Anal. Chem. 1992, 6 4 , 94-105
94
generally useful than those immobilized on hydrophobic substrates, such as XAD-2, for measuring free metal ion concentrations by the column equilibration technique. REFERENCES Cantwell. F. F.; Nielsen. J. S.; Hrudey, S. E. Anal. Chem. 1982, 54, 1498-1503. Chow, P. Y. T.; Cantwell, F. F. Anal. Chem. 1988, 60, 1569-1573. Swelleh, J. A.; Lucyk, D.; Kratochvll. B.; Cantwell, F. F. Anal. chem. 1987, 59, 586592. Product Bulletin, Pierce Chemical Co.,Rockford, IL, 1986. Sugawara, K. F.; Weetall. H. H.; Schucker, G. D. Anal. Chem. 1874, 46, 489-492. Marshal, M. A.; Mottola, H. A. Anal. Chim. Acta 1984, 158, 369473. Kdstad, A. K.; Chow, P. Y. T.; Cantwell, F. F. Anal. Chem. 1988, 60, 1565-1569. Wlllle, S. N.; Sturgeon, R. E.; Bennan, S. S. Anal. Chim. Acta 1983, 749. 59-66. Vernon, F.; Eccles, H. Anal. Chkn. Acts 1973, 63, 403-414. Parrish. J. R. Anal. Chem. 1982, 54. 1890-1892. Warshawsky, A.; Kallr. R.; Patchornick, A. J. J . Org. Chem. 1978, 43, 3151-3157. Warshawsky, A,; Kallr, R. J . Appl. P o i j ” Sei. 197s. 2 4 , 1125-1137. Trek. J.; Nielson, J. S.; Kratochvil, B.; Cantwell, F. F. Anal. Chem. 1983, 55. 1650-1653. Persaud, G. Ph.D. Thesis. University of Alberta, 1990.
(15) &Irdthelter, J. H.; Leib, R. I. J .
&T!y!; ~ N.
~
~
,
w .chem.1981, 26, 4078-4083.
~
- ~ , . ~ , 252-255, s ~ E.; Wlnget, G. D.; Winter, W.; Conndly, T. N.; Izawa. S.; Slngh, R. M. M. &bchemlptry 1988, 5 , 467-477. Smith, R. M.; Martell, A. E. Crltlcel Stability Constants; Plenum Press: New York, 1976; Vol. 4. Perrln, D. D.; Sayce, I.0. Talenta 1987, 14, 833-845. Martell, A. E.; Smlth, R. M. Criticel StebMty Constants; Plenum Press: New York, 1974; Vds. 1-5. Davles, C. W. Ion Assoclehlon, Butterworth: Toronto, 1962. Laltlnen, H.; H a d , W. E. Chemical Analysis, 2nd 4.;Mc(LawHI1I: Toronto, 1975; Chapter 11. Skoog,D. A.; West, D. M. Fundementals of Analytical Chmktry, 3rd ed.; Hdt, Rlnehert and Winston: Toronto, 1976; Chapter 4. Kragten, J. Atles of Metal-L@and EqdUbrk In Aquews SoluMons; John Wky and Sons: Toronto, 1978. Persaud, G.; Cantwell, F. F. Can. J . chem., In press. Myasoedova, G. V.; Sawin. S. B. Chelating Sorbents In Anelytlcal Chemistry; CRC Crltkal Revlews In Analytfcel chenWy; CRC Press: Cleveland. OH, 1987; Vol. 17(1).
(18) Good, (19) (20) (21) (22) (23) (24) (25) (26) (27)
RECEIVED for review August 1,1991. Accepted October 14, 1991. This work was supported by the Natural Sciences and Engineering Research Council of Canada and by the University of Alberta.
