Acoustic Wave Sensors: Theory, Design

Chapter 1

Why Acoustic Sensors?

Precise measurement tools are necessary parts of most successful scientific and engineering enterprises. The sensing devices that we consider in this volume are such tools, capable of measuring physical, chemical, and biological quantifies. What they have in common is that they all employ acoustic waves in their operation. The purpose of this introductory chapter is to provide an overview of these devices, and to answer the question: why use acoustic sensors?

1.1

What Is a Sensor?

The sensors we consider here produce an output signal in response to some input quantity, as indicated schematically in Figure 1.1(top). The output signal is usually electrical m an analog voltage or current, a stream of digital voltage pulses, or possibly an oscillatory voltage whose frequency represents the value of the input quantity. The range of input quantities covered in this book is large, including physical quantities such as the mechanical properties of thin films, and chemical and biological quantities such as the concentrations and identities of unknownspecies in air or liquid media. Inside the typical sensor of Figure 1.1(top), a process of transduction takes place, converting the input event into an electrical signal. The sensor may also contain circuitry that converts the often feeble electrical signal from the transduction process into a more robust form suitable for use outside the sensor itself. The output signal may be stored in a computer memory for later examination. Possible applications would have the signal activating an alarm to warn of the 1 ACOUSTIC WAVE SENSORS

Copyright 9 1997 by Academic Press All rights of reproduction in any form reserved. ISBN 0-12-077460-7

2

1.

Why Acousac Sensors? SENSOR i

i lll

i

i

i

i

I'! ,,<,c,.,c..,/ k

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[ i IJl

VI

Figure 1.1 Sensor principles. (Top) Schematic diagram of a sensor that produces an electrical output in response to the presence of an input quantity. (Bottom) Biosensor comprising the generic device shown at top with a molecular recognition layer that has a highly selective response.

presence of a toxic vapor, or combining with other signals to provide a physician with information on which to base a medical decision. Sensors are characterized in many different ways. Their sensitivity is a measure of the magnitude of the output signal produced in response to an input quantity of given magnitude; their resolution is a measure of the minimum change of input quantity to which they can respond; and their selectivity characterizes the degree to which they can distinguish one input quantity from another. However, with chemical sensors for vapors or gases, high selectivity is usually desired but often hard to achieve. A telling example is the commercial Taguchi gas sensor for natural gas or propane, which employs a fairly nonselective solid-state process

1.2 The Microsensor Revolution and the Role of Acoustics

3

that takes place at the surface of its heated sensing resistor. When a gas molecule reaches the sensor surface, it can strongly affect the electrical resistance of the element and thus trigger an alarm circuit. According to the manufacturer's instructions, however, the user achieves selectivity by mounting the device high on the wall if natural gas is to be detected, or near the floor if propane sensing is desired---propane being more dense than natural gas! In contrast, with certain biosensors selectivity can be very high. The biosensor may include as a "front end" a biorecognition element which responds to only one biological substance. As illustrated in Figure 1.1 (bottom), the molecular recognition element may contain particular molecules that react with only one other type of molecule. The example in the figure suggests using particular antibodies (the dark cloven objects) that bind to only one type of antigen (the triangularly shaped one). Exploiting this bioselectivity can permit detection of very low concentrations of substances in a very dense background of other molecules.

1.2

The Microsensor Revolution and the Role of Acoustics

The development of integrated circuits reduced the cost of computing, storing, and transmitting information from one location to another. It also made possible very sophisticated yet economical systems to deal with signals from sensors. But until recently, the sensors themselves had not evolved much, and were still fairy large and expensive devices. As an example, a standard device for determining the concentration and identity of unknown vapors was still a heavy, half-meterlong infrared spectrophotometer costing around ten thousand dollars. Sensor development lagged behind that of integrated circuits, and so increasing attention was directed toward the development of inexpensive microsensors. The success of this effort has resulted in the availability of a growing number of microsensors that are now moving from the research laboratories into development, commercialization, and use [1]. The effort worldwide engages many workers, and resulted in more than three thousand references to "chemical sensors" alone in the period from 1985 through 1989. One thread of this work has been the miniaturization of familiar potentiometric and amperometric chemical sensors [2]. Another is the use of optical sensors in which changes in optical index of refraction, amount of absorbance, or intensity of photoluminescence provide chemical or biological information. Yet another part of the effort has been based on acoustics, or more explicitly, the use of elastic waves at frequencies well above the audible range propagating in specially designed solid sensing structures.

4

1. Why Acoustic Sensors?

The first of the acoustic sensors was the so-called quartz crystal microbalance (Fig. 1.2a). The "QCM," as it has been known by chemists, employed a slightly modified quartz crystal made initially to stabilize the frequencies of radio transmitters. The modification that permitted it to be used for chemical sensing was the addition of a sorptive film on the crystal. This device was analyzed and improved by a succession of workers starting in the 1950s [3; 4]. Another advance was made in the late 1970s when Wohltjen and Dessy [5] realized that chemical vapor sensing could be accomplished with a device designed originally for processing purely electrical signals, the surface-acoustic-wave delay line (Figure 1.2b). In this device, acoustic waves are generated and detected with the comblike conducting structures shown at each end of the device; a piezoelectric material in the device substrate converts energy between electrical and mechanical forms at the comblike structures. More recently, two other sensors were introduced that employ similar principles but exploit different modes of elastic wave propagation- the acoustic-plate-mode device (Figure 1.2c) and the flexuralplate-wave device (Figure 1.2d). These devices are conveniently small, relatively inexpensive, quite sensitive, and inherently capable of measuring a wide variety of different input quantities. It is because of these far-reaching characteristics that we have written this book in order to bring a diverse audience of readers an understanding of acoustic sensor principles.

1.3

Where They Fit and How They Are Used

The four types of sensors that we discuss in this book operate over a frequency range of three orders of magnitude - - from less than one to more than one-thousand megahertz. In fact, the frequency spectrum of acoustic waves actually extends to more than eighteen orders of magnitude, as indicated by Figure 1.3 (page 6). This range is nearly as large as that commonly shown in charts of the electromagnetic wave spectrum. Incidentally, Figure 1.3 shows that there are many other types of acoustic sensors designed for purposes ranging from imaging the human heart to detecting cracks in airplane parts [6]. All of the sensors of Figure 1.2 "sense" by producing a change in the characteristics of the path over which the acoustic waves travel; the nature of these changes will be discussed in detail in later chapters. As suggested in Figure 1.4 (page 7), there are several ways of detecting such changes. One is the "active" approach in which one makes the sensor a part of an electronic oscillator circuit

1.3 Where They Fit and How They Are Used Cell with Liquid

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(c)

Output Transducer

Tra

El~trode

2

Piezoelectric Substrate

_..._~.~~ ~

P =__.._

5

Output

s,g..,

Piezoelectric Quartz Substrate

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.

(d)

Output

Membrane Sorptive Layer Silicon Substrate Figure 1.2

Schematic sketches of the four types of acoustic sensors discussed in detail in this book: (a) Resonant quartz crystal like that used in electronic communications systems (after Lu [6]); (b) Surface-acoustic-wave delay line with selective absorptive coating (after Wohltjen and Dessy [5]); (c) Acoustic-plate-mode delay line made from quartz crystal (after Ricco and Martin [7]); (d) Thin-membrane flexural-plate-wave delay line made by microfabrication techniques from a silicon wafer.

so that a change in the characteristics of the acoustic path cause a change in the frequency of the oscillator. This approach is a natural one for the quartz crystal resonator (Figure 1.2a), as the resonator was originally made for use in electronic oscillators. In a typical vapor-sensing application, the sorption of vapor molecules in a polymeric coating applied to one surface of the crystal increases the crystal's mass and lowers its resonant frequency and that of the circuit in which it is installed. The active approach is also illustrated with the surface-acousticwave and the flexural-plate-wave devices in Figures 1.2b and 1.2d, where electronic amplifiers are shown connected between input and output transducers of the devices. The alternative approach for getting information from these acoustic sensors is to measure the sensor characteristics passively; that is, to supply an external

6

1. Why Acoustic Sensors?

FREQUENCY

(Hz) 1013 10 I z

-'HIOHEST-FREOUENCY EL AST IC Y AVE OENIERATED P IEZOELECTRICALLY --THERPIOELASTICALLY OENERATED PHONONS

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Figure 1.3 Acoustic-wave spectrum. Almost as broad as the familiar electromagneticwave spectrum, the spectrum of acoustic waves that have been excited or detected covers frequencies that range over roughly eighteen orders of magnitude. The four sensors on which we concentrate, indicated by bold lines, range in operation from below 1 MHz to slightly above 1000 MHz.

electrical test signal and determine the response of the sensor to that signal. For example, as shown in later chapters, by measuring the attenuation of the test signal we can determine the viscosity of a fluid that contacts one of these sensors. In the following chapters we discuss these measurement options thoroughly. The most commercially developed of the acoustic sensors we will discuss is the quartz-crystal microbalance. This device is often used in vacuum deposition systems where it measures the thickness of deposited coatings. The commercial sensor shown in Figure 1.5 (page 8) includes a vacuum-tight water cooling system and a microprocessor-based controller that can be set for measuring and indicating the deposition rate and total thickness of films having different densities and sound speeds. Incidentally, hereafter we will refer to this device by the

1.4 About the Authors and the Rest of the Book ,

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more generic name "thickness-shear-mode" (TSM) sensor, since that name emphasizes the mode of propagation instead of the material from which the device is made. The surface-acoustic-wave sensor is also commercially available, either as a single sensor or as a part of an entire sensing system. The authors hope that informing potential users about acoustic sensors may stimulate the wider use of all the sensors that we discuss.

1.4

About the Authors and the Rest of the Book

It will be clear upon skimming through this book that we are dealing with a multidisciplinary subject. The disciplines involved include acoustics, electrical circuits, chemistry, some biology, and a lot of materials science and engineering. In view of this diversity, we have tried to provide plenty of supportive background material. The same multidisciplinary mix characterizes the authors: some are chemists (Ballantine, Ricco, Wohltjen, and Zellers); one is an electrical engineer (Martin);

8

1. Why Acoustic Sensors?

Figure 1.5 Commercial deposition thickness monitor (courtesy Sloan, Inc.) employing AT-cut, 5-MHz quartz crystal in the sensor head at left. Digital control and readout equipment is shown at right. one a chemical engineer (Frye); one an applied physicist working in an electrical engineering department (White); and one works in environmental and occupational health (Zellers, again). Three are academics (Ballantine, White, and Zellers); three work for the U. S. Government (Frye, Martin, and Ricco); and one started and runs his own sensor systems business (Wohltjen). We hope that this diversity is enriching. The remaining chapters have the following functions and contents: Chapter 2 introduces the acoustic principles common to all the devices; Chapter 3 describes the devices in detail and shows how specific input quantities affect the characteristics of the propagation path; Chapter 4 examines the ways these sensors can be used to determine materials properties; Chapter 5 looks at the chemical and biological applications of these sensors; and Chapter 6 deals with practical sensor systems.

References

9

References 1. Muller, R. S.; Howe, R. T.; Senturia, S. D.; Smith, R. L.; White, R. M. Eds., Microsensors; IEEE Press: New York (1990). 2. Madou, M. and Morrison, S. R. Chemical Sensing with Solid-State Devices; Academic Press, New York (1989). 3. Sauerbrey, G. Z. Phys., 155, 206 (1959). 4. King, Jr., W. H. Anal. Chem., 36, 1735 (1964). 5. Wohltjen, H. and Dessy, R. Anal. Chem., 51, 1458 (1979). 6. Lu, C.-S. J. Vac. Sci. Technol., 12, 578 (1975). 7. Ricco, A. J. and Martin, S. J. Appl. Phys. Letters, 50, 1474 (1987).

Chapter 2

Fundamentals of Acoustic Waves

This chapter explores the properties of elastic waves, as well as their electrical excitation and detection in piezoelectric materials. The types of elastic waves we consider, together with the velocities with which they travel, are shown in Figure 2.1. The bulk waves exist in a hypothetical medium which has no boundaries whatsoever. Introducing a single plane boundary that forms a semi-infinite solid permits surface elastic waves to propagate along that single boundary. Adding a second boundary parallel to the first forms a plate, and permits the propagation of plate waves that also have sensor applications. In this chapter we consider elastic wave fundamentals, and then in Chapter 3 we show how each of these wave types can be used in sensors.

2.1

Wave Propagation in an Elastic Medium

An elastic medium behaves as a distributed mass-spring system in which displacement of a single element results in the propagation of a disturbance throughout the medium. A particle at a free surface is different from one interior to the solid, in that it is constrained by adjacent particles from only one side. Thus, disturbances at a surface can behave somewhat differently from those in the interior of a solid. In fact, such boundary considerations give rise to unique modes of propagation that can only exist at the free surface of a solid. Before considering such so-called surface waves, it is instructive to examine plane waves that propagate far from any perturbing boundaries. Just as a mass/spring system oscillates due to the interplay of an inertial force associated with the mass with a restoring force from the spring, an elastic wave 10 ACOUSTIC WAVE SENSORS

Copyright 9 1997 by Academic Press All rights of reproduction in any form reserved. ISBN O-12-077460-7

2.1 Wave Propagation in an Elastic Medium

II

arises from the interplay of distributed elastic and inertial forces. While the mass/spring-system response is described by a differential equation involving mass, displacement and time, wave motion in the solid is somewhat more complex. Like the one-dimensional vibrating string, particle displacement in the solid is a function both of time and position, and the equation of motion must be a localized description. The waves that can propagate in a solid depend upon both the properties of the solid and its boundaries [1]. Figure 2.1 shows schematically the waves that can propagate in an unbounded solid, a semi-infinite solid having a single plane boundary, and in a solid plate that has two plane boundaries. The terminology, definitions and analyses that follow in this chapter are used to determine the nature of these wave motions and the phase velocities of the waves in particular solids.

Surface (Rayleigh) Wave

Bulk Longitudinal Wave

(a)

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(d)

(b)

Symmetric Vp = 2 0 0 0 - 6000 mls

Vp

2 0 0 0 - 1 2 , 0 0 0 m/s Anti Symmetric

Vp = 1 0 0 - 4000 mls

Figure 2.1 Pictorial representations of elastic waves in solids. Motions of groups of atoms are depicted in these cross-sectional views of plane elastic waves propagating to the right. Vertical and horizontal displacements are exaggerated for clarity. Typical wave speeds, vp, are shown below each sketch. (a) Bulk longitudinal (compressional) wave in unbounded solid. (b) Bulk transverse (shear) wave in unbounded solid. (c) Surface acoustic wave (SAW) in semi-infinite solid, where wave motion extends below the surface to a depth of about one wavelength. (d) Waves in thin solid plates.

12

2.1.1

2. F u n d a m e n t a l s of Acoustic W a v e s

DISPLACEMENT, STRESS, AND STRAIN

Application of force to a solid puts the solid under stress. Stress results in strain l within the solid: atoms or molecules of which the solid is composed are displaced from their unstressed locations. When a solid is deformed, the displacement of each particle from its original position is represented by a displacement vector u(x,y,z,t). In general, the displacement has components, which vary continuously from point to point in the solid, in the x, y, and z directions. A plane wave generates displacements that vary harmonically in the direction of wave propagation; if this is the x direction, for example, it may be represented as [ 1]:

u(x,y,z,t) = (ulx + ugy + u3z)e j(o't-kx),

(2.1)

in which u l, u2, and u3 represent particle displacements in the x, y, and z directions, respectively; x, y, and z are unit vectors in their respective directions; ~ is the angular frequency of the wave (to = 2wf, where f is frequency); j = X / - 1; and k is the wavenumber (k = 2or/A, where A is wavelength). As the name implies, the contours of constant displacement for a plane wave are planes perpendicular to the propagation direction. Figure 2.2 depicts a solid crystalline lattice perturbed by compressional and shear plane waves moving in the positive z direction. Because simple translation of the entire solid is not of interest, this class of motion is eliminated to give a parameter related only to local deformations of the solid; this parameter is the displacement gradient, Vu. The gradient of a vector field Vu is a second-rank tensor, specified by a 3 by 3 matrix. The elements of this displacement gradient matrix are given by (Vu)iy = Oui/Oxj, also denoted ui,j in which i denotes the ith displacement element and j denotes a derivative with respect to the jth spatial coordinate, 2 i.e. [ 1],

Vu =

OullOx OullOy au~laz OuJcgx OuJOy OuJOz ~ . Ou3/cgx Ou3/Oy Ou3/Oz/

(2.2)

The displacement gradient represents changes in interparticle distance as well as local rotations caused by the displacement. Just as the effect of simple translation was eliminated by taking the gradient of the displacement vector, the contributions due to rotations can be eliminated, resulting in a parameter that describes only the local stretching of the solid. This 1Stress is the force/unit area applied to the material, while strain is the dimensionless ratio of the length of the stressed material to its unstressed length. 2For the sake of simplicity in notation, xt, i = 1, 2, 3, will occasionally be used in place of x, y, and z, respectively.

2.1 Wave Propagation in an Elastic Medium

13

Figure 2.2 Plane waves propagating in a solid, showing (a) compressional and (b) shear waves. is accomplished by adding the displacement gradient to its transpose, the result being the strain matrix S, with elements given by [ 1] 1

su = -~(audOxj + Ouj/Ox~).

(2.3)

Strain is the change in length per (unstrained) unit length in the solid as a result of applied stress and can be calculated for any direction in the solid from the

14

2. Fundamentals of Acoustic Waves

strain tensor. For an infinitesimal element having direction cosines (ll, 12,/3), the strain is given by Siylily. Thus, the element Sll represents strain in the x direction, while $22 and $33 represent strains in the y and z directions, respectively. Thus, the diagonal strain terms (Sii) represent axial or compressional strains, while the off-diagonal elements (Sij with i ~:j ) are shear strains. The shear strains physically represent the angular change (in radians) between elements initially in the ith and jah directions. Example 2.1:

Suppose a bar of length L is rigidly fastened at one end at x = 0 and stretched in the x direction, as shown in Figure 2.3, being deformed according to u l --khx, u2 = u3 = O. Derive the elements of the strain matrix.

From Equation 2.2, ul,l = kh, while all the other elements of the displacement gradient are zero. As a result, the only non-zero strain element is S ll = kh. This represents a fractional change in length, in particular an elongation, in the x direction of magnitude kh.

Solution:

To represent completely the state of stress at each point in a solid requires use of a stress tensor, T. Each element of the stress tensor, Tij, represents the i th component of force per area acting on the j~h face of an infinitesimal volume element. T allows the determination of the stress in any direction on any plane

Figure 2.3

Bar under uniaxial strain.

2.1 Wave Propagation in an Elastic Medium

15

interior to the solid. The stress vector acting on a plane with a normal component as specified by the direction cosines (11,12,13) is given by Tijlj.

2.1.2

EQUATION OF MOTION

Definition of stress and strain permits derivation of the equation of motion for elastic deformations of a solid, in particular wave motion. Figure 2.4 shows an elemental volume of an elastic solid. The stresses that exert forces in the x direction of each face are indicated, with the assumption that stress has only changed a small amount ATi across the elemental lengths Ax, Ay, Az. The force exerted on each face is the product of the stress component indicated times the area over which the stress acts. The summation of all of the x-directed forces acting on the cube is thus F1 = [ ( T l l + ATt 1)A1 - Tl tA1] + [(T12 + AT12)A2- TI2A2] + [(TI3 + ATI3)A3- TI3A3].

(2.4)

The area of a face with a normal component in the xi direction is

Figure 2.4 Elemental volume of an elastic solid, showing forces exerted on each face.

16

2. Fundamentals of Acoustic Waves

Ai = Y.j.k ~YkAxJAxk( i , j , k) and its acceleration is i i i - 02ui/Ot2. Newton's law, F = m//, relates net force to acceleration in the x~ direction. The mass of the elemental volume is given by pAxAyAz, where p is the density (mass/volume) of the solid. From Equation 2.4 and Newton's law,

02Ul ATIIAyAz + ATIEAxAz + ATI3AxAy = pAxAyAz Ot2.

(2.5)

Dividing by the volume element AxAyAz and passing to the infinitesimal limit results in a one-dimensional partial differential equation, which can be generalized to all three coordinates:

3 OTij O2u~ Oxj = P " j= l Ot2

Equation of Motion (2.6)

This is the equation of motion for a solid (actually a set of three equations, corresponding to i = 1,2,3), relating inertial forces to the stress gradient. Completion of the characterization of a solid requires postulation of a relationship between stress and strain. It has been experimentally observed that for small deformations, the strain in a body is linearly proportional to the applied stress. In one dimension this is known as Hooke's law, relating the elongation of a spring or elastic material to the tensile force. A principle such as this, which relates stress to strain, is known as a constitutive relation, and can be generalized to three-dimensional, non-piezoelectric solids [ 1]: 3

Tij -" E

ciJkISkl'

Elastic Constitutive Relation (2.7)

k,l=l

in which the Ciykt, called elastic stiffness constants, serve as "microscopic spring constants" in describing what strain results from a given stress. The elastic constants completely characterize the elastic behavior of a solid in the smalldeformation limit. 3

2.1.3

REDUCED NOTATION (ENGINEERING NOTATION)

The four indices of the elastic stiffness constants, cokt, result in the possibility of as many as 34 = 81 elements in the stiffness tensor. Because the stress and strain 3A strain of 10-3 is a large strain, near the fracture strain for many substrate materials. Thus, one might consider that the linear behavior characterized by Hooke's law (or its three-dimensional analog, Equation 2.7) is limited to situations where the strain is 10-5 or less.

2.1 Wave Propagation in an Elastic Medium

17

tensors are symmetric, i.e., Tij = Tji and Sij = Sji, at most six of the nine elements of each tensor can be unique. For this reason, a system has been adopted for reducing the number of indices from two to one [ 1]. In reduced notation, the double index ij is replaced by a single index I as shown in Table 2.1. In equation form, the use of reduced index notation is signified by the use of capital subscripts, as St and/'!. The symmetry properties that allow reduction of the number of entries in the stress and strain tensors also allow reduction of the number of elastic stiffness constants cij~,i in the stiffness matrix. Replacing first the pair ij by a reduced index I from Table 2.1, then replacing kl by a reduced index J from Table 2.1, results in a stiffness matrix requiring only 36 entries in reduced notation: ctj, where I and J range from 1 to 6. Using reduced indicial (engineering) notation, Equation 2.7 is more succinctly expressed as [ 1] 6 T! = ~ cuSj.

Reduced Elastic Constitutive Relation

(2.8)

J = l

The stiffness matrix is also symmetric with respect to the interchange of reduced indices I and J so that for the most general solid, 21 distinct entries (6 diagonal and 15 triangular entries) are required to completely characterize the solid. Since the elastic properties reflect the symmetry of the material, as crystal symmetry increases the number of distinct elastic constants required to characterize the material decreases; for example, the number of independent constants is 6 for a trigonal material, 3 for a cubic material, and 2 for an isotropic material. The nonzero elastic constants for several materials are listed in Table 2.2. We note from Table 2.2 that isotropic materials, such as polyethylene and polycrystalline aluminum, are specified by only two independent elastic constants: c ll and c44. Alternatively, one may characterize the elastic properties of an isotropic medium by Young's modulus and Poisson's ratio, E and v respectively, or by the Lam6

Table 2.1

Reduced Index Notation i

Index ij

Reduced Index I

11

1

33 23or32 13or31

3 4 5

12 or 21

6

ill

|

Matrix Representation

/'6 T2 7'4 /'5 T4 T3

18

2. Fundamentals of Acoustic Waves Mechanical Properties of Selected Materials [1]

Table 2.2

Material Aluminum (polycrystalline) Lithium Niobate Polyethylene Quartz Zinc Oxide ii

ii

i

ii

i

ii

ii

Density

......

(kg/m 3)

Cll

Stiffness (10'~ newtonlmZ)

C33

2695

11.1

4700 900 2651 5680

20.3 0.34 8.674 20.97

i

i

i

ii

r

C12

.......

C13

C14

2.5

i

l

24.5

6.0 0.026 5.794 4.247

10.72 21.09 1

II

I

9

I

I

5.3

7.5

0.699 12.11

1.191 10.51

I

I

0.9

III

-1.791 I

I

II

constants, A and/x. See [2]. Anisotropic crystals, such as lithium niobate, quartz, and zinc oxide, require several more elastic constants.

Example 2.2: Calculate the stress distribution required to obtain the deformation specified in Example 2.1 in a cubic material. Solution: In Example 2.1 it was found that Sll = kh. In reduced notation, St = kh, while $2 through $6 are all zero. Using the reduced notation and the stiffness matrix for a cubic material [ 1]:

T1

Cll

7"2 T3 T4 T5 I'6

1r = |C12 /~ !

Cl2 C12 0 0 Cll Cl2 0 0 0 C12 Cll 0 0 0 0 0 C44 0 0 / 0 0 0 C44 0 0 0 0 0 C44

Sl 0 0 0 0 0

(2.9)

Performing the'matrix multiplication, Tl = cllSl = Cllkh, T 2 "- c12S1 = cl2kh, and/'3 = c 1 2 8 1 "- cl2kh, while all other entries are zero. The point is that in addition to the tensile stress Tl that caused the bar to stretch in the x direction, tensile stresses/'2 and/'3 acting in the y and z directions, respectively, are also necessary in order to obtain the simple strain field of Example 2.1. The reason is that a uniaxial strain typically results in contraction of the material in the transverse directions, causing non-zero strains in the y and z directions, rather than the simple strain field specified in Example 2.1. It is this tendency for a material to transmit strains in one dimension to strains in the other dimension that necessitates a tensorial stress-strain relation.

2.1.4

THE W A V E E Q U A T I O N

From the equation of motion (Equation 2.6) and the elastic constitutive equation (Equations 2.7, 2.8), it is a simple matter to derive the wave equation, which de-

2.1 Wave Propagation in an Elastic Medium

19

scribes the propagation of plane acoustic waves in a non-piezoelectric solid. The symmetry of the strain matrix means that Skl in Equation 2.7 is equivalent to Ouk/OXl. Differentiating Equation 2.7 with respect to xj gives 3

OT o

02uk

3

Z O X j - Z CijkltgxjOx, j= 1 j,k,l= 1

(2.10)

Equating the right-hand sides of Equations 2.10 and 2.6 results in the wave equation for non-piezoelectric, elastic solids [3]: 3

02u~.

02Ui = Z CijklOXJOXl P Ot2 j,k,l=l

Non-piezoelectricWave Equation (2.11)

It should be noted that Equation 2.11 represents a set of three wave equations (i = 1, 2, 3) in the particle displacements ul, u2, and u3, with summation over the indices j, k, and I. The polarization of a wave refers to the direction of particle displacement. In general, the solution to Equation 2.11 consists of three propagating wave types: a quasi-compressional wave, whose principal polarization lies along the direction of propagation, and two quasi-shear waves, each of whose principal polarizations is perpendicular to the propagation direction, as indicated in Figure 2.2. Equation 2.11 looks imposing, but in certain instances it reduces to a very simple set of equations, as illustrated in the next example. Example 2.3: Derive the set of wave equations corresponding to plane wave propagation along the x direction of a cubic crystal. The partial derivatives taken with respect to y and z are zero. Using the stiffness matrix corresponding to cubic symmetry in Equation 2.11 results in the following set of partial differential equations: Solution:

02ul 02Ul P Ot2 = Cll OX2 ,

02u2 p at 2

c92U3

(2.12)

02u2 =

c44 Ox 2 ,

(2.13)

02U3

P Ot2 = c 4 4 0 x 2.

(2.14)

Note that the equations for u; are decoupled in this case and may be solved independently. Equations 2.12-2.14 have three solutions representing acoustic waves having displacements along the x, y, and z directions, respectively. A solution to each of these equations is

20

2. Fundamentals of Acoustic Waves (2.15)

ui(x,t) = uioeJ(oJt-kx),

representing wave propagation in the + x direction: ui is the displacement in the ith direction and Uio is the amplitude. 4 Since Ul lies along the direction of propagation, the solution u i(x,t) represents a compressional wave propagating along the x axis, while u2 and u3 represent two shear waves propagating along +x, as indicated in Figure 2.2(b). Substituting Equation 2.15 into Equations 2.12-2.14 and noting that 02ui/Ox 2 = - k 2 u i and c92ui/c)t2 = -to2ui results in the dispersion relation for the compressional wave: poJ2 = Cl lk 2.

Dispersion Relation

(2.16)

A dispersion relation such as this allows one to calculate the phase velocity 5 of the waves, given by v = r For the compressional wave, Equation 2.16 implies that vl = "V'cll/p, while the dispersion relations for Equations 2.13 and 2.14 indicate that v2 = v3 = (c44/p) 1/2.

Example 2.4:

Calculate the compressional and shear wave velocities in aluminum

and polyethylene. Solution: Using values of c11, c44, and p from Table 2.2 in the equations given above for the compressional velocity (vi) and shear velocity (v2) yields the following:

,=,

Material

vl (m/s) compressional

v2 (m/s) shear

6420 1940

3050 540

Aluminum Polyethylene , .....

,

,

,

,

,

,

,

1,

,1,

,1,

In Example 2.2, Equation 2.11 reduced to equations for three uncoupled modes capable of propagating along the x axis of a cubic crystal. Under such conditions, the propagation direction is referred to as a p u r e - m o d e direction. In general, pure modes result when waves are propagating along a symmetry plane of a crystal and have polarization perpendicular to or parallel to this plane. Also, propaga4The quantity with physical significance is understood to be the real part of the expression for ui. The j in the exponent represents ( - 1)1/2.The exponential can be written in terms of sine and cosine functions using Euler's identity, ejx = cos(x) + jsin(x), from which the displacement is seen to be ui(x,t) = u~ocos(~- kx). A similar approach is used by electrical engineers to represent sinusoidally varying currents and voltages. 5The phase velocity is the speed at which one must travel to keep the phase of a sinusoidal wave at a constant value. The phase of the wave described by Equation 2.15 is the quantity ( ~ - kx).

2.1 Wave Propagation in an Elastic Medium

21

tion normal to or along a rotation axis of a crystal results in pure modes. When the propagation direction is in a direction with lower symmetry, quasi-modes are obtained whose longitudinal and shear components are coupled.

