Langmuir 2004, 20, 10617-10624
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Effects of Laterally Heterogeneous Slip on the Resonance Properties of Quartz Crystals Immersed in Liquids Binyang Du, Ilchat Goubaidoulline, and Diethelm Johannsmann* Institute of Physical Chemistry, Arnold-Sommerfeld-Str. 4, D-38678 Clausthal- Zellerfeld, Germany Received July 2, 2004. In Final Form: August 27, 2004 A formalism is presented which predicts the influence of laterally heterogeneous slip (for instance, induced by nanoscopic air bubbles) on the shift of the resonance frequency and bandwidth of quartz crystal resonators immersed in liquids. The lateral heterogeneities are decomposed into their Fourier components. The distribution of slip lengths provides a boundary condition, giving rise to a small, secondary flow field. The mean stress exerted by this secondary field induces a shift in resonance frequency and bandwidth. If the slip length is much smaller than the penetration depth of the shear waves and smaller than the lateral correlation length, one finds that the effects of the heterogeneities scale as n3/2, with n being the overtone order. The frequency, f, and half-band-half-width, Γ, decrease by the same amount. These calculations match experimental results obtained with a gold-coated resonator in contact with various hydrophilic liquids.
of eq 2, which is
Introduction Quartz crystal resonators are well-known tools for the determination of film thickness.1 According to the Sauerbrey equation,2 a film deposited on the crystal surface decreases the resonance frequency according to
δf -mf -2f0 ) ) m f mq Zq f
(1)
where δf is the frequency shift, f ) nf0 is the frequency, n is the overtone order, mf is the areal mass density of the film, mq ) Zq/(2f0) is the areal mass density of the quartz plate, f0 is the fundamental frequency, and Zq ) 8.8 × 106 kg m-2 s-1 is the acoustic impedance of a T-cut quartz. According to the Sauerbrey equation, the frequency shift scales as the overtone order, n. Equation 1 is the basis of the quartz crystal microbalance (QCM). In the 1980s, it was recognized that the QCM can also be operated in liquids, provided that the rather strong damping can be overcome.3 The presence of a Newtonian liquid also gives rise to a frequency shift. In this case, the Kanazawa equation holds:4
δf )
-f0 πnf0Fη πZqx
(2)
where F is the density and η is the viscosity. For liquids, the frequency shift scales as the square root of the overtone order. Liquids not only decrease the frequency but they also increase the dissipation as given by the bandwidth of the resonance. One can define a complex resonance frequency, f* ) f + iΓ, with Γ being the half-band-halfwidth (bandwidth, for short). There is a complex extension * Corresponding author. Phone: +49-5323-72 3768. Fax: +495323-72 4835. E-mail:
[email protected]. (1) Lu, C., Czanderna, A. W., Eds. Applications of Piezoelectric Quartz Crystal Microbalances; Elsevier Publishers: Amsterdam, The Netherlands, 1984. (2) Sauerbrey, G. Z. Phys. 1959, 155, 206. (3) Bruckenstein, S.; Shay, M. J. Electroanal. Chem. Interfacial Electrochem. 1985, 188, 131. (4) Kanazawa, K. K.; Gordon, J. C. Anal. Chim. Acta 1985, 175, 99; Borovikov, A. P. Instrum. Exp. Tech. 1976, 19, 223.
