Surface morphology and the quartz crystal microbalance response in liquids

COLLOIDS

ELSEVIER

Colloids and Surfaces A: Physicochemical and Engineering Aspects 134 ( 1998 ) 75-84

AND A SURFACES

Surface morphology and the quartz crystal microbalance response in liquids Michael Urbakh *, Leonid Daikhin School ~" Chemistry, Raj'mond and Be verly Sackler Faculty of Exact Sciences, TeI-Aviv University, 69978. Ramat-Aviv, Tei-Avil; Israel Received 15 July 1997; accepted 29 July 1997

Abstract

The effect, of surface roughness and the morphology of non-uniform surface films, on a quartz crystal microbalance (QCM j response in contact with a liquid has been investigated. We used a perturbation theory for the description of the effect of ~!ight roughness and an approach based on Brinkman's equation for the case of strong roughness. In the latter approach one treats the flow of a liquid through a nonuniform surface layer as the flow of a liquid through a porous medium. Our description includes both the effect of viscous dissipation in the interfacial layer and the effect of the liquid mass rigidly coupled to the surface. A relation between QCM response and the interface geometry has been found. The models discussed here can be used for the treatment of the QCM response of rough electrode surfaces, of porous deposited films and of surface polymer fihns. ~c,~1998 Elsevier Science B.V.

K¢:t'word~'." Microbalance: Quartz crystal; Roughness: Solid liquid interfaces; Thin films I. Introduction

In the last few years there has been an increased attention to the application of quartz crystal microbalance ( Q C M ) for studies of solid-liquid interfaces [!-3]. It was successfully demonstrated that Q C M can serve as a sensitive tool to probe interfacial friction [4], thin film viscoelasticity [5,6], polymer film properties [2,3] and bulk liquid viscosity and density [7-9]. The combination of the QCM-technique with electrochemical methods has made possible in situ measurements of minute mass changes which take place during adsorption, underpotential deposition, dissolution of surface films and other electrochemical processes [2, 3, 10]. in most of these investigations, frequency changes * Corresponding author. E-mail: urbakhfi~ ccsg.tau.ac.il 0927-7757/98;$19.00 C~1998 Elsevier Science B.V. All rights reserved. PIi S0927-7757( 97100334-8

were interpreted m terms of rigid mass changes, based on the Sauerbrey equation [11]. In liquids, however, the Q C M response depends on various factors such as an electrode microstructure, the morphology of surface films, interfacial liquid properties and on the solid-liquid coupling at the interface. In many cases frequency changes are not consistent with the predictions given by the Sauerbrey equations [ I, 12, ! 3] and a detailed theoretical analysis is needed to extract information from the experimental results. Progress has been made in characterizing the response of a smooth resonator operating in contact with a Newtonian liquid [7-9,14]. The oscillating surface generates a plane-parallel laminar flow in the liquid that causes a decrease in the resonant frequency and the resonator damping, proportional to (pl/) ~'2 where p and Jl are the liquid

76

M. Urbakh, L. Daikhin / Colloids Surfaces A: Physicochent Eng. Aspects 134 (1998) 75-84

density and viscosity. The influence of the surface microstructure on the QCM response in contact with liquids has only just begun to be investigated [15-23]. When the surface of the resonator is rough, the liquid motion generated by the oscillating surface becomes much more complicated than for the smooth surface. A variety of additional mechanisms of coupling between acoustic waves and a liquid motion can arise, such as the generation of a nonlaminar motion, the conversion of the in-plane surface motion into the surfacenormal liquid motion and trapping of liquid by cavities and pores. It has been experimentally demonstrated [16-18,20,21] that a roughnessinduced shift of the resonant frequency includes both the inertial contribution, due to the liquid mass rigidly coupled to the surface and the contribution due to the additional viscous energy dissipation caused by the nonlaminar motion in the liquid. Measurements of the complex shear mechanical impedance [20] have been used to analyze different contributions to the roughnessinduced response of the quartz resonator and to correlate the experimental results with the device rougAln,,, S. Nevertheless this subject is highly undeveloped and the interpretation of experimental results is ambiguous. This prevents the QCM technique from being widely employed as an analytical tool for studies of the solid-liquid interface. It is impossible at present to provide a unified description of the QCM response for nonuniform solid-liquid interfaces with an arbitrary geometrical structure. The limiting case of slightly rough surfaces has been considered in our recent works [22, 23]. The calculations demonstrated the generation of the liquid motion normal to the surface oscillations and the non-uniform distribution of the liquid pressure in the interfacial layer. In this case both inertial and viscous components contribute to the roughness-induced QCM response. The quantitative relation between the observable quantities (the resonance frequency shift and width) and the surface morphology have been found [22,23]. In the present paper we have combined a perturbation theory approach [22] to the description of the effect of slight roughness with the coarsegrained description for the strongly rough inter-