2.1.5

BULK-WAVE DAMPING

In the derivation of the acoustic wave equation (Equation 2.11), no consideration was given to energy loss mechanisms. In general, acoustic waves propagate with diminishing amplitude in a real crystal as a result of several loss mechanisms. Attenuation may arise from such mechanisms as phonon scattering, impurity (or Raman) scattering, and thermoelastic attenuation. Thermoelastic attenuation arises in compressional waves due to heat flow from compressed to expanded regions. The flow of heat is an irreversible process that increases entropy, which in turn can be related to attenuation. Thermoelastic attenuation is proportional to the thermal conductivity of the solid and, while important in metals, is not the dominant damping mechanism in insulators used in acoustic devices. In high-quality, insulating single crystals, the dominant damping mechanism is phonon scattering, which can be treated phenomenologically by introducing a viscous term into the elastic constitutive relation for the solid (Equation 2.8), 6

r~ = ~" (cu& + nuSs),

(2.17)

J=l

in which the viscosity tensor rltj has the same symmetry as the elastic stiffness tensor cu. The following example will serve to illustrate how attenuation may be predicted using this model. Example 2.5: Calculate the attenuation for a y-polarized shear wave propagating along the x axis of a cubic crystal, based on the elastic constitutive relation modified to include viscous damping, Equation 2.17.

In the case of harmonic motion, for which Sj = ja~Sj, Equation 2.17 implies that attenuation may be accounted for by representing the elastic constants cu by complex elastic constants cu + jo~u. (This is analogous to accounting for dielectric loss in electromagnetic and optical waveguides by the well-known method of postulating a complex dielectric constant or a complex index of refraction.) Equation 2.13, the lossless wave equation for this shear wave, becomes Solution:

02u2 p Ot: =

02u2 (c44 + jtor144) Ox2 9

(2.18)

22

2. Fundamentals of Acoustic Waves

The solution to this lossy wave equation is (2.19)

u2(x,t) = A + e j(~-~ri) e -ax,

in which a is the attenuation of the wave. Substituting Equation 2.19 into 2.18 results in the dispersion relation for shear wave propagation in the lossy material: --/90)2 = (C44 + jtor/44)(Ot + jk) 2.

(2.20)

Equating the real parts and then the imaginary parts of this equation yields --pr 2 "- C44(r 2 -- k2) - 2wakr/44

(real)

(2.21)

(imaginary)

(2.22)

and 0 = 2~kc44 + tor/44(ot2 -

k2).

The presence of the viscous damping term results in a second-order perturbation of the wave velocity and a first-order contribution to the attenuation. Since for most materials a ,~ k, Equatio n 2.22 enables solution for the attenuation coefficient a: to2'044 ot = 9 2pv 3 '

(2.23)

in which the viscous term 7144characterizes the damping of this y-polarized shear wave. The important point to note from this example is that the attenuation is proportional to the square of the frequency. This prediction has been borne out experimentally with both bulk and surface waves for a number of materials. Since the loss increases rapidly with frequency, it is important to use high-quality materials for acoustic sensors operating at high frequency.

2.2

Piezoelectricity

The coupling between strain and electrical polarization that occurs in many crystals provides a means for generating acoustic waves electrically. When the structure of a crystal lacks a center of inversion symmetry 6, the application of strain changes the distribution of charge on the atoms and bonds comprising the crystal in such a manner that a net, macroscopic, electrical polarization of the crys-

6A crystal structure with a center of inversion symmetryhas the property that a straight line drawn from any point in the lattice through the center of inversion will meet an equivalent point at an equal distance from that center.

2.2 Piezoelectricity

MECHANICALVARIABLES Strain ~ (2) ~

Stress I (2)

j

23

ELECTRICALVARIABLES ( Displacement

(Field k

(1)

Figure 2.5 Relations among mechanical and electrical variables for a crystal (after Nye [5]). The direct piezoelectric effect is the production of electric displacement by the application of a mechanical stress; the converse piezoelectric effect results in the production of a strain when an electric field is applied to a piezoelectric crystal. The relation between stress and strain, expressed by Equation 2.7, is indicated by the term "Elasticity." Numbers in square brackets show the ranks of the crystal property tensors: the piezoelectric coefficients are 3rd-rank tensors, and the elastic stiffnesses are 4th-rank tensors. Numbers in parentheses identify l st-rank tensors (vectors, such as electric field and electric displacement), and 2nd-rank tensors (stress and strain). Note that one could expand this representation to include thermal variables (see [5]) and magnetic variables.

tal results (see Figure 2.5). Crystals exhibiting this d i r e c t piezoelectric effect always exhibit the c o n v e r s e effect as well, whereby the crystal is strained by the application of an electric field. Of the 32 different crystallographic point groups to which crystalline materials belong, 21 are non-centrosymmetric (have no center of inversion); one of the 21, due to its high symmetry, has all zero piezo-

24

2. Fundamentals of Acoustic Waves

Table 2.3

Material

Symmetry Class

Gallium Arsenide Lithium Niobate Quartz Zinc Oxide

Trig. 3m Trig. 32 Hex. 6ram Cub. 43m

,

,

.,,

,

,

..

,

Piezoelectric Stress Constants

Piezoelectric Stress Constants (coulomb/m z) exl

ex4

exs

eyz

ea

ez3

3.7

2.5

0.2

1.3

-0.573

1.32

0.154 0.171

-.0436 -0.48

,

,

.

,=

i

ii

l

II

Ill

I

I II

I

electric stress constants 7. The remaining 20 piezoelectric point groups are spread among all seven of the crystallographic systems (cubic, hexagonal, etc.), making it impossible to generalize a priori about whether a given material will be piezoelectric unless its crystallographic point group is known. The coupling between electric field and strain can be accounted for by an appropriate modification to the elastic constitutive relation (Equation 2.8), as well as the electromagnetic constitutive relations,

TI = c~fl~ - etj Ey

(2.24)

Piezoelectric Constitutive Relations

and

Di = e~o.Ej + eijSj,

(2.25)

in which eli are the piezoelectric stress constants, having units of charge/(length) 2, Ei are the electric field components, Di are the electrical displacement components, and e0. are the permittivity constants. The above equations, known as the piezoelectric constitutive relations, completely describe the interplay of stress, strain, and electric field in a piezoelectric solid [6]. In these equations, summation over the repeated indices is assumed. The piezoelectric stress matrix (ety) is a third-rank tensor; reduced notation has been applied so that the subscript I (running from 1 to 6) replaces two lower-case subscripts. Thus, 18 piezoelectric constants are necessary to characterize a piezoelectric material in the most general case. The form of ety for various crystal classes is shown in Table 2.3. In general terms, as crystal symmetry increases, the number of unique piezoelectric constants decreases; as mentioned above, crystals possessing at least one center of inversion, including isotropic materials and many cubic crystals, are 7Other common forms of the constitutive relations for piezoelectric media, employing different piezoelectric coefficients, are [1]: E + a,jej s, = ~,jrj r~ = ~ s j - h,jOj o, = d,jTj + e~ej e, =-h,~sj + b~Oj The quantity stj is the compliance of the solid.

2.2 Piezoelectricity

25

non-piezoelectric and thus have piezoelectric constants equal to zero. The manner in which an electric field gives rise to strain in a crystal is best illustrated by an example.

Example 2.6:

Calculate the strain induced in a piezoelectric ZnO crystal by an electric field of I KV/cm applied along the z axis of the crystal, i.e., E = Ezz.

Solution: implies

If the crystal is unrestrained at its surfaces, then/'i = 0 and Equation 2.24 6

3

cuSj = ~ eljEj. y=l j=l

(2.26)

ZnO is a hexagonal material (symmetry class 6mm) with elastic constants as given in Table 2.2 and piezoelectric constants as given in Table 2.3. Applying the matrices appropriate to the crystal class results in the following equations: CllSl + C12S2 + c13S3 -"

el3Ez

cl2Sl + CllS2 + c13S3 = el3Ez c13S1 -I- c13S2 -I- c33S 3 "$4"- $5-

e33Ez

$6 = 0

Solving this set of equations by matrix inversion and using the appropriate elastic and piezoelectric constants for ZnO yields $1 = $2 = (-5.44 x 10 -12 m/V)Ez and $3 = (1.17 x 1O-II m/V)Ez. For the applied field Ez = 105 V/m, the corresponding strains are SI = $2 = - 5 . 4 4 • 10 -7 and $3 = 1.17 • 10 -6. Thus, an electric field along +z results in elongation of the crystal in the z direction and compression along the x and y axes, as indicated in Figure 2.6 (page 26). In fact, the lateral compression is such that the change in unit volume (known as the dilatation and equal to S l + $2 + $3) is nearly zero. These strains are reversed if the electric field direction is reversed. In general, the strains resulting from an applied field increase with the magnitude of the piezoelectric constants e 0 so that lithium niobate, for example, exhibits larger strains than does gallium arsenide for a given electric field. However, the particular strain pattern found depends on the form of the piezoelectric matrix (and hence, on the crystal class involved) as well as the electric field direction.

2.2.1

THE WAVE EQUATION FOR UNBOUNDED PIEZOELECTRIC MATERIALS

From the piezoelectric constitutive relations it is a simple matter to derive the wave equation for piezoelectric media. The piezoelectric wave equation is typi-

26

2. F u n d a m e n t a l s of Acoustic Waves

Figure 2.6

Strain induced in a ZnO crystal by an electric field applied along the z-axis.

cally written in terms of the displacements ui coupled with an electrical potential, if, rather than coupled with the electric field components Ei. Noting that Ek = -Otl~/Oxk, we can write Equation 2.24 as

3 Out 3 04, Tiy = ~ cijkt-~xk + ~1 e q , ~ . k,t= t

=

tgxk

(2.27)

Application of the equation of motion (Equation 2.6) to Equation 2.24 yields 3 y,k,t=;

Cqkl

d2Ui 3 02t~ t92Ui q- ~ eijk = p~. OXktgXj y,k= l OxkOxj tgt2

(2.28)

2.2 Piezoelectricity

27

If Equation 2.28 is compared with the wave equation in non-piezoelectric media (Equation 2.11), the presence of an additional term involving the electrical potential qb is apparent. This term may be considered as a source term responsible for the generation of an acoustic wave by an applied, time-varying electrical potential. 8 Conversely, the wave displacements generate an accompanying electrical potential through which the piezoelectric wave can be electrically detected. Equation 2.28 represents three equations in four unknowns: ul, u2, u3, and tk. A fourth equation may be obtained from Equation 2.26 by noting that in a region with no free charges, the electrical displacement is solenoidal (V 9 D = 0). This implies that OD~/Ox~ = 0 so that Equation 2.28 becomes 3

t92Rl

3

eikl i,k,l= 1 OXkOXi

02t~

~ 6.ik~ "- 0. i,k= 1 OXkOXi

(2.29)

The effect of coupling between displacement and potential in piezoelectric plane waves can best be illustrated by an example. Example 2.7:

Consider the

in a ZnO crystal. Write Solution:

propagation of a z-polarized, x-propagating shear electric field.

wave

the equation(s) describing the

Using the appropriate elastic and piezoelectric matrices, Equations 2.28 and

2.29 lead to

02u3

tgEt~

02u3

c 5 5 0 x 2 q- ex5 O--xT = p Ot2

(2.30)

and

t92U3

t92t~

ex5 0x 2 -- El l ~ x 2 -- 0.

(2.31)

Using the second equation, ~b is eliminated from the first, yielding C55 -t-

2

ex5) t92U3 Ell

t92U3

OX2 = p 0t2 '

(2.32)

while ul and u2 are zero. Note that Equation 2.32 is identical in form to Equation 2.11 with the following substitution for the term in parentheses:

c55 = c55 1 +

ex.__.___~5 = c55(1 + K2).

(2.33)

C55E11

8Note that the source term is non-zero only in regions where it varies spatially. This is an important consideration in the mechanism by which acoustic waves are launched in the thickness-shear mode resonators that we consider later.

28

2. Fundamentals of Acoustic Waves

The stiffness parameter c55 has, in effect, been increased by the factor (1 + K 2) - - - a n effect known as piezoelectric stiffening. The factor K 2 is the electromechanical coupling coefficient for the x-propagating, z-polarized plane wave: 2

ex5

(2.34)

K2 = ~ .

C551[11

Piezoelectric stiffening increases the wave velocity from that obtained in the non-piezoelectric case. Wave velocity is given by v = (c'55/p)It2. Since the factor K2 has' a value of 0.0717, which is much less than unity, the perturbation in wave velocity caused by piezoelectric stiffening can be approximated as: = [(1 -[- K2) 1/2 - 1] ~- K2~ ,

(2.35)

Z

V

leading to a 3.6% velocity increase for the wave considered in ZnO. The coupling factor K 2 is a measure of how strongly the electrical potential and mechanical displacements associated with a wave interact. From Equation 2.31, the electrical potential accompanying the shear displacement u3 is given by ex5 -- - - u ~'11

3.

(2.36)

Thus, the electrical potential is proportional to mechanical displacement, having a value of ~b = ( - 6 . 3 • 10 9 V/m)u3 for ZnO; a shear displacement of 10 A thus leads to a potential of - 6 . 3 volts. This potential has the same wave description as the displacement, = t~od(a't-~),

(2.37)

and is depicted in Figure 2.7. The electric field can be found by differentiating the potential: Ex = -O~/Ox = -jk~. This leads to an electric field described by E =

ik~exSu3x.

(2.38)

Ell

The electric field is along the direction of propagation and 90 ~ out of phase with the particle displacement, as shown in Figure 2.7.

2.2.2

ENERGY DENSITY AND POWER FLOW

Because passage of an elastic wave causes time-harmonic deformation of a crystal, each unit volume through which the wave passes has time-varying potential (strain) energy and kinetic energy. At any point in the crystal, the time-varying

2.2 Piezoelectricity

29

u(x,to)

Z

'~

X

r

X

l

E(x,to)

Z

9

~

9

~

,

-,~

~

--=~0.4.n=

"-

-=~o

4---

-

,~

~

~'

41--

9

-.~

-"

4-

9

~

~X .~

Figure 2.7 Electric field propagating along with mechanical wave in ZnO. potential energy is maximum when strain, or deformation, of the crystal at that point is maximum and the local particle velocity is zero; after one-quarter wave period has elapsed, local deformation of the crystal reaches zero and velocity (hence kinetic energy) reaches its maximum absolute value. Thus, there is an interchange of kinetic and strain energies, much as in a mass-spring oscillator. The instantaneous kinetic energy density (kinetic energy per volume) is given by 3

E

(2.39)

i=1

In this equation, the squares of the particle velocities are summed over the three directions (the subscript being repeated by squaring the quantity). The instantaneous strain energy density (energy per volume) is given by

6 Us =

~ l,J= 1

3 1

,

1 ijki' ~Oui ~ouk ,

ij,k,1 = I-~C

oxj ox,

(2.40)

30

2. Fundamentals of Acoustic Waves

where c' denotes the piezoelectrically stiffened elastic constants in piezoelectric materials. The similarity between this equation and the expression for the energy stored in a compressed spring, Us = -~kx 2, is apparent. In piezoelectric media, the strain energy is multiplied by the factor (1 + K 2) due to piezoelectric stiffening of the elastic constants. Thus, the strain energy includes electrical stored energy in this case. To calculate the magnitude of the electrical energy density Ue independent of the strain energy, the relation lie =

D" E _ 2

~

Di" Ei :~

- -=

(2.41)

is used. Example 2.8: Calculate the kinetic and strain energy densities for the shear wave examined in the Example 2.7. Solution: Since the shear wave is z-polarized and x-propagating, the only nonzero derivative in Equation 2.39 is Oua/Oxt = -jku3. Thus, the strain energy density is given by Us

=

1., { Ou3 ~2 \ Oxl]

=

1 c55k2u 3

(2.42)

and the kinetic energy density is ,2 UK = 1,,~t02 2v "3"

(2.43)

As mentioned previously, the energy stored in crystal deformation becomes kinetic energy one quarter of a cycle later. Thus, an energy conservation principle (known as the Rayleigh principle) applies to waves propagating in a lossless medium: peak strain energy density must equal peak kinetic energy density. One can apply this concept to calculate the wave velocity: an acoustic wave propagates at the velocity for which peak kinetic energy density exactly equals peak strain energy density. Setting Equation 2.42 equal to Equation 2.43 yields the phase velocity: v = to]k = (c'55/p) 1/2.

(2.44)

This agrees with the velocity found by solving the wave equation. It is instructive to calculate the electrical energy density for this problem. As found in Ex-

2.3 P e r t u r b a t i o n s in Acoustic W a v e P r o p a g a t i o n

31

ample 2.2, the only non-zero component of E is Ex. Thus, the peak electrical energy density is 1. E = Ue= - ~ - D

el l (jkexsu3 ]j 2 = /

- - -

2 \

r

k2e2u 2 x5

3

.

(2.45)

2r

Dividing the electrical energy density by the strain energy density yields UE

2 e x5

Us

c'55ell

K2

1 + K z"

(2.46)

The last term is approximately equal to K 2 when K 2 '~ 1. Thus, the electromechanical coupling coefficient (/(2) has a second interpretation: K 2 is approximately equal to the ratio of peak electrical energy density to peak strain energy density.

2.3

Perturbations in Acoustic Wave Propagation

In acoustic-wave sensor applications, one typically detects the change of wave velocity v and/or attenuation a that is induced as the device interacts with the environment. Consequently, it is important to understand how these interactions cause a sensor response. Changes in wave velocity and attenuation can be fundamentally related to changes in wave energy density and power dissipation, respectively. With regard to velocity changes, the power density P (power/area) carded by a wave can be related to the wave energy density U (energy/volume) stored in a lossless medium. Considering a unit cube (Figure 2.8, page 32) through which a wave is passing, the transit time for the wave across the cube is ~"= 1Iv. When the wave passes through the cube, the energy density in the cube increases by the incident power times the transit time: U = P~" = P/v. Thus,

P = Uv.

(2.47)

This relation can be used to relate changes in wave energy density to changes in wave velocity in a lossless medium, i.e., one in which P is constant. Implicitly differentiating Equation 2.47 yields Av V0

= --

AU U0

(2.48) '

where vo and Uo denote unperturbed propagation velocity and energy density,

32

2. Fundamentals of Acoustic Waves

Transit time --

x=l/v---

I -..,

Power --,

,-m

,,--

mm

Density mD

P

I ~

m

m

m

m

/

Stored Energy Density U Figure 2.8 Energy stored per unit volume is dependent upon incident power flow and propagation velocity.

respectively. Equation 2.48 represents a fundamental relation between wave velocity and energy density for a system excited at a given frequency: the fractional change in wave velocity is equal to the negative of the fractional change in wave energy density. This may seem reasonable if one considers that in a system excited at a given frequency, the wavelength adjusts so that the peak kinetic energy equals the peak potential energy of the wave. The result is that changes of the medium that affect the wave energy density will cause changes of the wave velocity. Example 2.9: If a plane wave propagates in a medium in which the mass density changes, how is wave velocity affected? Solution:

The peak kinetic energy density UK is 3 (2.49)

=

i=1

Implicit differentiation of the varying quantities leads to

AUK Ap = . UK Po

(2.50)

2.3 P e r t u r b a t i o n s in Acoustic W a v e P r o p a g a t i o n

33

Thus, from Equation 2.48, the fractional change in wave velocity is minus the fractional change in mass density of the medium: Av

Ap

= - --.

v0

(2.Sl)

190

The preceding analysis will be used extensively in Chapter 3 to derive relations between mass accumulation and wave velocity for acoustic modes other than plane waves. Next, the effect on wave propagation of power dissipation in a lossy medium is considered. From conservation of energy, the power dissipated by the wave Pd (power/volume) must be balanced by a reduction in power transmitted by the wave P. If the wave is propagating in the x direction, then

ed =

OP Ox

-- --'-.

(2.52)

Since energy density and power flow are proportional to the square of wave amplitude, in a lossy medium

P(x) = Po e-2ax,

(2.53)

OP - - 2t~P. ax

(2.54)

so that

Combining Equations 2.52 and 2.54 yields

ed a = 2P"

(2.55)

Equation 2.55 indicates the relationship between wave attenuation and power dissipation in the medium: attenuation is one-half the ratio of power dissipated to power transmitted by the wave. Note that in the derivations of this section, velocity and attenuation changes depend on ratios of energy and power, not on absolute levels. Consequently, in the small-signal limit, velocity and attenuation changes are independent of wave amplitude. It will be seen in Chapter 3 that many perturbations affect wave propagation. In general, perturbations change both energy storage and power dissipation and thus result in a combination of velocity and attenuation changes. The manner in which the propagation of the wave is described is therefore important and will be discussed here briefly.

34

2. F u n d a m e n t a l s of Acoustic W a v e s

The continuous propagation of a wave in the x direction is described generally as

u(x,y,z,t) = u(y,z)ey"-~x,

(2.56)

where y is a complex propagation factor representing both attenuation and wavenumber:

y = a + j k = a + j to

(2.57)

v

If frequency is constant, then changes in wave propagation can thus be represented by Ay = Aot

-

jko Av

(2.58)

v0

or, in normalized form, as [4] A7' =

A7 Aa Av - . , = ko k0 J v0

(2.59)

in which k0 is the unperturbed wavenumber. The latter makes clear that Aa/k and Av/vo are consistently normalized orthogonal components of changes in the complex propagation factor y caused by a generalized perturbation [4]. The dependencies of these quantities on a given perturbation therefore will be the subject of numerous theoretical calculations as well as experimental measurements to be discussed in later chapters. Since perturbations generally involve changes in both stored energy and dissipated power, it is useful to combine Equations 2.48 and 2.55 to define a complex power transfer Prfrom the wave, by the perturbation, that accounts for both power dissipation Pd and changes in stored energy AU:

PT = Pd + j2ooAU.

(2.60)

Using Equations 2.48, 2.55, and 2.59, changes in the complex propagation factor y are related to this complex power transfer [4], Aa ko

A v = Pr J v0 2koP'

(2.61)

giving a general relationship between power transferred from the wave Pr and the resulting changes in the wave propagation factors, Av/vo and Aa/k. It will be demonstrated in Chapter 3 that measuring (or calculating) both Aa/k and Av/vo and plotting these changes parametrically (i.e., one versus the other) often simplifies the representation of a wave perturbation. Since Aa/k and Avlvo

References

35

are orthogonal components of A3", parametric representation is a means of displaying the perturbation in the complex plane, a strategy that often yields a simpler and/or more distinctive form than is characteristic of individual plots of Av/vo or Aa/k.

References 1. Auld, B. A. In Acoustic Fields and Waves in Solids; Wiley: New York (1973). 2. Landau, L. D. and Lifshitz, E. M. Theory of Elasticity, 3rd ed.; Pergamon: New York, Ch. 1 (1986). 3. Brekhovskikh, L. and Goncharov, V. Mechanics of Continua and Wave Dynamics; Springer-Verlag: New York, (1982). 4. Martin, S. J.; Ricco, A. J.; Niemczyk, T. M.; Frye, G. C. Sensors and Actuators, 20, 253-268 (1989). 5. Nye, J. F. Physical Properties of Crystals; Clarendon Press: Oxford (1957). 6. IEEE Standard on Piezoelectricity 176-1987; IEEE Press: Piscataway, NJ (1986).

Chapter 3

Acoustic Wave Sensors and Responses

Devices based on piezoelectric crystals, which allow transduction between electrical and acoustic energies, have been constructed in a number of configurations for sensor applications and materials characterization. This chapter examines those devices most commonly utilized for sensing applications, including the thicknessshear mode (TSM) resonator, the surface acoustic wave (SAW) device, the acoustic plate mode (APM) device, and the flexural plate wave (FPW) device. Each of these devices, shown schematically in Figure 3.1, uses a unique acoustic mode. A distinction can be made between one-port acoustic devices, such as the TSM resonator, and two-port devices, including the SAW, APM, and FPW devices. In one-port acoustic devices, a single port serves as both the input and the output port. The input signal excites an acoustic mode which in turn generates charges on the input electrode. These signals combine to produce an impedance variation that constitutes the TSM resonator response. In two-port devices, one port is used as the input port and the other as an output port; these are typically interchangeable. The input signal generates an acoustic wave that propagates to a receiving transducer, which regenerates a signal on the output port. The relative signal levels and phase delay between input and output ports constitute two responses I.

IAs noted in Chapter 1, with all of these devices one has the option of using the device as part of an active c i r c u i t - one containing an energy source such as a battery m or in a passive manner where one measures the device characteristics by supplying a time-varying probe signal to the device. In the active device measurement mode, one monitors the oscillation frequency. In the passive measurement mode, one might monitor the resonant frequency and quality factor of the quartz crystal resonator of Figure 3.1a; correspondingly, one might measure the phase shift and the attenuation of the APM delay line of Figure 3.1 c.

36 ACOUSTIC WAVE SENSORS

Copyright 9 1997 by Academic Press All rights of reproduction in any form reserved. ISBN O-12-077460-7

t~ O me t~ I= .r

t~ C

t~ "O O t~ r~

Figure 3.1 Schematic sketches of the four types of acoustic sensors. (a) Thickness-Shear Mode (TSM) resonator; (b) Surface-Acoustic-Wave (SAW) sensor; (c) Shear-Horizontal Acoustic-Plate-Mode (SH APM) sensor; and (d) Flexural-PlateWave (FPW) sensor.

38

3. Acoustic Wave Sensors and Responses

Acoustic wave devices are sensitive, to varying degrees, to perturbations of many different physical parameters, both intrinsic to the device and extrinsic. Chemical sensitivity is typically imparted by attaching a thin film to an acoustically active region (i.e., a region having significant acoustic amplitude) of the device surface. Acoustic sensing is possible only when the film (or adjacent medium) interacts with the acoustic modes' Thus, the film must serve as a chemical-to-physical t r a n s d u c e r - one or more of its physical properties must change in response to the presence (and, optimally to an extent proportional to the concentration) of the chemical species to be detected. Clearly, important sensor parameters such as sensitivity, selectivity, reversibility, and durability are critically dependent on this chemically sensitive film, as detailed in Chapter 5. Commonly, the increased mass density of the film, arising from species accumulation, is relied upon for a sensor response. Changes in other film parameters, including elastic and electrical properties, however, can also contribute to the response and must therefore be considered as well. All the acoustic wave devices examined for sensing applications will function in a gaseous or vacuum ambient, but only a subset of them operate effectively when they are in contact with liquids. Acoustic devices that generate primarily shear motion in a liquid, e.g., the TSM resonator and shear-horizontal SH-APM device, can operate without excessive damping. In contrast, devices with substantial surface-normal displacement components (e.g., the SAW device), which radiate compressional waves into the liquid, suffer excessive damping. An exception to this rule occurs for devices utilizing waves that propagate at a velocity lower than the sound velocity in the liquid, e.g., the FPW device. Regardless of the disposition of the displacement components, such modes do not radiate coherently and are thus relatively undamped by liquids. Those modes that do propagate in contact with liquids, while not excessively damped, are nevertheless influenced by liquid properties. The mechanisms whereby all these devices interact with their immediate e n v i r o n m e n t - be it a thin film, a liquid, or both --as well as the resulting response, will be described in this chapter.

3.1

Thickness-Shear Mode (TSM) Resonator

The thickness-shear mode (TSM) resonator, widely referred to as a quartz crystal microbalance (QCM) 2, typically consists of a thin disk of AT-cut quartz with circular electrodes patterned on both sides, as shown in Figure 3.2. Due to the piezoelectric properties and crystalline orientation of the quartz, the application of a voltage between these electrodes results in a shear deformation of the crystal. The crystal can be electrically excited in a number of resonant thickness-shear modes.

3.1 Thickness-Shear Mode Resonator

Figure 3.2

39

Schematic of a thickness-shear mode quartz resonator.

For each of these, displacement maxima occur at the crystal faces, making the device sensitive to surface perturbations. The perturbations to be considered in this section include surface loading by (1) an infinitesimally thick mass layer, (2) a contacting Newtonian fluid, and (3) a viscoelastic layer of finite thickness. The T S M resonator was originally used in v a c u o to measure metal deposition rates [ 1]. More recently, the T S M resonator has been shown to operate in contact with liquids [2,3], enabling its use as a solution-phase microbalance. The device is typically incorporated in an oscillator circuit, where the oscillation frequency tracks the crystal resonance and indicates mass accumulation on the device surface. This microbalance capability has facilitated a n u m b e r of gas- and liquid-phase sensor applications that will be discussed in Chapter 5. W h e n an alternating voltage is applied across the T S M resonator electrodes, shear waves having opposite polarities are generated at the electrodes on the two 2The term "quartz crystal microbalance" is an unfortunate name for this device for several reasons: (1) The word "crystal" is redundant when it follows "quartz," a crystalline material; (2) the devices do not invariably act exclusively as microbalances, being subject to a number of other physical perturbations as well; (3) the name could also correspond to a SAW, APM, or FPW device fabricated from quartz. The term thickness-shear mode (TSM) resonator follows the convention used for the SAW, SH-APM, and FPW notations in that it describes the nature of the acoustic mode upon which the device is based.

40

3. Acoustic Wave Sensors and Responses

faces of the crystal resonator. The waves are generated only at the electrodes because those are the only regions in the crystal where the piezoelectric source term eqkE varies spatially. The displacement Ux in the crystal is a superposition of these waves: Ux(y,t) = (Ae yky + Be-Y~Y)e j'~,

(3.1)

where A and B are constants, r is the angular excitation frequency (r = 2,trf), k is the wavenumber, t is time, and j = (-1)1/2. The resonant condition for the TSM resonator can be determined by tracing the path of one propagating shear wave, shown in Figure 3.3, which may be thought to originate at the top electrode. After transiting the crystal, this wave is reflected (with phase shift due to reflection qbr = r at the bottom crystal face, returns to the top electrode where it again reflects (with ok,.= r phase shift) and is once again propagating downward having experienced a total round-trip phase shift of (2khs + 2fir), where hs is the thickness of the crystal. When this total phase shift is an integer multiple of 2r constructive interference (i.e., coherence) between the incident and return waves leads to crystal resonance. Noting that k = 2r = r where A is acoustic wavelength and vs is the shear wave velocity, this criterion for resonance leads to the following resonance conditions:

and

Nvs fN = 2hs'

(3.2b)

where N is an integer. Equation 3.2a indicates that resonances occur when the crystal thickness hs is a multiple of half the acoustic wavelength A. The shear wave phase velocity vs in the substrate is given by vs =

(3.3)

where/zq and pq are the shear stiffness and mass density, respectively. Equation 3.2b indicates the frequencies at which the family of resonances can be excited, corresponding to various values of the mode index N. A more detailed analysis indicates that the surface electrodes can excite only the odd harmonics, i.e., N = 1, 3, 5 . . . . [4]. Example 3.1: Calculate the fundamental resonant frequency for an AT.cut quartz crystal with hs = 0.033 cm, /xq = 2.95 X 1011 dyne/cm 2, and pq = 2.65 g/cm 3.