Four-Layer Theory for the Acoustic Shear Wave Sensor in Liquids Incorporating Interfacial Slip and Liquid Structure Wendy C. Duncan-Hewitt*
Faculty of Pharmacy, University of Toronto, 19 Russell Street, Toronto, Ontario, M5S 1Al Canada Michael Thompson
Department of Chemistry, University of Toronto, 80 St. George Street, Toronto, Ontario, M5S 1Al Canada
A theory Is developed which descrlbes the behavior of thickness shear mode acoustic wave sensors In llqulds. The arguments presented allow the lntroductlon of solld sensor, surfaceadjaoent lquld, transttbn, and bulk llquki layers. The rlgorous linear theory for plezoelectrlclty Is coupled to a treatment whlch describes lnterfaclal vkoslty In tenns of an actlvatlon energy barrler to Interfacial How. Surface free energy reflected In contact angle values can be correlated wlth the Interfacial s#p characteristks. Calculatlorw of serles resonance frequency, Impedance, and phase angle are presented for various condltlons.
INTRODUCTION The behavior of piezoelectric bulk acoustic wave devices type in liquids has been of the thickness shear mode (TSM) the subject of rising interest in recent times ( 1 ) . The significant body of experimental work, which has involved mostly the measurement of the series resonance frequency, has spawned a number of attempts to provide theories for coupling of the operating device to the liquid medium. One of the earlier empirical treatments demonstrated that a relationship between frequency and specific density and conductivity could be obtained for aqueous solutions (2). Again, on an empirical level it was shown that changes in frequency could be correlated with the density and bulk viscosity of liquids (3). In 1985, two simple models appeared which allowed the pre0003-2700/92/0364-0094$03.0010
diction of frequency changes on immersion of TSM devices in liquids using a number of sensor and bulk liquid properties. The treatment of Kanazawa and Gordon (4) introduces the quartz structure as a lossless elastic solid and the liquid as a purely viscous fluid. Frequency shifts arise from coupling the oscillation of the crystal, involving a standing shear wave, with a damped propagated shear wave in the liquid. An important element of this model is the assumption that the transverse velocity of the quartz surface is identical to that of the adjacent liquid layer. Here, the crystal does not drive the entire bulk of the liquid since transverse displacement dies exponentially. The decay length varies as the square root of the bulk liquid viscoeity and constitutes the effective thickness of the liquid treated as a rigid sheet. Using dimensional analysis,Bruckenstein and Shay (5) developed a similar model, which can be applied to the situations where the device is in contact with liquid at one or both faces. Both these theories appear to form a bridge between the classical Sauerbrey (6) gas-phase mass response approximation and the properties of a thin boundary film of liquid. Others have promoted a theory involving the influence of surface stress on frequency (7).Parabolic dependence of this parameter on the presaure between crystal faces can be related to the elastic energy stored in the quartz. Hager and coworkers (8) employed an analysis based on hydrodynamic coupling to evaluate fluid properties such as viscous energy losses and dielectric effects. The importance of the effects of surface rough” on the frequency found for the oscillating 0 1991 American Chemical Soclety
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~
ANALYTICAL CHEMISTRY, VOL. 64, NO. 1, JANUARY 1, 1992
device has been pointed out by Schumaker et al. (9). Again, the emphasis here is centered on the mass of liquid trapped in solid-phase cavities at the liquid-solid interface. Most recently, Martin et al. (10) have been able to combine an equivalent circuit model for the electrical behavior of the device with liquid-phase properties. The effect of interfacial viscosity on the response of the TSM device in water has been examined by Thompson and co-workers (11-13). Here, it is considered that perturbation of acoustic energy propagation could be caused by a partial slip boundary condition at the liquid-solid interface. This concept stands in sharp contrast to those described above where the so-called “no-slip” boundary condition is invoked universally. Obviously, for the interfacial viscosity argument to be accepted, it is necessary to introduce the possibility of slip at a solid-liquid interface in which one of the partners is in motion (an excellent review of this area was presented recently by Blake (14)). Furthermore, support for the slip argument comes from the work of Krim et al. (15) on layers adsorbed on piezoelectric devices and from studies of Israelachvili et al. (16) employing surface force microscopy on thin films. The authors believe that many of the discrepancies between theory and experiment arise because the microscopic size of the solid-liquid interface causes it to be overlooked the interface is viewed as a mathematical surface at which the properties of the system change abruptly from those of one bulk phase to those of another. A number of experiments and simulations (17-26) provide evidence that the properties of liquids (especially polar liquids) adjacent to solid interfaces are different from the bulk and that these properties may indeed influence the behavior of the entire system significantly. This paper describes the development and justification of a four-layer model of the TSM acoutic sensor in