faces. The last approach is based on Brinkman's equation for the velocity field in the interfacial region [24]. In this approach one treats the flow of a liquid through a non-uniform surface layer as the flow of a liquid through a porous medium [2529]. The morphology of the interfacial layer is characterized by a local permeability that depends on the "porosity" of the layer. The coarse-grained description gives a unified approximate description of the QCM response for various scales of roughness. It includes both the effect of viscous dissipation in the interfacial layer and the effect of the liquid mass rigidly coupled to the surface. The model discussed here can be also used for the treatment of QCM response of porous deposited films and of surface polymer films.

2. Description of the models We no,,~ consider a model for the coupling of shear waves in a piezoelectric crystal bounded by the rough surface with damped waves in a liquid. The solid surface profile may be specified by a single valued function ((x,y) giving the local height of the solid with respect to a reference plane. We take : axis pointing towards the fluid and the plane - = 0 to coincide with the unconstrained face of the quartz resonator. The second constrained face of the resonator coincides with the plane z = d. Here d is the average thickness of the quartz crystal film. For convenience, we choose the position of the reference plane : = d so that the spatial average of the solid profile function g(x,y) vanishes. (see Fig. 1).

Liquid

l a I T

.

×

Fig. 1. Schematic sketch of interfacial geometries for rough interfacial layer.

M. Urbakh, L. Daikhin / ColloidsSurfaces A: Physicochem. Eng. Aspects 134 (1998) 75-84 In order to determine the effect of roughness on the resonance frequency of the quartz crystal we have to find the solutions of a wave equation for elastic displacements in the quartz crystal and of the linearized Navier-Stokes equation for fluid velocities [30]. The solutions must satisfy the boundary conditions which include: (1) the absence of forces acting on the unconstrained crystal surface z = 0; (2) the equality of crystal and fluid velocities at the interface z = ((x.y); (3) the equality of the absolute values and the opposite directions of the shear stress on the fluid side of the interface and of the shear stress on the quartz side. The non-slip boundary condition Eq. (2) is one of the fundamental assumptions in fluid mechanics [30]. While experiments at macroscopic scales are consistent with this condition, the recent measurements which probe molecular scales indicate that the boundary conditions may he different [4,31]. Molecular dynamics simulations [32,33] have shown that the degree of slip at the interface is related to the fluid structure induced by the solid. At large interactions between fluid molecules and the substrate the first one or two fluid layers became locked to the wall. We will focus on this case only. The form of the dependence of the resonance frequency on the properties of the fluid and on the interface morphology is determined by the relations between the characteristic sizes of surface roughnesses and the length scales defined by the Navier-Stokes equation for liquid velocilies and by the wave equation for the elastic displacement in the crystal. These length scales are the decay length of liquid velocities, 6=(2tlltop)~/2 and the wavelength of the shear-mode oscillations in the quartz crystal, 2 = 27r( #q//Oq )!/2(0 - 1;

w h e r e llq and pq are the shear modulus and the density of the quartz crystal and to is the frequency of oscillations. For the frequencies used in QCM, to~ i-10 MHz, the lengths 6 and 2 are of the order of 0, l - l lam and 0. l cm, respectively.

77

The STM measurements show that the characteristic heights of the roughness of metallic films on the quartz crystal are in the region of tens or hundreds of nanometers [ 15,17]. The lateral scales of roughness can change over a wide range, from tenths of nanometers to hundreds and even thousands of nanometers. As a result we are confronted with both "slight" roughness, for which the characteristic height is less than the lateral sizes, and "strong" roughnesses (cavities, pores and bumps) for which the height is of the same order or larger than the lateral size.

