system with 1-nm Ti, 10-nm Au, 0.12-nm Au-S thiolate bond, and 2-nm hydrocarbon (HC) layers, approximating a hexadecanethiol SAM layer adsorbed onto a thin gold film on a silica support. These are the layer thicknesses used in calculations where other dimensions are not explicitly stated.
∆ h is )
( (
h i-1,s exp ∆ h i,i-1 + ∆
) x )
2ξnli-1si-1xm c
2ξnli-1si-1 m 1+∆ h i,i-1∆ h i-1,s exp c
(14)
for nonadjacent layers and, for any two adjacent layers i and j,
∆ h ij )
sji - pj sji + pj
∆ij )
sj - p sj + p
(15)
sj ) xp2 - 1 + j/m
(16)
In the limit n ) 0, using the same change of variables as in (3), the relation in (14) for two nonadjacent layers simplifies to
( (
) )
li-1x d ∆ h is ) li-1x 1+∆ h i,i-1∆ h i-1,s exp d h i-1,s exp ∆ h i,i-1 + ∆
(17)
This procedure does not introduce any formal difficulties for the n ) 0 term. For metals, the rule still applies that the limit f ∞ is taken before n is set to zero, so if layer li-1 in eq 17 is a metal layer, the exponential factors will equal 1 instead. It was noted in the previous section that in the limit n ) 0, the reflection coefficients ∆ h and ∆ for adjacent layers take different values depending on the order in which the limits ξ f 0 and f ∞ are taken. Is it reasonable to assume that f ∞ as ξf 0 for a thin metal film? Yes; because even 5 nm thick evaporated gold films are continuous,28 it seems appropriate to assume that these films respond to an external static field very much like “bulk” gold. Note that discontinuous metal films do not behave like dielectrics with high polarizability but must be treated with other methods; see for example ref 30 and references therein. The contributions to the Hamaker functions from the static (28) DiMilla, P. A.; Folkers, J. P.; Biebuyck, H. A.; Ha¨rter, R.; Lopez, G. P.; Whitesides, G. M. J. Am. Chem. Soc. 1994, 116, 2225-2226. As the thickness of a metal layer approaches the electron mean free path, the resistivity increases because of scattering at the surfaces. For gold films with thicknesses of 200, 10, and 5 nm, DiMilla et al. reported the following resistivities: 3.7, 4.8, and 7.1 × 10-8 Ω m. This is to be compared with the electron mean free path, ∼50 nm, and the bulk resistivity 2.2 × 10-8 Ω m (ref 29). See also de Vries, J. W. C. Thin Solid Films 1988, 167, 25. (29) Kittel, C. Introduction to solid state physics, 6th ed.; John Wiley & Sons: New York, 1986. (30) Smith, G. B.; Niklasson, G. A.; Svensson, J. S. E. M.; Granqvist, C. G. J. Appl. Phys. 1986, 59, 571.
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Ederth
D. On the Adequacy of Lifshitz Theory for Thin Films. Strictly, Lifshitz theory is valid only as long as the separation d . atomic dimensions, where the two bodies are so far apart as to appear as continua to each other. There seems to be some dissension as to whether Lifshitz theory adequately describes the interactions of thin films32,33 and on metal substrates in particular,34-37 although studies on the type of the SAMs considered here suggest that this might not be a problem extending to them. The van der Waals forces across a surface monolayer influence the wetting properties of a liquid in contact with the surface, and reports have demonstrated how contact angles on gold-supported SAMs depend on the length of the alkyl chain38-41 (and also how the depth to features within the SAM affects wetting behavior42). An investigation of this phenomenon was published a few years ago,43 where Lifshitz theory was successfully used to explain the effects of variations in contact angles with changes in the alkyl chain length of a gold-supported SAM. The length scales concerned in force measurements are considerably larger, ensuring that the applicability of Lifshitz theory is not a problem. III. Optical Properties
Figure 2. Static contributions to the Hamaker function in air (top) and water (bottom) for the model in Figure 1. In both figures, the solid line is the contribution under the assumption of ideal metals (which is used in the calculations), and the dashed line shows the relative contribution from this term to the total Hamaker function (right ordinate). The thick and thin dotted lines correspond to the contributions from the TM (∆ h ) and TE (∆) polarizations, respectively. Because ∆ t 0 for dielectrics, the thick dotted line (∆ h ) is also the resulting static contribution for dielectrics in the limit of large permittivity.