δf * δf + iδΓ i i i xFG ) xiωηF ) ) ) Z ) f0 f0 πZq l πZq πZq i-1 xωηF (3) x2πZq Here, Zl ) (FG)1/2 is the acoustic impedance of the liquid and G ) iωη is the complex shear modulus. Equation 3 predicts that the shift in the bandwidth is equal and opposite to the frequency shift. If one can be sure to deal with a strictly Newtonian liquid, this prediction can be used to check for certain artifacts. Newtonian behavior is, for instance, expected for water. The intrinsic relaxation times of water are in the gigahertz range.5 If, on the other hand, the liquid under study is viscoelastic, the complex viscosity, η*(ω) ) η′(ω) - iη′′(ω), can be explicitly derived from the shift of frequency and bandwidth by the relation
-(πZqδf*)2
η*(ω) ) η′(ω) - iη′′(ω) )
iFωf02
(πZq)2 2
Fωf0
)
(-2δfδΓ - i(δΓ2 - δf2)) (4)
There are several reasons for why the shifts in frequency and half-band-half-width often are not exactly the same, even if the liquid is perfectly Newtonian. First, there may be additional channels of dissipation such as coupling to the holder or emission of compressional waves.6 These effects would tend to make the shift in the half-bandhalf-width larger than the negative frequency shift because compressional waves withdraw energy from the resonator. A finite imaginary part of the viscosity, η′′(ω), would have the same effect (see eq 4). Experimentally, one frequently finds the opposite effect: the shift in the half-band-halfwidth is smaller than the negative frequency shift. (5) Hasted, J. B. Aqueous dielectrics; Chapman and Hall: London, 1973. (6) Lin, Z. X.; Ward, M. D. Anal. Chem. 1995, 67, 685.
10.1021/la0483515 CCC: $27.50 © 2004 American Chemical Society Published on Web 10/23/2004
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Figure 1. A data set taken with a 5 MHz quartz (Maxtek, Santa Fe Springs, CA) in water. The frequency shift, δf, and the shift of half-band-half-width, δΓ, are plotted vs the square root of overtone order, n1/2. The full line is a fit according to eq 3 with the single parameter η′. The imaginary part of the viscosity η′′ is zero in this model. The dashed line is a fit according to eq 4 with the two parameters η′ and η′′. The fit quality is much better but yields a negative value for η′′, which is physically impossible. The dotted line is a fit according to eq 5, where the fit parameters are the vertical scale of roughness, h, and the Newtonian viscosity, η′. Again, there remains a systematic difference between the data and the fit. This difference is caused by the fact that the Sauerbrey term scales linear with the overtone order, n, while the data scale with n in a different way. Equation 5 has no free parameter to fix this problem.
Figure 2. Experimental data of two quartz resonators in contact with water: (a) the frequency shift, δf; (b) the bandwidth shift, δΓ; (c) δΓ - δf; (d) δf + δΓ. The solid lines in parts c and d are fits of eq 49 to the data.
Actually, this often happens when the experimental conditions are particularly clean. In clear contradiction to the laws of physics, one often derives a negative value of η′′ by the application of eq 4 to the data. The classic interpretation of this discrepancy is trapped mass in conjunction with rough surfaces: the procedure of optical polishing as well as the thermal evaporation of electrodes results in rough surfaces.7 Liquid material located in the crevices moves rigidly with the crystal surface. It behaves like a thin film in the Sauerbrey sense. On a heuristic basis, one can decompose the frequency shift into a Sauerbrey term and a Kanazawa term according to
δf* -2nf0 i-1 ) Fh + πnf0Fη f0 Zq πZq x
(5)
where h is the vertical scale of roughness. Equation 5 can explain why the shift in the half-band-half-width may be smaller than the negative frequency shift. It suffers from two drawbacks: First, it is not clear how the parameter h is related to the geometrical roughness. Second, there (7) Martin, S. J.; Frye, G. C.; Ricco, A. J.; Centuria, S. D. Anal. Chem. 1993, 65, 2910-2922.