3. Results and discussion

3.1. Slightroughness First we discuss the effect of slight roughnesses on the QCM response. This problem has been recently considered by us in the framework of the perturbation theory [22] with respect to the parameters [V~(R)I< 1 and h/6 < 1. For the roughness described by a one-scale correlation function the fluid induced shift and the width of the resonance frequency has been written in the form [22]

(t"o)a/2(Pll)l/2 { } + h2 ! F(l/6) A(°=co--t°°=-

~(2pq/./q ) 1/2

F

'

(1)

(CO0)3/2(PlI)I/2 { AF=F-Fo=

/r(2pqlU q )1/2

l+

h2

)

F

} (2)

In writing Eqs. (1) and (2) we introduced the following notations: h and i are the root mean square (rms) of height, h and the lateral scale length, L which has the meaning of the correlation length for a random roughness and the period for a periodical corrugation, COo= (n/d)(llq/pq)l/2 and Fo are the frequency and the width of the resonance of the free quartz crystal film. The scaling

78

M. Urbakh. L. Daikhin / ColloMs Surfitces A: Physicochem. Eng. Aspects 134 (1998) 75-84

functions F(l/6) and ~(l/i)) have the following form

~ d2q F(l/i)) = V~(l/i)) j (21t)2 g(q)q( {[1 + 4(qi)/i ) 4] ~/2

of these functions for 1/i)>>1 and l/i)< ! are universal.

-

+ 1 }~/2_V~(l/&&)/q+V~cos 2 q~),

t" d2q

(3)

q~(l/6)

g(q)q{V~-(l/O)/q

- [( I +4(q611)-4)1!2 _ ! ]ii2 }

(4)

Here q = gl, where g is the two-dimensional wave vector, the function 12g(gl) is the two-dimensional Fourier transformation of the pair correlation function of roughness; ~b is the angle between the direction of shear oscillations and the vector K. The correlation function 12g(gl) is related to the Fourier component of the surface profile function ~(g) by the equation I~(K)]2 =S h 2 12g(Kl),

(5)

where S is the area of a crystal surface. It should be mentioned that the roughness factor R, i.e. the true-to-apparent surface ratio, is proportional to the square of the "slope" of the roughness, hi! and can be written as

h2 I d2q R= ! + ~ , ] ( 2 n ) 2 g(q)

q2.

F(I/i))= l/i)

(6)

The first terms in the braces ill Eqs. ( 1 ) and (2) define the shift, Ato~ and the broadening, AF,, of the shear resonance at the smooth crystal-liquid interface [8,34] respectively. The presence of surface roughness leads to the additional decrease, AtOrh and broadening, Af'rh, of the resonance frequency. We see that the roughness induced shift and width are proportional to the square of the slope of the roughness, h/I and to the one-parameter scaling functions, F(l/i)) and ~(I/6). The scaling parameter, I/i), is the ratio of the characteristic lateral length of the roughness to the decay length of the fluid velocity. The particular form of the scaling functions F(lfl~) and ~(I/i)) is determined by the morphology of the interface. However the asymptotic behaviors

S ~ + : q 6/i

at I/6 >>1

(1 1 + & l/i), at l/i)<<1'

= I/6(ylj i)/1, at i/6 >>1. G'2 l/i), at i/t)<< 1

(7) (8)

Eqs. ( 1), (7) and (8) demonstrate that in both limits, i/6> > 1 and l/&&<<1, the roughness induced shift of the resonance frequency is proportional to the fluid density and slightly depends on the viscosity. We also remark that in these asymptotic regions the shift is proportional to the square of the frequency of the free quartz crystal, too2, like in the case of mass loading [I]. However, our results show that the influence of slight surface roughness on the frequency shift cannot be explained in the terms of the mass of fluid "trapped" by surface cavities as was proposed in the previous papers [8,9]. This statement can be illustrated by the consideration of the sinusoidal roughness profile studied before. The mass of the fluid "'trapped" by the sinusoidal grooves does n o t depend on the slope of the roughness, h/l and is equal to hS/2. However Eq. ( 1 ) demonstrates that the frequency shift At,Jrh increases with the increase of the slope. For all values of the parameters (P, q, t,~o) the roughness induced width of the resonance is less than the roughness i:;.duced shift. Eqs. (2) and (8) show that in the regions I/6 > > 1 and //6<<1 the width, AF, h, is proportional to the factors (pll)l/2f03/2 and (p)3/2(Vl)-l/2(DSo/2, rcspectively. Fig. 2 presents the dependencies of the frequency shift on a liquid viscosity calculated for the Gaussian correlation function of roughness, "~ -~ ]2g(K[)= n] 2 exp(-l'.~-/4). We see that the roughness leads to new vi,,,-:~sity dependencies which do not appear in the case of a smooth surface. The effect of a roughness is most pronounced in the low viscosity limit when the fluid induced shift, Ato s, at a smooth surface is small. The roughness induced shift tends to a constant in the high viscosity limit when tl>12taop/2. The results obtained allow to estimate the effect of roughness on the QCM response if the surface profile function