contact with a polar liquid such as water. Following an initial and necessary review of the present state of knowledge about the state of water at the solid-liquid interface, the model will be developed from first principles and justified on the basis of the experimental facta presented earlier. Ita behavior will be explored through the use of simulations and compared with the measured responses of well-characterized experimental systems. LIQUID-SOLID INTERFACE Surface-Adjacent Layers of Polar Liquids. The nature of a surface-adjacent layer of liquid has been evaluated experimentally when the surface is in contact both with bulk liquid and with air saturated to varying extents with the liquid under consideration. The latter case, while being expected to differ somewhat from the former experiment, is generally easier to perform and interpret because interference from the bulk is circumvented. Thus, infrared spectroscopy of quartz plates that have been equilibrated with saturated water vapor provides evidence that surface-adjacentlayers of water possess strengthened hydrogen-bonding (27, 28) and ellipsometric studies have provided estimates of the multimolecular film thickness of surface-associated water as a function of vapor pressure and the surface wettability (29). Additionally, calorimetric measurementa show that the specific heat capacity of surface-adjacent layers of water on quartz (30)is different from that of the bulk. The experimental results vary a great deal, and there is no generally accepted theory to explain and predict the observed behavior, perhaps because the peculiarities arise from the forces acting in the boundary zone which depend strongly on the lyophilic nature of the material comprising the solid (21, 24). Theories of wetting provide the framework in which the structure and behavior of the interfacial region may be evaluated and predicted. The equilibrium configuration of
95
a system consisting of three immiscible phases can take one of two configurations, one corresponding to the case where Yl2 713 + 7 2 3 (1) and one in which the right and left sides of eq 1 are equal:
where rijis the interfacial free energy between phases i and j (a complete list of symbols is attached in Appendix I). A n
inequality in the sense opposite to that expressed by eq 1is never expected to hold at equilibrium because, in that case, the free energy of the system could always be decreased by interposing a completely wetting layer of phase 2 between phases 1and 3 (31-33). Consider the situation in which phase 1is a solid and phases 2 and 3 are the liquid and vapor phases of a one-component fluid. Further, assume that the liquid completely wets the solid surface so that eq 2 holds. The thermodynamics of the situation implies that a macroscopically thick layer of the liquid is interposed between the solid and the vapor. When the vapor is unsaturated, the interposed layer remains but becomes microscopic in dimension. The associated adsorption isotherm is of the type 1form. Partial wetting, on the other hand, implies a type 2 isotherm. Under these conditions, an adsorbed layer forms but can only attain some finite microscopic thickness. Equilibrium analysis of liquid adsorption onto partially wetting surfaces shows that the adsorbed layer must be at equilibrium with bulk water at 100% relative humidity (34)and, thereby by inference, with the bulk when the surface is actually submerged in water. In other words, a surface-adjacent layer possessing a nature different from that of the bulk must exist even when the surface is submerged. Dash (31, 32) also defined a type 3 adsorption isotherm in which no measurable adsorption occurred before saturation of the vapor and which corresponded to complete nonwetting with a contact angle of 180°. For this system, a negative surface adsorption with respect to the bulk liquid phase would be observed. The interface would possess the same density as the vapor, and the vapor would perfectly wet the solid phase. The interaction of water with a hydrophilic wall should cause the liquid to become more ordered, giving rise to anisotropy of the liquid adjacent to the surface and birefringence as a consequence. This phenomenon has been observed experimentally (35).The dieledric constant of this more ordered layer is intermediate between that of the substrate (for example, that of quartz is approximately 4),and that of bulk (for example, the dielectric constant of water is approximately 80). McCafferty et al. (3s)showed that the dielectric constant of thin layers of water adjacent to surfaces increases rapidly to a thickness of about three molecular layers and then levels off at approximately 30. Palmer et al. (37) found that the dielectric constant of a 2 Nm thick layer of water between mica sheets is only 10. The theory of van der Waals forces shows that under theae conditions, the bulk water and the underlying surface repel one another. For an intermediate layer which is sufficiently thin, this repulsive force causes the bulk to exhibit a finite contact angle with the surface. The exact nature of the relationship between surface forces and thickneas has not been elucidated; however Adamson and co-workers (38-40)proposed a potential of exponential form: where ylvis the liquid surface tension, 6 is the contact angle, V,,,I~ is the molecular volume, y, is the layer thickness, and ao, ul, and a2,are material constants which characterize the behavior of the system under consideration. Frumkin-Derjaguin theory (41,42) provides a comparable approach in which the contact angle is given as a function of the integral of the disjoining pressure