terms for the system in Figure 1 in air and water are illustrated in Figure 2, under the assumption of metal behavior in the static limit. In air, the static term is relatively small, increasing from 3% at 1 nm to 17% at 100 nm. In water, the static contribution dominates for small and large separations, being about 50% at 1 nm, whereas at intermediate separations, 5-50 nm, it is less than or approximately equal to 15%. As a metal film becomes thinner, its bulk properties change and the permittivity is a function of both frequency and the thickness of the film (in addition to the wavevector dependence, which is ignored here). An electromagnetic field decays exponentially inside the metal, with the characteristic decay length (the “skin depth”) given by δ ) x2/µ0σω for a nonmagnetic material, where σ is the static conductivity and µ0 is the permeability of free space. For gold at the frequency of the first nonzero term in the summation in (1), this length is ∼10 nm. This is about the thickness of the thinner of the metal films considered here; thus, these films appear to be thin also to the incident electromagnetic field.31 From this, it is clear that for frequencies where n * 0, applying the permittivity as obtained from thick samples to thin films is also an approximation, and the permittivity for a very thin film is actually lower than for the corresponding bulk sample. (31) Strictly, in the frequency range where δ is smaller than the mean free path, the electric field acting on an electron is not constant between two collisions, and the penetration depth is rather described by the “anomalous skin-effect”; see for example: Abrikosov, A. A. Introduction to the theory of normal metals; Academic Press: New York, 1972.
In principle, the Lifshitz theory requires the optical response (iξ) at all (imaginary) frequencies to be known, though this is rarely the situation. In systems where the materials have similar densities, it has been demonstrated that actually very little spectral information is required.44,45 For many materials, oscillator models of the dielectric function along the imaginary axis describe the optical response with enough precision, because absorption occurs only at a few frequencies. For metals, whose electron transitions cover a continuum of frequencies, such models are less accurate or lose the simplicity which is their appeal. Further, in high- and low-frequency regions where optical data are missing, extrapolations might be required, which introduces additional sources of error. The dielectric function is a complex function of the frequency: (ω) ) ′(ω) + i′′(ω). For a given frequency, ′ + i′′ ) n2 - k2 + i2nk, but only the imaginary part of the dielectric function is required to calculate (iξ) along the imaginary axis, using the Kramers-Kronig (K-K) relationship:
(iξ) ) 1 +
x′′(x)
∫0∞x2 + ξ2 dx
2 π
(18)
Extensive tables of n and k for many materials are (32) Beaglehole, D.; Radlinska, E. Z.; Ninham, B. W.; Christenson, H. K. Phys. Rev. Lett. 1991, 66, 2084. (33) Panella, V.; Chiarello, R.; Krim, J. Phys. Rev. Lett. 1996, 76, 3606. (34) Lando, D.; Slutsky, L. J. J. Chem. Phys. 1970, 52, 1510. (35) Lando, D.; Slutsky, L. J. Phys. Rev. B 1970, 2, 2863. (36) Vukusic, P. S.; Bradberry, G. W.; Sambles, J. R. Surf. Sci. Lett. 1992, 277, L34. (37) Bradberry, G. W.; Vukusic, P. S.; Sambles, J. R. J. Chem. Phys. 1993, 98, 651. (38) Bain, C. D.; Troughton, E. B.; Tao, Y.-T.; Evall, J.; Whitesides, G. M.; Nuzzo, R. G. J. Am. Chem. Soc. 1989, 111, 321. (39) Walczak, M. M.; Chung, C.; Stole, S. M.; Widrig, C. A.; Porter, M. D. J. Am. Chem. Soc. 1991, 113, 2370. (40) Laibinis, P. E.; Whitesides, G. M.; Allara, D. L.; Tao, Y.-T.; Parikh, A. N.; Nuzzo, R. G. J. Am. Chem. Soc. 1991, 113, 7152. (41) Tao, Y.-T. J. Am. Chem. Soc. 1993, 115, 4350. (42) Bain, C. D.; Whitesides, G. M. J. Am. Chem. Soc. 1988, 110, 5897. (43) Miller, W. J.; Abbott, N. L. Langmuir 1997, 13, 7106. (44) Parsegian, V. A.; Ninham, B. W. Nature 1969, 224, 1197. (45) Ninham, B. W.; Parsegian, V. A. Biophys. J. 1970, 10, 646.