3.1 Thickness-Shear Mode Resonator

41

Figure 3.3 Transit of a shear wave, illustrating the resonant condition.

Solution: From Equation 3.3,1:, = 3.34 • 10s cm/s; substituting this into Equation 3.2b gives fl = 5.06 MHz. Having derived the condition for crystal resonance allows the displacement profile at resonance to be calculated. When the crystal is operated in air or vacuum, the crystal faces experience no external restoring force and are considered to be stress-free boundaries; this implies that OUx/Oy = 0 at the upper and lower crystal faces. Applying this boundary condition to Equation 3.1 yields the shear displacement profile across the crystal:

Ux(y,t ) = Uxo cos(kNy) ejoJt,

(3.4)

where Uxo is the surface displacement amplitude and kN = Nrr/hs. Equation 3.4 represents a standing shear wave with maxima at the crystal surfaces. Since displacement varies only across the thickness of the crystal in this one-dimensional model, Equation 3.4 describes a family of thickness-shear modes, each having a unique standing shear-wave pattern across the crystal thickness. Figure 3.4 (page 42) illustrates the mode profiles for the fundamental (N = 1) and third-harmonic (N = 3) resonances. These modes are referred to as "thickness-shear" because the shear displacement varies across the thickness of the device. Note that these modes do not involve any change in the thickness of the substrate. For a typical crystal thickness of 0.33 mm, these modes are excited at approximately 5 and 15 MHz, respectively.

42

3. Acoustic Wave Sensors and Responses

Figure 3.4 Shear displacement profiles across the resonator thickness for the fundamental and the third-harmonic resonances. (Reprinted with permission. See Ref. [9]. 9 1991 American Chemical Society.)

3.1 Thickness-Shear Mode Resonator

3.1,1

43

TSM RESONATOR MASS SENSITIVITY

The presence of displacement maxima at the crystal surfaces makes the thickness-shear mode very sensitive to surface mass accumulation. Mass that is rigidly bound moves synchronously with the crystal surface, perturbing the TSM resonant frequency. The relation between surface mass accumulation and resonant frequency change can be derived from a simple variational principle proposed by Rayleigh [5]: resonance in a mechanical system occurs at frequencies at which the peak kinetic energy Uk exactly balances the peak potential energy Up. This principle is a consequence of the fact that energy is periodically exchanged between kinetic and potential forms at resonance. The accumulation of an ideal mass layer at the crystal surface, which is an anti-node or maximum of displacement, causes an increase in the kinetic energy with no change in potential energy This assumes that the mass layer is sufficiently thin and/or rigid and that displacement is uniform across its thickness, i.e., it acts as an acoustically thin layer. The Rayleigh hypothesis indicates that when mass accumulates on the surface, the resonant frequency must change to rebalance kinetic and potential energies. The peak kinetic energy density Uk (energy per surface area) in the TSM resonator occurs at the instant when particle velocities are maximum and displacements are zero. This energy is found using Equation 2.39 and summing the kinetic energies from infinitesimal slices taken across the crystal thickness, beginning with the infinitesimally thin surface mass layer (electrode mass is neglected): hs

Uk -- ---~

psUxo -t- pq

l Ux(y)

dy ,

(3.5)

where Ps is the areal mass density (mass/area) of the surface mass layer and pq is the volume mass density (mass/volume) of the quartz substrate. Substituting Equation 3.4 into 3.5 and integrating yields

Uk-

u:o( 2

- Ps 4- p q h s

)

.

(3.6)

2

The peak potential energy density Up in the TSM resonator occurs at an instant (depicted in Figure 3.4) when displacement is maximum in the crystal and velocity is zero. From Equation 2.40, this is given by

1. k2uZ fh~ 22 Up = ~l~q XOJo sin2(ky) dy = tzqk 4uxohs "

(3.7)

44

3. Acoustic W a v e Sensors and Responses

Invoking the Rayleigh hypothesis by balancing peak kinetic and potential energy densities (Equations 3.6 and 3.7) gives a relationship between resonant frequency to and surface mass density ps: = 1 +

h,pq

,

(3.8)

where a~o = (NTr/hs)(/Xq/pq) 1/2 is the unperturbed resonant frequency, i.e., that obtained when Ps = 0. For Ps "~. hspq, Equation 3.8 is approximated by a linear relationship:

af fo

=

h,pq

,

(3.9)

indicating that the fractional shift in resonant frequency is equal to the fractional change in mass contributed by the mass layer. This linear approximation to Equation 3.8 has been found to hold for mass fractions up to about 2% [6]. Combining Equations 3.2, 3.3, and 3.9 gives the Sauerbrey equation commonly used to relate changes in TSM resonant frequency to surface mass density Ps [1]:

2f ps Af =

-

([.Lqtoq) l/2 .

(3.10)

E x a m p l e 3.2: (a) If the sensitivity S is defined by S = dfldps, find the sensitivity of a 5 MHz TSM resonator having thickness hs = 0.033 cm and quartz density pq = 2.65 g/cm 3. (b) If the limit of mass resolution is defined as the mass density that gives a frequency shift three times larger than the oscillator fluctuation, and TSM resonator stability is O.1 Hz, calculate the limit of mass resolution.

Solution: (a) From the definition of sensitivity and Equation 3.9, S = dfldps = -fol(pqh) = -57 Hz-cm2/~g. This means that for each 1/zg/cm 2 of mass accumulation, the resonant frequency will decrease by 57 Hz. (b) The limit of mass resolution is the mass density that causes a frequency shift of 0.3 Hz: Rm = 3(AJ)/S = (0.3 Hz)/(57 Hzcm2//zg) = 5 ng/cm 2.

The high mass sensitivity calculated in the previous example justifies the term "microbalance" in describing the sensing capabilities of the quartz resonator. Equation 3.9 can be used to calculate frequency shifts for surface accumulations that behave as ideal mass layers. A real film behaves as an ideal mass layer if it is sufficiently thin and rigid so that it moves synchronously with the oscillating device surface. On a TSM resonator, this condition is realized if the acoustic phase shift ff across the film is small, i.e., ~b '~ ~'. The phase shift is

3.1 Thickness-Shear Mode Resonator

45

where p, G, and h are the film density, shear modulus (assumed real for now), and thickness, respectively. Under this condition, Equations 3.9 and 3.10 can be used to predict the change in frequency contributed by the film, using Ps = ph. Example 3.3:

Does a 1 ton film on a 5 MHz TSM resonator that has G = 101~ dyne/cm 2 and p = 1 g/cm 3 behave as an ideal mass layer? If so, what frequency shift does it cause?

Solution: Substituting film parameters and frequency into Equation 3.11 gives tk--0.03. Since this phase shift is small compared with w, the dominant contribution to the resonator response is due to the film mass and it may be considered an ideal mass layer. Using hq = 0.033 cm and pq = 2.65 g]cm3, Equation 3.9 indicates Af = -5.7 kHz. The previous example illustrates the calculation of frequency shifts caused by films that behave as an ideal mass layer. Section 3.1.9 will describe the treatment when films do not meet this criterion.

3.1.2

ELECTRICAL

CHARACTERISTICS

OF AN UNPERTURBED

TSM RESONATOR

In the previous section we considered the conditions under which mechanical resonances would occur in a TSM resonator. In considering only the mechanical properties of the crystal, however, we neglected consideration of how these resonances would actually be excited or detected. The device uses a piezoelectric substrate material in which the electric field generated between electrodes couples to mechanical displacement. This allows electrical excitation and detection of mechanical resonances. In constructing a practical sensor, changes in resonant frequency of the device are measured electrically. The electrical characteristics of the resonator can be described in terms of an equivalent-circuit model that describes the impedance (ratio of applied voltage to current) or admittance (reciprocal of impedance) over a range of frequencies near resonance. There are two general types of equivalent-circuit models that can be used to describe the resonator: the distributed (or transmission-line) model and the lumped-element model. The distributed model for the TSM resonator is shown in Figure 3.5a, page 46 [7]. This distributed model uses a transmission line to represent the propagation of acoustic energy across the device thickness. The acoustic variables, stress T and particle velocity v, are coupled, via a transformer, to an electrical port. The model thus has two acoustic ports and one electrical port. By terminating the acoustic ports in a mechanical impedance Zs, representing the surface "loading condition," the electrical response of the device is obtained from the model.

46

3. Acoustic Wave Sensors and Responses

(a)

v2

vI

!!';AZ ta,'iI"='-! ~;cJil!

/

-jAZ

sinll(d) T1

T2

-c o

V

(b)

L1

c~

c~ ~:: R 1

m m

Figure 3.5 Equivalent-circuit models to describe the near-resonant electrical characteristics of the resonator: (a) distributed model; (b) lumped-element model. (Reprinted with permission. See Refs. [7;14]. (a) 9 1994 American Institute of Physics and (b) 9 1993 AmericanChemical Society.)

3.1 T h i c k n e s s - S h e a r M o d e R e s o n a t o r

47

With an unperturbed device the surfaces may be considered stress-free boundaries (neglecting electrode mass): T1 = T2 = 0, so that Zs = 0. Thus, the unperturbed condition corresponds to short-circuiting the acoustic ports in the equivalent-circuit model of Figure 3.5a. The distributed (transmission-line) equivalent-circuit model of Figure 3.5a can be reduced to a simpler lumped-element model, shown in Figure 3.5b, to describe the near-resonant electrical characteristics [7]. This lumped-element model is called the "Butterworth-Van Dyke" equivalent circuit [8]. A static capacitance Co arises between the electrodes located on opposite sides of the insulating quartz. A capacitance Cp is included to account for parasitic capacitance that is found to arise in the test fixture: C ~ = Co + Cp. Since the quartz is also piezoelectric, electromechanical coupling gives rise to an additional motional contribution (L l, C1, R l), i.e., one that is associated with the motion of the resonating crystal in parallel with the static capacitance. The static capacitance dominates the electrical behavior away from resonance, while the motional contribution dominates near resonance. This model simulates the TSM resonator electrical characteristics over a range of frequencies near resonance. The elements of the circuit are given by [9]"

Co-

~2A hs ' 8K'2Co

C1-- (Nqr)2, L1 =

1

2

tosC1

,

(3.12a)

(3.12b) (3.12c)

r/q el

-

I&qC1

(3.12d)

where A and hs are electrode area and substrate thickness; tos = 2"n'fs, where fs is the series resonant frequency for the unperturbed TSM resonator; K 2, E2z,/Xq, pq, and r/q are the square of the quartz electromechanical coupling coefficient, dielectric permittivity, shear stiffness, mass density, and effective viscosity, respectively. TSM resonator electrical characteristics are typically described in terms of electrical admittance, defined as the ratio of current flow to applied voltage (the reciprocal of impedance). The total TSM resonator admittance can be determined from inspection of the equivalent circuit model:

Y(to) = jtoC o -~ Zm 1 '

(3.13)

48

3. Acoustic Wave Sensors and Responses

where the motional impedance for the unperturbed resonator is 1

Zm = R1 + jtoLI + -jas ----.

(3.14)

The series resonant frequency fs is defined as the frequency at which the motional reactance is zero, i.e., 1

jtosL~ + . . . . = 0. ja,,C~ Solving Equation 3.15 for tos, and noting that tos = 2r

(3.15) gives

1

fs = 2r

(3.16)

for the unperturbed device. When f > fs, Equations 3.14 and 3.15 indicate that the motional branch has a net inductance. This resonates with the parallel capacitance C o, causing a "parallel resonance." The parallel resonant frequencyfp is defined as the frequency at which the total reactance (motional plus static admittances) is zero:

,[1(,

_,

Figure 3.6 shows the admittance magnitude and phase angle measured near the fundamental resonance of a TSM resonator in air. The figure indicates the series (fs) and parallel (fp) resonances, where the admittance phase angle ( / Y ) is zero and device reactance is zero. The admittance magnitude Irl is near maximum at f~ and near minimum at fp. The solid lines in Figure 3.6 are the admittance magnitude and phase calculated from the equivalent-circuit model (Equations 3.12-3.14) using best-fit values of Co, Cp, fs, and Rl.

3.1.3

ELECTRICAL CHARACTERISTICS OF THE TSM RESONATOR WITH AN ARBITRARY SURFACE PERTURBATION

Piezoelectric coupling between mechanical displacement and electrical potential in the quartz causes the mechanical interactions of a surface perturbation with the TSM resonator to influence the electrical characteristics of the TSM resonator, i.e., the impedance or admittance. By using a continuum model that describes the coupled mechanical displacement and electrical potential, the electrical characteristics can be related to the properties of the perturbing mass or liquid layer [ 10]. The continuum model can be solved to obtain an equivalent circuit that ap-

Admittance Mag., IYI (mS)

Admittance Phase, LY (deg) I

I

I

0

0

0

0

0

0

O

0

r

o

~

~t'rl ! q

q

I

q q

=.

)i,

~'~

(1)

m--

t~

m.

~D ml mlt

~D

O

0

O

01

b

G.

O

4~ ~D

50

3. Acoustic Wave Sensors and Responses

proximates the electrical characteristics of the TSM resonator for excitation frequencies near resonance [9,11 ]. The equivalent circuits (Figure 3.5) can be used to describe the electrical response of the perturbed device. The lumped-element model, Figure 3.5b, is most convenient to use. When the resonator has a surface perturbation, the motional impedance increases, as represented by the equivalent-circuit model of Figure 3.7. This model contains the elements Co*, L1, Cl, and R1 corresponding to the unperturbed resonator. In addition, the surface perturbation causes an increase in the motional impedance Zm, as described by the complex electrical element Ze in Figure 3.7a. This element is given by [12] Ze --

Nrr ( Zs ~, 4K~wsCo \ZqJ

(3.19)

where Zq = (pqlZq)1/2 is the quartz shear-wave characteristic impedance and Zs is the shear mechanical impedance at the device surface [ 13]:

z~ = Txy [

~a.2o)

Vx lY=O

where Txy is the sinusoidal steady-state shear stress (force per area in the xdirection on a y-normal plane) imposed on the contacting medium by the resonator, and Vx is the resulting x-directed surface shear particle velocity. Zs is a complex quantity: the real part, Re(Zs), corresponds to the component of surface stress in phase with the surface particle velocity and represents mechanical power dissipation at the surface; the imaginary part, Im(Zs), corresponds to the stress component 90 ~ out-of-phase with particle velocity and represents mechanical energy storage at the surface. Letting Ze = R2 + jtoL2 allows the complex element Ze to be represented by a real motional resistance R2 and inductance L2, as indicated in Figure 3.7b. From Equation 3.19, the motional impedance elements L2 and R2 can be related to the components of the surface mechanical impedance as [ 14]

N'n" Re(Zs) R2 = 4K2tosCo Zq N~r Im(Zs) L2 = 4K2~2sCo ---:---. Zq

(3.21a)

(3.21b)

The electrical characteristics of the TSM resonator with a generalized surface perturbation can be described by the equivalent-circuit model of Figure 3.7b [ 14]

3.1 Thickness-Shear Mode Resonator

(a)

(b)

__L_k. 1

51

,,, T

ILl

/r

t

Co : =

C1

Co =

==C 1

R1

R1 L2 R2

J, Lumped-elementequivalent-circuit models for the perturbed resonator [ 14]" (a) with complex impedance element Ze, and (b) with motional inductance L2 and resistance R2. (Reprinted with permission. See Ref. [14]. 9 1993 American Chemical Society.) Figure 3.7

with L2 and R2 related to the mechanical impedance contributed by the surface perturbation through Equations 3.21. The electrical admittance of the loaded resonator is as given in Equation 3.13, but with the motional impedance given by: Zm :

(RI + R2) +

jm(L1 +

L2) 4-

1

jmCl

.

(3.22)

Since the series resonant frequency is defined as the point where motional inductance and capacitance resonate, the motional inductance L2 causes a shift in series resonant frequency (relative to the unperturbed case) given by" Afs = -

L2f~ 2(LI + L2)

~"

L2A 2LI

.

(3.23)

The formalism outlined above will be applied to determine equivalent-circuit models for a TSM resonator with (1) an ideal mass layer, (2) a contacting semiinfinite liquid, and (3) a viscoelastic film. By determining the mechanical impedance Z s associated with each perturbation, the equivalent-circuit model arising from each can be obtained. In cases where the perturbation cannot be easily modeled, the procedure can be reversed: the resonator response is used to determine Z s and thereby characterize the perturbation.

52

3.1.4

3. Acoustic Wave Sensors and Responses

E L E C T R I C A L C H A R A C T E R I S T I C S OF T S M R E S O N A T O R W I T H AN IDEAL MASS LAYER

An ideal mass layer is assumed to have an infinitesimal thickness, yet contribute a finite areal mass density to the device surface. In Section 3.1.1, we noted that this criterion holds as long as the acoustic phase shift across the film ff is small compared with ~r. The equivalent-circuit model for the mass-loaded resonator can be determined from the surface mechanical impedance Zs contributed by a surface perturbation. The surface stress required to sinusoidally accelerate a mass layer is [14] Txy = Ps Vxo = jtopsV~o

(3.24)

where Ps is the areal mass density (Ps = ph, where p and h are film density and thickness) contributed by the mass layer, and vxo is the surface particle velocity. From Equation 3.20, the surface mechanical impedance associated with the mass layer is [14]: Z mass layer =

Txy I = jtops. Vx [y=O

(3.25)

Combining Equations 3.19 and 3.25 gives the motional impedance elements arising from the ideal mass layer [9,14]' 2 tosL l ps

N zrps

L2 = NTrN/ t.~qpq = 4g2tosCoN/ ij,qpq R2 = 0.

(3.26a) (3.26b)

Equations 3.26 reflect

the fact that moving surface mass leads to energy storage (L2) but no power dissipation (R2 = 0). Energy stored in the inductance arises from the kinetic energy of the mass layer moving synchronously with the resonator surface. The ideal mass layer causes a shift in the series resonant frequency that can be determined from Equation 3.23 [9]: A.fs ~

L2fs

2

2f sPs = ---'--'--'2LI N~v/12,qpq

(3.27)

reproducing the Sauerbrey equation [ 1] derived in Section 3.1.1. When a mass layer is added to one side of the TSM resonator, the electrical characteristics are changed, as described by the element L2. Figure 3.8 shows the effect of mass loading on TSM resonator admittance near resonance. It is apparent that the major effect of the mass layer is to translate the admittance curves

3.1 Thickness-Shear Mode Resonator

~

53

A

C---t

3.0

L

I~]---- A

B

~. 2.5

B m

9 C m

E

9o

2.0

D 1.5

~.o 0.5

0 i

i

ii1|

i

i

i

t

t

i

I A

90 ~

-

=-;-

=

-

60

'

j

,,

~3o .~

a,,

' 0

B ~

I~

-3o

'

-90 !

i !

, '

4.98

I

'

4.99

I

'

5.00

Frequency

A

._.j-'

'

5.01

(MHz)

Figure 3.8 Electrical admittance vs frequency before and after deposition of a 124 nm Au layer: (A) in air, (B) in water; after Au deposition: (C) in air, (D) in water. (Reprinted with

permission.

See

Ref.

[9]. 9

1991

American

Chemical

Society.)

54

3. Acoustic Wave Sensors and Responses

toward lower frequency without affecting the admittance magnitude. The admittance of the TSM resonator under mass loading can be obtained from the unperturbed case by the addition of the inductance L2 to the motional arm of the equivalent circuit. This element represents the increased kinetic energy contributed by the mass layer moving synchronously with the TSM resonator surface. The solid lines in Figure 3.8 are admittances calculated from the equivalent-circuit model (Equations 3.12-3.14) after adding an inductance value L2 = 188/xH corresponding to the surface mass density of 225/xg/cm 2.

3.1.5 ELECTRICAL CHARACTERISTICS OF THE TSM RESONATOR CONTACTED BY LIQUID The TSM resonator can be operated in liquid to measure either (1) the accumulation of mass onto the surface from the liquid phase, or (2) properties of the contacting liquid itself. In this section we derive the equivalent-circuit model for the resonator contacted by a semi-infinite Newtonian fluid. A Newtonian fluid is one in which the shear stress and the gradient in fluid velocity are related by a constant, independent of amplitude or frequency [15]:

Ovx T x y = - r l Oy

(3.28)

where rl is the shear viscosity of the fluid. The velocity field, vx, generated in a contacting liquid by the in-plane oscillation of the TSM resonator surface is determined by solving the Navier-Stokes equation for one-dimensional plane-parallel flow [ 15,16]:

O2Vx

rl Oy2 = pVx,

(3.29)

where p and r/are the liquid density and shear viscosity, respectively, and ~x = Ovx/Ot. The solution to this equation with an oscillatory shear driving force at the solid-liquid boundary is [15]

Vx(y,t) = Vxoe-Y/S cos ( ~ -

o~t),

(3.30)

where y is the distance from the surface, Vxo is the surface particle velocity and 8 is the decay length for the envelope of the liquid velocity field. Equation 3.30 represents a critically damped shear wave radiated into the liquid by the oscillating TSM resonator surface (Figure 3.9).

too

,.q me t~ gr r gO m/

gh t~ O =

Figure 3.9

Cross-sectional view of a TSM resonator contacted on one side by a liquid. (Reprinted and adapted with permission.

O

See Ref. [14]. 9 1993 American Chemical Society.)

tall L~

56

3. Acoustic Wave Sensors and Responses

The decay length, 8, is [ 16] =

.

(3.31)

The shear stress imposed by the surface on the liquid to generate the velocity field of Equation 3.30 is [14]

OVx [

_ *lVxo (1 + j).

,=o

(3.32)

8

Application of Equation 3.20 and 3.31 to Equation 3.32 yields the surface mechanical impedance due to a semi-infinite liquid [ 14]:

Zs=

topt/)1/2 2

(3.33)

(1 +j).

The motional impedance elements arising from liquid loading are found from Equations 3.21 [9,14,17]:

L2 = ~ R2-

I~qpq

= 4K2tosCo

2tos/Xqpq

(3.34a)

tOsLl ( 2tosprl ) ,/2 Nzr ( p r l ) , / 2 N~r /Xqpq - 4K2Co 2tos/Xqpq

(3.34b)

where we note that R2 --- tosL2 for loading by a Newtonian fluid [ 14]. Equations 3.34 were derived for one-sided liquid contact; for two-sided, L2 and R2 are doubled. The motional inductance L2, representing the kinetic energy of the entrained liquid layer (with effective thickness 8/2), leads to a decrease in the series resonant frequency [14,17] from Equation 3.23 in agreement with the prediction of Kanazawa and Gordon [ 18]:

Af s _~

L zf s _ 2Ll

f 3/2 ( . . p rl .)'/2 N 7rld~qpq

(3.35)

The motional resistance, R2, represents power radiated into the contacting liquid by the oscillating device surface. It can be considered a shear-wave "radiation resistance." This motional resistance leads to resonance damping. Muramatsu et al. [19] and Beck et al. [20] have shown experimentally that the motional resis-

3.1 Thickness-Shear Mode Resonator

57

tance arising from liquid contact is proportional to (pr/) 1/2. Martin et al. have shown that Equation 3.34b accurately predicts the magnitude of the motional resistance for devices with sufficiently smooth surfaces [ 14]. Yang and Thompson [21] have noted that when a TSM resonator is operated in a liquid, fringing electric fields can enter the liquid, making C o sensitive to the dielectric properties of the liquid. This sensitivity, which can be considered to arise from changes in the parasitic capacitance Cp, is especially pronounced when both electrodes are immersed. Tiean et al. [22] have noted that under these circumstances, a parallel conductance must be added to the equivalent-circuit model to account for conduction through the liquid between electrodes. Example 3.4: Calculate the liquid decay length 8, motional resistance R2, and change in series resonant frequency Afs caused by placing water in contact with one face of a 5 MHz TSM resonator having Co = 5 pF. For quartz [231: K 2 = 7.74 • 10-3, pq = 2.65 g/cm 3,/.tq = 2.95 • 1011 dynelcm2; for water: p = 1 g/cm 3, and r / = 0.01 P. Solution: From Equation 3.31, the liquid decay length in water at 5 MHz is 8 = 0.25 ~m. From Equation 3.34b, the motional resistance is 290 ohms; From Equation 3.35, Afs = -713 Hz. The sensitivity of the TSM resonant frequency to liquid properties, illustrated in the previous example, necessitates close control of liquid properties when trying to measure mass accumulation from solution [24]. Liquid viscosity, in particular, varies exponentially with absolute temperature and must be closely controlled to avoid spurious TSM resonator responses. When liquid contacts one face of the TSM resonator, the electrical response of the TSM resonator changes, as described by the elements R2 and L2. Figure 3.10 (page 58) shows admittance-vs-frequency data (points) measured as the density-viscosity product (pr/) of a solution contacting the TSM resonator varies. With increasing pr/, the admittance magnitude plot shows both a translation of the series resonance peak toward lower frequency, and as a diminution and broadening of the peak. The solid lines in Figure 3.10 are admittances calculated from the equivalent-circuit model when best-fit L2 and R2 values are included. The model accurately produces the admittance-vs-frequency curves measured under liquid loading using parameters determined from the unloaded TSM resonator. The translation of the admittance curves arises from the inductance contribution L2; this element represents the kinetic energy of the entrained liquid layer. The broadening and diminution of the resonance peaks arises from the resistance contribution R2; this element represents power dissipated due to radiation of a damped shear wave into the liquid.

58

3. Acoustic Wave Sensors and Responses

3.0

i

A

A

E 2.5 "O .IBr elml

2.0

r

m

3i

1.5 C

C

rm 1.0 e.--.

i

I I I I i.~I I I I I I I !1

i

ii

i

Dry Device

E <

0.5

0.0 90 A

O~

"O

60

O U} .r

a.

30

r

Cl

r e a

E

qD

-30 A -60 4.965

4.970

4.975 Frequency (MHz)

4.980

Figure 3.10 Electrical admittance vs frequency near the fundamental resonance with glycerol (in water) solutions contacting one side of a TSM resonator: (A) 0% glycerol, (B) 40% glycerol, (C) 60% glycerol, (D) 70% glycerol. (Reprinted and adapted with permission. See Ref. [ 14]. @ 1993 American Chemical Society.)

3.1 Thickness-Shear Mode Resonator

3.1.6

59

EFFECT OF SURFACE ROUGHNESS ON TSM RESONATOR LIQUID LOADING

In Section 3.1.5 we noted that in-plane oscillatory motion by a smooth TSM resonator surface generates plane-parallel laminar flow in a contacting fluid. We describe this fluid that is dragged along by the oscillating surface as "viscously coupled." A textured surface, with either random roughness or lithographically defined features, exhibits an enhanced interaction with a contacting fluid. This is evidenced by an increase in motional resistance (R2) and inductance (L2) measured upon liquid contact. Schumacher [25] and Beck et al. [26] have identified one source of this increased solid-liquid interaction: vertical features on the surface constrain or "trap" a quantity of fluid (in excess of that viscously coupled), forcing it to move synchronously with the oscillating surface. This trapped fluid thus behaves as an ideal mass layer, as opposed to a viscously entrained liquid that would undergo a progressive phase lag with distance from the surface. The kinetic energy of trapped fluid leads to an increase in the motional inductance (L2) and frequency shift over that measured with a smooth device. Researchers have also shown that a textured device exhibits increased motional resistance (R2) over a smooth device [27,28]. This is believed to be due to the generation of compressional waves and surface-normal fluid motion by surface asperities that increase power dissipation in the liquid. The electrical response of a liquid-loaded TSM resonator can be related to the , shear mechanical impedance, Zs, at the device surface. This mechanical impedance serves as a quantitative measure of the strength of the interaction between the solid and a contacting liquid. The electrical characteristics of the TSM resonator with a generalized surface perturbation can be described by the equivalent-circuit model of Figure 3.7b. Measurements can be made on a dry TSM resonator to determine C o, LI, Cl, and R1. Fixing these parameters and fitting the equivalent-circuit model to data measured on an immersed device determines R2 and L2. Equations 3.21 can then be used to determine the components of Zs from L2 and R2. Figure 3.11 shows the components of the surface mechanical impedance measured vs the liquid parameter (pr/) 1/2 for several values of surface roughness [ 14]. The real part of Zs represents power dissipation in the liquid by the oscillating device surface; the imaginary part represents energy storage. The dashed line is the mechanical impedance calculated for an ideally smooth surface in contact with a Newtonian liquid (Equation 3.33). For the smooth surface, Re(Zs)= Im(Zs), indicating that peak energy storage is equal to power dissipation. For smooth devices, Equation 3.33 indicates that both real and imaginary parts

60

3. Acoustic Wave Sensors and Responses

of Zs are proportional to (pT/)1/2. This dependence arises from viscous coupling of liquid to the surface. The data in Figure 3.11 show that even for rough devices, these components continue to vary as (pr/) I/2, indicating that viscous coupling occurs even in the presence of surface roughness. For devices with roughness much less than the liquid decay length 8 (0.25 gm in water at 5 MHz), Zs is very close to that predicted for an ideally smooth surface (dashed line). As the roughness scale approaches 8, both real and imaginary parts of Z~ increase, indicating an enhanced solid-liquid interaction. The imaginary part, in particular, shows an offset that increases roughly proportionally with the average surface roughness 3. Since Im(Zs) is associated with energy storage, and this offset has been shown to be proportional to the density of trapped fluid, this effect is attributable to liquid trapping in surface features. Re(Zs) increases more erratically with surface roughness, indicating increased power dissipation by the rough surface. The origin of this increased dissipation is believed to be due to a conversion from plane-parallel liquid flow to surface-normal flow by surface asperities [14]. When the scale of surface roughness is small compared with the liquid decay length 8, roughness has a negligible effect on liquid coupling. In this case, the surface can be considered hydrodynamically smooth [29], contributing a negligible influence on device response. When the roughness scale approaches the liquid decay length & however, the additional response caused by roughness is quite significant. The frequency shift observed with water contact, for example, is more than doubled by an average surface roughness of 240 nm. The role of surface roughness in device response has been frequently overlooked. Several researchers have reported device responses that greatly exceeded that predicted for a smooth device. Rejakovic et al. [30] reported frequency shift enhancements of 1.8-2.6 for 5 MHz devices and 7.1-7.4 for 9 MHz devices. To account for this "excess" response, Thompson and coworkers [31-34] and Haardt [37] have postulated liquid ordering in a layer adjacent to the surface, giving rise to greatly enhanced liquid density and viscosity. Haardt claims viscosity enhancements near the surface of 4.2 times that of the bulk liquid. The results of Figure 3.11, however, indicate that for devices having hydrodynamically smooth surfaces, the measured responses agree well (in comparison with the discrepancies noted above) with those calculated for an ideally smooth surface using bulk values of density and viscosity. Within experimental uncertainties, there is no evidence for enhanced liquid properties near the surface. The changes in device response caused by surface texture can be used to advantage in constructing sensors to measure liquid properties. The response of a 3Average surface roughness was measured using a scanning optical interferometer. See [14].