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ANALYTICAL CHEMISTRY, VOL. 64, NO. 1, JANUARY 1, 1992
(4)
where II(y) is the disjoining pressure and y, is the thickness of the wall-adjacent film at zero disjoining pressure. In light of the fact that the structural component of the disjoining pressure has been shown to vary exponentiallyas a function of the interlayer thickness (43-46),the expressions can be shown to be of the same form. Strictly speaking, it is impossible to compare the behavior of systems of different composition in this framework unless the material parameters are known a priori, in which case, all systems appear to fit isotherms of identical form. In the present situation, the system is being used to measure interfacial changes and, presumably, to evaluate the parameters themselves. Some polar liquids such as nitrobenzene (47) form distinct liquid crystalline phases adjacent to a solid surface. So far, a similar phenomenon has not been observed for water (48). In interfaces between a solid and water it appears as though there is a transition zone in which the properties of the liquid gradually approach those of the bulk with increasing distance from the solid-liquid interface. The calculations of Ninham et al. (25,26,49)indicated that water close to hydrophilic surfaces should be more dense than the bulk but that, adjacent to a hydrophobic surface, the density should be smaller, predictions which are corroborated by the Monte Carlo and Molecular Dynamics simulations of water next to hydrophobic walls by a number of groups (5&52) and experimental evidence (53-55). Viscosity measurements of thin liquid layers adjacent to surfaces are fraught with problems such as contamination by dust and grease and long equilibration times, but the work of Churaev et al. is free of most of the associated errors. The viscosity of water in thin capillaries (48) and between montmorillonite particles (56) is increased. In a quartz capillary of radius 0.05 pm, the apparent viscosity is approximately 1.5 times that of bulk water. Kiseleva et al. (57) interpreted the data in a manner that predicted a surface adjacent layer 150-200 A thick. Similar behavior was not observed for either benzene or carbon tetrachloride. The authors concluded that the effect arose from structural changes of the liquid adjacent to the wall of the capillary. Increasing the temperature to approximately 70 “C annihilated the effect. Many other phenomena, such as irregularities in the surface tension temperature profile, also disappear at approximately the same temperature and all have been attributed to changes in water structure. Solid-Liquid Interfacial Energetics, Contact Angles, and Wetting. The interfacial tension between a solid and another phase (liquid-vapor) is defined as the reversible work required to form a unit area of new surface at constant temperature, volume, and mass of the system. Owing to the often insurmountable difficulties associated with measuring this parameter directly, attention has been focused on alternate methods based on the measurement of contact angles (58,59) which can be measured readily on most solids and are often used simply as empirical parameters to quantify wettability (60).
Thomas Young proposed that the contact angle between a liquid and a solid arises from the requirement that the interfacial tensions acting at the three-phase line must be balanced at equilibrium (61, 62): Yav
- Tal
Y I COS ~ 6
(5)
where yavis the solid-vapor interfacial tension, yal is the solid-liquid interfacial tension, ylvis the liquid surface tension, and 0 is the contact angle. General thermodynamic (63,64) and the hydrostatic (65) derivations of the Young equation
(eq 5 ) have been published more recently. The underlying assumptions of all these derivations are that (1)the surface is rigid and (2) the system is at equilibrium. Under normal conditions the effects of surface deformation will be small (66). The thermodynamic status of contact angles has been a subject of considerable debate (67-70) because it is difficult to establish whether the system is in equilibrium, a problem which arises because solids are able to sustain shear stresses. It is then necessary to differentiate between surface tension and surface stress (71) which are not generally considered to be equal in a solid except at high temperatures where surface diffusion mechanisms are relatively rapid. The interpretation of contact angles is complicated further by contact angle hysteresis, a phenomenon in which the contact angle formed by a liquid advancing a c r w an unwetted surface is generally larger than the contact angle of the same liquid as it recedes across the previously wetted surface. Several models have been advanced that predict the existence of metastable states and contact angle hystereais arising from surface heterogeneity and roughness (72-74). In 1937 Bangham and Razouk (75,761 called attention to the fact that the solid-vapor interfacial tension is d e r than the solid surface tension due to adsorption effects. While it is generally agreed that this factor must be considered, there is disegreement about the magnitude of the spreading pressure which gives the energetic change due to adsorption. For example, in the work of Good (77) and Fowkes (781, the spreading pressure is assumed to be negligible while more recent studies claim that the spreading pressure is considerable (79).