Computation of Lifshitz-van der Waals Forces
Langmuir, Vol. 17, No. 11, 2001 3333
available (see e.g. ref 46), in which cases the calculation of (iξ) is straightforward. For dielectrics, the Ninham-Parsegian representation is frequently used to model the permittivity along the imaginary axis:44,47
(iξ) ) 1 +
di
∑i 1 + ξτ + ∑j
fj
ωj2 + ξ2 + gjξ
i
(19)
Although it is, for a given form of the permittivity, of minor importance precisely which function is used to represent it, this form has a meaningful physical interpretation48,49 and can easily be fitted to optical data for many materials. Beyond the limits of tabulated data, approximate models have to be used for the permittivity. For dielectrics, the absorption is normally decreasing quickly as the lowfrequency limit is approached, and no optical transitions occur below the microwave region. For metals, however, intraband transitions cause increasing absorption as the frequency decreases. The simplest model of a metal in this region is the plasma model, with (ω) ) 1 - ωp2/ω2, or (iξ) ) 1 + ωp2/ξ2, where ωp ) Ne2/0m* is the plasma frequency of the material, where N is the number density of free electrons, e is the unit charge, 0 is the permittivity of free space, and m* is the effective mass of a free electron. The plasma model does not account for dissipative mechanisms, and the Drude model is preferred,
(ω) ) 1 -
ωp2 ω(ω + iγ)
(20)
giving along the imaginary axis
(iξ) ) 1 +
ωp2 ξ2 + ξγ
(21)
As the frequency increases from zero, the variation with frequency changes from 1/ω to 1/ω3, but at some frequency interband transitions add to the excitations, and the Drude model is no longer a good description of the behavior. The parameters ωp and γ have to be adjusted so as to fit the tabulated data in an overlap region and should ideally produce also a plausible static conductivity, given by σ ) 0ωp2/γ. The fitting of these parameters and the effect of the choice of models have been discussed in fair detail recently,23,22 and the method adopted here is that suggested in ref 23, where the parameters are fitted to both the tabulated data and the static contribution. At high frequencies, under the assumption that all matter behaves like an electron gas, the plasma model provides an appropriate description of the permittivity, and if required, the tabulated data have been extrapolated from the last data points with a plasma model. The tables extend to ω ≈ 2 × 1018 rad/s, and the error introduced by this procedure is believed to be small, because in this frequency region the absorption is decreasing rapidly and the contribution to the interaction is small. For air, (iξ) ) 1 was used throughout. (46) Handbook of optical constants of solids I-III; Palik, E. D., Ed.; Academic Press: Orlando, FL, 1985-1998. (47) Parsegian, V. A. In Physical chemistry: Enriching topics from colloid and surface science; van Olphen, H., Mysels, K. J., Eds.; Theorex: La Jolla, CA, 1975; pp 27-72. (48) Fano, U. Phys. Rev. 1956, 103, 1202. (49) Daniel, V. V. Dielectric relaxation; Academic Press: London, 1967.