M. Urbakh, L. Daikhin / Colloids Surfaces A: Physicochem. Eng. Aspects 134 (1998) 75-84

-•~

8

~,,,.~

..,.,~"P

6

alj al~

an

am

at'

ill

~llml°'

mum

mem

~

4

~

2

e

i

0.0

0.5

i

1.0

i

i

i

1.5

2.0

2.5

79

meability ~ can change with the distance from the quartz surface [36]. Here, for simplicity, we do not consider this effect, in our model gH ~ presents an average permeability of the layer. We assume that the solid phase (surface roughnesses, a porous deposit, a polymer film) is rigidly coupled to the crystal surface and oscillates with the velocity Iio exp(kot). The liquid flow through the interfacial layer, vx(z,t)=vx(z,to)exp(itot), is described by Brinkman's equation [25-29] d2

imp vx(:, m)=~l ~

vx(:, oJ)+ ~l~2[Vo- vx(:, ~o)].

q ,¢P

Fig. 2. Dependence of the resonance frequency shift on the viscosity of a liquid for a randomly rough surface. The solid, dashed and dotted lines are the roughness induced shifts, Aw~h, for correlation lengths equal to I = 100 n m , / = 250 nm and / = 5 0 0 n m , respectively; the fluid induced shift, &o~× 10 -~ for a smooth interface is shown by the squares. The calculations were carried out for the following values of parameters: i1---2.947 x 10H dyne cm -z. pq=2.648 g c m -3. fo=¢Oo/2n=5 MHz, p = I g c m -3, h = 3 0 nm.

~(R) can be found from the independent measurements, for instance via STM. On the other hand we believe that the QCM measurements in liquid can provide valuable information on the geometrical structure of the interface [22,23].

3.2. Strong roughness We now discuss the effect of a strong roughness on the QCM response. In this case the theoretical approach based on the perturbation theory cannot be applied and we have to use a simplified model [24]. We consider the liquid side as a two-layer system: the non-uniform interfacial layer located in the region d < z < L and the bulk liquid occupying the semi-space z > L , (see Fig. 1). For hydrodynamic purposes we treat the interfacial layer as a two-phase (solid-liquid) porous medium [25-29,35-37] with a permeability of ~ . Physical meaning of the permeability length scale, ~H, depends on the nature of the interfacial layer. For instance, in the case of rough surface layers ~ is related to their porosity and for an entangled polymer layer ~H is of the same magnitude as the equilibrium correlation length [38]. The local per-

(9) In this equation the effect of the solid phase on the liquid flow is given by the resistive force that has a Darcy-like form, ~l~(~ "-(Vo - v A : , co)).

Brinkman's equation presents a variant of the effective medium approximation which does not describe explicitly the generation of nonlaminar liquid motion and conversion of the in-plane surface motion into the normal-to-interface liquid motion. These effects result in additional channels of energy dissipation which are effectively included into the model by the introduction of the Darcylike resistive force. It should be noted that QCM response discussed here is sensitive to the energy dissipation rather than to the details of the velocity field in the liquid. While the limitation of Brinkman's equation are apparent, there is no alternative equation in the literature which has been accepted unconditionally. In the frame of the approx---nation discussed here the system is treated as macroscopically homogeneous along the surface plane x-.v. The only inhomogeneity takes place in the :-direction normal to the surface of the quartz crystal. In writing Eq. (9) we have taken into account that the pressure is constant for systems which are homogeneaus along the plane of oscillation. Eq. (9) shows that the effect of the interfacial layer is most pronounced for small values of ¢H/L. The velocity gradient in the layer decreases with the decrease of ?~a/Land for ?,H/L< 1 the majority of