Most surfaces do not meet the requirements listed above, nevertheless many investigators have become convinced not only that contact angles are thermodynamically significant quantities but that a fundamental relationship exists which relates the liquid surface tension with the solid-liquid interfacial tension and the solid-vapor interfacial tension, the difference between the latter two parameters being calculated using Young’s equation (80-88). We do not adopt such approaches because of the difficulties listed above. Nevertheless, we view contact angles as rough guides of interfacial energetics. Furthermore, we subscribe to the view that the spreading pressure is not negligible (in fact, the assumption that a wetting surfaeadjacent layer forms for some partially wetting surfaces is inherent in the model described below). Surface Chemistry of Electrode Materials. Gold and aluminum are commonly used as electrode materials to excite piezoelectric structurea. While the latter material is commonly known to oxidize readily, it is often assumed that the former is relatively inert. In fact, many clean metallic surfaces (e.g. W, F‘t, Cu, Ni, Fe) (89) are generally fairly reactive with respect to water many decompose water to hydrogen and oxygen and then chemisorb oxygen or a hydroxide ion. The resulting surface is similar to those of oxides and therefore is hydrophilic. Although gold electrodes are the standard for TSM sensors,they have limited usage in most practical applications because of their susceptibility to passivation through surface oxide formation and their fouling and poisoning tendencies. In order to clean the surfaces, they are often cycled through anodic/cathodic triangular cycles (90,91). The relative magnitude of the surface free energy of gold and hence its water wettability has been in dispute for some time: while some authors insist that the clean gold surface is hydrophobic due to stress relaxation in the f i t few layers, others suggest that the material is inherently hydrophilic although it usually appears hydrophobic due to surface contamination by hydrocarbons. The lability of the surface character of the metal electrodes is a facile explanation for some of the variability observed in
ANALYTICAL CHEMISTRY, VOL. 64, NO. 1, JANUARY 1, 1992
97
The first and last layers of the four-layer model are the piezoelectric quartz resonator and the liquid overlayer. The necessity for the two intervening layers, a denser, surfaceassociated region and a rarified, partially wetting layer, was discussed above. Tiersten (92) provides the following set of equations which must be satisfied according to the linear theory of piezoelectricity:
= pqiij
(6)
=0
(7)
= CijkPkl - e k i j E k
(8)
Uijj
Dij 6ij
Ii
Flgwe 1. Fw-layer model. under conditkns of complete wetting (A), the sufa-djacent region is rdatively thick and the transitional region possesses the density and properties of the bulk IiquM. Under partial wetting conditions (B) the surface-adjacent region is thinner and the transitional region becomes rarified and less viscous. The relative density and viscosity of the layers Is indicated by the deqee of shading. PARTICLE DISPLACEMENT: u2= u3= 0 '%+d11+d12c
&%+d11+d12
dq+ dIi
db
ELECTRIC DISPLACEMENT: Dl=D3=0 ELECTRIC FIELD: EI = E?= 0
The analysis which follows is based on the solution of this set of equations when the deformation is restricted to pure strain in the plane of the quartz plate. Under these circumstances eH (the strain) is equal to (the particle displacement gradient) because the rotation tensor is equal to zero. While Tiersten's analysis is restricted to elastic deformations, we assume that a viscous element is added in parallel with the elastic element, the effect of which is to stiffen the system in a manner which depends upon the strain rate. Finally, it is assumed that the electric displacement, Di, is zero in the x and z directions so that by eq 7 Dw must be identically zero. Furthermore, the electric field is in the y direction and equal to zero in the x and z directions. Substituting eq 11in eqs 8 and 9 yields the relevant constitutive equations for the geometry under consideration: 621,2
= PqCl
(12)
Flgure 2. Geometry used to describe the four-layer model.