A. Silica. Optical data (n and k) for amorphous SiO2 were obtained from ref 50, and the Kramers-Kronig relation (eq 18) was used to produce the permittivity along the imaginary axis. The silica substrate will be given very little attention in the following and is only included to provide a realistic “background” interaction from the substrate in the case of thin metal films. B. Gold and Titanium. Although copper and silver can also be used as substrates for thiol self-assembly, for this purpose gold is by far the most popular of the coinage metals. However, the wetting of silica by gold is poor, and to promote adhesion between the gold layer and the silica substrate and to ensure that the gold layer does not form “islands” on the silica during evaporation, an adhesion layer is used. Typically, this is a thin (1-5 nm) layer of chromium or titanium, preceding a thicker layer of gold (10-200 nm). In this analysis, titanium was chosen as the adhesion layer (chromium users: an oscillator model for chromium is discussed briefly in ref 51, and optical data can be found in ref 52, though existing low-frequency data are conflicting, and extrapolation of data is even more prone to error than for titanium, as will be described in the following). Tabulated spectroscopic data (n and k) for gold53 and titanium54 were used to calculate (iξ) using (18) for each frequency ξn. In the low-frequency regime, the dielectric functions were extrapolated using a Drude model (from approximately 2 × 1013 rad/s for Ti and 2 × 1014 rad/s for Au). Following ref 23, the fits were subjected to the constraints that they both fit the spectroscopic data in the low-frequency regime and that the values of the plasma frequencies ωp and the relaxation parameters γ produce correct static conductivities, that is, 4.5 × 107 (Ω m)-1 for gold and 2.3 × 106 (Ω m)-1 for titanium.29 For gold, this procedure resulted in the parameters ωp ) 1.5 × 1016 and γ ) 4.3 × 1013 rad/s. For titanium, a Drude model could not be made to fulfill both constraints, but with ωp ) 3.8 × 1015 and γ ) 5.6 × 1013 rad/s, the transition from the tabulated data is quite smooth, though it does not fit the very lowest tabulated frequencies. The potential deficiencies of this procedure are not a major source of error, because the greatest contributions to the sum in (1) come from frequencies ξ ∼ c/d. For separations between 1 and 100 nm, this corresponds to frequencies of approximately 1015-1017 rad/s (note that the first nonzero frequency ξ1 ≈ 2.4 × 1014 rad/s) and changes in the details of the extrapolation models become significant at frequencies outside this range. Figure 3 shows the relative contributions to the energy from the integral in (2) for 1, 10, and 100-nm separations. Also, using the Drude approximation for transition metals at low frequencies is in itself rather doubtful, because unfilled d-bands can cause transitions at very small energies, and it is not possible to determine with certainty a frequency below which the approximation is valid. Again, this is a minor problem, because the low-frequency contributions to the Lifshitz result are small for the separations in question. The resulting permittivities for gold and titanium are shown in Figure 4. Included in this figure is also the (50) Philipp, H. R. In Handbook of optical constants of solids; Palik, E. D., Ed.; Academic Press: Orlando, FL, 1985; pp 749-764. (51) van Blokland, P. H. G. M.; Overbeek, J. T. G. J. Chem. Soc., Faraday Trans. 1 1978, 74, 2637. (52) Lynch, D. W.; Hunter, W. R. In Handbook of optical constants of solids II, Palik, E. D., Ed.; Academic Press: Orlando, FL, 1991; pp 374-385. (53) Lynch, D. W.; Hunter, W. R. In Handbook of optical constants of solids; Palik, E. D., Ed.; Academic Press: Orlando, FL, 1985; pp 275-368. (54) Lynch, D. W.; Hunter, W. R. In Handbook of optical constants of solids III; Palik, E. D., Ed.; Academic Press: Orlando, FL, 1998; pp 233-286.
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Ederth Table 1. Parameters for the Thiolate Layer Modela
a
ωj (eV)
Fj (eV2)
Γj (eV)
2.2 4.3
1.0 1.44
0.62 2.0
From ref 58.
model for imaginary frequencies: 2
(iξ) ) n02 +
Figure 3. The relative contribution from different frequencies to the interaction energy for the model in Figure 1. Note that the summation in eq 1 starts at ξ ≈ 2.4 × 1014 rad/s.
Figure 4. The permittivity of gold and titanium along the imaginary axis, as calculated using the Kramers-Kronig relationship (eq 18) and experimental data. Also included is the oscillator model for gold (using DESY data) from ref 55.