3.1 Thickness-Shear Mode Resonator

'1 A

O

1

- ".

I

I

I

I

0"(ca~c.)

O <10 nm Z~ 197 nm v 525 nm

61

1

)< V

o"

N" n-

~A

3

O )< V

o"

.~ N E l

2

O

0.0

0.1

0.2

0.3

0.4

0.5

0.6

(p~)V2 (g cm-2 s-1/2) Normalized components of the surface (shear) mechanical impedance Zs (at 5 MHz) vs liquid properties for several surface roughnesses. (Reprinted and adapted with permission. See Ref. [14]. @ 1993 AmericanChemical Society.) Figure 3.11

62

3. Acoustic Wave Sensors and Responses

smooth device depends only on the liquid density-viscosity product (pr/), making it difficult to separately measure fluid density and viscosity with a smooth device. However, the presence of surface texture causes an additional response (due to trapping) proportional to density. By combining smooth- and texturedsurface devices in a single sensor, researchers have demonstrated the determination of both fluid density and viscosity [38]. 3.1.7

SOLID~LIQUID B O U N D A R Y C O N D I T I O N S A N D W E ~ I N G TEXTURED SURFACES

OF

In calculating liquid displacements generated by a resonator surface in contact with a liquid, a non-slip boundary is typically assumed at the interface. This relies on having sufficient interfacial force between the solid and liquid to ensure that the liquid layer adjacent to the solid moves with the solid and that continuity of displacement is maintained across the interface. Since the TSM resonator response is very sensitive to liquid loading, the device is particularly sensitive to the details of solid/liquid coupling. Recent results by Thompson and coworkers [31-34] have called into question whether the non-slip boundary condition rigorously holds under all conditions. They have shown that surface treatments can affect the response of the TSM resonator to liquid loading. In particular, coating the electrode surface with a monolayer of an alkane thiol resulted in significantly smaller frequency shifts due to water loading. The critical parameter, it was believed, was the decrease in the surface's affinity for water: liquid contact angles increased indicating increased hydrophobicity for the treated surface. To explain the change in response with surface treatment, they argue that with a hydrophilic surface, the solid-liquid interaction is sufficiently strong to ensure a non-slip boundary, while a hydrophobic surface gives a weak solid-liquid interaction so that interfacial slip occurs, i.e., a discontinuity in displacement arises at the solid/liquid interface. Interfacial slip would lead to a diminished liquid displacement, resulting in less device response, as observed. While this explanation is plausible, recent results suggest an alternate mechanism for this change in solid/liquid interaction with surface treatment. Martin et al. [14] measured the effect of surface treatments on the liquidloading response with devices having various surface roughnesses. With hydrodynamically smooth devices, the surface treatments used by Thompson et al. had no significant effect on device response. With devices having roughness on the order of the liquid decay length, however, the liquid-loading response was diminished by the hydrophobic surface treatment, consistent with the observations of Thompson et al. The fact that surface treatments only modified the liquid loading of rough devices indicates that the treatments are modifying a response con-

3.1 Thickness-Shear Mode Resonator

63

tributed by surface roughness. The response contributed by surface roughness to the frequency shift (from an increase in motional inductance L2) arises from liquid trapping. Thus, the liquid trapping process is apparently being modified by changes in the liquid contact angle. It is reasonable that liquid trapping by concave surface features should depend on the microscopic liquid contact angle. Capillary forces tend to draw a contacting liquid into concave surface features present in a rough surface, such as pits, crevices, or pores. These capillary forces are stronger for hydrophilic (low contact angle) surfaces than for hydrophobic (high contact angle) ones. Whether the volume fraction of these features that does not contain liquid is filled with air compressed by liquid entry (Figure 3.12a) or only the vapor of the liquid (if the surface is filled from a vacuum or by liquid displacement, Figure 3.12b), the result is qualitatively the same. The extent of liquid trapping is determined by a competition between forces, of which the capillary force is dependent on the microscopic liquid contact angle. Hence, liquid trapping in a rough surface is greater with a hydrophilic surface than with a hydrophobic one. To model the dependence of liquid trapping by a rough surface on the liquid contact angle, we consider a sinusoidally corrugated surface, approximating the surface texture actually produced by a polishing procedure. We further assume that the groove periodicity is equal to the crest-to-trough amplitude. When the surface is initially wet by a liquid, the rapid initial contact of the solid by the liquid can result in the liquid spanning small concave asperities and trapping air. To simplify the analysis, we assume that the entire volume of air below the crests is trapped in the initial wetting process. Upon bridging the crests, the trapped air is compressed by capillary forces, tending to draw liquid into the crevice. The shape of the trapped air bubble after compression is shown for several contact angles in Figure 3.12 (page 64) (dashed lines); liquid lies above the line while air is below. These shapes were determined by simultaneously satisfying the contact angle requirement and balancing the capillary pressure across the interface [39] (AP = y/r where y is the liquid surface tension and r is the radius of curvature of the liquid meniscus) with the pressure increase arising from isothermal compression of the trapped air (AP = Pamb (Vi/Vf- 1) where Pambis ambient pressure and Vi and Vf are the initial and final volumes of the trapped air). With high-energy surfaces (low contact angle), the capillary forces can result in substantial void pressures ~ 20 atm for 0 = 0 in Figure 3.12. It is clear from Figure 3.12 that the amount of liquid penetration depends strongly on the liquid contact angle. In addition, since capillary pressures vary inversely with asperity size, the amount of liquid penetration also depends on the size of the asperity. These trends are shown in Figure 3.13 (page 65) where the volume fraction of the pore filled by liquid is plotted vs the contact angle for

I a_. 120 I m m I m I I I I I I I I I I I I I I I I I

I I I I I I I I I I I I I O I I I I I T I I I I I I I I I I I I I I I I I I

\

I

i

tO8

9O

l l l w n w w w ~ I

Om

l l w w w w w w m w m w m m m

I ill

I

III

idl I I

II

0

II

III

#0

i IIi

~ i I

<108

i

iiiI

I i

i

I

ii

III i I

i

I

i

ii iii i i

i

iii

III

ii

III

III

.

iiiiiiii ii i

Illl

.

.

.

I

.

.

.

.

.

b

~

,

__ J. I _ _ - -- I-

~

lI?

I. . . . .

.J. __ - -- J - - - -- - - - - . 7 '

__/

Figure 3.12 Cross-section of a liquid meniscus formed as liquid penetrates into a sinusoidally-textured surface, illustrating contact angle-dependent trapping [14]: (a) with trapped air, and (b) without trapped air. The dashed lines indicate the initial (nonequilibrium) penetration of liquid for the indicated microscopic contact angle. The solid lines indicate the equilibrium penetration, becoming complete for 0 < 108 ~ (upper) or 0 < 117 ~ (lower). (Reprinted with permission. See Ref. [141. 9 1993 American Chemical Society.) 64

3.1 Thickness-Shear Mode Resonator I

1.0

_

~

--"

_

m'

"=

i

I

_

"="

,.

q=,

~

-i

iq=

q~

~

i

iI

i

I i

9

I

0

-I"

"-

i

I

""'

3000

I

-.

I:

3000

.,=4

i

- - 1 0 0 , 1000,

I!

- -

i

I

1000

0.8

i

_

~

100--1 .,,.4

i

65

-.,

0.6

I

%

I I

% I

%

I I

0.4

I %

0

I %

0.2

%%

I I U

0.0

..

. ... I ii

ii Iillll

, i

i

I. . . .

I

I

I

i

~,

,

i I

1.0 I.==4

I=..4 o,=.I

0.8 0

.,=4

0.6 3000 q~

0

100

0.4

0.2

0.0 0

30

60

90

120

150

180

Liquid Contact Angle (deg) Figure 3.13 Variation of liquid penetration into a sinusoidal surface feature (Figurc 3.12) with liquid contact angle [14]. Different curves are obtained for different groove dimensions: peak-to-peak amplitude is indicated in nanomctcrs next to each. (Reprintedwith permission. See Ref. [14]. 9 1993 American Chemical Society.)

66

3. Acoustic Wave Sensors and Responses

various sized asperities. The results demonstrate that, especially for small asperities, there is a critical contact angle where the volume fraction filled drops dramatically. 3.1.8

D Y N A M I C S OF A TSM R E S O N A T O R WITH A VISCOELASTIC F I L M

This section examines the dynamic behavior and the electrical response of a TSM resonator coated with a viscoelastic film. The elastic properties of viscoelastic materials must be described by a complex modulus. For example, the shear modulus is represented by G = G' + jG", where G' is the storage modulus and G" the loss modulus. Polymers are viscoelastic materials that are important for sensor applications. As described in Chapter 5, polymer films are commonly applied as sorbent layers in gas- and liquid-sensing applications. Thus, it is important to understand how polymer-coated TSM resonators respond. 9A film deposited on the TSM resonator surface is subjected to an oscillatory driving force at the TSM resonator/film interface. Typically, the film is bonded to the TSM resonator surface sufficiently well that the base of the film moves synchronously with the resonator surface. However, the upper portions of the film may lag behind the driven surface, as shown in Figure 3.14. The dynamic behavior of the film's shear displacement vs position across the film thickness hf can be predicted from a continuum electromechanical model described by Reed et al. [ 10]. Several distinct regimes of dynamic behavior can be identified [40], determined by the acoustic phase shift, q~, across the film. If the film is sufficiently thin and/or rigid so that ~b ,~ ~'/2, then the entire film tends to move synchronously with the resonator surface, as shown in Figure 3.15a (page 68). In this case, displacement ux is uniform across the film thickness and a negligible strain (proportional to the displacement gradient, Vux = OUx/Oy) occurs in the film. In this regime the device responds, as described in Section 3.1.1, only to the surface mass density ps (product of film thickness and density) contributed by the film. This is because the elastic energy stored in the film, proportional to G'lVUxl 2, and power dissipated, proportional to triG" I V//x [ 2, remain negligible when ~b ,~ 7r/2. This can be referred to as the microbalance regime of operation m where changes in resonant frequency can be monitored to unambiguously indicate the mass/area accumulated on the TSM resonator surface. If the film coating is compliant, resulting in an appreciable phase shift through the thickness of the film, the upper regions of the film tend to lag behind the driven resonator/film interface. Significant shear deformation is induced in the film causing elastic energy to be stored and dissipated. In this regime, film displacement is not synchronous with the driving resonator surface, but varies

3.1 Thickness-Shear Mode Resonator

67

Figure 3.14 Cross-sectional view of a thickness-shear mode resonator with a polymer film coating the upper surface [40]. Shear displacement profiles are shown at maximum excursion. (Reprinted with permission. See Ref. [40]. 9 1991 IEEE.) across the film thickness, as depicted in Figures 3.15b and 3.15c. In this case the resonator no longer functions as a simple microbalance, but resonant frequency and damping depend upon film thickness, density, and shear elastic properties. The sensitivity of the device to these parameters enables the device to be used to extract film properties in this regime [40]. For tk ~< ,r/2, displacement at the upper surface of the film exceeds that at the resonator surface (i.e., overshoot occurs), as shown in Figure 3.15b, but remains essentially in-phase with the driving surface displacement. In this case, the frequency is lower than for the uncoated resonator. For ~b near rr/2, film resonance occurs and the interaction between the resonator and film exhibits characteristics of coupled resonant systems [40]: displacement in the film/resonator system exhibits in-phase and out-of-phase modes, with a concurrent splitting of resonant frequency into two corresponding branches. For th -> ,r/2, the upper film surface is 180 ~ out of phase (Figure 3.15c) and frequency is higher than for the uncoated device. In addition, the system is highly damped in the vicinity of resonance, making sensor operation more difficult. Although it is difficult to directly observe the dynamic behavior of the film, which varies across the film thickness, its influence on the TSM resonator elec-

68

3. Acoustic Wave Sensors and Responses

b

a ,

I

c

|,

ii

p/ I I I I

I I I

/

I

I l l l I I l,

Figure 3.15 The dynamic film response generated by the oscillating resonator surface varies with the acoustic phase shift tk across the film [40]: (a) for ck < < ~'/2, synchronous motion occurs; (b) for tk ~ r overshoot of the upper film surface in-phase with the resonator surface occurs (film resonance occurs when ck = Ir/2); (c) for tk > ~'/2, the upper film surface is 180~ out-of-phase. The film is the thin region at the top; the crystal is below. (Reprinted with permission. See Ref. [40]. 9 1991 IEEE.)

trical characteristics can be more readily determined. By considering the mechanical coupling between the resonator and a film overlay, an equivalent circuit model can be derived that relates the near-resonant electrical characteristics to the film properties. This model allows prediction of how film properties influence the resonant frequency and damping. In addition, in the regime where the film is deformed, measurement of the electrical characteristics of a film-coated resonator can be interpreted to obtain the film's shear storage and loss moduli [40].

3.1.9

ELECTRICAL CHARACTERISTICS OF A TSM RESONATOR COATED WITH A VISCOELASTIC FILM

Chemically sorbent films are commonly coated on TSM resonators to construct gas or vapor sensors. The absorption of species by these films leads to a change in the areal mass density as well as plasticization or softening of the film. Corn-

3.1 Thickness-Shear Mode Resonator

69

monly, however, polymer films are either sufficiently soft to begin with, or become softened by temperature or vapor absorption, that the criterion for considering the film as an ideal mass layer, namely that tk "~ ~r/2, is not satisfied. Then, a more detailed model for the resonator-film interaction must be considered. The equivalent circuit model of Figure 3.7 can be used to describe the nearresonant electrical characteristics of the quartz resonator coated by a viscoelastic film. The surface film causes an increase in the motional impedance, denoted by the complex element Ze. From Equation 3.19, this element is proportional to the ratio of the surface mechanical impedance Zs contributed by the film to the characteristic shear wave impedance Zq of the quartz. The oscillating resonator surface may be considered as a source for shear waves that are radiated into the contacting film. The upper film surface reflects these radiated shear waves downward, so that the mechanical impedance seen at the quartz surface is dependent upon the phase shift and attenuation undergone by the wave in propagating across the film. When the film is rubbery, significant phase shift across the film occurs. Consequently, the coupling of acoustic energy into the film depends upon thin-film interference. The finite thickness of a film on the resonator surface makes the calculation of the mechanical impedance at the surface analogous to that of an appropriately terminated transmission line [41 ]. Noting the correspondence between stress and voltage and between particle velocity and current, the stress-free upper film surface is analogous to a short-circuited electrical transmission line. From this analogy, the input impedance seen at the resonator/film interface is [40]

Zs = Txy [

= Zo tanh ('yhf),

(3.36)

Vx [y=hq where Zo = (Gpf)112 and 3' = Jto(pf/G)ll2; G and pf are film shear modules and density; hq is quartz thickness. Equations 3.19 and 3.36 can be combined to find the change in (electrical) motional impedance that arises from a viscoelastic film on a thickness-shear mode resonator [40]:

Nzr

(Gpf)l/2

. . . . . Id, . qpq Ze . . 4K2tosCo

tanh (Thf).

(3.37)

For lossless films, G" = 0 and Ze is imaginary; in this case, Ze represents energy stored in the film, becoming infinite at film resonance when th = mTr12 (m odd). For lossy films, G" > 0, and Ze becomes complex, with the real part (R2) representing power dissipation in the film and the imaginary part (L2) representing energy storage. The dependence of Ze in Equation 3.37 on ~/hfmakes it difficult to resolve Ze into real elements R2 and L2, except in a few limiting cases.

70

3. Acoustic Wave Sensors and Responses

A condition of film resonance occurs when the acoustic phase shift ~ across the film reaches an odd multiple of 7r/2. This enhances the coupling of acoustic energy into the film, resulting in a greater extraction of electrical energy from the source. Consequently, dramatic changes in the motional impedance occur at film resonance (these arise from the complex Ze contribution (Equation 3.37)). These changes lead to changes in the resonant frequency, Af, and damping, R2, for the coated resonator that can be determined from Ze using Equations 3.21 and 3.23. Figure 3.16 shows the changes in resonant frequency, Af, and damping, R2, as a function of film phase shift tk and loss tangent (G"/G') calculated from Equations 3.21, 3.23, and 3.36. The behavior of Af and R2 with ~ is distinct in each of the regimes of dynamic film response outlined previously: (a) For ~b ,~ 7r/2, Af decreases linearly with ~b and damping is nearly fixed at the uncoated resonator value. (b) For ~b ~ zr/2, Af decreases more rapidly with ~b, while R2 increases from the uncoated resonator value. In this regime, dynamic calculations indicate overshoot of the upper film surface, leading to significant deformation in the film. (c) For ~b ~ 7r/2 (film resonance), Af increases rapidly, while R2 is maximum. The discontinuity that occurs in resonant frequency can be attributed to the abrupt change in mode shapes shown in Figure 3.15 (b and c). Energy dissipation in the film diminishes away from resonance. 3.2

Surface Acoustic Wave (SAW) Devices

The stress-free boundary imposed by the surface of a crystal gives rise to a unique acoustic mode whose propagation is confined to the surface and is therefore known as a sulface acoustic wave (SAW). In 1887 Lord Rayleigh discovered this mode of propagation in which acoustic energy is confined very near the surface of an isotropic solid [5]. This mode, now known as the Rayleigh wave [5], is of interest to seismologists because it is excited by earthquakes. The utility of Rayleigh waves in sensor applications is also due to the surface confinement of energy, allowing them to be excited by surface electrodes [42] in piezoelectric materials and also making the wave extremely sensitive to surface perturbations. In order to satisfy the stress-free boundary condition, coupled compressional and shear waves propagate together in a SAW such that surface traction forces are zero (i.e., T..~ = 0, where .~ is normal to the device surface). The generalized surface acoustic wave, propagating in the z-direction, has a displacement profile u(y) that varies with depth y into the crystal as u(x,y,z,t) = (ux(y)eJ4"l,~ + Uy(y)eJ4~2~+ Uz(y)eJ4J3~)eJ~t-Tz,

(3.38)

3.2 Surface Acoustic Wave (SAW) Devices

71

3500

3000 2500

C

A

A

2000

B

1500 1000 500 0 C

A

-20

N 3: _v

B -40

A -60

-80

-100

I . . . . . . . . . .

0

,

. . . . . . . .

'r

,

rJ2

,,

'3~4

~rdn) Figure 3.16 Variation in resonant frequency (Af) and damping (R2) vs the film phase shift ~b for various values of the film loss tangent (G"/G'): (A) 0.1; (B) 0.25; (C) 1.0. (Reprinted with permission. See Ref. [40]. 9 1991 IEEE.)

where to is the angular frequency (2,n'f); 3' is the complex propagation factor; Ux, Uy, and Uz represent displacement components in the x-, y-, and z-directions, re-

72

3. Acoustic Wave Sensors and Responses

spectively, and ~ the phases of the components with respect to Uz. The component Uy is perpendicular to the surface, Uz is in the direction of propagation, and Ux is transverse to the yz plane (i.e., the sagittal plane). The displacement components ui(y) vary approximately as e -2'ry/x, where )t is the SAW wavelength along the surface and y is distance into the substrate; amplitude thus decays rapidly with distance into the bulk of the crystal. A crosssectional view of the strain field generated by a surface wave propagating along the surface of a crystal is shown in Figure 3.17. The strain energy density, also shown in the figure, indicates that the majority of wave energy is contained well within one wavelength of the surface, which thus acts as a waveguide. At higher frequencies (i.e., shorter wavelengths), acoustic energy is confined more closely to the surface and wave sensitivity to surface perturbations increases. The sensitivity of SAW devices to surface perturbations is dependent upon the wave amplitude at the surface. The wave amplitude can be represented by the surface particle velocities Vxo, Vyo, and Vzo in the x-, y-, and z-directions, respectively. These are listed in Table 3.1 (page 74) for several different substrate materials. For propagation in an isotropic medium or along a pure-mode direction of a crystal (e.g., a plane of symmetry), Equation 3.38 reduces to a Rayleigh wave, characterized by having no transverse component: Ux = 0. Since Uy and Uz are 90 ~ out of phase, the particles move in an elliptical orbit in the sagittal plane; the surface motion resembles that of the ocean under the influence of a passing wave. The presence of the surface-normal displacement component makes the SAW poorly suited for liquid sensing applications. When the SAW medium is contacted by a liquid, this component generates compressional waves in the liquid; the power thus dissipated leads to excessive attenuation of the SAW. 3.2.1

S A W E X C I T A T I O N AND DETECTION

The discovery by R. M. White of the University of California at Berkeley that surface acoustic waves could be excited and detected by lithographically pattemed interdigital electrodes on the surface of piezoelectric crystals [42] has led to widespread use of SAW devices in a number of signal-processing applications. These include frequency filters, resonators, delay lines, convolvers, and correlators [43,44]. A surface acoustic wave (SAW) is most conveniently excited on a piezoelectric crystal using an interdigitated electrode pattern, or interdigital transducer (IDT), as shown in Figure 3.18 (page 75). Application of a voltage between alternately connected electrodes causes a periodic electric field to be imposed on the crystal. When an altemating voltage is applied, a periodic strain field is gen-

3.2 Surface Acoustic Wave (SAW) Devices Probed Film

I~,

"~

73

.._1

STRAIN ENERGY

== 13

Figure 3.17 Deformation field due to a SAW propagating to the right along a solid surface (top) and the associated distribution of potential energy (bottom).

74

3. Acoustic Wave Sensors and Responses Properties of Several SAW Substrate Materials

Table 3.1 9

,

,

,

,,

,

,

,

,,

,

,

,,

i

,,

,

==

,

,

,

,,

,,,

,

,

Substrate Cut

Propagation Direction Quartz ST X Lithium Niobate -y Z Gallium Arsenide Z X + 22.5 ~ ii

i ill

V~o

Vyo

V~o

Propagation Velocity

coP

coP

coP

4~

4Jz

3.158

0.13

1.34

0.88

90

90

0

3.488

0

0.83

0.56

--

90

0

2.763

0.16

1.22

0.91

0

90

0

( x l 0 s cmls)

( x l O-6 cml/Z gt/Z)

(degrees)

ill

erated in the piezoelectric crystal that produces a standing surface acoustic wave. This standing wave gives rise to propagating waves that are launched in both directions away from the transducer; the wavefronts are parallel to the transducer fingers. The transducer operates most efficiently when the SAW wavelength, A, matches the transducer periodicity, d. This occurs when the transducer is excited at the synchronous frequency, defined by fo = vo/d, where Vo is the SAW propagation velocity. As discussed in Section 2.2.1, propagation of a mechanical wave in a piezoelectric medium is accompanied by an associated wave potential, ~b. When the wave is incident on a receiving transducer, this potential induces a current flow in each transducer electrode; these currents combine to produce a current flow in the external detection circuit. The addition of current contributions in the receiving transducer is also optimized when the transducer periodicity matches the acoustic wavelength. Thus, a reciprocity relation holds, as it must for a passive linear device, between the wave and external signals.

3.2.2

INTERDIGITAL TRANSDUCER FREQUENCY RESPONSE

Each transducer finger may be considered to be a discrete source for the generation of surface waves in a piezoelectric medium because the piezoelectrically generated stress varies with position near each transducer finger. A simple trans-

3.2 Surface Acoustic Wave (SAW) Devices

75

fer function relates the continuous wave (CW) voltage V1 applied to a finger and the electrical potential associated with the waves radiated in each direction [43]"

d~+-

I,*sV],

=

(3.39)

where/Xs is a substrate-dependent constant, (h+ is associated with the rightward propagating SAW, while ~b- is a leftward propagating SAW. The parameter/Xs may be considered frequency independent: the frequency response of the transducer arises mainly from interference between finger contributions, and is relatively insensitive to the frequency response of the individual elements. This

r (a) . ,.. V" 9

--!

(b)

' i. x......

o

i

i

i

. . . .

V1

2

II

"3

ii

Transmitter

(c)

i

'

Receiver

,T,TTT

t-

T~~T,

Piezoelectric Substrate .

.

.

.

.

.

.

.

.

.

.

Ill

.

.

.

.

.

.

-!

Figure 3.18 Interdigital transducer, formed by patterning electrodes on the surface of a piezoelectric crystal, for exciting surface acoustic waves: (a) SAW electrical potential, (b) plan view, (c) side view.

76

3. Acoustic Wave Sensors and Responses

approximation is typically made in analyzing wave scattering from an array of elements: the "element factor" is typically considered frequency-independent compared with the "array factor." When an array of fingers is excited, as occurs with an interdigital transducer (IDT), the wave potential for a rightward propagating wave ~+ evaluated at position z is a vector sum of the contributions from each finger: Nf-I

dP+(z)=l~s E

VneJkCz-zn)'

(3.40)

n=O

where zn is the position of the nth finger excited with voltage Vn; Nf is the total number of fingers. Equation 3.40 has the form of a discrete Fourier transform [45] of the sequence Vn. Consequently, the frequency response of the device is proportional to the Fourier transform of the sequence of transducer finger contributions. Schemes have been devised to vary the individual finger contributions in order to achieve a desired frequency response. The interested reader is referred to excellent books on SAW filter design by Datta [43], Morgan [44], and Ristic [46]. If Nf identical fingers are spaced periodically with period d and excited with alternating voltages Vn = (-1)n Vo, Equation 3.40 becomes Nf-I

~b+(0) = IxsVo ~

( - 1)ne-jnkd/2.

(3.41)

n=O

The sum in Equation 3.41 is a geometric series whose elements become unity, and add constructively, when kd/2 = mTr, where m is an odd integer. This condition defines the relationship between SAW wavelength, A, and transducer periodicity, d, for coherent addition, as shown in Figure 3.19. The IDT excites odd harmonics at odd multiples of the synchronous frequency: fm= mfl. Moving away from the synchronous frequency, the addition of components from individual fingers becomes incoherent, giving rise to the frequency response [~+(f)[=

sin (X) X

(3.42)

in which X=

Np~(f - fo)

fo

0.43)

where fo is the transducer's synchronous frequency and Np is the number of IDT periods: Np = Nf/2. The wave potential as a function of the detuning parameter

3.2 Surface Acoustic Wave (SAW) Devices 4-

I-

--

4-

'i,,i i

,ii

--

I ........I I, , ""I, .i

i.~ +x

I

I

I

I

I

I

I

I

I

I

~lJ

I

\lJ

I I I Figure 3.19

+

I

I

77

"-

I

I I

I I

I I

I I

Relationship between transducer periodicity and coherently excited waves.

X, described by Equation 3.23, is shown in Figure 3.20 (page 78). Note that when X is a multiple of 11",th+ is zero ~ a result of complete cancellation between finger contributions. Consequently, the frequency interval B between the first nulls on either side of the synchronous frequency is B -

2

Np

.

(3.44)

Thus, the transducer bandwidth B is inversely proportional to the number of IDT fingers. As will be described in Chapter 4, a narrow bandwidth is desirable for oscillator applications in order to avoid spurious oscillations and to improve stability. The frequency response measured between a pair of transducers having Ao = 32/xm and Np = 50 finger pairs is shown in Figure 3.21 (page 79). The amplitude, shown on a log (decibel) scale, shows the characteristic sin(X)/X behavior. The delay line phase shift q0 is

2 ~rfL q~(f) = k L -

Vo

,

(3.45)

78

3. Acoustic Wave Sensors and Responses

10

X s

m,/I,

i

__._

X-Ir

f,i

--..

Figure 3.20

... L

X

v

The calculated transducer response, sin(X)/X, vs the "detuning parameter,"

X. (Reprinted with permission. See Ref. [46a].)

where L is the path length (center-to-center distance) between transducers. Differentiation of Equation 3.45 shows that the phase slope dq~/dfis proportional to L/A, the transducer separation in wavelengths.

3.2.3

SAW PERTURBATION MECHANISMS

When SAW devices are used for sensors or thin-film characterization, the measured responses arise from perturbations in wave propagation characteristics, specifically wave velocity and attenuation, resulting from interactions between the SAW and a surface layer. Because a SAW propagating in a piezoelectric medium generates both mechanical deformation and an electrical potential, both mechanical and electrical coupling between the SAW and surface film are possible. Consequently, a number of interactions between surface waves and a surface film have been found that give rise to velocity and attenuation responses. SAW-film interactions that arise from mechanical coupling between the wave and film include mass loading caused by the translation of surface mass by the SAW surface displacement, and elastic and viscoelastic effects caused by SAWinduced deformation of a surface film. SAW-film interactions that arise from electrical coupling between the wave and film include acoustoelectricinterac-

50i

--"

-r

"

"'

'~

.......~ . . . . . .

Calculated

' ............ " ........

Insertion

Measured

Insertion

~ ........

Loss

Loss ....... ; .........." ' ' | . . . . . .

80 ~

................

"

t

70

0 8O 0

B~

-r,=r .4..)

(D

r

90

0

,A

100bD--

1

II

,I

t I

4.'1

I

"l

m.

|

r%

gO

"- ,,,,"

110 90

95

100

105

Frequency (MHz) Figure 3.21 See Ref. [46a].)

The frequency response measured between a pair of interdigital transducers. (Reprintedwith

,~---"

#B m.

80

3. Acoustic Wave Sensors and Responses

tions between electric fields generated by the SAW and charge carders in a conductive film. This section examines the velocity and attenuation changes caused by several interactions between SAWs and surface layers. This survey is by no means exhaustive---new interactions are being discovered all the time. 3.2.4

SAW MASS LOADING

The simplest interaction, and the one most utilized for SAW sensor applications, is the response due to changes in the areal/mass density (mass/area) on the device surface. The harmonic motion of the crystal surface caused by the passing surface wave causes particles bound to the surface to be translated in an elliptical orbit in synchronism with the SAW surface displacement. The effect on wave velocity and attenuation of this interaction may be derived from energy considerations. Movement by the wave of a surface layer that is sufficiently thin or rigid that it moves synchronously with the wave causes an increase in the kinetic energy density, U,, of the wave without dissipating any wave energy. From the discussion in Section 2.3, this is expected to change the wave propagation velocity without affecting attenuation. The change in average kinetic energy per area of surface is AUk =

p,.V2

+ V2 yo +

V2zo),

(3.46)

where Vxo, Vyo, and Vzo are the SAW particle velocities at the surface and Ps is the surface mass density. Particle velocities are related to particle displacement u by vi = jtoui. This increase in kinetic energy density results in a decrease in wave velocity, according to Equation 2.48. Combining Equations 2.47, 2.48, and 3.46 yields an expression for the change in wave velocity arising from surface mass loading: ~_~V ~.