M. Urbakh, L. Daikhin / CoUoids Surfaces A: Physicoc/u,m. Eng. Aspects 134 (1998) 75-84

80

the layer moves with a velocity equal to the velocity of the quartz surface, This behavior reflects the trapping of the liquid by the surface roughness. For higher values of the parameter in/L> 1, the influence of the interfacial layer goes down and the velocity field approaches the velocity field at the smooth solid-liquid interface. The liquidinduced shift, AoJ and the width, F, of the resonance frequency may be written in the form [24]

A(o=

(o2op g e / l

L

~2 .------~ \ q o + ~Hql

7[(flqPq) 1/2

!

1 =2 2 W ~=Hql

xt2q°[cosh(q,L)-l]+ sinh(qlL)}>, (10) Lql

r =

tamp

lm

ff(/l~/pq)t/2

+ .,---3-

~;hqt

W ~nqt'22

shift due to a mass loading. The effect results from the inertial motion of the liquid trapped by the inhomogeneities in the interfacial layer. The effective thickness of the liquid film rigidly coupled to the oscillating surface is equal to L - Ca and is less than the thickness of the inhomogeneous layer, L. This point must be taken into account in interpreting the QCM data. In the limiting case considered here, Ca <6 and Ca < L, the layer-induced correction to the width of the resonance is much less than the correction to the shift. The increase of the permeability 42 leads to the enhancement of the velocity gradient in the layer, which results in the decrease of the mass loading shift and in the increase of the width caused by the energy dissipation. When the layer thickness is the shortest length of the problem, L < 6, L < Ca and ~H < 6 Eqs. (10) and (11 ) can be rewritten in the form

x t2q° [cosh(q,L)-l] + sinh(q,L) }>. (11) Here qo =(io)p/tl) t/2 and ql =q~ + ~ 2 the first terms in the rhs of Eqs. (10) and ( I l ) describe the QCM response for the smooth quartz crystal-bulk liquid interface [7]. It should be mentioned that in this case the liquid-induced shift and width are equal. The additional terms present the shift and the width of the QCM resonance caused by the interaction of the liquid with the nonuniform interfacial layer. Let us focus on the limiting behaviors of Eqs. (10) and ( l 1 ). When the permeability length scale is the shortest length of the problem, ~H <6 and ~a < L, Eqs. ( ! 0) and ( I ! ) reduce to ,,o

Ato~- d- T~(/lqpq)t/2

]

L\2cOo/ +P(L--~H)

'

(12)

,:o F=

/t(flqpq) 1/2

(\2(Oo /

+pL ~lptoo ~l -tPl 2L

\pt%/ d)"

(13)

We see that the layer-induced shift is proportional to the liquid density and does not depend on the viscosity. It has the form of the frequency

F(.,,y,21

,oo

Lqt

7f(~tlqpq) 1/2

L\2- o/

+ ~ pL (L/~u) 2

].

(14)

[(.,,7

F = 7~(/lqpq)1,,2 L\2(Oo)/

l

(p~Jo~':2 L ] .

+ -~ pL (L/¢H) 2 \-~-1 j

(15)

Ill this limit the layer-induced shift is also proportional to the liquid density and does not depend of viscosity. However in contrast to the previous case it cannot be related to the mass of trapped liquid. The correction to the width of the resonance depends on the viscosity and it is substantially less than the layer-induced shift. We would like to stress that in both limiting cases discussed above the corrections to the shift and to the width of the resonance differ considerably. This difference has been ignored by Beck et al. [ 16] in interpreting of QCM measurements at rough surfaces. The usual form of the description of the experimental data in liquids is the representation of the real and imaginary components of QCM response as functions of liquid density p and of the parameter (p~l)t/2. Figs. 3-6 show the dependencies of the layer-induced shift and the width of the resonance

• ['r~ M. Urbakh, L. Daikhin / Colloids Surfi~ces A: Phys,cc, d,~r,t. ,~t.g Aspects 134 (1998) 75-84

.-,,,

0 <~

P.

81

o

-2 -1

_.

N m

-4

-2 gJ Z:

¢z,]

~Q

-3 C~ grJ

i

-4 O.S

2.

(a)

~-.5

2

2.5

(no)ta , to"tg sin%''a

0.$

1.