the response of TSM sensors and will be used in this paper to justify the modification of wetting parameters while bulk elastic and viscous properties of the sensor are held constant. WAVE EQUATIONS FOR THE AT-CUT QUARTZ TSM PIEZOELECTRIC DEVICE The model of the four-layer TSM sensor (Figures 1 and 2) will be developed for the AT-cut piezoelectric quartz crystal. The fmt layer consists of the quartz/electrode system. It will be assumed that the electrodes deposited onto the surfaces are thin and elastic so that they resemble a layer of quartz, albeit with a smaller quality factor. All changes in wettability are assumed to occur as a result of chemical changes on these electrode surfaces. The second layer is an ordered surfaceadjacent layer of liquid possessing a greater density and viscosity than the bulk, the thickness of which is a function of the solid surface-liquid interaction (see below). The third layer is a thin transition region between the surface-adjacent region and the bulk liquid. Its composition and behavior is influenced by the wetting characteristics of the system and is predicted from contact angle data. The fmal region is bulk liquid approximately 3 pm thick. In the development of the differential equations for the four-layer model the following conventions will be employed: (1) subscripts 1, 2, and 3 correspond to x , y, and z, (2) a variable subscripted thus u1,2 gives the partial derivative of u, with respect to y; (3) a dot over a variable indicates its partial derivative with respect to time.
The compressed notation for the elastic constant has been used. It is now assumed that a harmonic motion is imposed on the system so that the displacement can be represented as the product of the amplitude and an exponential function of the frequency:
u1 = uloe x p ( i o t )
(15)
When the partial derivativeswith respect to y of eqs 13 and 14 are taken and the results combined with eq 12, the following equations of motion may be derived: E66u1,2,2
-
= pqiil = - w 2 p q u 1
e26u1,2,2
- €224.22 =
(16) (17)
where = Ca + e&/tz2+ i q qis an effective complex shear modulus. Quation 16 is recognized as a harmonic differential equation whose characteristic equation and complex propagation vector are given by eqs 18 and 19, respectively:
ea
If A and B are undetermined coefficients,then the general solution of eq 16 is u1 = (A exp(ikqy) + B exp(-ikqy)) exp(iwt) 0 Iy Idq (20)
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ANALYTICAL CHEMISTRY, VOL. 64, NO. 1, JANUARY 1, 1992
The displacement and the electrical potential can be related by integrating eq 17 twice: (21) 4 = e26/t22u1 + EY+ F where E and Fare constants of integration ( i.e., independent of Y ) . EQUATIONS OF MOTION FOR SUBSEQUENT LAYERS Although in principle the complex_shearmodulus for each layer should have the same form as c66, we assume that the relaxation time for each of these subsequent layers is sufficiently short that they behave in a viscous manner. Because the subsequent layers are not exposed to the electric field, the electrodes being only on the two sides of the quartz, the first and thud terms on the right side in the equations possessing the form of eq 13 are equal to zero (712
= 9iu1,2
dq
IY
(22)
vhere vi is the viscosity of the layer in question. When these assumptions hold, kIl, k12, and kb are equal to (wfi1)0.5(i~1)4.5, and (~Pb)"~(ilm)~.~, respectively. A major part of this paper provides the justification and derivations of the relationships between k11, k12, 711, and 712, with measurable properties of the bulk liquid phase such as its viscosity, surface tension, and contact angle with the surface of the material constituting the resonator. Similar derivations for the subsequent layers result in eqs 23-25. u1 = (G exp(ikIly) + H exp(-ikIly)) exp(iwt) d, Iy Id, dI1 (23)
+
u1 = (J exp(ikI0) + K exp(-ikI0)) exp(iwt) u1 =
(24)
+ D exp(-ikby)) exp(iwt) dq + dI1 + dI2 5 y Idq + dI1 + dI2 + db
(25)
(C exp(ik0)
The boundary conditions for this system consist of those quantifyingthe electrical potential in the quartz, displacement of the mathematical dividing surfaces between the layers, and the shear stresses in all the layers. The first boundary condition stipulates that the potential at the bottom of the quartz resonator (y = 0) is determined by the applied voltage #o exp(-iwt) so that when eq 20 is substituted into eq 21, and both sides of the equation are divided by exp(iwt) (t >> 0) the result is e26