Vo

tO

ps

V xo +

v yo +

v zo

tOP

toP

toP

.

(3.47)

Due to the greater confinement of wave energy near the surface that occurs as operating frequency increases, surface particle velocities increase in proportion to (pto)l/2. Thus, the quantities in parentheses (Vio2/toP), being independent of wave amplitude and depending only on the substrate material, remain constant. Slobodnik et al. have tabulated these normalized surface particle velocities for a large number of substrates [47]; parameters for the most commonly used SAW substrates are listed in Table 3.1. Note that for propagation along a crystalline axis of lithium niobate (LiNbO3), two components of par-

3.2 Surface Acoustic Wave (SAW) Devices

81

ticle velocity are generated (in the y- and z-directions). X propagation in the ST cut of quartz (a rotated cut chosen for its desirable temperature characteristics), however, results in three components of particle velocity because of the lack of crystal symmetry. Grouping all the substrate-dependent constants together results in the expression for the mass-induced change in SAW propagation velocity" Av =

Vo

-Cmfop,,

(a.4a)

where the mass sensitivity factor Cm is

Cm=T

2 + .1)2o ~rrVo Vxo . . . . + .V. zo .. toe top toe

(3.49)

Note from Equation 3.48 the frequency dependence of the SAW mass sensitivity: the fractional velocity change Av/vo varies with operating frequency fo. Because the mass layer is assumed (in this case) to be lossless, Equation 2.55 implies that attenuation is unchanged by mass loading. Example 3.5: (a) Calculate the mass sensitivity factor CmfOr a IO0-MHz SAW device on ST-cut quartz. (b) If a SAW device is incorporated in an oscillator loop, so that fractional frequency changes track fractional velocity changes (i.e., Aflfo = Av/vo), calculate the sensitivity S = dfldps. (c) Calculate the limit of mass resolution for a typical SAW oscillator stability of 1 Hz.

(a) Using Equation 3.49 with normalized surface particle velocities (V2xo/oJP, etc.) obtained from Table 3.1, Cm = 1.29 • 10-6 cm2-s/g. (b) The sensitivity calculated for the 100-MHz SAW device is S = dAf/dps = -Cmf2o = - 13 Hz-cm2/ng. (c) The limit of mass resolution is Rm = 3AflS = 3 Hz/(13 Hz-cm2]ng) = 0.23 ng/cm2.

Solution:

The previous example illustrates the superior mass sensitivity of the SAW device in comparison with the TSM resonator: sensitivity is some 200 times larger for the 100-MHz SAW device than for the 5-MHz TSM resonator. Part (b) of the Solution also reveals that mass sensitivity, when expressed in the form df/dps, increases with f2. The velocity and attenuation changes resulting from depositing a mass layer on a 97-MHz SAW device using an ST-cut quartz substrate are shown in Figure 3.22 (page 82). Velocity decreases linearly in this thickness regime, yielding cm = 1.32 • 10 -6 cm2-s[g, in good agreement with the mass sensitivity factor calculated above for a 100-MHz SAW. As predicted from the model, the relative attenuation change (Aa[k, where a is the attenuation and k = 2~r/A is the wavenumber) due to mass loading is negligible in comparison with Av/vo (shown on the same scale).

100

.

.

.

.

=

'

=

.....

= '

i

"=

. . . . . .

~

800.

. . . . .

0

~

>0 "~

<1

700

-100

600

-200

500

-300

400

-400

3oo

-500

200

-600

100

-700

A

_

-1

_

A

&

A

A

A

A

A

t

J

0

1

A

,,

A

&

A

A

I

2

A

,

A

,

A

A

A

I

3

A

A

A

, ,

A

A

I

4

,,

A

A

A

A

0

A

,I,

5

,

,

6

Ps (/gg/cm2) Figure 3.22

Fractional change in SAW propagation velocity and attenuation vs thickness of an evaporated metal film.

(Reprinted with permission. See Ref. [48]. 9 1989 IEEE.)

3.2 Surface Acoustic Wave (SAW) Devices

3.2.5

83

SAW ACOUSTOELECTRIC RESPONSE

When a SAW propagates in a piezoelectric material, it generates a layer of bound charge at the surface that accompanies the mechanical wave. This bound charge is the source of the wave potential ~b discussed previously and also generates an evanescent electric field, shown in Figure 3.23. When a conductive film is deposited onto the SAW medium, charge carriers in the film redistribute to compensate for the layer of bound charge generated by the passing surface wave. The effect of wave/charge-carrier coupling on SAW propagation can be determined from a model that accounts for wave-generated conduction currents in the film and displacement currents in the adjacent dielectric media. The interaction can be visualized by reference to an equivalent-circuit model [48], shown in Figure 3.24. The time-varying surface charge generated by the wave is represented by an alternating-current source. The current generated per area of surface, Io, is

[48] 12 = 2K2 tok2( Eo + Es)P,

(a.5o)

where K 2 is the electromechanical coupling coefficient squared (see Table 3.1), and Eo and ~s are air and substrate dielectric permittivities, respectively; k and P

Figure 3.23 Evanescent electric field generated by a surface acoustic wave propagating along the surface of a piezoelectric crystal. (Reprinted with permission. See Ref. [54]. 9 1989 Elsevier Publishers.)

84

3. Acoustic Wave Sensors and Responses

|

i

ill

9 I1

k(o I F

I

_~l

i

A

'

11

'

'

'IF

io ej(ut AIR SOLID

kE1 i

......... II

........

13 Figure 3.24 Equivalent-circuit model to describe the acoustoelectric interaction between a SAW and charge-carriers in a film overlay. (Reprinted with permission. See Ref. [48]. 9 1989 IEEE.)

are the acoustic wavenumber and power density (power per beam width). Note that the current generated is proportional to K2 and P. Displacement currents generated in the substrate and air arise from capacitances (per area of surface) of k~s and k~, respectively, Conduction currents in the film overlay are accounted for by the shunt conductance k2o's, where ors is

3.2 Surface Acoustic Wave (SAW) Devices

85

the sheet conductivity of the film. The sheet conductivity, ors, of a film is related to the bulk conductivity, or, and film thickness, hf, by ors = orhf. It is simple to derive the changes in velocity and attenuation arising from SAW/thin-film acoustoelectric coupling using the formalism outlined in Section 2.3. In the absence of a conductive film, energy is stored in the evanescent electric field generated by the wave. The complex power flow in this case (into the capacitors k~s and k~o in Figure 3.24) is Parl =

2jo,k(Eo + E,)

9

(3.51)

With a conductive film on the surface, this power flow becomes e r 2 = 2[k2ors + jtok(Eo + Cs)]"

(3.52)

The difference in power flows is the acoustoelectric effect, which is measured as

film conductivity is changed: PT = P r 2 -

12 k2 ors Prl = - - - ~ jtokcs(k2ors + jtokcs) '

(3.53)

where cs = Go + r Substituting Equation 3.50 for Io2 into Equation 3.53 gives the complex power flow, Par. This may be substituted into Equation 2.61 to obtain the change in complex propagation factor, % Equating real and imaginary parts indicates the partitioning of this effect between AV/Vo and A a l k [48]: Av

Vo = Aa k -

2

K2

cr~

2

orzs + (VoCs)2

(3.54a)

K2 VoC~tr~ 2 or2$ + (VoCs)2.

(3.54b)

Equations 3.54 agree with those derived by Datta from field considerations [43]. These equations, plotted vs ors in Figure 3.25 (page 86), have the form of a relaxation response [48]: as ors varies from much less than vocs to much greater, SAW velocity decreases monotonically while attenuation goes through a peak. The magnitude of the acoustoelectric response is proportional to K 2, and is thus substrate dependent. Table 3.2 (page 87) shows that K 9 is much larger for LiNbO3 than for quartz. The peak in attenuation, as well as the maximum rate of velocity decrease, occurs at a critical sheet conductivity defined by orc - VoCs. Referring to the equivalent circuit of Figure 3.24, this behavior can be interpreted physically" the SAW can be regarded as a current source, with a reactive source impedance given by k(~ + Es). Regarding the "load" impedance as kEors, maxi-

1

",

,

'

,"

'

,

'

~'

'

,

9

,

i.

16 .,

14 0

12

r/l mt

-2

10

qp~ |

E

,e

o

0 =1

0 .m

8

ilmm

6

el :3

4


o -5 -6

-7 !,,

-9

-8

I --7

Log(sheet Figure

3.25

film overlay.

I -6

,

,

I "5

conductivity,

,,

I -4

ohm

......

1 "3

,,,

I -2

"1

"1 c m " 1 )

Calculated acoustoelectric velocity and attenuation changes vs the sheet conductivity of a

dk~

r

3.2 Surface Acoustic Wave (SAW) Devices

Table 3.2

87

Acoustoelectric Properties of Several SAW Substrate Materials

Substrate Cut Propagation Direction

Vo ( x l O s cm/s)

Kz (%)

cs (pF/cm)

3.158

0.11

0.5

3.488

4.8

4.6

2.763

0.022

1.2

Quartz ST X Lithium Niobate -Y Z Gallium Arsenide Z X + 22.5 ~

mum power dissipation occurs when the magnitude of the load impedance matches the magnitude of the source impedance, i.e., when ors = ore.

Example 3.6: Deposition of a 100 nm-thick Al film on a LiNb03 SAW device causes sheet conductivity vary from as "~. VoCs to ors ~> vocs. (a) What acoustoelectric velocity and attenuation changes arise from this film? (b) What is the maximum acoustoelectric attenuation (in dB) for a IO0-MHz LiNb03 device with a path length of 100 h?

Solution:

(a) From Figure 3.25 or Equation 3.54a, note that as ors varies from much less than VoCs to much greater, Av/vo decreases by K2/2 = 2.4% = 24,000 ppm. (b) The maximum acoustoelectric attenuation, occurring when ors = VoCs, is Aa/k = K214 = 0.012 or Ac~A = .0754. The attenuation per wavelength in dB is 20 log(e '~A) = 0.65 dB per wavelength. For a device with a 100A propagation path length, the acoustoelectric attenuation is 65 dB, a very substantial loss! The previous example illustrates that the acoustoelectric response can be very significant with SAW devices, particularly those using strongly piezoelectric substrates such as LiNbO3. In fact, the acoustoelectric response can be much greater than the mass loading response in certain instances. Shown in Figure 3.26 (page 88) are the SAW velocity and attenuation responses measured as a nickel film is deposited onto a quartz SAW device [48]. Deposition of the film causes both an acoustoelectric and a mass response. The mass response causes a linear decrease in velocity, with no attenuation change, over the entire thickness range; the acoustoelectric response causes a rapid drop in velocity and a peak in attenuation over the 10 to 3 0 / ~ thickness range.

i,i

tj

'1

'

I

"

|

I

I

'

'

I1'

'~ I

II

i

I I

3.0

&

0

o

2.5 E t:3.

t

~o

&

-200

o

&

&o o& AO&

2.0

f

-400

t~ tXl

l"

m

tt~

to

1.5

o =j

o t~

o r

t

-600

1.0

9

&

o

O m

t~ r~

&

C:T (I) LL

0.5

A &

C

&

<

-800

0

&A&&A&AA,tU~AAAA&AAA A A A J

.....

I

0

_

I

2

......

I

I

4

6

,L

I

8

|

II

10

12

9 I1,, ,,

14

I1

16

II

II

18

20

_~

Time, min Figure 3.26

"0 0

&

m

Measured changes in SAW velocity and attenuation vs metal film thickness, illustrating the acoustoelectric interaction.

(Reprinted with permission. See Ref. [48]. 9 1989 IEEE.)

3.2 Surface Acoustic Wave (SAW) Devices

3.2.6

89

PARAMETRIC REPRESENTATION OF THE SAW ACOUSTOELECTRIC RESPONSE

An interesting parametric relationship exists between Av/vo and At~/k arising from thin-film acoustoelectric perturbations. From Equations 3.29 (a and b) [48],

vo

4=

(3.55)

It is apparent from Equation 3.36 that the acoustoelectric interaction takes on a particularly simple form when At#k is plotted vs AV/Vo, with sheet conductivity as the variable parameter: as Ors increases from zero, a semicircle centered at (AV/Vo, Aa/k) = (-K2/4,0) and having radius K2/4 is traced out [48]. The angular position along the semicircle corresponding to ors is [48]

ors = VoCst a n ( 2 ),

(3.56)

where 0 is the angular displacement along the semicircle with respect to the ors = 0 point. The acoustoelectric response can be isolated from the data of Figure 3.26 by subtracting the mass contribution from the AV/Voresponse; this is accomplished by simultaneously measuring the mass accumulation using a quartz crystal microbalance (which, due to the relatively thick metal electrodes covering both its active surfaces, does not exhibit an acoustoelectric response). This acoustoelectric response, shown in the 7'(Aa/k vs Av/vo) plane in Figure 3.27 (page 90), takes the form of a semicircle as predicted by Equation 3.55. The radius R of the best-fit semicircle is 3.50 x 10 -4, indicating a K 2 value of 1.4 • 10-3, slightly higher than the literature value of 1.1 • 10 -3 for surface acoustic waves on ST-cut quartz [43].

3.2.7

ELASTIC AND VISCOELASTIC FILMS ON A SAW DEVICE

In Section 3.2.4 we considered the effects of an ideal mass layer on SAW response. In the model used to derive the mass-loading response, the layer was assumed to be (1) infinitesimally thick, and (2) subject only to translational motion by the SAW. Translational motion was found to induce a change in SAW velocity proportional to the areal mass density (ph) contributed by the film the mass loading response. Since no power dissipation arises in film translation, no attenuation response was predicted. With an actual film having finite thickness and elastic properties, it is important to also consider the effects of SAWinduced film deformation. Energy storage and power dissipation due to film deformation cause additional contributions to SAW velocity and attenuation that were neglected in the earlier treatment.

('3331 686I @ "[817]"J;~I ~ S -uo!ss!uu;xt ql!~ p~ugd=~) "osuods~).l 3.ul:);)[~)olsno;:)l~ oql j o

(todd) ~ 0

00;~-

00~-

uog~u;sozdaz ~.~oumred

V 009-

008-

i•-

== == "=

/.Z's a.m~;A

0

I 9 -.,.

i

1

~-

,-...

I L. 0 q,) r~

-

'

--

I I

-

'

I

I

-OOE

i

3 SO

-

'

00s

0 f,#

<

_.

I I

' 0",

"0 "O

OOl~

3.2 Surface Acoustic Wave (SAW) Devices

91

The mechanical properties of a linear, isotropic material can be specified by a bulk modulus, K, and a shear modulus, G. For an ideal elastic solid, these moduli are real-valued. For real solids undergoing sinusoidal deformation, these are best represented as complex quantities [49]: K = K' + jK" and G = G' + jG". The real parts of K and G represent the component of stress in-phase with strain, giving rise to energy storage in the film (consequently K' and G' are referred to as storage moduli); the imaginary parts represent the component of stress 90 ~ out of phase with strain, giving rise to power dissipation in the film (thus, K" and G" are called loss moduli). The wave energy stored and dissipated in the film depends upon the strain modes generated by the SAW. Two distinct regimes of film behavior can be identified. Films that are thin and rigid behave as acoustically thin: the entire film moves synchronously with the substrate surface, resulting in uniform displacement across the film thickness. Figure 3.28a (page 92) shows the deformation arising in an acoustically thin film. Displacements ui are constant across the film thickness, and only gradients 4 in the plane of the film arise. In Figure 3.28a, the gradient in Uz leads to regions of compression and tension in the film, while the gradient in uy lead to bending. Films that are thick or soft behave as acoustically thick: the upper film portions lag behind the film/substrate interface, causing non-uniform displacement across the film thickness. Figure 3.28b shows the deformation in an acoustically thick film. Displacement varies not only in the plane of the film but also across the film due to inertial lag of the upper film regions. The deformation in an acoustically thick film arises from a combination of both in-plane and cross-film gradients, with cross-film gradients dominating. As shown in Figure 3.28b, the gradient in Uz across the film leads to shear deformation of the film. The regimes in which in-plane and cross-film gradients dominate and the corresponding velocity and attenuation changes will be considered below. Since the strain modes generated in each regime are different and result in distinct device responses, it is essential to understand the regime of operation for any particular measurement. The regime of film operation can be determined from the ratio R of cross-film to in-plane gradients induced by the SAW [50]:

R = Afvoph

Icl

(3.57)

where p, h, and G are the film density, thickness, and shear modulus and A is a substrate-dependent parameter having a value of 1.9 for ST-cut quartz [50]. When the film coating is sufficiently thin (small h) and rigid (large I G I ) in compari4A gradient is defined as the magnitude and direction of the maximal spatial rate of change.

92

3. Acoustic Wave Sensors and Responses

Figure 3.28 Deformation generated by a SAW (a) in an acoustically thin (R < < 1) film, in which in-plane displacement gradients (due to sinusoidal wave variation) dominate, and (b) in an acoustically thick (R - l) film, where cross-film gradients (due to inertial film lag) also arise. (Reprinted with permission. See Ref. [50]. 9 1994 American Chemical Society.) son to the oscillation frequency, such that R ,~ 1, in-plane gradients dominate over surface-normal gradients and the film is acoustically thin. When film properties are such that R -> 1, inertial lag becomes significant so that surface-normal gradients are dominant (Figure 3.28b), and the film is acoustically thick. E x a m p l e 3.7: For a film on a IO0-MHz ST-quartz SAW device, what film thickness h marks the transition from acoustically thin to thick for (a) a glassy polymer film with G' = 10 l~ dyne/cm 2, and (b) a rubbery polymer film with G' = 107 dyne/cm2? Assume p = 1 g/cm 3, G" ~ G ', and A = 1.9.

3.2 Surface Acoustic Wave (SAW) Devices Table 3.3

93

Moduli Associated with the Strain Modes (Figure 3.29) Generated by a SAW in an Acoustically Thin Film (R ,~ 1) [50]. ,

,

,,

,

,

,

,,

,,,

1,

Modulus E~

Strain

Displacement Gradient

Modulus Definition

(in terms of ~, it)

(in terms K, G)

Transverse Shear Bending

OUx/OZ

E (l) = Tt3/(2Sl3)

t~

G

Ouy]Oz

E (2) = T23](2S23)

Longitudinal Compression

OUz/OZ

E (3)

Mode

=

T33]$33

~0

~0

4~(A + p) A + 2/x

4G(3K + G) 3K + 4G

Setting R = 1 in Equation 3.57 and solving for h gives the film thickness at the transition between the acoustically-thin and thick regimes: (a) For ] G I = 10l~ dyne/cm2, h = 1.7/zm; (b) for lG ] = 107 dyne/cm2, h = 1.7 nm.

Solution:

If the film is acoustically thin (R ,~ 1), then displacements are constant across the film thickness, and only gradients in the plane of the film arise. The SAWinduced film deformation can be decomposed into three translations (in the x-, y-, and z-directions) and three strain modes, as shown in Figure 3.29 (page 94). An important parameter in determining the contribution of each strain mode in perturbing SAW propagation is the modulus E (i) - - t h e ratio of stress to strain associated with each strain mode in Figure 3.29. These moduli are listed in Table 3.3 in terms of the intrinsic elastic properties of the film, represented in terms of the Lam6 constants (A,/z) and the bulk and shear moduli (K, G). These sets of moduli are interrelated [51 ].

3.2.8

SAW RESPONSE FROM ACOUSTICALLY THIN FILMS

Changes in SAW propagation velocity and attenuation are determined by the mechanical impedances at the substrate/film interface resulting from film translational and strain modes. The impedance associated with each film translation is floph, and with each strain mode is -joohE~i)/V2o, where E (i) are taken from Table 3.3. Thus, from a perturbation analysis, the change in SAW propagation arising from acoustically thin films is [50]: A T = Ao~ ko ko

Av =jcoh Ci p J Vo i=1

~

Vo

(3.58)

94

3. Acoustic Wave Sensors and Responses

Y a

. . . . . . . .

i

,

IU x

-Z

X

Y b

-Z

X

C

L

!

-Z

X

Figure 3.29 Strain modes generated in an acoustically thin (R < < 1) film due to sinusoidal variation in the direction of propagation of the three displacement components. (Reprinted with permission. See Ref. [50]. 9

American Chemical Society.)

3.2 Surface Acoustic Wave (SAW) Devices

95

where the SAW-film coupling parameter ci v z/(4koP): Vio denotes the surface particle velocity (related to surface displacement Uio by Vio = jtOUio) in each direction, and P is the SAW power density (power flow per beam width). The ci parameters, given in Table 3.4 for X-propagation in the ST-cut of quartz, are determined for SAW propagation in the uncoated substrate and are assumed to be unchanged by the film. Each displacement component contributes two terms to Equation 3.58: one, proportional to p, arises from the kinetic energy associated with film translation; another, proportional to E (i), arises from the potential energy associated with film strains (modes shown in Figure 3.29). The kinetic contribution causes a SAW velocity decrease in proportion to film mass per area ( p h ) ~ the mass loading contribution. Equating real and imaginary parts of Equation 3.58 yields velocity and attenuation changes resulting from the SAW/film interaction. With elastic films, the intrinsic elastic moduli are real, resulting in real Ea~ and imaginary A3,/ko, so that Aa/ko = 0. Substituting the E a) expressions written in terms of the Lam6 constants (A,/z) from Table 3.3 into Equation 3.58 yields the Tiersten formula [52] --"

Av •o

io

z)] oJh[ Cl( /9 ) _~Vo + c2p + c3( /9 _ 4/z2 AA++ /2/x 11o

(3.59)

that gives the SAW velocity change contributed by an acoustically thin (R ,~ 1), elastic (K" = G" = 0) film. While Equation 3.59 does not apply to viscoelastic films, the velocity and attenuation arising from acoustically thin, viscoelastic films can be determined from Equation 3.58 by inserting complex moduli (e.g., K and G) into the Er expressions.

Table 3.4 SAW-Film Coupling Parameter ci and Phase Angles ~i for SAW Propagation in the X-direction of ST-cut Quartz [47] , , . ,

,.

,

,

Displacement Component

,

ci (•

i,

.

,

,..,

i

VZol(4koP) -z cmZ-s/g)

Uz .

.

.

.

.

.

.

.

.

.

.

.

.

, ,

,,

. . . . .

~, (deg)

0.013 1.421 0.615

Ux Uy

.

90 90 0 ,

,

,

,,,

96 3.2.9

3. Acoustic Wave Sensors and Responses S A W R E S P O N S E F R O M A C O U S T I C A L L Y THICK F I L M S

Next, we focus on a portion of the film that is small in lateral extent compared with the SAW wavelength. As the wave passes a fixed point, the lower surface of the film oscillates in response to the sinusoidal SAW surface displacement. If the film is acoustically thick (R > 1), the upper portions of the film tend to lag behind the driven substrate/film interface, inducing strains across the thickness of the film. This inertial deformation of the film results in nonuniform displacement across the film. The perturbations of SAW velocity and attenuation caused by the film are related to the mechanical impedances experienced by the surface displacement components in deforming the film. In the acoustically thick regime, film displacement may be thought to arise from a superposition of waves generated at the substrate/film interface and radiated into the film (Figure 3.30). The upper film surface reflects the radiated waves downward so that the mechanical impedance seen at the substrate/film interface is dependent upon the phase shift and attenuation undergone by the waves in propagating across the film. Consequently, a distributed model is used for the film, in which the impedance at the lower film surface depends upon the nature of the interference between the waves generated at the lower film surface and those reflected from the upper (film/air) surface. From the aforementioned considerations, a perturbational formula can be derived for the SAW velocity and attenuation changes, applying to either acoustically thin or thick viscoelastic films [50]:

A'y_ A~

~V

~ ci[3iMi

. . . . . . = ~., ko ko J Vo i=1

where

to

tanh (jflih).

2 ( P- E(i)/Vo)l/2

fli = to "

Mi

(3.60a)

(3.60b)

with Ml = M3 = G, while M2 = K; qJi and Ea~ values are as given in Tables 3.3 and 3.4. Calculating the sum in Equation 3.60a and equating real and imaginary parts determines Aa/ko and AV/Vo. The fli factors in Equations 3.60 represent propagation constants for waves with polarization direction xi propagating across the film (Figure 3.30): fl~ and f13 are associated with shear waves generated by the in-plane surface displacements Uxo and uzo, respectively;/32 is associated with a compressional wave generated by the surface-normal displacement Uyo. The response given by Equation 3.60 is largely determined by the phase shifts 4~i = Re{ flih}, where Re{ } denotes the real part of the complex expression. For

3.2 Surface Acoustic Wave (SAW) Devices

97

Air h Film

131

13 3

~2

#

Uy all

m,m

m,m

IBm

UZ m,,,,

>z

Figure 3.30 Displacement in the film is a superposition of waves generated at the substrate/film interface by the surface displacements Uio and radiated across the film. The surface-normal component Uyo generates compressional waves, while the in-plane components (Uxo, Uzo) generate shear waves. (Reprinted with permission. See Ref. [50]. 9 1994 American Chemical Society.)

elastic films, ffi ~ tlr3 ~ toh(p/G') 1/2 and t]~2~ toh(p/K') t/2. For polymer films, K' remains in the vicinity of 10 l~ dyne/cm 2 in both glassy and rubbery states, so that the compressional wave phase shift 4~2 remains small (<'r r/2) for typical polymer thicknesses. Thus, polymer films remain acoustically thin with regard to compressive displacement (Uyo) applied to the lower film surface. Since G' can be as low a s 1 0 7 - 1 0 8 dyne/cm 2 in rubbery polymer films, with ~bl and th3 reaching at/2 for films as thin as 80 nm (at 100 MHz), films do not remain acoustically thin with regard to shear displacements applied to the lower film surface. Since ca >> cl, in ST-cut quartz, ~b3 dominates the viscoelastic contribution to the response. Figure 3.31 (page 98) shows the attenuation and velocity responses calculated from Equation 3.60 vs the shear wave phase shift q~3 for various values of a film loss parameter r3. The loss parameter ri -- -Im(fli)/Re(fli) describes the intrin-

3. Acoustic Wave Sensors and Responses

98

sic lossiness of the film as experienced by the xi-polarized wave. This parameter, a ratio of power dissipation to energy storage, is analogous to a loss tangent, taking on values between 0 and 1 for all viscoelastic materials: ri = 0 for an elastic film and increases toward 1 for purely viscous films.

0.1 A olam

:3

0.2

es lb.

0.4

0.1 e u

0.4

c

.0 Im

0 o

-1

<3

-2 -3 -4 0.00

0.25

0.50

0.75

1.00

~31/I; Figure 3.31 Contributions to SAW velocity and attenuation vs the shear wave phase shift ~ for several values of the film loss parameter r 3. The dashed line is predicted from the Tiersten formula (Equation 3.59). (Reprinted with permission. See Ref. [50]. 9 1994 American Chemical Society.)

3.3 Acoustic Plate Mode (APM) Devices

99

We note in Figure 3.31 that velocity decreases linearly with tk3 for ~b3 "~ 7r/2, following the dashed line predicted from the acoustically-thin model (Equation 3.58). The response deviates substantially from this linear behavior near dr3 = ~r/2: attenuation goes through a maximum while velocity undergoes an upward transition. This combination of responses arises from film resonance, arising from interference between waves generated at the lower film surface and those reflected from the upper (film/air) interface. The resonant response at ~b3 = 7r/2 is predicted to recur at 3zr/2, 57r/2, etc., constituting harmonics of the fundamental film resonance. Martin et al. [50] have shown how the above model can be used to predict velocity and attenuation responses as polymer-coated SAW devices undergo temperature changes or are exposed to vapors. The response arises from temperature- or vapor-induced variations induced in the shear modulus G, bulk modulus K, film thickness h, and mass density p.~ Figure 3.32 (page 100) shows the velocity and attenuation responses measured (points) and calculated (lines, Equation 3.60) as pentane and TCE vapors were absorbed by a polyisobutylene-coated SAW sensor. The dashed lines show the predictions based on mass-loading by the absorbed vapors. At low concentrations, the mass-loading contribution accounts for 40% of the measured (low concentration) Av/vo response for pentane and 67% for TCE. The remainder of the response is due to film plasticization (softening) by the absorbed vapor molecules. This is consistent with a comparison of SAW and chromatography measurements by Grate et al. [53] that indicated that the mass response was a small fraction (25%) of the total Av/vo response. At high vapor concentrations, film plasticization due to acquired solvent volume causes the film to behave as acoustically thick. Film resonance occurs when solvent plasticization causes tk3 to reach Ir/2.

3.3

Acoustic Plate Mode (APM) Devices

This section describes a class of acoustic wave sensors utilizing a shearhorizontal (SH) acoustic plate mode (APM), which has been developed for sensing in liquids. SH modes have particle displacement predominantly parallel to the device surface and normal to the direction of propagation, as shown in Figure 3.33 (page 101). The absence of a surface-normal component of displacement allows each SH plate mode to propagate in contact with a liquid without coupling excessive amounts of acoustic energy into the liquid. By comparison, when surface acoustic waves are propagated at a solid-liquid interface, the surface-normal displacement radiates compressional waves into the liquid and severely attenuates the wave.

100

3. Acoustic Wave Sensors and Responses

2.5

2.0 A

O

1.5 X

2

1.0 II B

0.5

0.0

-I A t~

6 -2 X

al O

9 .

-3

~ =lww J

P

II B 9 ,

-4

.... |

-5

0

I

,

,, I ......

2

I ............

I

3

4

-,,j

II II]

5

Concentration (mole/I) Figure 3.32 Velocity and attenuation changes measured (points) and calculated (solid lines) vs absorbed (O) pentane and ( 1 ) trichloroethylene concentrations for a 97-MHz SAW device coated with a 0.70-/.tm polyisobutylene film. A QCM coated with an identical film was used to measure the mass of absorbed vapor, from which film concentration was determined. (Reprinted with permission. See Ref. [50]. 9 1994 American Chemical Society.)