is a slight but systematic disagreement between eq 5 and the experimental data found when comparing data from many overtones. Figure 1 illustrates a feature which has been seen for many different surfaces. When plotting the shift in frequency, δf, and the shift in bandwidth, δΓ, versus the square root of the overtone order, one finds a slight downward curvature. The Kanazawa equation would predict a straight line. The frequency and bandwidth deviate from the Kanazawa line by the same amount. The quantities δf - δΓ and δf + δΓ are well approximated by a function proportional to n1/2 and n3/2, respectively (Figure 2). The experiments can be well represented by functions of the form
δf -1 ) πnf0Fη(1 + Bn) f0 πZqx
(6)
1 δΓ ) πnf0Fη(1 - Bn) f0 πZqx with one single dimensionless parameter, B. We propose laterally heterogeneous slip as the source of these deviations of the experimental data from the Kanazawa line. The occurrences of slip at solid-liquid interfaces as well as its physical reasons have been intensely discussed in the recent literature.8-11 Various cases need to be distinguished: First, sliding often occurs between solid surfaces under large lateral stress.12 This is a nonlinear phenomenon, which is much different from what we consider here. Second, slippage and even plug flow is common for polymer-solid interfaces because the entanglement network provides for a very high viscosity in the bulk. Since this network does not include the substrate, the rate of shear strain at the surface often is much larger than that in the bulk. This phenomenon also is outside the scope of this paper. With regard to simple fluids, surface slip was considered the exception rather than the rule for a long time, because the interaction between the molecules and the surface should be about as strong as the interaction between the molecules.13 It is not easy to see why the near-surface region should have a decreased viscosity. There is a class of experiments where the slip of monolayers of simple liquids has been investigated with the QCM.14,15 For monolayers, the local van der Waals pressure is small, which generates a situation much different from the case of the bulk. Even though this result came as somewhat of a surprise, it is now widely accepted that slip does occur at the solidliquid interfaces under certain conditions. At low frequencies, this kind of slip has been experimentally investigated with the surface force apparatus16-18 as well as with dynamic colloidal probe experiments.8,19-21 (8) Cho, J.-H. J.; Law, B. M.; Rieutord, F. Phys. Rev. Lett. 2004, 92, 166102. (9) Migler, K. B.; Hervert, H.; Leger, L. Phys. Rev. Lett. 1993, 70, 287-290. (10) Pit, R.; Hervet, H.; Le´ger, L. Phys. Rev. Lett. 2000, 85, 980. (11) Brochard, F.; de Gennes, P.-G. Langmuir 1992, 8, 3033-3037. (12) Persson, B. N. J. Sliding Friction; Springer: Heidelberg, Germany, 1998. (13) See, for example: Larson, R. G. The Structure and Rheology of Complex Fluids; Oxford University Press: 1999; Chapter 1.5.2. (14) Krim, J.; Widom, A. Phys. Rev. B 1988, 38, 12184. (15) Rohdahl, M.; Kasemo, B. Sens. Actuators, A 1996, 54, 448. (16) Kumacheva, E. Prog. Surf. Sci. 1998, 58, 75-120. (17) Zhu, Y.; Granick, S. Phys. Rev. Lett. 2002, 88, 106102. (18) Horn, R. G.; Vinogradova, O. I.; Mackay, M. E.; Nhan, P.-T. J. Chem. Phys. 2000, 112, 6424-6433. (19) Craig, V. S. J.; Neto, C.; Williams, D. R. M. Phys. Rev. Lett. 2001, 87, 054504. (20) Bonaccurso, E.; Butt, H. J.; Craig, V. S. J. Phys. Rev. Lett. 2003, 90, 144501.
Effects of Laterally Heterogeneous Slip