(a)

1_5

2

2.5

3

p, g sm"a

-4 _~ "

m

-10

N

-8

Z

Z

r~a

-~.2 ~ r~

-20

ee k,h

0.5

(b)

'~

!.~.

2

0

2.$

(qp)la lO.~g sm-Zs.V2

Fig. 3. The layer induced shift of the resonance frequency versus 01p) j~2 (at fix value p = l gcm -~} for two layer thicknesses: (a) L=0.56o and (b) L=26o, where 60=250 nm is the decay length of shear waves in water when the frequency is 5 MHz. The calculations were carried for the following values o f the parameters: J~=t,~o~2n=5 MHz and ( 11 ~./L =0.5, 12)

(b)

0.$

I.

]..5

2

2 . "~

~, g sm-3

Fig. 4. The layer induced shift of the resonance frequency versus

p (at fixed value t / = l c P ) for two layer thicknesses: (a) L =0.56o and (b) L =260, where 30=250 nm is the decay length of shear waves in water when the frequency is 5 MHz. The calculations were carried for the following values of the parameters:.fo=~Oo/2n=5 MHz and ( 1 ) ~tl/L =0.5. (2) ~,/L=0.2.

~H/L=0.2.

frequency on p and (pr/) 1/2, which have been calculated according to Eqs.(10) and (11). The common features of all curves are the increase of the QCM-response with the increase of the film thickness and with the decrease of the ratio ~H/L. The frequency shift changes rapidly with (p~/)'J-" in the region of (P/Ill/2 < t*(L/6,~a/L), where the limiting value t* is of the order of unity and slowly increases with the increase of the parameters LIb and ~H/L. For higher values of (Pill t/2 the function Aco ((p~l) t/2) tends to a constant (see Fig. 3). This behavior reflects the fact that in the high viscosity limit, when L/i3 < 1, the gradient of the velocity in the interfacia[ layer is small and the layer influences only slightly the velocity field in the bulk liquid. As a result the viscosity dependent contribution to the frequency shift, which is proportional to the velocity gradient, would remain the same as for

the smooth solid-liquid interface. This point explains also the decrease of the layer-induced width of the resonance with increasing ( p i l l t/2 for (p~/)112> t*(L/6,~.n/L) (see Fig. 5). We remark that the plots AoJ versus (Pill t/2 presented on Fig. 3 closely resemble the plots on Fig. 2 obtained for slightly rough surfaces within the perturbation approach. The predicted dependencies of Ato on (p~l) t/'- agree qualitatively with experimental data [ 15] found for rough solid surfaces in contact with methanol-water mixtures and alcohols. Fig. 4 shows that the frequency shift is linear in the liquid density for small thicknesses of the interfacial layer, L/3 <1 and the deviations from the linear proportionality arise as the thickness increases. The deviations are caused by the increase of the contribution of the energy dissipation processes to the QCM response with the increase of the thicknesses.

M. Urbakh, L. Daikhin / Colloids Surjbces A: Physicochem. Eng. Aspects 134 (1998) 75-84

82

1...5

m

>,

>, r~ Z Q~

0.5

0 =¢

0

7 r~

0.5

¢? ;Q =¢ 0 0

0.5

L

(a)

2

1.5

2.5

0

(qp)m io-t g sm-Zs-ta

o . 0

(b)

. O.S

. 2.

. L . .g

.1.

(a)

.

~ 2

0.5

2.5

(qp)l~ lO-I g sm%-Irz

Fig. 5. The layer induced width of the resonance frequency versus Olp)~;'-(at fixed value p = 1 g cm-3) for two layer thicknesses: (a) L=0.5 60 and (b) L = 2 6 o , where 6o=250 nm is the decay length of the shear waves in water when the frequency is 5 MHz. The calculations were carried for the following values of the parameters: 1o = ~,~,~/2:r= 5 MHz and ( 1) ~u/L =0.5, (2)

(b)

Z

2..5

3

2

2.5

3

J

p , g sm "3

oi 0

:3

1..5

0.5

J.

?..5

p,gsm "3

Fig. 6. The layer induced width of the resonance frequency versus p ~'at fixed value Pl= I cP) for two layer thicknesses: (a) L = 0.5 6o and (b) L = 260, where c~o= 250 nm is the decay length of shear waves in water when the frequency is 5 MHz. The calculations were carried for the following values of the parameters:J~o~o/2n=5 MHz and ( 1 ) ~x/L=0.5, (2) ~jl/L=0.2.

ev,..L =0.2.