-(A
+ B) + F = 4 0
(26)
€22
The second boundary condition applies to the potential at the top of the quartz resonator (y = d,) which is determined by the applied voltage (-#o exp(-id)) e26
-(A
exp(ik,d,)
+ B exp(-ik,d,)
= G exp(ikIld,)
+ H exp(-ikIld,) G exp(ikIl[d, + dI1]) + H exp(-ikIl[d, + dI1]) = J exp(ikIz[d,
+ B exp(-ik,d,)) + Ed, + F = -&
€22
(27)
The third to fifth boundary conditions ensure the necessary mathematical continuity at the mathematical dividing surfaces that define the length of each layer in the system and is often described as the "no-slip" condition. This latter terminology is imprecise and is often interpreted as though slip or viscous flow is impossible in the interfacial regions. In fact, it is our argument that it is precisely such slip (or lack of slip) that gives rise to many of the apparent anomalies in the behavior of real systems. The mathematical continuity required to solve the differential equations implies that the displacement in the y direction of the two layers that meet at a given interface must be equal:
(28)
+ d111) + K exp(-ikIz[d, + d1~1)(29)
J exp(ik12[d, + dI1+ d12])+ K exp(-ikIz[d, + dI, + d1~1)= C exp(ikb[d, + dI1 + d121) + D exp(-ikb[dq + 4 1 + d121) (30) The shear stresses, which must also be equal at each interface, can be calculated by differentiating eqs 20 and 23-25 with respect to y and time and eq 21 with respect to y and inserting the resultant equations into eqs 13 and 22. After rearrangement, the following equations result: ik,&(A exp(ik,d,) - B exp(-ik,d,)) + e2$ = ikIm(G exp(ikIld,) - H exp(-ikIld,)) (31)
+ d111) - H exp(-ikIl[d, + ~ I J ) = ~~IZT exp(ikdd, I Z V + d111) - K exp(-ik~~[d,+ d1~1))
i k m 1 ( G exp(ikIl[d,
(32) ikIZtlIZ(J
exp(ik12[dq + 4 1 + 4 2 1 ) + dI1 + d12]))= ikb'lb(C exp(ikb[d, dI1 + d121) - D exp(-ikb[dq + dI1 + dI21)) (33)
+
K exp(-ik12[d,
Finally, the shear stresses at the two free surfaces (y = 0 and y = d, + d11 + d12 + db) must be equal to zero. i k , C & ( ~- B ) + e 2 a = o (34)
c exP(ikb[dq + 4 1 + dI2 + 4 2 1 ) D exp(-ikb[d,
+ dI1 Iy Id, + dI1 + dI2
d,
A exp(ik,d,)
+ dll + dIz + d12]) = 0
(35)
Equations 26 through 35 are 10 equations in 10 unknowns which may be solved by matrix methods using Cramer's rule. Of the unknowns calculated by this method, E is the most important because it is directly related to the electrical parameters measured by the circuit analyzer or resonance methods. The displacement current density is defied as follows (93): JD,
= aDi/at
(36)
Because there can be no moving charges in the dielectric, the conduction current density is equal to zero so that the total current density is equal to J D . When eq 21 is differentiated with respect to y and time and substituted into the time derivative of eq 14 and using the fact that E = Eo exp(iwt), it can be shown that J D 2 = J = -iweZ2E (37) The admittance per unit electrode area is the ratio of the current density over the potential difference between the top and bottom electrodes (240)and the impedance (Z)is simply the inverse of this quantity:
Z = 240/JD, = -2&/(-i~~22Eo)
(38)
Explicit equations for the admittance and impedance are shown in Appendix I1 of this article. For the purpose of demonstrating the behavior of this model, we assume that series resonance occurs when the impedance is a minimum and equal to the mimimum resistance. An equivalent method of locating resonance is to determine the minimum frequency at which the phase angle is equal to zero. The entire impedance curves and the phase behavior near resonance will be explored as a function of the wetting behavior. We require estimates of the thicknesses, densities, and viscosities of the two interfacial regions. These parameters
ANALYTICAL CHEMISTRY, VOL. 64, NO. 1, JANUARY 1, 1992
00
thermal activation with applied stress (forward direction )
no applied stnss
thermal activation with applied stress
distance (reaction or deformation coordinate)
c
Figure 3. Activation diagram for stress-induced flow. The details of the plot are discussed in the text.