A number of researchers have also investigated highfrequency slip with quartz crystal resonators.22-30 There is agreement thatsto first order in the slip lengthsslip increases the frequency, which looks like a negative Sauerbrey mass.26 This makes sense intuitively: if the plane of extrapolated zero-shear is located inside the crystal, this has the same consequences as a hypothetical film with a negative thickness. Daikhin et al. have experimentally investigated the adsorption of pyridine onto gold and could explain their results if they assumed that the adsorbate introduced slip.24 Ferrante and coworkers have investigated the slip in a variety of different liquids and found that the slippage was larger in liquids of low viscosity.23 Thompson et al. have also looked into the question of whether trapped gas can explain the apparent negative Sauerbrey mass by virtue of its reduced density alone and concluded that this is not the case.28 Most recently, Ellis and Hayward determined the slip values for a number of different polar and nonpolar liquids with the QCM and found agreement with the literature values within 1 order of magnitude.30 A physical picture has emerged where there are several sources of slip.31,32 One potential explanation of slip is the formation of small gas bubbles at a surface which is not wetted well by the liquid.33,34 Actually, the formation of gas bubbles may be viewed as wetting of the surface by the gas.35 The comparison of the effects of hydrophobic and hydrophilic liquids indicates that this mechanism is at work in our experiments, as well, even though we do not claim that it is the only one. It is well-known that wetting on rough surfaces is a process which does not proceed in a laterally homogeneous way. The valleys and crevices of the surface are wetted first due to capillary condensation. Note that, in this case, the wetting phase is a gas rather than a liquid. One could term the phenomenon “inverse capillary condensation” in the sense that the crevices are the dry parts of the surface.22,36 If one adopts the view that gas bubbles are the source of slip, laterally heterogeneous slip is a natural consequence. The dry parts of the surface are expected to slip, whereas the wet parts should obey a no-slip condition. As a first attempt for modeling, one might argue that one can average over the heterogeneities and use the average slip length for viscoelastic modeling in the usual way. However, this picture does not explain our findings. The same is true for the more complicated expressions applying to larger slip lengths. We have therefore used a heterogeneous slip boundary condition in conjunction with (21) Neto, C.; Craig, V. S. J.; Williams, D. R. M. Eur. Phys. J. E 2003, 12, s71-s74. (22) Duncanhewitt, W. C.; Thompson, M. Anal. Chem. 1992, 64, 94105. (23) Ferrante, F.; Kipling, A. L.; Thompson, M. J. Appl. Phys. 1994, 76, 3448. (24) Daikhin, L.; Gileadi, E.; Tsionsky, V.; Urbakh, M.; Zilberman, G. Electrochim. Acta 2000, 45, 3615-3621. (25) Hayward, G. L.; Thompson, M. J. Appl. Phys. 1998, 83, 2194. (26) McHale, G.; Newton, M. I. J. Appl. Phys. 2004, 95, 373. (27) McHale, G.; Lucklum, R.; Newton, M. I.; Cowen, J. A. J. Appl. Phys. 2000, 88, 7304-7312. (28) Thompson, M.; Nisman, R.; Hayward, G. L.; Sindi, H.; Stevenson, A. C.; Lowe, C. R. Analyst 2000, 125, 1525-1528. (29) Lu, F.; Lee, H. P.; Lim, S. P. J. Phys. D: Appl. Phys. 2004, 37, 898-906. (30) Ellis, J. S.; Hayward, G. L. J. Appl. Phys. 2003, 94, 7856-7867. (31) Ponomarev, I. V.; Meyerovich, A. E. Phys. Rev. E 2003, 67026302. (32) Lichter, S.; Roxin, A.; Mandre, S. Phys. Rev. Lett. 2004, 93, 086001. (33) Vinogradova, O. I. Langmuir 1995, 11, 2213-2220. (34) Andrienko, D.; Dunweg, B.; Vinogradova, O. I. J. Chem. Phys. 2003, 119, 13106-13112. (35) Kipling, A. L.; Thompson, M. Anal. Chem. 1990, 62, 514. (36) Theisen, L. A.; Martin, S. J. Anal. Chem. 2004, 76, 796.