3.3 Acoustic Plate Mode (APM) Devices

101

SH-APMs utilize thin single-crystalline quartz plates that serve as acoustic waveguides, confining acoustic energy between the upper and lower surfaces of the plate as the wave propagates between input and output transducers. This is in contrast to the SAW, for which nearly all the acoustic energy is concentrated within one wavelength of the surface. The consequences of this difference are two-fold: (1) the sensitivity of the SH-APM to mass loading and other perturbations depends on the thickness of the substrate; (2) both faces of the crystal undergo displacement, so that detection can occur on either surface of the device. SH plate modes may be thought of as a superposition of plane waves with inplane (shear horizontal) displacement reflected at some angle between the upper and lower faces of the quartz plate. These upper and lower faces impose a transverse resonance condition that results in each APM having displacement maxima at the surfaces, with sinusoidal variation between surfaces. In studying these devices as sensors, many salient features can be inferred by modelling the plate as an isotropic medium. This approximation greatly simplifies the analysis of APM sensors and leads to reasonable predictions regarding mode resolution and perturbation mechanisms acting on the device.

Figure 3.33 Schematic of an acoustic plate mode (APM) device showing the shear horizontal (SH) displacement of the mode as it propagates between input and output transducers. (Reprinted with permission. See Ref. [54]. 9 1989 Elsevier Publishers.)

102

3. Acoustic Wave Sensors and Responses

The particle displacement associated with the n th order SH plate mode (propagating in the z-direction) has only an x-component, given by [54] (nzry) Ux(y) = Uxo cos b e(Ja#-YNZ)' (3.61) where b is the plate thickness, Uxo is the particle displacement at the surfaces, n is a transverse mode index, and t is time. Since, unlike the TSM, modes are excited from only one side of the crystal, modes corresponding to all integer values of N can propagate. The exponential term in Equation 3.61 describes the propagation of the displacement profile down the length of the waveguide (along the z-direction) with angular frequency ~o and wave number YN, where

YN = J

Vo

-- --if-

(3.62)

where Vo is the unperturbed propagation velocity of the lowest-order mode. The cross-sectional (in the xy-plane) displacement profiles for the four lowest-order isotropic SH plate modes are shown in Figure 3.34. The mode index n corresponds to the number of nodes in the displacement profile. As illustrated in Figure 3.34, each mode has equal displacement on both surfaces of the APM sensor, allowing the use of either side for liquid-sensing applications. As with the SAW, a lithographically-patterned interdigital transducer on one side of the piezoelectric plate can be used to excite and detect APMs in quartz. Both the ST- and AT-cuts of quartz have been used; the latter has the advantage of greater temperature stability, wider spacing between modes, and without allowing SAW excitation. An IDT couples most efficiently to a plate mode when the transducer periodicity matches the mode wavelength along the surface, An. The frequency at which each mode is most efficiently coupled by a transducer is thus given by fn = vn/d, where vn is the phase velocity of the n th SH mode. Hou et al. [55] have shown that transducer coupling efficiency also varies inversely with the ratio of plate thickness to wavelength. Modeling the quartz plate as an isotropic medium, the Nth order SH plate mode will be generated most efficiently by a transducer of period d at a frequency approximated by [54]

fN=~

1+

-~

(3.63)

Equation 3.63 gives an approximation for the SH plate-mode spectrum found in an unperturbed quartz plate. The presence of surface features, including transducers, perturbs the wave velocity, and hence the excitation frequency, of each mode.

3.3 Acoustic Plate Mode (APM) Devices

103

Figure 3.34 Cross-sectional displacement profiles for the four lowest-order shear horizontal plate modes. These profiles are normalized for equal power flow per width of the plate. (Reprinted with permission. See Ref. [54]. © 1989 Elsevier Publishers.)

In order to avoid problems of mode interference, it is desirable when using APM devices as sensors to excite only a single mode. This can be accomplished by using a transducer whose bandwidth is less than the frequency separation between modes.

3.3.1

APM/LIQUID INTERACTIONS

A number of interactions can affect plate-mode propagation characteristics, particularly in a liquid environment. In the following sections, models of several of the important modes of interaction will be outlined and compared to experimental results. These include: (1) mass accumulation on the device surface, (2) viscous entrainment of the contacting liquid medium by the oscillating device surface, and (3) acoustoelectric coupling between evanescent plate mode electric fields and the liquid.

104

3.3.2

3. Acoustic Wave Sensors and Responses

MASS SENSITIVITY OF THE APM

When mass is bound strongly to either surface of the APM device, it oscillates synchronously with the quartz surface under the influence of the passing plate mode. The extent to which mechanical surface perturbations will influence APM propagation is proportional to V2xo/P, the ratio of surface particle velocity, Vxo, squared to acoustic power flow, P, down the "wave guide." The larger the surface particle velocity is in comparison with the power flow in the guide, the more sensitive the mode will be to surface mechanical perturbations. The crosssectional displacement profiles, shown in Figure 3.34, are normalized for equal acoustic power densities. It is apparent from the figure that the lowest-order mode (n = 0) has a smaller relative surface displacement (and particle velocity), and hence less sensitivity, than the higher-order (n >-- 1) modes. An ideal mass layer deposited at the surface results in increased kinetic energy which is offset by a decrease in propagation velocity [54]:

AV Vo

JnPs bpq

Cfps,

(3.64)

where cf is the mass sensitivity, pq is the density of the quartz substrate, Ps is the surface mass density (mass/area on one surface), Jo = 1/2, and Jn = 1 for n -> 1. Equation 3.64 predicts that velocity will decrease linearly with accumulated mass density and that the lowest-order mode will be half as sensitive to surface mass as higher-order modes. We note the similarity of Equation 3.64 to Equation 3.9 for TSM resonator mass sensitivity; the only difference is the Jn factor that accounts for the reduced sensitivity of the zero-order mode (which is not excited in the TSM resonator). Rayleigh wave devices exhibit sensitivity, when calculated on a relative basis (AV/Vo), that is proportional to frequency. In contrast, SH plate mode sensitivity displays no significant frequency dependence. This difference can be attributed to the fact that with surface waves, the acoustic energy becomes distributed closer to the surface as frequency increases. Like the TSM resonator, the energy density for each propagating SH mode is, dependent upon plate thickness rather than frequency, The mass sensitivity of an ST-quartz APM device was determined by depositing metal onto the unelectroded quartz surface, i.e., the side opposite the transducers. The plate mode velocity shift is plotted vs the surface mass density of deposited silver in Figure 3.35. As expected from the discussion above, the device is approximately twice as sensitive when higher-order modes (n > 1) are excited than with the lowest-order (n = 0) mode. The mass sensitivity measured

I

'

"1

"

I

I

A I

0 X

,I=,,

"

!

" '

,, n=O, cf= 9.5 <>n - l , cf=18.5 T n=2, cf=19.7 o n=3, cf=20.O

0 (,O

'1

-20 -40

I'

"

cm2/g cm2/g cm2/g cm2/g

-60 -80 <3 -100 0

@ m. t~

m t~ @

-120 0

2

4

6

8

Ps(l~g/cm 2) Figure 3.35 Fractional change in APM propagation velocity vs areal/mass density of silver deposited onto the device surface. (Reprintedwith permission. See Ref. [54]. 9 1989 Elsevier Publishers.)

"t ~

.r m. t~

t.~

106

3. Acoustic Wave Sensors and Responses

for the device is 9.5 cm2/g for the lowest-order mode and 19.4 cm2/g on average for the next three higher-order modes. The corresponding sensitivities estimated from Equation 3.64 (9.3 cm2/g and 18.6 cm2/g, respectively) are in good agreement with the experimental values. It is reasonable to ask how accurately the mass sensitivity in vacuum reflects the sensitivity when the device has liquid contacting the surface. This was investigated by monitoring the frequency shift of a single device both during vacuum deposition of a metal film and removal of the same film in an etching solution. The sensitivity in the liquid was approximately 6% less than the value measured in vacuum, a discrepancy that lies within our estimates of experimental uncertainty in this case [54]. The high mass sensitivity of the APM device enables it to function as a general purpose detector, serving as a microbalance in a number of sensor applications. Submonolayer accumulations of surface species can be readily measured. Through suitable chemical modification of the surface, the APM device can be sensitized to the presence of specific species or classes of species present in solution. One approach is to derivatize or chemically modify the properties of the sensing surface, enabling it to selectively bind species from solution. Species that are strongly bound move synchronously with the oscillating APM device surface and perturb the oscillation frequency, according to Equation 3.64, leading to a sensor response. Application of the high mass resolution of the APM to chemical sensing is discussed in Chapter 5.

3.3.3

APM LIQUID LOADING

The in-plane oscillation of the quartz surface contacting the liquid sensing environment leads to entrainment of a thin layer of liquid near the interface, similar to operation of the TSM resonator in contact with a liquid. This viscous coupling of liquid to the plate mode causes both a change in the propagation velocity, V, and attenuation a of the mode. The changes in APM velocity are analogous to changes in TSM resonant frequency, while APM attenuation is analogous to changes in TSM resonator admittance magnitude. In modeling the interaction of a liquid with plate modes, the high frequency of operation necessitates the consideration of viscoelastic response by the liquid. For the simple liquids examined, good agreement was obtained by modeling the liquid as a Maxwellian fluid with a single relaxation time r When the Maxwellian fluid is driven in oscillatory flow with to~-~ l, it responds as a Newtonian fluid characterized by the shear viscosity, T/. For to1-~> l, the oscillation rate approaches the rate of molecular motion in the liquid and energy ceases to be dissipated in

3.3 Acoustic Plate Mode (APM) Devices

107

viscous flow, being stored elastically instead [56]. Consequently, when driven at high frequencies, a Maxwellian fluid behaves as an amorphous solid with shear properties characterized by a shear modulus,/z. The relaxation time, ~', associated with the transition from viscous to elastic behavior in a Maxwellian liquid is related to these parameters by [57] r/ ~"= --. /z

(3.65)

Viscous coupling of plate modes to an adjacent fluid results in both attenuation of the plate mode and a change in propagation velocity. These can be estimated from a perturbation analysis and are given by [54] Aa =

k

Av Vo in which

~2 .

cfrIRe(') 2Vo 1 + jto'r c/film ( -2to

'

(3.66~)

!

)

1 + jto'r

(k . 2. . t~ /x

+j

top 7/

(3.66b)

(3.66c)

where cf is the mass sensitivity of plate mode velocity (see Equation 3.64). It is interesting to note that attenuation and velocity shift arise as the real (Re) and imaginary (Ira)parts of a single complex function describing the viscoelastic fluid response. The velocity shift arises from mass loading by the entrained liquid layer, while attenuation arises from power dissipation in the liquid. Since liquids couple to the surface displacement, it is reasonable that the viscous sensitivity of a device is proportional to mass sensitivity. The plate mode velocity shift and attenuation arising from liquid entrainment are shown vs 771/2in Figure 3.36 (page 108). At low values of viscosity, the fluid behaves as a Newtonian liquid with AV/Voand Aalk proportional to ('!7) 1]2. For viscosities exceeding about 10 cP, relaxation times become comparable to the wave period (6 ns) for the 158-MHz device and viscoelastic behavior results. The solid line is the calculated change in propagation characteristics based on the viscoelastic model of the liquid (Equations 3.66). A shear stiffness/x - 3.1 • 108 dyne/cm 2 was chosen to give a best fit of these equations simultaneously to the experimental velocity shift and attenuation measurements. Interestingly, this value of/x implies a shear relaxation time (from Equation 3.65) that agrees fairly closely with liquid dielectric relaxation times [57]. The dashed line is the pre-

108

3. Acoustic Wave Sensors and Responses

0 A

E -100 v

>o

e

-200

I -400

,

-

S

\

.

:

~ .....

|

i

, ....

,,I

,

.

I

~

, I

, .....

,,,

, ....

I

,.,

. . . .

i ....

,

......

i

,,,

~

,

_~

, . . - S"

I

,ot ,dh

~

~

.

0.5 0.0 0.0

0.2

0.4 V~

0.6

0.8

(Poise) 1/2

Figure 3.36 Change in APM velocity and attenuation vs liquid viscosity. The solid line is calculated using a Maxwell model for the liquid (Equations 3.66); the dashed line is calculated using a Newton]an model for the liquid. (Reprinted with permission. See Ref. [54]. 9 1989 Elsevier Publishers.)

diction when viscoelastic effects are neglected and the liquid is treated as a Newton]an fluid over the entire viscosity range, i.e.,/z = oo in Equations 3.66. Apparently, it is the viscoelastic nature of the fluid that leads to saturation in attenuation and a reversal of the velocity shift at higher viscosity levels [54,58]. The effect of liquid entrainment on APM propagation has several implications. First, the sensitivity of the device to liquid viscosity enables the device to

3.3 Acoustic Plate Mode (APM) Devices

109

be used as a microviscometer [54,56] in the Newtonian regime (co~",~ 1). The ability to detect perturbations on the unmetallized side makes the device attractive for in-situ monitoring of viscosity in engines, chemical process streams, etc. Second, increasing the mass sensitivity of the APM device (e.g., by using a thinner plate) results in a correspondingly higher viscous loss, resulting in a tradeoff between the mass sensitivity and the output signal level of the device. This trade-off is made less severe by decreasing the viscosity of the sensing environment and/or decreasing the operating frequency of the device. When operating the device in the oscillator configuration, the amplifier gain must be increased to offset the viscous attenuation. Third, since changes in liquid properties, such as density or viscosity, lead to changes in plate mode velocity or oscillator frequency, solution properties must be controlled or monitored closely; the relatively high sensitivity of the viscosity of water to temperature changes, for example, means that the temperature of test solutions must be well controlled.

3.3.4 APM ACOUSTOELECTRIC COUPLING Another sensing mechanism of APM devices results from acoustoelectric interactions. Like the SAW, a plate mode propagating in a piezoelectric waveguide generates a layer of bound charge at the surface (Figure 3.23). The evanescent electric field associated with this layer of bound charge extends into the adjacent liquid, coupling to ions and dipoles in solution [58,59]. This interaction is analogous to that found previously between SAWs and charge carriers in a conductive film overlay. Like the viscous entrainment of liquid by the surface, the electrical entrainment of ions decays exponentially with distance from the solid/liquid interface. Since the electric field has a decay length of A/27r, however, ion coupling extends several micrometers into the liquid. Ion, dipole, and induced-dipole motion resulting from this acoustoelectric coupling lead to a perturbation in plate mode velocity and attenuation. The effect of wave/ion coupling on APM propagation can be determined from an equivalent-circuit model analogous to that of Figure 3.24. The perturbation in APM velocity and attenuation arising from a given ion concentration can be related to the solution conductivity cr by [59]

Av Vo -

K2(~.s+~o) 2

o.2

Es + e---"---~ o"2 + oj2(Es + et)2

Aot K2(~.s+Eo) tOO'(Es + ,l) ko - T as q- ai 0 -2 q- to2(es + ai) 2 '

(3.67a)

(3.67b)

110

3. Acoustic Wave Sensors and Responses

in which K 2 is the square of the electromechanical coupling coefficient for the APM; Es, E~, and 6o are the dielectric permittivities of the substrate, liquid, and free space, respectively. Equations 3.67 are very similar to Equations 3.54 derived for the SAW/thin film acoustoelectric interaction. In the present case, however, o" refers to a bulk, rather than a sheet, conductivity. Furthermore, an additional term appears in the present equations that scales the acoustoelectric response inversely with liquid permittivity. This term was unity in the SAW calculation, under the assumption that the medium above the film was vacuum or air. Josse has treated the APM acoustoelectric interaction and obtained a slightly different result from that presented here [58]. The acoustoelectric interaction between plate modes and ions in solution was investigated by measuring variations in APM propagation velocity as a function of solution conductivity or, as shown in Figure 3.37 for four solvents [59]. Attenuation of the APM due to ionic conductivity is too small to measure with precision when using quartz devices, but may be significant with more highly piezoelectric plates such as lithium niobate. The dielectric constant of the solvent significantly affects both the rate at which velocity changes with conductivity and the overall magnitude of the change. This is consistent with Equations 3.67, which predict that velocity changes most rapidly when or = ages + el). Thus, lower dielectric solvents such as ethanol and methanol cause the velocity transition to occur at a lower conductivity than do higher dielectric constant liquids. The magnitude of the overall velocity shift is also diminished, as predicted from Equations 3.67, with higher dielectric liquids, e.g., water. The solid lines in Figure 3.37 are velocity shifts calculated using Equations 3.67 from the dielectric constant for each solvent s and a best-fit value of K 2 = 3.2 • 10 - 4 (same k 2 value used to fit all data). The model works well in the range of conductivities or < 2ore, where ore = co(Es + 61). In many sensor applications, responses resulting from conductivity variations are an unwanted interference. In the addition or removal of a conductive film in an aqueous solution, for example, the acoustoelectric velocity shift accounts for approximately 14 ppm of the overall shift. This would be even larger in most solvents; the large dielectric constant of water helps to minimize the acoustoelectric signals. In aqueous solutions, the acoustoelectric response from a conductivity variation between zero and crc is comparable to the mass response from an adsorbed monolayer. However, there are several approaches that can eliminate signals arising from ion coupling [59]: (1) solution conductivity can be main5The dielectric constants used for the solvents were values estimated at 158 MHz from literature values of the low-frequencydielectric constant and the dielectric relaxation time "r.

3.4 Flexurai Plate-Wave (FPW) Devices

111

-2

-4

-6

~E

A

E

o. o. o

<1

1 = 79.3

-8 -10

...~~r

g-"

: 52.5

-12

09

00

9 - KNO 3 IN H 2 0

-14 (1 : 3 2 . 9

-16

I

- LICI IN H 2 0

9

- LICI IN H 2 0 /

ETHANOL 9 - LICI IN M E T H A N O L

-18

~)~- LICI IN ETHANOL

(:1 = 2 4 . 2 O

0.5

1

1.5

2

2.5

(T (~--~-1 m - l ) Figure 3.37 Change in APM velocity due to the acoustoelectric interaction between electric fields generated in the piezoelectric plate and ions in solution; solid lines are calculated from Equations 3.67, while points are measured. (Reprinted with permission. See Ref. [59]. 9 1988 American Institute of Physics.)

tained much below or above o'c, where conductivity effects are small; (2) the conductivity can be maintained at a constant value throughout the measurement interval; or (3) a sufficiently conductive metal layer can be deposited on the quartz surface to shunt evanescent electric fields, thereby decoupling ions in solution from the APM without affecting mass sensitivity.

3.4

Flexural Plate-Wave (FPW) Devices

In a flexural plate wave (FPW) membrane, Figure 3.38 (page cussed---the TSM, SAW and device can sense quantities that

device, an acoustic wave is excited in a thinned 112). As with the other acoustic sensors disAPM devices m the flexural-plate-wave (FPW) cause its phase velocity, Vp, to change. A unique

112

3. Acoustic Wave Sensors and Responses

F L E X U R A L PLATE WAVE OSCILLATOR FREQUENCY < 1-200MHZ INSERTION LOSS OF ONE TRANSDUCER II

II

FREO r ~ fosr SHIFT DUETO LIQUID LOADING

floaded ---J ....

MEMBRANE USUALLYTHIN COMPAREDWITH WAVELENGTH VEL. f . ~ . - - SAW PIEZOELECTRIC-COATEDSUBSTRATE SILICON ALUMINUM VELOCITY<100-600m/s NITRIDE J SoMEMANY MODESMoDE OFDI S sPERSIvEPROPAGATION THICKNESS/WAVELENGTH

ZINC OXIDE

Figure 3.38 Schematic diagram of a flexural-plate wave (FPW) device.

feature of the FPW is that it can be dimensioned so that its phase velocity is lower than that of most liquids, which lie in the range from 900 to about 1500 m/s. When the FPW device contacts or is immersed in such a liquid, a slow mode of propagation exists in which there is no radiation from the plate. Thus, the FPW device functions well in a liquid environment, and is therefore a good candidate for biosensing and chemical sensing in liquids. Because the "plate" of a FPW device may be only a few micrometers thick, the mass per unit area of the thin plate can be increased significantly by mass-loading produced by the adsorption of chemical vapor molecules on the plate. This causes the phase velocity of an ultrasonic wave propagating on the plate to decrease. Other mass-loading effects that can be measured in fluid-loaded FPW devices by monitoring decreases in the real part of the phase velocity are increases in the density of a fluid on the plate, and the attachment onto the plate of protein molecules, cells, and bacteria from a liquid that contacts the plate. An opposite effect occurs if the tension in the thin plate is increased, for example, by establishing a differential gas pressure across it, by applying a force on the plate, or by bending the frame that surrounds it. The increased tension causes the real part of the phase velocity to rise. One finds experimentally that the amplitudes of the displacements associated with a flexural wave of given total power are large relative to those of the other acoustic sensors discussed here. Peak-to-peak normal displacements up to 100 nm have been measured, using laser diffraction, when only a few milliwatts of wave energy were propagating in a plate a few micrometers thick. As a result of

3.4 Flexurai Plate-Wave (FPW) Devices

113

this large-amplitude motion, one can produce useful kinetic effects, such as the transport of granular solids and the pumping and mixing of fluids. Thus, the FPW device can be both a sensor and an actuator. We will now discuss in detail the sensing and actuating properties of these waves.

3.4.1

FLEXURAL PLATE WAVES

The characteristics of flexural plate waves are found by a process similar to that used for surface acoustic waves: 1. Write the constitutive relations for the medium to relate stress to strain, assuming elastic linearity (Hooke's Law); 2. Formulate the relation between acceleration of the medium and the spatial variation of stress (Newton's First Law); 3. Express the conditions on stress and particle displacement at the boundaries of the plate; 4. Obtain expressions for the displacement components of the waves that satisfy all the relations above. One finds in the simplest case of an elastically isotropic plate that there is an infinite set of waves that can exist. These are known as Lamb waves, after Sir Horace Lamb who published their first detailed description [60]. Figure 3.39 (page 114) shows the phase velocities of the waves as a function of the product ktd, where kt -" 2"rr]At, At is the wavelength of the bulk transverse (shear) wave in the medium of which the plate is made, and d is the plate thickness. The waves divide naturally into two sets: symmetric waves (denoted by So, S l . . . . ) whose particle displacements are symmetric about the neutral plane of the plate, and antisymmetric waves (A0, Al . . . . ), whose displacements have odd symmetry about the neutral plane. Figure 3.38 shows that for sufficiently thin plates (ktd < 1.6), only two waves exist m the lowest-order symmetric mode (So) and the lowest-order antisymmetric mode (Ao). These are the modes shown earlier in Figure 2.0d. The plate mode that we will emphasize here is the A0 mode, in which the elements of the plate undergo flexure as the wave propagates. The shape of a plate during propagation of this flexural mode has been likened to that of a flag waving in the wind. The characteristics of Lamb waves have been derived and explored by a number of authors [61,62,63,64]. Here we will focus on the aspects of the waves that are most relevant for sensing and actuating in devices made by micromachining techniques. These techniques include many of those customarily employed in making silicon integrated circuits. Examples are processes for depositing thin

.

s~

A~,

A2

.

.

.

.

O

[

....9

.r

II

I_/

g~ O ='1

t~ g~ t~

Q

t

"g

3$

t,

3"

6

l

a

s

so n

/z I j

ktd Figure 3.39 Symmetric (S) and anti-symmetric (A) Lamb wave characteristics. Vertical axis: Lamb-wave velocity normalized to transverse bulk-wave velocity. Horizontal axis: Product ktd where kt = transverse wavenumber = 27flAt, where At is transverse wavelength and d is plate thickness. (Reprinted with permission. See Ref. [61]. 9 1967 Plenum Press.)

3.4 Flexural Plate-Wave (FPW) Devices

115

films, such as insulators and conductors consisting of metallic or doped semiconductors. For acoustic-wave transduction, one utilizes piezoelectric films that are typically deposited by sputtering. Key etching techniques for making microstructures are bulk and surface micromachining. In the former, one etches away silicon lying under a deposited film to leave a beam or membrane. In surface micromachining, one deposits and shapes films deposited on a so-called sacrificial layer. One then shapes these films by photolithography and etching, and finally, dissolves the sacrificial layer to free the shaped film. These micromachining processes are described in a recent book on semiconductor sensors [65] and in articles in a reprint volume about microsensors [66]. 3.4.1.1

Behavior of the FPW Velocities and Particle Motions

Figure 3.40 shows the phase velocity of the Ao and So Lamb waves in tensionfree silicon nitride, as calculated by a numerical analysis program by Nassar and Adler [67]. Silicon nitride, an amorphous glasslike substance, is a practical material to use in micromachined flexural-wave sensors since thin layers of it can easily be made on a silicon wafer by LPCVD (low-pressure chemical-vapor deposition). The underlying silicon can later be etched to leave a thin unsupported plate, as mentioned above. A silicon-rich formulation having nearly equal numbers of silicon and nitrogen atoms is preferred over stoichiometric silicon nitride, Si3N4, because it can have a very low residual stress. The behavior shown in Figure 3.40 (page 116) is typical: 1. For very small thickness-to-wavelength ratios, the phase velocity of the Ao mode approaches zero asymptotically. 2. As d/A increases, vp increases, finally becoming asymptotic from below to the surface-wave velocity for the medium. 3. The phase velocity of the So mode is maximum for a very thin plate, and it falls as d/A increases, finally becoming asymptotic from above to the surface-wave velocity for the medium. The rise in (2) is due to the increased effective stiffness of the medium as an ever-thickening plate is required to assume the sinusoidal wave shape. The approach to the surface-wave velocity as the plate becomes thick is to be expected, as a surface wave can be represented as a superposition of antisymmetric and symmetric plate waves. The particle motions that are indicated schematically by the ellipses near the curves of Figure 3.40 are predominantly normal to the plate for the Ao mode and predominantly tangential to the plate for the So mode. Fortunately, the most interesting applications of the FPW arise for very thin plates, for which d/A "~ 1. In this regime, we can fabricate plate-mode delay lines

SO ~~'

:

"-'-':, -

i

............ i .................................

i

Q

I= .< r

................ . ~ . ~

A

.......................................................... O

Q, =I gl. r "O O r~ r 9 . . . .

0.0

0.5

. . . o o . , . . . , , . o , o , , P , o , , , e , , i

d/~,

, e t o t o , t e t o e t e e ~ t t o * t e e e e ~ e o ~ e e e o

9

o

1.0

1.5

Figure 3.40 Calculated phase velocity of flexural plate waves vs ratio of plate thickness, d, to wavelength, A, for silicon nitride. Material is assumed to have the elastic properties of SiaN4 and no residual tension. The mode shapes are illustrated at the right with a greatly enlarged vertical scale for clarity. Ellipses at left show the retrograde elliptical particle motions of the lowest symmetric and antisymmetric modes for d/A = 0.03. (Reprinted with permission. See Ref. [62].)

3.4 Flexurai Plate-Wave (FPW) Devices

117

whose phase velocity is below the velocity of sound in water, and so one can use them for biosensing. Furthermore, because of the low phase velocity, the frequency of operation, f, of the device for a given wavelength is low, since

vp f = ~. A

(3.68)

A low operating frequency is an attractive feature as it implies relatively inexpensive associated electronic circuitry. Finally, it is possible in the thin-plate regime to approximate the phase velocity quite well by the simple asymptotic expression [68]

Vp =

(3.69)

where B is the bending stiffness of a homogeneous, elastically isotropic plate and M is the mass per unit area of the plate. For an Ao Lamb wave in a tension-free plate, the bending stiffness takes the form B =

12

(3.70)

where E'-

1-

E

v2

(3.71)

is the so-called effective Young's modulus, E being the actual Young's modulus and u the Poisson's ratio for the material. As do the SAW and APM sensors, the FPW device employs interdigital transducers and piezoelectric coupling to generate and detect the waves. Figure 3.41 (page 118) shows a cutaway view of an FPW device in which the supporting membrane is a low-stress silicon-rich silicon nitride layer formed by the LPCVD process. After the layer is deposited, the silicon is etched away to leave an unsupported membrane. A conducting groundplane is deposited, followed by the RF-magnetron sputtering of piezoelectric zinc oxide and then the sputtering or evaporation of a second conducting layer. The transducer electrodes may be formed in either conducting layer. The dimensions of a typical device are indicated in Figure 3.41. Note that in contrast with the SAW and APM transducers, for the dimensions given each transducer finger is quite close to the opposite groundplane, and so the electric fields in the piezoelectric are directed primarily perpendicular to the plate. (In the thicker SAW and APM devices, significant transverse electric fields are also produced by the transducers.) The electronic measurement techniques used with SAW and APM sensors can also be used with FPW devices. Thus, often a delay-line oscillator is formed by

118

3. Acoustic Wave Sensors and Responses

Figure 3.41 Typical flexural-wave device. Thickness of the surrounding silicon frame, typically 250-500 microns, is reduced here for clarity. P is the IDT periodicity. (Reprinted with permission. See Ref. [621.)

connecting an electronic amplifier between output and input transducers. Changes of wave attenuation arising from changes of viscosity are reflected in changes of the signal amplitude at the output transducer. The frequencies employed in FPW

3.4 Flexural Plate-Wave (FPW) Devices

119

devices are usually in the 1-10 MHz range, significantly lower than those used with SAW and APM sensors. Example 3.8: Find the mass per unit area, M, of the composite plate of a typical FPW device whose layer thicknesses are: SixNy, 1.8 I~m (density 3100 kg/m3); ZnO, 0.2/xm (density 5665 kg/m3); Al ground plane, 0.2 Ixm (density 2700 kg/m3).

Solution:

M, the sum of the thickness-density products for each of the layers, is 11.8 •

10 -3 kg/m 2.

Before examining the interactions between the plate wave and the ambient, we should note two important details: (1) The surface of the silicon nitride typically oxidizes and so it is actually silicon dioxide, a substance that lends itself well to use in biosensing. (2) In the fabrication process, usually a tensile stress develops in the plate that can affect the phase velocity. Thus, the plate has one aspect of a membrane: it has in-plane stress. This stress affects the phase velocity somewhat. Its effect can be included in the thin-plate approximation for the phase velocity, Equation 3.69, as v~ =

(Tx+B),/2 M

(3.72)

where Tx is the component of in-plane tension in the propagation direction (taken here as the x direction) per unit width in the y direction, perpendicular to the direction of propagation. Explicitly, /

.d

Tx = J ~'yy(~)d~

(3.73)

o

where ~'yy is the y component of tensile stress in the plate and ~: is a variable of integration normal to the plate whose bounding surfaces are at z = 0 and z = d. In typical FPW devices operated in air, the in-plane tensile stress may account for about ten percent of the phase velocity (in other words, Tx/B ~ 0.2).