will be expressed as products of the relevant bulk properties and parameters for two reasons: (1)The approach facilitates the mathematical simulations which reveal the relationship between the response of the TSM system and ita structure. (2) The model is useful only if all its components can be expressed as functions of measurable quantities. It is for this re88on that equationsthat give relative viscosities and relatwe densities are developed below.
PROPERTIES OF THE RARIFIED INTERFACIAL REGION Thickness. The thickneas of the transition region between two phases has been estimated both theoretically and experimentally to range anywhere from approximately 1 molecular diameter to 5 molecular diameters (94-97). Most experimental and theoretical work favors the thinner of these 80 for the purposes of this manuscript, it will be assumed that the thickness of the interfacial region is equal to 1 molecular diameter. Viscosity. Tolstoi (98)and later Blake (14) have explored the possibility of correlating solid-liquid interfacial viscosity with bulk liquid viscosity using the Frenkel theory of liquid bulk viscosity (99) and contact angle data. In Frenkel’s theory, flow occurs as a result of the motion of molecules into preexisting vacancies under the action of the applied shear stress so that the activation energy for flow is proportional to the energy required for vacancy nucleation. Likening the formation of a vacancy to the production of a spherical liquid surface, the model then equates this activation energy with the surface free energy of the liquid. Although Tolstoi and Blake’s adaptation of Frenkel’s theory is intuitively attractive, as it is described it possesses the following deficits: (1)The relaxation time is assumed to be a constant, independent of the properties of the system. Of course this kinetic parameter is inversely proportional to the rate constant which is characteristic of the system. (2) The interfacial properties are assigned to a i”ensional dividing surface in the Gibbsian sense (loo),yet flow is a three-dimensional process. The extent of the interfacial region strongly influences the predictions of the modeL For example, if the transitional region is several molecular layers thick, then flow in this region should be affected by the interfacial free energy alone. On the other hand, the behavior of a monomolecular layer would be influenced by the presence of the bulk phasea on each side. (3) The role that the equilibrium concentration of vacancies plays in deformation kinetics is omitted. All these points are addressed in the following modified model which is developed from first principles.
According to reaction rate theory, the rate of an arbitrary firsborder reaction (chemical, plastic or viscous deformation, transport, etc.) is equal to the product of the rate constant and the concentration of the reactants. Both these variables vary as a function of the strength of intermolecular forces.
R = k[lreac
(39)
k = KkBT/h exp(-AG*/kBT) (40) where R is the reaction rate, k is the first-order rate constant, K is the transmission coefficient (usually assumed to be approximately equal to one), T is the absolute temperature, kB is Boltzmann’s constant, h is Planck’s constant, and AG* is the activation free energy which can be calculated from the intermolecular potentials via the partition function (101). The manner in which the rate constant varies as a function of the intermolecular potentials (hence AG*) will be examined first. Consider Figure 3. For an unstressed mechanical system at equilibrium AG* is the same for both the forward (from (a) to (e)) and backward (from (b) to (e)) directions along any arbitrary flow direction so that the rate of activation is the same in both directions and there is no net flow. When stress is applied it does work, simultaneously decreasing the height of the activation barrier in the forward direction (from (d) to (e)) and increasing the height of the activation barrier in the backward direction (from (0 to (e)). Although the net deformation rate constant is the sum of the forward and backward constants, that for deformation in the forward direction quickly becomes overwhelming because the rate constants are exponential functions of the height of the activation barrier. Thereforewe need only consider the forward reaction in subsequent analyses. The rate constant can then be described by kdef = KkBT/h exp(-[AG* + wf]/kBT) (41) It can be shown that for steady flow, the work done by the applied stress is equal to the product of the stress and the activation volume, the latter of which is proportional to the size of a “flow unit” (a dislocation in a solid or a vacancy or hole in a fluid). If uVact