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the formalism proposed by Urbakh and Daikhin (originally put forward to predict the consequences of geometric roughness).37,38 If the bubbles have a lateral extension larger than the penetration depth, the outcome of the calculation is surprisingly simple: The parameter B in eq 6 turns out to be twice the mean-square variability of the slip length divided by the square of the penetration depth on the fundamental resonance frequency. We assume that the slip is small in the sense that the flow field above the quartz plate is similar (but not identical) to a pure shear flow. The boundary conditions introduce a secondary flow field. Since we assume linear acoustics, the superposition principle holds and the distribution of slip lengths can be Fourier-decomposed into sine waves of the wave vector q. The flow field can be calculated for each wave vector separately. The field induces a nonzero average stress at the surface, which in turn affects the resonance frequency and the bandwidth. Calculation of the Effects of Slip on Frequency and Bandwidth We start out from the general equation
i 〈σ〉 δf * δf + iδΓ i Zload ) ) ) f0 f0 π Zq πZq vs
(7)
where Zload is the impedance at the surface of the quartz. Zload is the ratio of stress, 〈σ〉, and lateral speed at the surface, vs. Zload is not a material constant. For liquids, one has Zload ) Zl and the Kanazawa relation results. For films, one has Zload ) iωmf, leading to the Sauerbrey equation. Note that we use the stress balance at the interface to derive the frequency shift rather than the integrated kinetic energy and the integrated dissipation. Urbakh and Daikhin have used the latter approach.39 In the following, we calculate the average stress above a planar and a structured surface on the basis of the Navier-Stokes equation and the boundary conditions provided in detail below. We assume that the Reynolds number is much less than unity and nonlinearities can therefore be neglected. We can decompose the flow field into its Fourier components and solve the linearized hydrodynamic equations for each Fourier component, separately. The Navier-Stokes equation is given by
F
dv - η∇2v + ∇p ) 0 dt
(8)
where v is the speed, F is the density, η is the viscosity, and p is the pressure. The lateral motion occurs along the x-direction at an angular frequency, ω. The surface normal is z. The fluid is assumed to be incompressible, leading to volume conservation as expressed by
dvx dvz + )0 dx dz
(9)
Note that, under conditions of incompressibility, the pressure, p, is not a true pressure but rather a variable which always assumes a value ensuring volume conservation. Smooth Surfaces and Pure Shear. A smooth, laterally oscillating surface with a no-slip boundary condition (37) Urbakh, M.; Daikhin, L. Phys. Rev. B 1994, 49, 4866. (38) Urbakh, M.; Daikhin, L. Langmuir 1994, 10, 2836. (39) Both approaches entail an implicit approximation because the displacement pattern inside the resonator is assumed to be given by a plane shear wave. By virtue of continuity, roughness and heterogeneous slip will launch more complicated wave patterns not only in the liquid but also in crystal.
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induces a pure shear flow, whose x-component varies along the z-direction according to
vps(x, z, t) ) vs exp(i(ωt - kz))
(10)
with k ) (-iωF/η)1/2 ) k′ - ik′′ ) (1 - i)/δ being the wave vector of shear sound and vs being the lateral speed of the substrate. The index ps stands for “pure shear”. The shear gradient dv/dz is proportional to vs/δ, with δ being the penetration depth of the shear wave in the liquid:
δ)
2η xωF
(11)
The stress exerted by the quartz onto the liquid is given by
(
)|
∂vx ∂vz + σ ) -η ∂z ∂x
z)0
) iηkvs )
1+i xFωηvs x2
(12)
From eq 12, one derives the Kanazawa equation (eq 3) by noting that δf*/f0 ) i/(πZq) σ/vs. Structured Surfaces and Secondary Flow Fields. For a structured surface, the field vps does not fulfill the boundary conditions. Before treating the boundary conditions in detail, we introduce a secondary flow field, v(x, z, t), and write vtot ) vps + v. Even without specifying the boundary conditions, certain statements on this field can be made: Since we assume linear hydrodynamics, the field v(x, z, t) can be decomposed into the Fourier series v(x, z, t) ) ∑ vq(z) exp(iqx) exp(iωt). The index q denotes the wave vector of the respective Fourier component. Each component vq(z) exp(iqx) exp(iωt) must satisfy linear hydrodynamics: We omit the index q from vq in the following. For the flow field with the lateral wave vector q along the x-direction, we use the ansatz
Equation 15 is a linear homogeneous equation system which only has a solution if the determinant vanishes
-(iωF + (q2 - R2)η)R2 + (iωF + (q2 - R2)η)q2 ) 0 (16) Using iωF/η ) -k2, this reads
(-k2 + q2 - R2)(q2 - R2) ) 0
(17)
Equation 17 has the following two solutions:
RI ) xq2 - k2 ) xq2 + 2i/δ2
(18)
RII ) q The plus sign has been used in the upper line in order to ensure that the wave decays at infinity. RI and RII correspond to two separate modes. The two modes have the amplitudes v0xI and v0xII. The parameter v0z is fixed by eq 14c. For mode I, we find
v0zI ) -
q v RI 0xI
(19)
For mode II, one has
v0zII ) -v0xII
(20)
We now use the boundary condition of the z-component at the surface (vz(z)0) ≡ 0) to fix the relative amplitudes of modes I and II. The condition of vanishing vertical flow at the surface implies that
[v0zI + v0zII] sin(qx) ) 0
(21)
Using eqs 19 and 20, this leads to
vx(x, z) ) v0x cos(qx) exp(-Rz) vy(x, z) ) 0 vz(x, z) ) v0z sin(qx) exp(-Rz)
v0xII ) (13)
p(x, z) ) p0 sin(qx) exp(-Rz) where R is a complex decay constant. Inserting this ansatz into eqs 8 and 9, we arrive at the following system of linear equations:
iωFv0x cos(qx) - η(-q2 + R2)v0x cos(qx) + qp0 cos(qx) ) 0 (14a) iωFv0z sin(qx) - η(-q2 + R2)v0z sin(qx) Rp0 sin(qx) ) 0 (14b) -qv0x sin(qx) - Rv0z sin(qx) ) 0
(14c)
q v ) -(1 + 2i(qδ)-2)-1/2v0xI RI 0xI
(22)
v0xI is the only free amplitude. For simplicity, we term it v0 in the following. v0 is fixed by the x-component of the boundary condition. The x-component of the flow field is given by
[
vx(x, z) ) v0 exp(-RIz) -
]
q exp(-qz) cos(qx) (23) RI
If the q-vector is perpendicular to the direction of shear (that is, along the y-direction), the situation is simpler. We make the ansatz
vx(x, z) ) v0 cos(qy) exp(-Rz) vy(x, z) ) 0 vz(x, z) ) 0
(24)
p(x, z) ) 0
or equivalently:
[iωF + (q2 - R2)η]v0x + [q]p0 ) 0 [iωF + (q2 - R2)η]v0z + [-R]p0 ) 0 [-q]v0x + [-R]v0z ) 0
(15)
where, again, R is a complex decay constant. Inserting this ansatz into the Navier-Stokes equation (eq 8), we find
iωFv0 cos(qy) - η(-q2 + R2)v0 cos(qy) ) 0
(25)
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or equivalently
[iωF + (q2 - R2)η]v0 ) 0
(26)
Remembering that iωF/ηt ) -k2, eq 26 is solved by
RI ) xq2 - k2 ) xq2 + 2i/δ2
(27)
If the q-vector is perpendicular to the direction of shear, there is just one mode, the amplitude of which again is determined by the boundary condition of the lateral flow component. Boundary Condition with Finite Slip. Up to eq 27, our calculation is equivalent to the small-roughness calculation done by Urbakh and Daikhin.37,38 In the following, we use a different boundary condition because we claim that slip, rather than roughness, determines the deviations from the Kanazawa line. For the lateral wave vector q along the x-direction, the boundary condition reads