3.4.2

FPW P E R T U R B A T I O N M E C H A N I S M S

We will now consider the response of the lowest-order flexural mode to various perturbations. For generality, we assume that there is some initial tension, T~, in the plate. For simplicity, we assume that the plate is quite thin (d/A ,~ 1), and so the approximate phase velocity expression of Equation 3.72 can be used as a basis for discussion. Figure 3.42 (page 120) summarizes the perturbations that we will discuss in Sections 3.4.2.1 to 3.4.2.5

120

3. Acoustic Wave Sensors and Responses

iii

i

i iii i

i

I

I U N L O A D E D PLATE one Flexural Mode: "AO Lamb" Mode

,,

I

Real Phase Ve'iocltg

I

i ii

,,

,,

ii

.

i

ii

I

FLUID-LOADED PLATE Two Flexural Modes: "AO Lamb" Mode --Always Lossy .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

"Scholte" ,M~

Non-viscous Fluid [

.

-- Vp < vF

!

Viscous Fluid

Real Phase Velocity. Complex Phase'velocltg I .=

Mess- I oedl nq c hange: Absolute pressure Chemical vapors Cells, bacteria ....

Mass- 1oedi ng r hanoe: E:Iect ro pl eft ng FIutd density change Protetn adsorption Cell growth Stiffness chanae: [lectropleting (thick)

Stiffness chenaeDifferential pressure Force Acceleration

Mes_s-loadlno r F1uld within vtscous decay distance

Attenuationcher~ge. Ytscosttychange

Figure 3.42 FPW perturbations for unloaded and fluid-loaded plates. The mode names are in quotes because in-plane tension modifies the behavior from that of conventional Lamb and Scholte modes. (See text for details.)

3.4.2.1

Effects of Increasing the Mass/Area of the Plate

Intuition correctly suggests that if we increase the mass per unit area of the plate by an amount m', the phase velocity for a thin-plate FPW device will become just

vp

( T x + B , ) '/2 M+m

=

(3.74)

Note that if m'/M ~ 1, we can expand the square-root expression in a Taylor series and show that the fractional change in velocity, Avp/vp, is approximately AVp __ (l,p)fina I -- (Vp)o,.iginal ~ Vp

(Vp)o,.iginal

m' 2M

(3.75)

where (Vp)fina I is the phase velocity after the mass increase and (Vp)o,.iginal is that before the mass increase. As was done earlier with the APM device, we can define a mass sensitivity, Sin, and obtain its value for the FPW device as S/71

=

1 AVp

m

1

vp

~

1

2M

The units of the mass sensitivity are [area/mass].

(3.76)

3.4 Flexural Plate-Wave (FPW) Devices Example 3.9:

121

Find the gravimetric sensitivity of the 3txm-thick FPW device of Ex-

ample 3.8. Solution:

As Sm ~ -1/2M, Sm= --42.4 m2/kg = 424 cm2/g.

Note that the value of Sm obtained for this typically dimensioned FPW device is much higher than those for the TSM and APM devices. If one wishes to increase the FPW mass sensitivity even further, one should make the device thinner, reducing the velocity (and incidentally reducing the frequency of operation). This increasing sensitivity with a lowering of the frequency is opposite to the situation for the TSM, SAW and APM devices, for which one increases the sensitivity by raising the operating frequency. Example 3.10: Assume that the FPW device of Exercise 3.8 is operated as a delayline oscillator whose unperturbed oscillation frequency is f, and suppose that the random fluctuation of frequency (noise) is Afnoise. Find the minimum detectable added mass per ! P unit area, m rain, that could be detected with a 3: I signal-to-noise ratio. Calculate m rain assuming f = 5 MHz and Afnoise = 0.4 Hz. Solution: For the delay-line oscillator, Avp/Vp = Aflf. Setting the minimum significant frequency shift, Af, equal to 3Afnoise, from Equation 3.57 we obtain

mmin = - 2 M ( Avp Vp )min = - 6 M ( Aff~

(3.76)

For the values given, m'mi~ = 5.7 X 10-9 kg/m2 = 5.7 • 10 --l~ g/cm2 (570 pg/cm2). If one wishes to reduce this value, one can reduce the membrane thickness, and hence, M. With the device of Figure 3.41, a lower limit on thickness of the composite membrane is set by the minimum thickness of the piezoelectric layer, as extremely thin ZnO layers tend to be randomly oriented and hence have low piezoelectric coupling. Much thinner membranes can be achieved if one dispenses with the piezoelectric and instead drives a very thin membrane with electrostatic forces produced by an opposing interdigital electrode array [62,69].

Example 3.11: For the sensor of Exercise 3.10, what fraction of a monolayer coverage could be detected of molecules having a specific gravity of 1.35 ? (This is the specific gravity of a typical protein molecule.) Assume that the molecules are spheres having a diameter of 1 nm located on centers that are one diameter apart. Solution: The fractional coverage, F, equals m'min/(mass of monolayer per unit area). For the values given, F = 5.7 • 10-9/7.1 • 10-7 = 0.008 (or 0.8% of a monolayer coverage).

Detecting Chemical Vapors with the F P W Sensor A practical FPW chemical vapor sensor typically employs a sorptive film on the plate, as do the TSM, SAW and APM devices (Figure 3.43, page 122). When calculating the mass sensitivity of the coated FPW sensor, we simply include the mass per unit area of

122

3. Acoustic Wave Sensors and Responses

Figure 3.43 Schematic cross-sectional view of polymer-coated FPW device for vapor sensing. The inset, with exaggerated vertical scale, shows the wave propagating along the membrane. (Reprinted with permission. See Ref. [62].)

the sorptive film together with that of the plate itself. Thus, if the density of the sorptive film is Ps and its thickness is ds, the approximate mass sensitivity of the coated FPW vapor sensor is sm= -

1 9 2(M + psds)

(3.77)

Figure 3.44a shows FPW sensor response to toluene vapor [68]. The sorptive coating was a 1.5-b~m-thick layer of poly(dimethylsiloxane), PDMS. The sorptire polymer was in its rubbery state at the measurement temperature of 24~ Very linear response was observed (Figure 3.44b). A different FPW device operating at 2.85 MHz and having a 0.5-/xm-thick coating of ethyl cellulose, which was expected to be better suited to toluene detection [70], had a measured mass sensitivity of Sm = 1064 cm2/g. Its estimated minimum detectible concentration was 70 ppb, based on an assumed 3' 1 signal-to-noise ratio and the measured short-term frequency instability of +0.04Hz. Example 3.12: Determine the mass sensitivity of the FPW device of Example 3.8 when it is covered with a O.1 lain layer of a sorptive polymer having a specific gravity of 1.05. By what percentage has the sensitivity been reduced from the value for the uncoated device?

Solution: For the values specified, psds = 1.05 x 10 -4 kg/m 2, yielding Sm = -42.0 mi/kg = -420 cmZ/g, a sensitivity reduction of 0.9%.

Figure 3.44 FPW response to toluene vapor. (a) Temporal responses of infrared spectrometer (MIRAN) and PDMS-coated FPW delay-line oscillator for different concentrations of toluene vapor in nitrogen carrier gas. Delay of FPW response was due to delay in traveling through a length of piping. (b) Concentration vs frequency shift for toluene vapor. Diameters of data symbols indicate probable experimental error. (Reprinted with permission. See Ref. [62].)

123

124

3. Acoustic Wave Sensors and Responses

3.4.2.2

Effects of Loading with a Non-Viscous Fluid

Let us suppose that we put a fluid having a speed of sound vF in contact with a FPW delay line whose phase velocity satisfies the inequality Vp < vF. Based on experience with light and radio-frequency electromagnetic waves, we might expect that the ultrasonic wave energy would not radiate away into the fluid, because the plate would act as a slow-wave propagation medium. The case is similar to the energy confinement that occurs in fiber optics: optical energy does not radiate from the high-refractive-index, low-velocity fiber core into the lowindex, higher velocity cladding. The wave in the faster medium (the fluid) becomes evanescent, its amplitude diminishing as an inverse exponential function of the distance from the plate (Figure 3.45). FPW experiments support this simple picture for the most part, but detailed differences must be noted. When a semi-infinite fluid contacts the FPW device, new modes of propagation exist that are not allowed in unloaded FPW devices. The lowest-order mode, which is useful for sensing, is similar to the Scholte wave of geophysics [71], which is guided along the boundary between the semi-infinite seabed and the liquid layer, the ocean. As shown in Figure 3.46 (page 126), the phase velocity for this wave in the thin-plate case ranges from zero to the velocity of sound in the fluid for plates that are thick compared with the wavelength. If the fluid is inviscid, this wave is lossless. It appears that this mode is the one that is responsible for most reported flexural-wave sensing results. In addition, there is a fluidloaded asymmetric Lamb wave that can have a very large attenuation factor, even for frequencies well below the speed of sound in the fluid. (Calculated attenuation factors for this Lamb wave for an inviscid water-like loading of a 2-micronthick plate at a 100-micron wavelength exceed 800 dB per centimeter!) It has been shown [68,72] that when a semi-infinite body of fluid contacts one side of a thin FPW device, the mass loading that the fluid produces can be described by simply adding an additional term to the mass per unit area, M, in the phase velocity expression of Equation 3.74:

Tx+B )1/2 Vp =

(3.78)

M + pFc$e

where Pr is the density of the fluid and Be, the so-called evanescent decay length, is given by

.

(3.79)

t~

m_ i

.r

--o

t~

Figure 3.45 FPW device in contact with a semi-infinite fluid showing evanescent disturbance excited in the fluid after [62]. Lengths of arrows in fluid represent amplitudes of particle velocity locally. Note that amplitudes decrease rapidly within a fraction of a wavelength into the fluid, showing that the disturbance in the fluid is localized near the membrane. (Reprinted with permission. See Ref [62].)

t,~ t~

126

3. Acoustic Wave Sensors and Responses

4000

] ..................................................................... ~ ~ ~

~ ~

i

3000

.I

.....

....,......****..,b

A

E 2000

............................................

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i .................. ............................

(a)

=,

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,,.,b.

~ . . . . . . . . . . .

.~:

.

.

.

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,, CPT .................... /

1000

.... ;.,.:.:

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, . . . . . . . . . . . . . . . . . . . . . . . . . .

.,..,.,,

.....

, ....

. ....

,~ ...........

,..,~

....

,

Iossless (Scholte)

0 0.00

0.04

0.08

O. 12

O. 16

0.20

d/k 70000 1 .....................'......................................................................................... I

"E'

~.

i oooo ,

..........

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........ '"

..................... ' ..................... ~..................... i"....................

(b)

30000 z.,

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0.16

0.20

Figure 3.46 Phase velocity (a) and attenuation (b) of fluid-loaded flexural plate modes plotted vs thickness/wavelength, d/A. The Iossless (Scholte) mode velocity approaches from below the velocity of the fluid, assumed to be water. CPT denotes result of classical plate theory for an unloaded plate. (Reprintedwith permission.See Ref. [62].)

127

3.4 Fiexurai Plate-Wave (FPW) Devices

For plates and fluids that satisfy vp ~ VF, the term in the brackets is approximately unity, and we have A

(3.s0)

Physically, Equation 3.80 means that one may consider the mass-loading effect of an entire half-space of fluid as being equivalent to that of a layer of the fluid having a thickness of only BE. Example 3.13: Determine the approximate and the exact values of the evanescent decay length for water, assuming that the wavelength is 100 microns, vp = 200 m/s and Vwater -" 1480 m/s.

Solution: Approximate evanescent decay length is t~Elapprox "~" (A/2~r) = 15.92 microns. Exact evanescent decay length 8F.lexact= ( A / 2 " t r ) [ 1 - - (Vp/VF)2]-1/2 = 16.06 microns.

Equation 3.80 has been verified in measurements made on a flexural-wave oscillator operated successively in air at STP (essentially zero loading) and then with various liquids on one side of the plate. Table 3.4.1 lists the liquids and their properties, and compares measured frequencies with those calculated from Equation 3.80. The mass-loading effect of a gas was evaluated in a room-temperature pressure cell filled with helium (whose speed of sound, VF, is 965 m/s at 0~ at from 1 to 4 atmospheres gauge (Figure 3.47). A linear dependence of velocity on gas density was observed, as expected [62]. The discrepancy between calculated and measured values was thought to be due to insufficient thermal equilibration between measurements. In this application of the FPW device, there is no differTable 3.4.1 FPW Density Determinations for Low-Viscosity Liquids [62]. Velocity of Sound in Fluid is Denoted by v~. The Measured Frequencies for Air Loading and Methanol Loading Yielded the Device Parameters M = 6.411 x 10 -3 kg/m 2 and Tx + B = 539.2 N/m Used in Calculating Predicted Responses for Other Liquids. Liquid

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128

3. Acoustic Wave Sensors and Responses

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Variation of FPW delay-line oscillator frequency vs pressure in helium-

filled

vessel.

pressure

( R e p r i n t e d with p e r m i s s i o n . See R e f [62].)

ential force on the plate, and hence the device has no obvious upper pressure limit. At high enough gas pressures, we would expect the velocity shift to depend nonlinearly upon gas pressure (or, actually, gas density) because of the nonlinearity of the mass-loading relationship and because ultimately the elastic stiffness of the m e m b r a n e could be affected. In spite of this, the sensor could be useful so long as the response was monotonic and reproducible. Let us consider the ability of the F P W device to measure small changes in fluid density. The basic gravimetric sensing expression of Equation 3.76 can be rewritten for a delay-line-oscillator F P W sensor as

af --

f

=

SmAm

(3.81)

3.4 Flexural Plate-Wave (FPW) Devices

129

where Am is the change of mass per unit area due to a change of fluid density, Ap. If the change is small '~ 1), we obtain

(SEAp/M af

(3.82a)

-7--- Sm~EAp = Spi~to where

~e Sp = - 2(M +

PFSE)"

(3.82b)

Measurements have shown [72] that for liquids whose density is near that of water, the discrepancy between literature density values and those deduced from FPW measurements are in the range of _.+0.2%. Strictly speaking, one needs to know the speed of sound in the liquid under test in order to determine accurately the value of ~ , and hence, the density. This can be done by using two FPW sensors operating at different wavelengths. Also, designing the FPW device to have the lowest possible phase velocity will reduce the importance of knowing the speed of sound in the liquid. 3.4.2.3

Gravimetric Detection of an Added Mass in a Liquid

There may be instances where one wishes to detect the absorption of molecules from a liquid into a polymeric film on the plate surface. And in biosensing, one may wish to detect the adsorption or attachment of a layer of protein molecules, cells, bacteria or other organisms on a surface. Figure 3.48 (page 130) shows an exploded view of a three-chip FPW liquid flow cell made for such applications. We can determine the velocity or frequency shift caused by such types of loading, which is in addition to the loading due to the liquid, by substituting into the denominator of the phase velocity expression of Equation 3.78 all the relevant masses per unit area: that of the plate itself, M; the equivalent loading of the liquid, that of any selective biological or chemically sorptive layer, and that due to the unknown itself, Am. If Am is much smaller than the sum of the other terms, as is usually the case when one is seeking the minimum detectable added mass, we can use the first term of the Taylor series expansion of the denominator of the square-root in the velocity expression, and write the approximate phase velocity as

PF~e,;

msorptive;

vp~( •

Tx+B M + pFt~E + msorptive +

)1/2 ~ (

Am

( 1 -- 2(':'M + pF6eAm-t-msorptive) ) "

Tx+B

)1/2

M "k- PF~E dr msorptive (3.83)

130

3. Acoustic Wave Sensors and Responses

Figure 3.48 Exploded schematic view of a flow-cell FPW liquid sensor. The silicon chip containing the thin silicon-nitride membrane, piezoelectric film and transducers is sandwiched between two etched silicon chips, The upper chip is a cap with fluid inlet and outlet fittings. It also provides vias for contact to a temperature-sensing polysilicon resistor deposited on the FPW chip below it. The lower chip introduces transducer contact leads and protects the underside of the membrane from contact with the fluid. (Figurecourtesy of Ben Costello,BerkeleyMicrolnstruments,Inc.)

3.4 Flexural Plate-Wave (FPW) Devices

131

For such a device the gravimetric sensitivity is

1

Sm

-- - 2(M +

PF~E + msorptive)"

(3.84)

We then find that the minimum detectable added mass, Ammin, in this fluid-loaded case with a sorptive film is just Ammin

-- am(

?)

Afmin

(3ss)

\

where Afmin is the minimum significant frequency shift.

Example 3.14:

For an FPW device having in a vacuum vp = 200 m/s, a lO0-micron wavelength, and the mass per unit area, M, of the device of Example 3.8, .find the fractional velocity decrease that would be produced by loading (a) one side with water (Pwater = lOakg/m3), (b) both sides with water, and (c) one side with methanol (Pmethanol = 0.89 X 103/kg/m3). Solution: The effective areal mass added to the membrane mass per unit area by liquid loading on one side of the membrane is pFSE, and twice that for two-sided loading. The values of this added areal mass for the three cases here are 15.77 • 10 -3, 31.54 X 10 -3, and 14.04 • 10 -3 kg/m2 respectively. These are comparable with M (= 11.8 • 10-3 kg/m2 from Example 3.8), so the fractional decreases of velocity are substantial. Using Equation 3.78 we find that the phase velocity is reduced from its original unloaded value by the factor 1/(1 + PFSE/M)1/2 for one-sided loading and 1/(1 + 2pFSE/M)1/2 for two-sided loading. The percentage fractional decreases are 34.6%, 47.8%, and 32.4%, respectively. The effect of liquid loading can be decreased substantially by reducing the FPW wavelength.

3.4.2.4

Effects of Loading with a Viscous Liquid

Analytical and experimental studies of FPW device response to loading with a viscous liquid show many similarities with the effects already described in this chapter in connection with the three other acoustic devices [62,72]. Representing the operating angular frequency as to and the viscoelastic relaxation time as ~', one can show that if toy ,~ 1, the attenuation of the FPW is proportional to (r/) 1/2, where 7) is the shear viscosity. There is also a mass-loading effect that may be thought of as being due to the mass of the entrained liquid that is viscously coupled to the plate. This loading can be accounted for analytically by adding a viscous mass-loading term in the approximate expression for the FPW phase velocity, viz., ( Vp =

Tx + B M +

pFtSE + M n

),/2 (3.86a)

132

3. Acoustic W a v e Sensors and R e s p o n s e s

where Mn -

~Sv 2

(3.86b)

and the quantity 3v is the so-called viscous decay length, the distance over which a shear disturbance in a viscous medium decreases to 1/e of its amplitude at the point of excitation: 3v(2"rl) 1/2 = ~ . COpE

(3.87)

3.15: Calculate the viscous decay length and the value of Mn for a 1GOmicron-wavelength, 5 MHz FPW device of Example 3.50 that contacts water on one side (the viscosity of water, ~, is one centipoise = 10-3 Pa-s). Example

Solution: The viscous decay length is 250 nm, which is much smaller than the evanescent decay length (16 microns, from Example 3.55). The value of M,~ is 1.3 x 10-4 kg/m2, or about one percent of the areal mass of the membrane itself. The effect of the viscosity on the real part of the FPW phase velocity is obtained by substituting M,7 for added mass per unit area m' in Equation 3.74. The dependence of the attenuation coefficient of the FPW, a, is given by [72] -

T + 2B + ~p~8~/(~/2~) 2

(3.88)

As a viscosity sensor, the FPW device has an advantage relative to APM sensors because of its low operating frequency" the viscosity for which ~o~"= 1 and the attenuation saturates is much higher than that for the SAW and APM devices. Also, the sensitivity of attenuation to viscosity should be higher than that of a TSM resonator operating at the same frequency because of the larger surface-tovolume ratio of the FPW device. Figure 3.49 shows results of measurements [72a] of the temperature dependence of viscosity for water and aqueous solutions of DMSO (dimethylsulfoxide). These tests were made in preparation for measuring the viscosity of a rare and expensive glycoprotein. In this case, measuring viscosity acoustically is a distinct advantage, as fewer than six microliters of sample are required. Experiments with aqueous solutions of DNA and of other polymers suggest that polymers having high molecular weights - - and hence large viscoelastic relaxation t i m e s - do not take part in the acoustic motion and thus do not contribute to the wave dissipation [72]. For operation at 5 MHz, the maximum molecular weight for which reliable viscosity measurements can be made acoustically is about 15,000 daltons.

3.4 Flexural Plate-Wave (FPW) Devices .

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curve

133

fit

water 5 N e

~

r~

o

r~

<>

0 -10

0

10

20

30

40

Temperature (oC) Figure 3.49 Viscosity as a function of temperature for 1M, 3M and 5M aqueous dimethylsulfoxide (DMSO) solutions. Solid curves are curve fits to the data based on the exponential expression r / = rlph exp [AEact'ffr ( l k T - D,Tph)], where T is absolute temperature in Kelvin, R is the universal gas content, AE,,ct is the activation energy, Tph is the phase change temperature, and r/ph is the viscosity of the solution at the phase-change temperature. The dashed curve is the exponential curve fit for water, and the dashed circles indicate the region of temperature where phase change occurred. (Reprinted with permission. See Ref. [72a]. 9 1993 ASME International.)

3.4.2.5

Other Perturbations

The ability to utilize an electrically conducting film on the FPW plate permits one to keep electric fields produced by the transducers from reaching a fluid in contact with the plate. Hence, the device can be made insensitive to the electrical conductivity of the medium, if desired. Unless temperature compensation is employed, the FPW velocity will depend fairly strongly upon temperature. The measured temperature coefficient of the relative phase velocity of the device of Figure 3.48 was - 1 2 0 ppm/~ This re-

134

3. Acoustic Wave Sensors and Responses

suits from a combination of thermal expansion of the frame (silicon) and the elements of the composite plate (silicon nitride, aluminum, zinc oxide). Owing to the small heat capacity of the thin plate, the phase velocity can change rapidly in response to heating from either an electric current in a resistive film on the plate or from absorbed thermal radiation [68]. A possibly useful feature of the FPW device is its ability to sense while coated with a layer of gel that is much thicker than the plate [73]. Water-based gels such as gelatin, poly(acrylamide), poly(vinylalcohol), agarose, and alginate having less than 2-5% solids content have shear moduli that are small compared with the elastic moduli of the plate's components (around 1011 Pa). Hence, the gels merely mass-load the plate. Recalling the evanescent FPW acoustic field structure, we note that the phase velocity will only be affected by loading within the evanescent decay distance, Be. Thus, one can use the gel to screen out large particles while responding to any small particles that diffuse to the gel-plate interface. Examples are shown in [73].

3.4.3

FPW KINETIC EFFECTS: PUMPING AND MIXING

We conclude this section with a brief overview of the kinetic functions that the FPW device can perform, in addition to the sensing just discussed. 3.4.3.1

Experimental Observations

The following experiments were made with FPW devices like that sketched in Figure 3.41.

Granule transport: In room air tests, when the lefthand FPW transducer was driven by a 10 V peak-to-peak RF source tuned to the center of its transducer response, granular particles sprinkled on the plate moved to the right at velocities of at least 3 cm/s [74]. The direction of motion was initially surprising, as one might have expected frictional forces to drive granules in the opposite direction owing to the retrograde flexural-wave particle motion (as indicated in Figure 3.40). This frictional drive is utilized in most macroscopic rotary ultrasonic motors. Liquid with marker spheres: In a similar experiment, deionized water containing 2.3-micron polystyrene spheres was transported to the right at velocities up to 300 microns/s. The experimental set-up is sketched in Figure 3.50. Initial measurements were made with marker spheres whose specific gravity, 1.05, differed from that of the liquid; thus the spheres settled onto the membrane. The speed

3.4 Flexural Plate-Wave (FPW) Devices

135

Figure 3.50 Experimental set-up for measuring flow velocity. (a) Cross-section view of silicon-based flow cell with FPW transducer on left side of bottom plate and glass slide to cover top of cell. (b) Cell on stage of optical microscope that is fitted with video camera and VCR for recording. (c) Schematic cross section of flow cell showing marker spheres and light rays converging from lens having small depth of field and large focal distance. (Neutrally buoyant spheres are depicted here; see text for details.) (Reprinted with permission. See Ref. [75]. 9 1994 IEEE.)

of fluid motion measured near the membrane was proportional to the square of the drive voltage, and to the square of the amplitude of the plate's normal motion, as measured by laser diffraction. The pumping speed measurements shown in Figure 3.51 (page 136) were made with 2.5-micron spheres that were near the membrane of the FPW device. Figure 3.52 (page 137) shows results obtained at various distances from the membrane of a device containing a dilute sugar-water solution in which such marker spheres were neutrally buoyant [75]. The observed recirculation is due to the closed-cell configuration. Detailed analytical study of the flow at heights below one micron suggest that flow velocities there could be as high as 250 microns/s under these experimental conditions.

136

3. Acoustic Wave Sensors and Responses

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Amplitude [ Angstroms I Figure 3.51 Pumping with flexural-wave device. Data: velocity measured from video recorded motion of polystyrene spheres vs amplitude of wave motion. Solid line: theoretical values based on acoustic streaming theory. (Reprinted with permission. See Ref. [74]. @ 1989 American Institute of Physics.)

Liquid with dye: In experiments where a dye (consisting of permanganate ions) was introduced into a covered water-filled well on the device, circulation of the water was evident. This test was performed to eliminate the possibility that the flow seen earlier was being caused by radiation pressure acting on the marker spheres (whose acoustic impedance differs from that of the liquid).

Electrochemical cell: The limiting current in an electrochemical cell was found to increase substantially when one electrode was a platinum-coated FPW device driven so as to produce either propagating or standing flexural waves [76]. The fractional increase in cell current was proportional to the square of the drive voltage, and hence to the square of the wave amplitude (Figure 3.53, page 138). 3.4.3.2

Phenomena Involved

These experimental observations can be accounted for by an approximate analysis based on the nonlinear phenomenon of acoustic streaming [77] in the fluid that contacts the FPW device. Briefly, the phenomena involved are the following:

3.4 F l e x u r a l P l a t e - W a v e ( F P W ) D e v i c e s

137

1. Because of its small volume, the displacement of the surface of the FPW plate is relatively large when even a small amount of acoustic power is used. For example, a few milliwatts at a few megahertz in a 3mm-wide, 2-micron-thick plate produces 100nm peak-to-peak displacements. 2. If the disturbance in the fluid that contacts the plate is evanescent (that is, if vp < VF), the energy in the fluid is concentrated within the evanescent decay distance, Be, from the plate. For the device illustrated in Figure 3.41, having a 100/xm wavelength, this distance is only 16/~m. Thus, the acoustic intensity can be relatively large near the membrane. 3. Because of the high amplitudes of particle motion in the fluid due to (l) and (2), nonlinear acoustic effects can be important. In particular, acoustic streaming can occur, so that a propagating sinusoidal wave produces a steady ("zero frequency") force in the direction of wave propagation. This steady force causes fluids in contact with the membrane to move. 4. In the electrochemical cell, circulation of the entire fluid (propagating wave) or local circulation (standing wave) stirs the electrolyte near the

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Figure 3.52 Flow velocity measured at different heights in closed flow cell of Figure 3.50. Below 170/zm liquid flows to the right in the direction of wave propagation. Above that height liquid flows to the left due to recirculation in the closed cell. (Reprinted with permission. See Ref. [75]. 9 1994 I E E E . )

138

3. Acoustic Wave Sensors and Responses

Figure 3.53 Electrochemical cell employing flexural-wave device as one electrode. Top: Cell with RF source to drive platinum-covered flexural-wave device in bottom of cell. Middle: Limiting cell current vs time as both transducers on FPW device are switched on and off, producing standing waves, at the different drive amplitudes shown as parameter. Bottom: Square-law dependence of increase of limiting current upon transducer voltage that produces mixing of the liquid in the cell. (Reprintedwith permission.See Ref. [76]. 9 1991 IEEE.)

3.4 Flexural Plate-Wave (FPW) Devices

139

electrode and so reduces the current limitation caused by relatively less effective mass transport by diffusion. An approximate analysis of FPW fluid pumping [74,75,78] indicates that acoustic streaming accounts well for the observed pumping speeds observed in the experiments with water and the marker spheres. The transport of granular particles is believed to result from pumping of the ambient air. In addition, the FPW wave intensity and relatively low frequency favor the production of sonochemical effects [79].

Applications

3.4.3.3

The ability to include FPW pumping and mixing along with sensing in microfabricated devices suggests that one might realize useful "microflow systems.,' The simplest such system might be a compact planar micropump included in the package with an integrated circuit to circulate coolant over the chip so as to reduce the tendency for hot spots to occur and degrade circuit performance. Another example is including an FPW micropump as a gas sampler in an FPW vapor sensing module, which could be used as an air-quality monitor or for sensing accidental release of toxic vapors in an industrial plant. Figure 3.54 shows schematically a generic microflow system that might be used to synthesize fine chemicals from reactants on demand, or to produce individual doses of a reconstituted lyophilized drug for medical treatment. A multiPROCESSING CHAMBER PARTICI~ TR'ANsF~RT"L--ih~ INPUT PUMP #1. I

|

i |

iJ

in

i

n

i

LIQUID INPUT PUMP #I |

IMI~R'I PROCESS SENSOR I HEATER ]

Figure 3.54 Conceptual view of integrated microflow system employing FPW pumps, mixer, process sensor and insonicator to produce ultrasound-assisted chemical reactions. Heater would be deposited metal or polysilicon meanderline formed on a surface of the chamber.

140

3. Acoustic Wave Sensors and Responses

plicity of input pump sections could be employed to bring a number of reactants together. In the processing chamber, FPW elements could be used to mix reactants or solvents and solutes. Other FPW elements could monitor liquid density and viscosity. The insonicator indicated could be used to produce sonochemical effects, such as in situ cell lysis as a first step in biochemical analysis of DNA [80] and access to alternative reaction pathways [81]. The mixing produced by a propagating flexural plate wave could be used to augment the effect of diffusion in bringing reacting ions to an electrode in an amperometric chemical sensor; the result would be to reduce the response time and effectively increase the signal-to-noise ratio for the sensor. As an indication of the magnitude of the improvements that ultrasonic mixing could bring, note that researchers working with an ELISA immunoassay found that insonication with bulk ultrasonic waves could produce increases in the rate of reaction of more than 500 times [82].

3.5

Comparison of Acoustic Sensing Devices

It may be helpful to close this chapter by discussing what the TSM, SAW, APM, and FPW sensors have in common and in what ways they differ. We will also describe several additional types of acoustic sensors.