permittivity for gold as obtained with an oscillator model55 with parameters based on the “DESY” data (see ref 56 and references therein), which has been used in some previous Lifshitz calculations involving gold surfaces.43,57 Comparison of the two data sets for gold reveals that the permittivity as obtained using the DESY data is smaller than that obtained through the Kramers-Kronig analysis over the whole range of frequencies used in the summation in (1), with the result that calculated values of Hamaker constants/functions for interactions between gold surfaces are higher in this study than those obtained using the DESY data (note that the tabulated data used in the K-K analysis include the DESY data on which the model from ref 55 is based). C. The Metal-Thiolate Interface. Shi et al.58 used ellipsometric data to confirm that the metal-thiolate layer forming the bonds between the gold surface and the thiols terminating the hydrocarbon chains possesses unique optical properties and determined the effective thickness of this layer to 1.2 Å. Ellipsometric data were fitted to two damped harmonic oscillators, resulting in the following (55) Parsegian, V. A.; Weiss, G. H. J. Colloid Interface Sci. 1981, 81, 285. (56) Hagemann, H.-J.; Gudat, W.; Kunz, C. J. Opt. Soc. Am. 1975, 65, 742. (57) Rabinovich, Y. I.; Churaev, N. V. Colloid J. 1990, 52, 256; Kolloidn. Zh. 1990, 52, 309. (58) Shi, J.; Hong, B.; Parikh, A. N.; Collins, R. W.; Allara, D. L. Chem. Phys. Lett. 1995, 246, 90.
∑ j)1
Fj
ωj2 + ξ2 + ξΓj
(22)
with parameters as given in Table 1. The high-frequency index of refraction n0 was not determined, because “[it] might be larger than unity due to non-resonant contributions and to undetected resonances at high energies”.58 In the following, n0 is assumed to equal 1. The resulting thicknesses for n-alkanethiol SAMs formed from CH3(CH2)nSH were found to be 3.13 + 1.19n Å (including the 1.2 Å thiolate layer) with a refractive index of 1.50 for the hydrocarbon layer. Other ellipsometric measurements of the thickness of n-alkanethiols adsorbed onto gold indicate that for chains with about 10 or more methylene units, the layer thickness increases with approximately 1.41.5 Å per methylene unit;38,59,60 Porter et al. found that the total thickness of the SAM increases with n as 3.8 + 1.5n Å,59 using a hydrocarbon refractive index of 1.45. Because the conclusions inferred from ellipsometric data are to a large extent model-dependent, the additional interfacial metal-thiolate layer should preferably be used only in conjunction with the simultaneously obtained result for the hydrocarbon layer thickness, even though the effective thickness of a methylene unit is smaller than that observed in other studies. (Presumably, the discrepancy might be caused not only by different models but also by differences in the SAM structure of the studied surfaces, caused by variations in the substrate and preparation conditions.) D. Crystalline Hydrocarbon. The optical properties of the hydrocarbon layer were modeled with a single oscillator:17
(iξ) ) 1 +
(n2 - 1) 1 + (ξ/ωUV)2
(23)
where n ) 1.50 (unless stated otherwise) and ωUV ) 1.9 × 1016 rad/s for a solid hydrocarbon.18 This simple model provides a fairly good description of the hydrocarbon. Extensive data for polyethylene are available,61 but there are reasons to hesitate about their use for the present purpose; if it is true for metals that the method of preparation influences the optical properties, this is even more so with polymers. Because very little data are available for the type of solid hydrocarbon layers formed by self-assembling alkylthiols, there is little to gain from using an elaborate model, the details of which might not apply to this particular system. Further, considering the data provided in ref 61, it is also clear that absorption occurs only in a rather narrow band in the ultraviolet, thus justifying the use of an oscillator model with a single UV relaxation and avoiding the unnecessary complexity of the Kramers-Kronig analysis. (59) Porter, M. D.; Bright, T. B.; Allara, D. L.; Chidsey, C. E. D. J. Am. Chem. Soc. 1987, 109, 3559. (60) Atre, S. V.; Liedberg, B.; Allara, D. L. Langmuir 1995, 11, 3882. (61) Ashok, J.; Varaprasad, P. L.; Birch, J. R. In Handbook of optical constants of solids II; Palik, E. D., Ed.; Academic Press: Orlando, FL, 1991; pp 957-987.
Computation of Lifshitz-van der Waals Forces
Langmuir, Vol. 17, No. 11, 2001 3335
Table 2. Parameters for the Water Model fj (eV2)
ωj (eV) 2.07 × 10-2 6.9 × 10-2 9.2 × 10-2 2.0 × 10-1 4.2 × 10-1
6.25 × 10-4 3.50 × 10-3 1.28 × 10-3 5.54 × 10-4 1.35 × 10-2
1.5 × 10-2 3.8 × 10-2 2.8 × 10-2 2.5 × 10-2 5.6 × 10-2
Ultravioletb 3.26 3.87 12.0 63.6 114.0 24.3
8.21 10.0 11.4 13.6 17.8 25.2 a
gj (eV)
Infrareda
0.63 0.84 2.05 3.90 7.33 5.43
From ref 65. b From ref 66.