∆b(x) ) ∆b0,q cos(qx)
(29)
where ∆b0,q is the amplitude of the slip length variation at the wave vector q. For reasons of continuity, the wave vector q used here must be the same as the wave vector q in the flow field (eq 13). Again, we omit the index q from ∆b0,q in the following. According to eq 29, the average of ∆b is zero. We have moved the reference plane to a position such that the average slip length is zero; that is, we eliminated uniform slip from our analysis. The reference plane is slightly displaced with respect to the quartz surface. We only consider fluctuations of the slip length around its average. Inserting eqs 29 and 23 into eq 28, we find
{
]
}|
d v (x, z) - ikvs exp(-ikz) ∆b0 cos(qx) dz x
RI(j) ) xjq2 - k2, j ) 0, 1, 2
z)0
A complication has appeared: In the form of eq 23, vx(x, z) contains the factor cos(qx). As a consequence, eq 30 contains both a term proportional to cos(qx) and a term proportional to cos2(qx). It cannot be fulfilled with a secondary flow field of the form given in eq 23. This problem can be stated in another way: since cos2(qx) ) 1/2(1 + cos(2qx)), eq 30 contains terms with different wave vectors, which are 0, q, and 2q. The boundary condition couples different Fourier components to each other. There is no one-to-one correspondence between the Fourier components of the surface heterogeneity and the Fourier components of the flow field. Equation 30 can only be fulfilled with a secondary flow field containing multiple Fourier components:
(32)
Inserting eq 31 into eq 30, we find
(
v0(0) + v0(1) 1 -
)
q
RI(1)
cos(qx) +
(
{
[
∆b0 cos(qx) -RI(0)v0(0) + -RI(1) +
[
-RI(2) +
4q
2
]
RI(2)
q2
2q
]
(1)
RI
)
RI(2)
cos(2qx) ≈
v0(1) cos(qx) +
v0(2) cos(2qx) - ikvs
}
(33)
Using the trigonometric identities cos2(qx) ) (1 + cos(2qx))/2 and cos(qx) cos(2qx) ) (cos(qx) + cos(3qx))/2 and collecting the coefficients to the various wave vectors, one finds a set of the following three equations:
(
)
1 q2 v0(0) ) - ∆b0v0(1) RI(1) - (1) 2 R
(
v0(1) 1 -
[
q
)
RI(1)
(0)
- ik∆b0(v0
(
I
)
v0(2) 1 (30)
]
We now deal with the three amplitudes v0(0), v0(1), and v0(2), where the superscripts indicate the wave vector. In principle, one should include an infinite number of Fourier components. However, it turns out that the stress resulting from the higher components eventually averages out. It suffices to consider three components only. For the complex decay constants, eq 18 becomes
(28)
z)0
where b(x) is the slip length. We again decompose the variability of the slip length into its Fourier components and only consider a certain wave vector, q:
vx(x, z)|z)0 )
[ [
v0(2) 1 -
|
dvtot,x(x, z) vx(x, z)|z)0 ) b(x) dz
vx(x, z) ≈ v0(0) exp(-RI(0)z) + q v0(1) exp(-RI(1)z) - (1) exp(-qz) cos(qx) + RI 2q (2) (2) v0 exp(-RI z) - (2) exp(-2qz) cos(2qx) (31) RI
( (
1 4q2 + vs) + ∆b0v0(2) RI(2) - (2) 2 R
2q
)
(2)
RI
I
)] )
(34)
1 q2 ) - ∆b0v0(1) RI(1) - (1) 2 R I
where the relation RI(0) ) ik has been used. Solving this equation set, we obtain
1 v0(0) ) i∆b02k(RI(1) + q)vs 2 (1)
v0
v0(2) )
)-
i∆b0kRI(1)
vs
(RI(1) - q)
(35)
i∆b02k(RI(1) + q)RI(2) vs 2(RI(2) - 2q)
We again write down the full form of the secondary flow field induced by the laterally heterogeneous slip at the surface
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1 vx ) i∆b02k(RI(1) + q)vs exp(-ikz) 2 i∆b0kRI(1) q vs exp(-RI(1)z) - (1) exp(-qz) cos(qx) + (1) (RI - q) RI
[
2
(2)
]
[
(1)
+ q) vs exp(-RI(2)z) 2(RI - 2q) 2q exp(-2qz) cos(2qx) RI(2) (36) i∆b0kRI(1) q (1) vs - (1) exp(-RI z) + vz ) (RI(1) - q) RI i∆b0 kRI (RI (2)