Our calculations predict an interesting feature of the layer-induced width of the resonance as a function of (p~/)t'2 (see Fig. 5). For L_> ~ the curves for F versus (t,l) t'' have the maxima located at (p~l)t"2~t*(L/6, ~H/L). The appearance of the maxima results from the fact that the energy dissipation induced by the interracial layer diminishes for both high and low viscosities. This effect has been already discussed above. The dependencies of the resonance width on the liquid density shown on Fig. 6 reflect the influence of both the mass loading and the energy dissipation contributions. For the thick layers the curves F versus p have the quadratic form, as predicted by Eq. (13). Our model demonstrates that in the case of polymer films being in contact with a liquid, the QCM response strongly depends on the thickness of the film and on the correlation length. These

parameters are changed drastically under phase transitions. The QCM response of the amphoteric polymer film undergoing phase transitions between the isoelectric and the cationic and anionic fonns has been studied by Wang et al. [39]. Large resonance frequency shifts, which are not consistent with the mass changes, have been observed for polymer films in the vicinity of the phase transition point [39]. The phase measurement interferometric microscopy demonstrated that phase transitions were accompanied by dramatic changes in the film thickness. Our model explains the observed effect by the variation of the thickness or/and correlation length of the polymer film under phase transition. In order to estimate the changes of the film thickness and of the correlation length both the real and the imaginary part of QCM response should be measured.

M. Urbakh, L. Daikhhz / ColloMs Surfaces A: Physicochem. Eng. Aspects 134 (1998) 75 84

4. Conclusions

We have considered the effect of both slight and strong surface roughnesses on the QCM response in contact with a liquid. The problem has been studied within the perturbation theory approach and with the use of Brinkman's equation for the velocity in the non-uniform interfacial region. We have found that the frequency changes due to both the inertial motion of a liquid rigidly coupled to the surface and due to the additional viscous energy dissipation induced by roughness. The results obtained are as follows: ( 1) For the slightly rough surfaces the roughness induced shift and the width are proportional to the square of the "'slope" of the roughness and depend on the height-height correlation function. The effect of the roughness is most pronounced for the low fluid viscosity. The rms height and the lateral correlation length of the slight roughness can be estimated from the dependencies of the shift and the width on the fluid viscosity and on the frequency of the free quartz crystal. (2) For the strongly rough surfaces the roughness induced shift and width of the resonant frequency are determined by two parameters: 6/L and g~,/L. They are the ratios of the decay length of the shear wave in the liquid and the perrneabilitv length scale in the interfacial layer to the thickness of this layer. In the case of thin rough layers, L< f, the resonant frequency decreases rapidly with the increase of (pll) ~'z. When L approaches 6, resonant frequency loses its dependence on (pq)t;2 and tends towards a constant. This behavior agrees with the results derived within the perturbation approach [22]. The linear dependence of the frequency shift on the liquid density has been found for L < & The slope of the lines At,~ versus p increases with the decrease of the parameter ?,~l/'L. For, L > 6, the dependence of the shift on the liquid density becomes more complicated. Our calculations predict that the roughness induced width of the resonance, F, has a maximum as a function of (pq) ~2. The position of the maximum moves to lower valueLof (pq)V2 with the decrease of the thickness of the interfacial layer. The roughness induced width increases with the

83

increase of the liquid density. For thin interfacial layers, L<6, curves of F versus p have a quadrafic form. (3) Our calculations demonstrate that while for smooth surfaces, liquid induced, shift and width of the resonance are equal, they differ essentially for rough surfaces. This effect, ignored by Beck et al. [16], has been observed by Martin et alo [20]. Our results show that empirical representation of the QCM response in a liquid as a linear combination of (pq)l/2 and p terms [20] can be applied for the description of the shift of the resonant frequency in the case of thin rough layers, L < 6.