The alkyl chains in the thiolate monolayer form a closepacked, crystalline layer with an approximately 30° tilt from the surface normal,62 making the layer birefringent, as well. The required formalism for including this effect in the Lifshitz analysis is available,63,64 but no such anisotropy effects have been included in the following, because in practice, the evaporated films that are commonly used in experiments are polycrystalline and rough, and the molecular orientation in the adsorbed monolayer is very unlikely to be the same over large surface areas. E. Water. The importance of interactions in aqueous systems in colloidal and biological systems can hardly be overestimated, and accurate representation of the dielectric properties of water is thus of significant scientific interest. The Ninham-Parsegian representation is commonly used, where Debye relaxations in the microwave region are added to damped harmonic oscillators in the infrared and ultraviolet regions. Data for the different contributions have been described and discussed by Gingell et al.65 for the IR contributions, and by Parsegian47 and Roth et al.66 for the UV regions, with the following result:
(iξ) ) 1 +
d 1 + ξτ
11
+
∑ j)1
fj
ωj2 + ξ2 + gjξ
(24)
One Debye relaxation term is included, for which d ) 74.8, and 1/τ ) 6.5 × 10-5 eV. Parameters for the five IR and six UV terms are listed in Table 2. This representation is popular because of the simplicity, and it also shows good agreement with calculations using spectral data67 and the Kramers-Kronig relationship (eq 18) (see Figure 5.) Bostro¨m et al. used the data in ref 67 and the temperature dependence of the surface tension to construct a temperature-corrected six-oscillator model of the dielectric function, with two Debye relaxation terms, three vibrational modes, and a single electronic absorption,68 which is included in Figure 5 for comparison. The differences between the three representations are rather small, but the Ninham-Parsegian representation described above (with data from Parsegian and Roth et al.) (62) Nuzzo, R. G.; Dubois, L. H.; Allara, D. L. J. Am. Chem. Soc. 1990, 112, 558. (63) Smith, E. R.; Ninham, B. W. Physica 1973, 66, 111. (64) Parsegian, V. A.; Weiss, G. H. J. Adhes. 1972, 3, 259. (65) Gingell, D.; Parsegian, V. A. J. Theor. Biol. 1972, 36, 41. (66) Roth, C. M.; Lenhoff, A. M. J. Colloid Interface Sci. 1996, 179, 637. (67) Querry, M. R.; Wieliczka, D. M.; Segelstein, D. J. In Handbook of optical constants of solids II; Palik, E. D., Ed.; Academic Press: Orlando, FL, 1991; pp 1059-1078. (68) Bostro¨m, M.; Sernelius, B. E. Appl. Surf. Sci. 1999, 142, 375.
Figure 5. The permittivity of water for imaginary frequencies, calculated using the full spectral data (ref 67) and the KramersKronig relationship (eq 18), the Ninham-Parsegian representation (eq 19) with parameters as described by Parsegian (ref 47) and Roth et al. (ref 66), and the oscillator representation by Bostro¨m et al. (ref 68).
has been used for calculations, on the ground of simplicity and a slightly better fit to the Kramers-Kronig analyzed optical data. IV. Numerical Procedures Because of the exponential factors in the integral in (2), the major contribution to the sum in (1) comes from frequencies ξ ∼ c/d. As was noted earlier, this corresponds to approximately 1015-1017 rad/s. For gold, titanium, and silica, (iξn) was calculated by integration of (18) between 1012 and 1020 rad/s for each frequency ξn, that is, over a frequency range extending considerably beyond the frequencies which contribute noticeably to the interaction. Interpolation of the spectral data was performed on a loglog scale, as recommended by Lambrecht et al.22 The integration was performed with an adjustable step length, kept at ≈x/100, and the accuracy was checked by changing the length to x/1000 for some randomly selected frequencies ξ, which consistently resulted in a change of