q RI(1)
]
[
2
exp(-qz) sin(qx) + 2q
exp(-RI2z) + (2)
RI
]
(2)
(1)
[
+ q) vs (2) 2(RI - 2q)
i∆b0 kRI (RI
]
2q
exp(-2qz) sin(2qx) (2)
RI
This secondary flow field solves linear hydrodynamics and the partial-slip boundary conditions. For uniform slip or a no-slip condition (∆b0 ≡ 0), this secondary flow field vanishes. The secondary flow field is now fully specified, and we can proceed to the calculation of the shear stress induced at the interface. Importantly, we only have to consider the space-averaged stress. This much simplifies the equations. Only the component of the flow field independent of x does not average out. We have
〈σ〉 ) -η
〈(
)| 〉
∂vx ∂vz + ∂z ∂x
∂ ) -η (vx(0) exp(-ikz))|z)0 ∂z
(37)
1 ) - ∆b02ηk2(RI(1) + q)vs 2 According to eq 7, the secondary flow component induces a complex frequency shift
δfs + iδΓs i 〈σ〉 i ) )∆b 2ηk2(RI(1) + q) (38) f0 πZq vs 2πZq 0 where the index s indicates the part of the shift related to slip. In the large-q limit (q . k, RI(1) ∼ q), this result is approximated by 2 δfs + iδΓs i 2 η ∆b0 ≈∆b02ηk2q ) qδ ) f0 πZq πZq δ δ2 2 Fωη∆b0 qδ (39) 2 δ2
x
where k2 ) -2i/δ2 has been used. For large q, the secondary flow field only affects the frequency shift and the effect scales as n. In the limit of a large lateral scale (q , k, RI(1) ∼ ik), eq 38 approximates to 2 δfs + iδΓs ∆b02ηk3 1 + i η ∆b0 ≈ )) f0 2πZq πZq δ δ2
-
2 Fωη∆b0 (40) 2 δ2
x
1+i πZq
∆b(y) ) ∆b0 cos(qy)
(41)
The boundary condition reads
v0(0) + v0(1) cos(qy) + v0(2) cos(2qy) ≈ ∆b0 cos(qy)[-RI(0)v0(0) - RI(1)v0(1) cos(qy) (2)
(2)
(42)
cos(2qy) - ikvs]
RI v 0
Collecting the coefficients to the different Fourier components and solving this equation system, one finds
1 v0(0) ) i∆b02kRI(1)vs 2 v0(1) ) -i∆b0kvs
(43)
1 v0(2) ) i∆b02kRI(1)vs 2 For the calculation of the stress, we again only need the component independent of x.
δfs + iδΓs i σ i ) )∆b 2ηk2RI(1) f0 πZq vs 2πZq 0
z)0
2 πZq
Since ω ∝ n and δ ∝ n-1/2, eq 40 contains the n3/2 dependence which is also found in experiment. Also, the frequency and the bandwidth decrease by the same amount. They are independent of q. If the q-vector is perpendicular to the direction of shear, there is no vertical flow. Also, there is only one mode with the decay constant RΙ. The equivalent of eq 29 is
(44)
In the large-q limit (q . k, RI(1) ∼ q), this result is approximated by 2 δfs + iδΓs i 1 η ∆b0 )∆b02ηk2q ) qδ ) f0 2πZq πZq δ δ2
-
2
x
Fωη ∆b0 qδ (45) 2 δ2
1 πZq
Apart from a numerical factor of 1/2, eq 45 is the same as eq 39. In the small-q limit, we find 2 δfs + iδΓs ∆b02ηk3 1 + i η ∆b0 ) )) f0 2πZq πZq δ δ2
x
1+i πZq
2
Fωη ∆b0 (46) 2 δ2
Equation 46 happens to be the same as eq 40. We now focus on the small-q limit. As eqs 40 and 46 show, the frequency shift in this limit does not depend on q. The above discussion was limited to one single q-vector. However, as long as q is smaller than δ-1, we may consider a random superposition of all Fourier components, rather than a single component, and use the mean-square variability of the slip length, 〈∆b2〉, instead of ∆b02. Replacing the amplitude squared by the mean square introduces a factor of 2. With these substitutions, we find
δfs + iδΓs 1+i )f0 πZq
2 Fωη 2〈∆b 〉 2 δ2
x
(47)
Effects of Laterally Heterogeneous Slip
Langmuir, Vol. 20, No. 24, 2004 10623
Table 1. Parameters Obtained by Fitting eq 49 to the Experimental Data Shown in Figures 2, 4, and 5a
water #1 water #2 ethanol 2-propanol toluene cyclohexane a
F (g/cm3)41
ηlit (cp)41
ηmeas (cP)
∆brms (nm)
kf (cP/nm)
0.997 0.997 0.790 0.785 0.865 0.778
0.89 0.89 1.074 2.038 0.56 0.894
1.000 1.003 1.229 2.338 0.570 0.873
11.0 9.5 12.0 17.8