References [ll M. Thompson, L.A. Kipling, W.C. Duncan-Hewitt, L.V. Rajakovic, B.A. Cavic-Vlasak, Analyst 116 ( 1991 ) 881. [2] D.A. Buttry, in: A.J. Bard (Ed.), Electroanalytical Chemistry, Vol. 17, Marcel Dekkcr, New York, 1991, p. 1. [3] D.A. Buttry, M I L Ward, Chem. Rev. 92 (1992) 1355. [41 E.T. Watts, J. K,'~r,i, A. Widom, Phys. Rev. B 41 (1990) 3466. [5] D. Johannsmann, K Mathauer, G. Wegner, W. Knoll, Phys. Rev. B. 46 (1992) 7808. [6] C.E. Reed, K.K. Kanazawa, J.H. Kaufman, J. Appl. Phys. 68 (1990) 1993. [7] S. Bruckenstein, M. Shay, Electrochim. Acta. 30 (1985) 1295. [8] K.K. Kanazawa, J.G. GordonlI, Anal. Chim. Acta 175 (1985) 99. [9] T. Nomura. M. Okuhara, Anal. Chim. Acta 142 (1982) 281. [10] R. Schumacher, Angew. Chem., Int. Ed. Engl. 29 (1990) 329. [! I] G. Sauerbrey, Z. Phys. 155 (1959)206. [12] A.L. Kipling, M. Thompson, Anal. Chem. 62 (1990) 1514. [13] L.V. Rajakovic, B.A. Caviv-Vlasak, V. Ghaemmaghami, M.R.K. Kallury, A.L. Kipling, M. Thompson, Anal. Chem. 63 ( 1991 ) 615. [14] R. Beck, U. Pittermann, K.G. Weil, Ber. Bunsen-Ges. Phys. Chem. 92 (1988) 1363. [15] M. Yang, M. Thompson, W.C. Duncan-Hewitt, Langmuir 9 (1993) 802. [16] R. Beck, U. Pittermann, K.G. Well, J. Electrochem. Soc. 139 (1992) 453. [17] M. Yang, M. Thompson, Langmuir 9 (1993) 1990. [18] R. Schumacher, G. Borges, K.K. Kanazawa, Surl: Sci. 163 (1985) L621. [19] R. Schumacher, J.G. Gordon, O. Melroy, J. Electroanal. Chem. 216 ( 19871 127.

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[20] S.J. Martin, G.C. Frye, A.J. Ricco, S.D. Sentaria, Anal. Chem. 65 (1993) 2910. [21]S. Bruckenstein, A. Fensore. Z. Li, A.R. Hiliman, J. Electroanal. Chem. 370 (1994) 189. [22] M. Urbakh, L. Daikhin, Phys. Rev. B. 49 ( 1994j 4866. [23] M. Urbakh, L. Daikhin, Langmuir 10 (1994) 2839. [24] L. Daikhin, M. Urbakh, Langmuir 12 (1996) 6354. [25] H.C. Brinkman, Appl. Sci. Res. A 1 (1947) 27. [26] M. Sahimi, Rev. of Modern Phys. 65 (1993) 1393. [27] R. Hiller, in I. Prigogine and Stuart A. Rice (Eds), Advances in Chemical Physics, Vol. 92, John Wiley & Sons, Inc., New York, 1996, P.299. [28] C.K.M. Tam, J. Fluid. Mech. 38 (1969) 537. [29] T.S. Lundgren, J. Fluid. Mech. 51 (1972) 273. [30] L.D. Landau, E.M. Lifshitz, Fluid Mechanics, 2nd ed., Pergamon, New York, 198~.

[31]J.N. Israelashvili, P.M. McGuiggan, A.M. Homola, Science 240 (1988) 189. [32] P.A. Thompson, M.O. Robbins, Phys. Rev. A. 41 (1990) 6830. [33] P.A. Thompson, M.O. Robbinso Science 250 (1990) 792. [34] M.O. Robbins, D. Andelman, J.-F. Joanny, Phys. Rev. A 43 ( 1991 ) 4344. [35] P.G. de Gennes, Macromolecules 14 ( 1981 ) 1637. [36] Glenn H. Fredrickson, P. Pincus, Langmuir 7 ( 1991 ) 786. [37] L.G. Jones, C.M. Marques, J.-F, Joanny, Macromolecules 28 (1995) 136. [38] For entangled polymer the correlation length is the average distance between entanglement points. [39] J. Wang, M.D. Ward, R.C. Ebersole, R.P. Foss, Anal. Chem. 65 (1993) 2553.