3.5.1 3.5.1.1

R E C E N T L Y INTRODUCED ACOUSTIC SENSORS Thin-Film Compressional Bulk-Wave Sensor

As an outgrowth of research on piezoelectric filter elements incorporated into high-frequency integrated circuits, Lakin and co-workers [82a] have realized a thin-film compressional bulk-wave resonator that functions as a gravimetric sensor. Like the conventional TSM, this resonator is operated at the frequency for which its thickness is one-half a wavelength. The piezoelectric film is chemicalvapor-deposited aluminum nitride, having a thickness of approximately four micrometers, formed on a silicon or gallium arsenide wafer that is subsequently etched to leave an unsupported resonant diaphragm. Resonant frequencies near 1 GHz result. Because the membrane vibrates in a compressional mode, rather than shear as in the TSM device, this sensor is most likely to be useful in a gaseous rather than liquid ambient. A gravimetric sensitivity, Sin, of 555 cm2/g has been reported [83] for one of these 1 GHz devices; this value is in good agreement with what one would predict from the analysis cited in Section 3.5.2 below.

3.5 Comparison of Acoustic Sensing Devices 3.5.1.2

141

Surface Transverse Wave (STW) Sensor

In this device, a thin film or periodic grating on the surface of the crystal slows the wave and prevents radiation of energy into the interior of the crystal [84]. With proper choice of crystal orientation, a purely transverse particle motion at the surface can be obtained, permitting the sensor to operate successfully in liquids.

3.5.1.3

Love Wave Sensor

Another approach to obtaining a sturdy sensor substrate and a transverse particle motion to permit operation in liquids is the use of a structure that supports Love waves [85,86]. These waves are guided in a layer that is thin compared with the wavelength on a thick substrate. With the proper choice of materials one may also be able to achieve a significant degree of temperature compensation in this device.

3.5.1.4

Thin-Rod Flexurai.Wave Sensor

Just as flexural waves can propagate at low speeds in a plate whose thickness is much less than the wavelength, a low-speed flexural wave can propagate in a cylindrical rod whose diameter is much smaller than the wavelength [87]. Because of the low wave speed, operation as a gravimetric sensor in liquids is possible, as with the flexural plate-wave sensor. The gravimetric sensitivity for this sensor is typically Sm = -1/(2PFa), where a is the radius of the rod.

3.5.2

COMPARISON OF GRAVIMETRIC ACOUSTIC SENSORS

The gravimetric sensitivity, Sin, of each of our four main sensors can be expressed in a simple form that permits easy comparison of device sensitivity. The key is to focus on the dimensions of the "active region" of each device where wave energy is present. For the TSM, APM, and FPW devices, the dimension, d, of the active region is the device thickness. In the TSM resonator, the thickness, d, must be an integer multiple, n, of the wavelength, A: nA dTSM ......... 9 2

(3.89)

In the SAW, APM, and FPW devices, the wavelength is determined by the spatial periodicity of the interdigital transducer, while the thicknesses of the APM

142

3. Acoustic Wave Sensors and Responses

and FPW plates are chosen for mechanical strength or ease of handling. In the SAW device, the wave energy is concentrated within a wavelength or less of the surface, even though the SAW substrate may be tens or hundreds of wavelengths thick. It has been shown [87,88] that one can start from traditional expressions for the gravimetric sensitivities of these four devices and derive the simple approximate expressions for gravimetric sensitivity that are given in Table 3.5.1. The reader should consult the reference for the TSM, SAW, and FPW derivations, or Section 3.3.2 for the corresponding APM result. To interpret these expressions, recall that the gravimetric sensitivity Sm is defined as Sm -- ~ m \ Vp ]

(3.90)

where Avp = change of wave velocity from its unperturbed value, vp, produced by adding a mass per unit area, Am, onto the surface of the device. For delayline oscillator sensors in which Am causes a frequency shift, Af, from the unperturbed resonant frequency f, one can show that (Af/f) = (Av/v), and so Sm = -~m

"

(3.91)

By focussing attention on wavelength rather than frequency, one arrives at the simple expressions for gravimetric sensitivity tabulated at the left side of Table 3.5.1. The expressions show that to obtain high sensitivity for the TSM and SAW, one must reduce the operating wavelength; this can be done by increasing the operating frequency, since A = Vp/f. In the TSM, mechanical fragility limits the amount by which one may reduce the wavelength, since the plate thickness is related to the wavelength by drsM = nM2. (Overtone operation at odd multiples of a half-wavelength, or use of deposited thin-film resonators permits making moresensitive devices.) In the APM and FPW, higher sensitivity is obtained by reducing the plate thickness. Fragility sets a limiting minimum thickness of perhaps 100 micrometers for APMs made from mechanically lapped and polished oriented single-crystal material. In contrast, the FPW may be made by thin-film depositions, and so the thickness of its plate can even be less than one micrometer. Reducing the FPW plate thickness reduces the thickness-to-wavelength ratio, if the wavelength is held constant, and so reduces both the wave velocity (Figure 3.40) and the operating frequency. The gravimetric sensitivity is only one factor that affects the minimum added

Table 3.5.1

Gravimetric Sensitivities of Acoustic Sensors

Experimental Devices

Sensor Bulk-wave

Theoretical Sensitivity Sm -2/pA

or

Experimental

Calculated

Frequency (MHz)

Sm Value (cmZ/g)

Sm Value (cmZ/g)

AT-cut quartz

6

-14

-14

97

- 130 a

- 130b( - 133) c

97

--9.5 (zeroth

- 9 . 3 (zeroth

TSM

resonator Surface-wave

n/pd

Operating

Device Description

-K(tr)/pA

ST-cut S A W delay line

device (assumed elastically isotropic) Acoustic plate-mode

- 1/pd

mode); - 19.4

APM

device Flexural plate device

ST-cut quart

- 1/2pd

mode); - 18.6

ZnO-on-silicon

4.7

(higher modes) -442 d

(higher modes) -450 c

nitride FPW

2.6

-990 d

-951 e

delay line aExperimental value determined by deposition of nickel film. bExpected value for mass loading alone. CMass-loading value calculated for SAW in isotropic solid using from Sm= K(tr)/pA assuming Poisson's ratio tr = 0.35. dLiquid loading experiments verified functional dependences and gave value of M, membrane mass per unit area; experimental value of Sm is then Sm= - 1/2M. eValue of M is based on composite membrane thickness and densities. Sm is then given by Sm = -1/2M. (Reprinted with permission. See Ref. [88]. 9 1991 American Institute of Physics.)

Table

Qualitative Comparison of Acoustic Sensors

3.5.2

ii

Device p.t

4~

TSM SAW

Sm

Temperature Stability Uncoated

Motion at Surface of Device

Low Med

High High (Med) a

Transverse Normal and Transverse Transverse

APM

Low-

High

FPW

Med High

Med

i

Normal and Transverse i

i

i

ii

i

Wave Velocity Relative to that Liquid Fast (V > Cl) Slow (v < cD

Immersible?

Fast Fast

Yes

Low

Med

Discrete

R

No

Med-High

High

Discrete or

D or R D D or R

Frequency

of Operation

Mechanical Strength

Discrete or Multiple Fabrication

Fast

Yes

Med-High

Med

Multiple Discrete

Slow

Yes

Low

Low-Med b

Multiple

,

i

i

i

Delay-line or R_esonator

i

aST-quartz is a highly temperature stable single-crystal SAW cut. SAWs made with piezoelectric films deposited on silicon or other substrates typically have lower temperature stability. bFPW devices utilize thin membranes that are mechanically rugged if their transverse dimensions are not too large.

References

145

mass that these sensors can detect. Other important factors that are dealt with in later chapters are the instabilities (noise) of the device in its operating condition - - bare, coated with a sorptive layer, in contact with a liquid - - and the noise contributed by the associated electronic measurement equipment.

3.5.3

Q U A L I T A T I V E C O M P A R I S O N OF A C O U S T I C S E N S O R S

Table 3.5.2 summarizes qualitatively the characteristics of the four sensor families discussed. The reasons for many of the entries should be apparent from the preceding discussion. Additional points to note are: (1) The thermal stability of any of the devices made from temperaturestable crystal cuts is degraded considerably when the device is coated with a polymeric film used for vapor sorption. Contact with a liquid may also introduce temperature variations that affect the short-term noise of the entire system. (2) The devices whose particle motions are transverse only, or whose phase velocities are lower than the speed of sound in the liquid, can be immersed in a liquid without suffering excessive radiative loss. (3) A high frequency of operation may lead to a high gravimetric sensitivity, but at the expense of more costly electronics. In viscosity sensing, the higher the operating frequency the lower the maximUm viscosity that can be sensed. (4) Discrete devices can, of course, be connected in arrays to obtain better selectivity or higher accuracy. Devices fabricated concurrently may have more similar characteristics than discrete devices made at different times, and so be better suited for use in arrays.

References 1. 2. 3. 4.

Sauerbrey, G. Z. Phys. 155, 206-222 (1959). Numura, T. and Minemura, A. Nippon Kagaku Kaishi, 1621 (1980). Konash, P. L. and Bastiaans, G. J. Anal. Chem. 52, 1929-1931 (1980). Tiersten, H. In Linear Piezoelectric Plate Vibrations; Plenum: New York, Chap. 10 (1969). 5. Rayleigh, Lord Proc. London Math. Soc. 17, 4-11 (1885). 6. Benes, E. J. Appl. Phys. 56, 608 (1984). 7. Granstaff, V. E. and Martin, S. J. J. Appl. Phys. 75, 1319-1329 (1994).

146

3. Acoustic Wave Sensors and Responses

8. Rosenbaum, J. F. Bulk Acoustic Wave Theory and Devices; Artech: Boston, Sect. 10.5 (1988). 9. Martin, S. J.; Granstaff, V. E.; and Frye, G. C. Anal. Chem. 63, 2272-2281 (1991). 10. Reed, C. E.; Kanazawa, K. K.; and Kaufman, J. H. J. Appl. Phys. 68, 1993-2001 (1990). 11. Mecea, V. and Bucur, R. V. Thin Film Solids, 60, 73-84 (1979). 12. Granstaff, V. E. and Martin, S. J. J. Appl. Phys. 75, 1319-1329 (1994). 13. Cady, W. G. Piezoelectricity; McGraw-Hill: New York, 1946. 14. Martin, S. J.; Frye, G. C.; Ricco, A. J.; Senturia, S. D. Anal. Chem. 65, 2910-2922 (1993). 15. White, F. M. Viscous Fluid Flow; McGraw-Hill: New York (1974). 16. Glassford, A. P. M.; J. Vac. Sci. Technol. 15, 1836-1843 (1978). 17. Martin, S. J.; Frye, G. C.; and Wessendorf, K. O. Sensors and Actuators A44 209-218 (1994). 18. Kanazawa, K. K. and Gordon II, J. G. Anal. Chem. 57, 1770-1771 (1985). 19. Muramatsu, H.; Tamiya, E.; Karube, I. Anal. Chem. 60, 2142-2146 (1988). 20. Beck, R.; Pittermann, U.; Weil, K. G. Ber. Bunsenges. Phys. Chem. 92, 1363-1368 (1988). 21. Yang, M.; Thompson, M. Anal. Chem. 65, 1158-1168 (1993). 22. Tiean, Z.; Liehua, N.; Shouzhou, Y. J. Electroanal. Chem. Intelfacial Electrochem. 293, 1-18 (1990). 23. Ballato, A. IEEE Trans. Sonics UItrason. SU-25, 185-191 (1978). 24. Bruckenstein, S.; Shay, M. Electrochimica Acta 30, 1295-1300 (1985). 25. Schumacher, R. Angew. Chem. Int. Ed. Engl. 29, 329-343 (1990). 26. Beck, R.; Pitterman, U.; Weil, K. G. J. Electrochem. Soc. 139, 453--461 (1992). 27. Beck, R.; Pittermann, U.; Weil, K. G. Ber. Bunsenges. Phys. Chem. 92, 1363-1368 (1988). 28. Martin, S. J.; Frye, G. C.; Ricco, A. J.; Senturia, S. D. Anal. Chem, 65, 2910--2922 (1993). 29. Schlichting, H. Boundary-Layer Theory; McGraw-Hill: New York, 1979; Ch. 11. 30. Rajakovic, L. V.; Cavic-Vlasak, B. A.; Ghaemmaghami, V; Kallury, M. R. K.; Kipling, A. L.; Thompson, M. Anal. Chem. 63, 615-621 (1991 ). 31. Kipling, A. L.; Thompson, M. Anal. Chem. 62, 1514-1519 (1990). 32. Rajakovic, L. V.; Cavic-Vlasak, B. A.; Ghaemmaghami, V; Kallury, M. R. K.; Kipling, A. L.; Thompson, M. Anal. Chem. 63, 615-621 (1991). 33. Thompson, M.; Arthur, C. L.; Dhaliwal, G. K. Anal. Chem. 58, 1206-1209 (1986). 34. Thompson, M.; Dhaliwal, G. K.; Arthur, C. L.; Calabrese, G. S. IEEE Trans. Ultrason. Ferroelec. Freq. Contr. UFFC-34, 127 (1987). 35. Mecea, V. M. Sensors and Actuators A, 41-42, 630-637 (1994). 36. Mecea, V. M. Sensors and Actuators A, 40, 1-27 (1994). 37. Haardt, H. Dissertation, Universit~it Kiel (1971).

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38. Martin, S. J.; Wessendorf, K. O.; Gebert, C. T.; Frye, G. C.; Cemosek, R. W.; Casaus, L.; Mitchell, M. A. Proc. of the 1993 IEEE International Frequency Control Symposium; IEEE: New York, 603-608 (1993). 39. Adamson, A. W. Physical Chemistry of Surfaces; Wiley: New York, p. 349 (1982). 40. Martin, S. J. and Frye, G. C. Ultrasonics Symposium Proceedings; IEEE: New York, 393-398 (1991). 41. Ramo, S.; Whinnery, J. R.; Van Duzer, T. Fields and Waves in Communication Electronics; Wiley: New York Sect. 1.18 (1965). 42. White, R. M. Proc. IEEE, 58, 1238-1276 (1970) 43. Datta, S. Sulface Acoustic Wave Devices; Prentice-Hall: Englewood Cliffs, NJ (1986). 44. Morgan, D. P. Sulface-Wave Devices for Signal Processing; Elsevier: New York (1985). 45. Frederick, D. K. and Carlson, A. B. Linear Systems in Communication and Control; Wiley: New York (1971). 46. Ristic, V. M. In Principles of Acoustic Devices; Wiley: New York, p. 127 (1983). 46a. Pfeifer, K. B.; Martin, S. J.; Ricco, A. J. "Surface Acoustic Wave Sensing of VOCs in Harsh Chemical Environments," Sandia Report, SAND93-0070, June 1993. 47. Slobodnik, A. J.; Conway, E. D.; Delmonico, R. T. Microwave Acoustic Handbook, Vol. IA. Surface Wave Velocities; National Technical Information Service, U. S. Dept. of Commerce (1973). 48. Martin, S. J. and Ricco, A. J. 1989 Ultrasonics Symposium Proc.; IEEE, New York, 621-625 (1989). 49. Ferry, J. D. Viscoelastic Properties of Polymers, 3rd ed; Wiley: New York, Ch. 1, (1980). 50. Martin, S. J.; Frye, G. C.; Senturia, S. D. Anal. Chem. 66, 2201-2219 (1994). 51. Landau, L. D.; Lifshitz, E. M. Theory of Elasticity, 3rd Ed.; Pergamon: New York, Ch. 1, (1986). 52. Tiersten, H. F.; Sinha, B. K. J. Appl. Phys. 49(1), 87-95 (1978). 53. Grate, J. W.; Snow, A.; Ballantine, D. S.; Wohltjen, H.; Abraham, M. H.; McGill, R. A.; Sasson, P. Anal. Chem. 60, 869-875 (1988). 54. Martin, S. J.; Ricco, A. J.; Niemczyk, T. M., Frye, G. C. Sensors and Actuators 20, 253-268 (1989). 55. Hou, J. and van de Vaart, H. Proc. IEEE Ultrasonics Symp.; Denver, CO, 573-578 (1987). 56. Ricco, A. J. and Martin, S. J. Appl. Phys. Lett. 50, 1474--1476 (1987). 57. Matheson, A. J. Molecular Acoustics; Wiley: New York, pp. 82-83, (1971). 58. Josse, F. Z.; Shana, A.; Radtke, D. E.; Kelkar, U. R.; Haworth, D. T. Electronics Letters 25, 1446-1447 (1989). 59. Niemczyk, T. M.; Martin, S. J.; Frye, G. C.; Ricco, A. J. J. Appl. Phys. 64, 5002-5008 (1988). 60. Lamb, H. Proc. Roy. Soc. (London), Ser. A, 93, 114 (1917).

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61. Viktorov, I. A. Rayleigh and Lamb Waves; Plenum: New York (1967). 62. Wenzel, S. W. Applications of Ultrasonic Lamb Waves, Doctoral Dissertation, EECS Department, University of California, Berkeley, CA (1992). 63. Grate, J. W.; Martin, S. J.; White, R. M. Anal. Chem., 65, Part I: 940A-948A; Part II: 987A-996A (1993). 64. Auld, B. A. In Acoustic Fields and Waves in Solids; Wiley: New York (1973). 65. Sze, S. M.Ed., Semiconductor Sensors; Wiley: New York (1994). 66. Muller, R. S.; Howe, R. T.; Senturia, S. D.; Smith, R. L.; White, R. M. Microsensors; IEEE Press: Piscataway, NJ (1991). 67. Nassar, A. A. and Adler, E. L. Proc. IEEE Ultrasonics Symp., 369 (1983). 68. Wenzel, S. W. and White, R. M. IEEE Trans. Electron Devices, ED-35, 735 (1988). 69. White, R. M. and Wenzel, S. W. U. S. Patent No. 5,189,914 (1992); U. S. Patent No. 5,129,262 (1992). 70. Personal communication, Jay Grate, Battelle Pacific Northwest National Laboratory. 71. Scholte, J. G. Mon. Not. Royal Astronom. Soc., Geophys. Suppl., 5:120 (1947). 72. Costello, B. J.; Wenzel, S. W.; White, R. M. Technical Digest, 7th International Conference on Solid-State Sensors and Actuators, Transducers '93, Yokohama, Japan, pp. 712-715 (7-10 June 1993). 72a. Eto, T. K.; CosteUo, B. J.; Wenzel, S. W.; White, R. M.; Rubiusky, B. J. Biomech. Eng., 115, 329-331 (1993). 73. Costello, B. J.; Wenzel, S. W.; Wang, A.; White, R. M. Proc. IEEE Ultrasonics Symp., 279 (1990). 74. Moroney, R. M.; White, R. M.; Howe, R. T. Appl. Phys. Lett., 59, 774 (1991). 75. Bradley, C. E. and White, R. M. Proc. IEEE Ultrasonics Symposium (1994). 76. Tsao, T. R.; Moroney, R. M.; Martin, B. A.; White, R. M. Proc. IEEE Ultrasonics Symposium, 937-940 ( 1991). 77. Nyborg, W. L. Acoustic So'earning, in Physical Acoustics, Mason, W. P. Ed., 2B, Academic Press 265, (1965). 78. Moroney, R. M.; White, R. M.; Howe, R. T. DSC-32, Symposium on Micromechanical Sensors, Actuators and Systems, ASME Winter Annual Meeting, 181-90 (1991 ). 79. Suslick, K. S. Ultrasound: Its Chemical, Physical, and Biological Effects; VCH Publishers: New York (1988). 80. Northrup, M. A.; Ching, M,; White, R. M.; Watson, R. Technical Digest, 7th International Conference on Solid-State Sensors and Actuators, Transducers '93, Yokohama, Japan, 924-6 (1993). 81. Mason, T. J. Ed., Chemistry With Ultrasound; Elsevier Applied Science: London (1990). 82. Chen, R.; Wenz, L.; Sizto, N. C.; Osoria, B. C.; Hsu, J.; Rodgers, R.; Litman, D. J. Clin. Chem., 30, 1446-1451 (1984). 82a. Lakin, K. M.; Wang, J. S.; Landin, A. R. Proc. 36th Ann. Symp. Freq. Coutr., 517-524 (1982). 83. Personal communication, Mark Porter, Iowa State University.

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Chapter 4

Materials Characterization

The field of materials science has grown dramatically in the past decade, with new materials being synthesized and/or developed for applications such as lubrication, corrosion protection, electronics, paints and coatings, and chemical separations. Many of these materials have complex properties quite different from those associated with simple "ideal" substances. Since the chemical and physical properties of a material determine its ability to meet the often stringent specifications required for a given application, characterizing the properties of materials plays a vital role in materials science. Thin film technology is an excellent example. Thin film materials are currently used in a wide variety of industrial applications. For example, thin films are used as protective or passivating layers [1-3], as conductive or photoactive (i.e., photoresist) layers [1], as dry lubricants [3], as catalysts [4], as gas separation membranes [5], and as optical layers [6]. Thin films can be formed by a variety of processes [ 1-8], including spraying, spin-coating, dip-coating, chemical vapor deposition (CVD), evaporation, and sputtering. To effectively optimize thin film properties, techniques to directly characterize thin film materials are critical. These techniques can be utilized as research and development tools to characterize new materials or, at the other extreme, as on-line probes of film properties during production. A major challenge in developing techniques for characterizing film materials is the limited amount of material present. For example, in a one-micrometer-thick film, there is only 10 -4 cm 3 of material for each cm 2 of film area. Thus, a 10cm 2 film has a volume of only one microliter and a mass on the order of one milligram. Many material characterization instruments do not have sufficient sensitivity to analyze these small volumes or masses [9]. In addition, those tech150 ACOUSTIC WAVE SENSORS

Copyright 9 1997 by Academic Press All righls of reproduction in any form reserved. ISBN 0-12-077460-7

4.1 Overview of Applications

151

niques with the required sensitivity (e.g., reflectance spectrometry, X-ray fluorimetry) have other disadvantages such as excessive cost, extensive sample preparation, long analysis times (no real-time monitoring), and restrictive sample environments (i.e., vacuum) [6,9]. Acoustic wave (AW) devices are ideally suited to thin film characterization due to their extreme sensitivity to thin film properties [10]. The sensitivity of AW devices to a variety of film properties (see Chapter 3), such as mass density, viscoelasticity and conductivity, makes them versatile characterization tools. The ability to rapidly monitor changes in device responses resulting from changes in thin film properties permits their use for monitoring dynamic processes such as film deposition, chemical modification (e.g., photo-polymerization, corrosion), and diffusion of species into and out of films. In this chapter, we explore the current and potential future applications of AW devices for materials characterization and process monitoring. Because of the limited mass of material that can be applied to the AW device surface, the majority of these applications deal with the chemical and physical characterization of thin-film properties. This thin film focus should not be thought of as a limitation of AW devices, but rather as a useful capability - - the direct measurement of properties of materials in thin-film form. Since material properties can depend on the physical form (e.g., film, bulk) of the material (see Section 4.3.1.3), AW devices are uniquely suited to directly characterize thin-film materials. These considerations also indicate that even though it is possible to use AW thin-film data to predict bulk material properties, such extrapolations should be performed with care.

4.1 4.1.1

O v e r v i e w of Applications C H A R A C T E R I Z A T I O N OF T H I N F I L M M A T E R I A L S

The development of AW thin-film characterization techniques has occurred largely because of the interest by various research groups in developing chemical sensors based on coated AW devices (see Chapter 5). Thus, many of the film characterization techniques described here were developed in an effort to characterize sensor coatings or to interpret the observed responses from AW chemical sensors in operation. As described in Chapter 3, mass detection limits for AW devices are typically at or below one ng/cm 2. These low detection limits translate into hundredths of a monolayer of atoms and film thicknesses of hundredths of nanometers. This

152

4. Materials Characterization

sensitivity permits quantitative detection of submonolayer mass changes in thin films formed on AW devices. This extreme mass sensitivity can be used to advantage in the characterization of film properties such as film thickness (Section 4.4.1) and surface area and pore size distribution (Section 4.3.1.2). In addition, it is useful for real-time monitoring of processes such as film deposition (Section 4.4.1), materials modification (Sections 4.4.2 and 4.4.5), corrosion (Section 4.4.3), and diffusion (Section 4.2.2). It can also be used to monitor adsorption at surfaces from both gases and liquids (Section 4.3). Using AW devices to monitor dynamic processes such as diffusion and corrosion can dramatically reduce the time required to quantify these processes. For example, as discussed in Section 4.2.2, diffusion equilibration times typically increase with the square of the diffusional length. For a thin film, this length scale, the film thickness (h), is very small. This enables the quantification of diffusion coefficients as low as 10-15 cm2/sec in less than one day, whereas months would be required using many conventional techniques that use thick films or bulk samples. For corrosion monitoring, the dramatic decrease in mass detection limits obtainable using coated AW devices, as compared with conventional balances and sample coupons, allows detectable mass changes to be achieved in minutes or hours rather than days or months (Section 4.4.3). AW device sensitivity to viscoelastic parameters and electrical properties can be used to advantage in some film characterization techniques. In these situations, a comparison of the AW device response to a model of the AW/thin film interaction is often crucial to the effective evaluation of thin film parameters. These additional interaction mechanisms typically involve changes in both the wave velocity and the wave attenuation for SAW, APM and FPW devices, and changes in both resonant frequency and admittance magnitude in TSM devices. In contrast, mass loading does not contribute to wave attenuation or decreases in admittance since moving mass involves no power dissipation (see Chapter 3). Having detectable changes in two sensor responses allows the amount of information that can be extracted regarding film properties to be increased, since agreement between both responses and predictions from the model aids in the discriminating power of the characterization technique. A demonstration of this can be found in the ability to determine viscoelastic parameters based on monitoring both sensor responses during a temperature cycle for a polymer-coated device (Section 4.2.1.2). These responses are also useful in elucidating the changes occurring during such processes as polymer cross-linking (Section 4.4.2), or the absorption of species in polymers (Section 4.2.1.3).

4.1 Overview of Applications 4.1.2

153

CHARACTERIZATION OF FLUID PROPERTIES

Another area of materials characterization involves characterizing the properties of a contacting fluid. Since the fundamentals of acoustic wave/liquid interactions are covered in detail in Chapter 3, this topic will not be repeated here. However, it seems relevant to provide a brief summary of some of the fluid properties that can be measured. Since SAW devices are excessively damped with liquids, these characterization techniques generally involve only APM, FPW, and TSM devices. Once again, the utility of using two sensor responses can be important. Two key properties that can be probed are viscosity (7/) and density (p). As discussed in Sections 3.1.5 (TSM), 3.3.3 (APM), and 3.4.2.4 (FPW), the responses are often proportional to the square root of the product (pr/); data showing trends vs (pr/) 1/2 have been reported using TSM (see Figure 3.10) [11-15], APM (see Figure 3.35) [16], FPW (see Figure 3.48) [17-19], and Love wave devices [20]. In some cases, one property is held constant to probe the other, for example probing viscosity at constant density [18,21 ]. In many cases, simple trends are shown such as the maximum in viscosity at intermediate concentrations of water/ethanol mixtures [15,22,23], or increasing response with increasing sugar content [22]. It has been observed with TSM devices that rough surface features result in liquid trapping and a term proportional to/9 and an ability to separate out p and 7) (see Section 3.1.6 and Figure 3.11) [24,25]. Similarly, since FPW devices have a velocity dependence proportional to density (see Section 3.4.2.2) and velocity and loss terms proportional to (pr/) 1/2 (see Section 3.4.2.4), it may be possible to use FPW device responses to characterize both p and r/simultaneously. Wave velocity in a fluid, which is a strong function of density, has been probed using longitudinal-mode resonators for analyzing gases (e.g., pressure or changes in composition) [26-28]. SAW devices have also been used with thin liquid layers and a reflector plate for probing liquid properties (e.g., changes in density due to changes in salt concentration) [29,30]. Both of these devices rely on probing the reflected compressional wave, and depend on the separation of the AW device and the reflector. Acoustoelectric interactions enable solution electrical properties to be probed with AW devices. It should be noted that these acoustoelectric interactions can be "shorted out" using a conductive (e.g., metal) layer between the substrate and the solution for APM and FPW devices. Similarly, for TSM devices, if the grounded electrode is placed in contact with the solution, no acoustoelectric effect should be present. The key parameter that has been monitored is solution conductivity. For example, measurements of AW responses vs conductivity have been reported using TSMs [ 11,15,31,32] and APMs (see Figure 3.36) [ 16,33-35].

154

4. Materials Characterization

The APM interaction is described in Section 3.3.4 while the TSM interaction is probably due to parasitic conduction through the solution. In one APM study, porous thin films were found to have an effect on conductivity trends, proposed in the study to be due to changes in solution conductivity in the porous regions [35]. Another explanation could be changes in the dielectric constant in the region of the film, since the dielectric constant has an effect on conductivity responses for APM devices (see Section 3.3.4) [16,34]. In another study, a TSM was used with a conductivity electrode to make a sensitive probe of conductivity that had little dependence on solution viscosity and density [36]. In addition, the parasitic contribution to the static capacitance in TSM devices has been correlated with solution dielectric constant [11,12]. Some sensors for extrinsic properties have also been demonstrated. For example, sensors for mass-flow rate using SAW [37,38] and APM [38] devices combined with either on-chip resistors [38] or acoustic absorbers [37] for device heating have been demonstrated. These devices use the temperature sensitivity of the devices to probe temperature changes induced by convective cooling by the flowing gas. Another investigation showed that the magnitude and direction (relative to the wave velocity) of an imposed shear stress could be monitored with a SAW device. This was proposed to be useful in developing a sensor for local and global turbulence [39]. Finally, a capacitance-dependent TSM sensor system has been demonstrated for measuring liquid volumes in the 0-1 ml range [40]. The demonstrations cited above illustrate how AW devices can be used to probe intrinsic and extrinsic fluid properties. This capability can be useful for providing in-situ probes of critical solution properties such as viscosity, density, and conductivity. This capability should prove useful in the monitoring of process streams or critical fluids (e.g., automotive oil condition monitoring [41 ]).

4.2

Characterization of P o l y m e r s

A polymer can be defined as a compound consisting of a large number of repeating units, called monomers. These monomers are joined together by covalent bonds to form a long chain. The degree of polymerization is defined as the number of repeating units in the chain. The properties of the polymer depend on the overall size of the polymer chain (i.e., average molecular weight) and on the inter- and intra-molecular forces that hold the polymer together [42--44]. The intramolecular forces consist of the covalent bonds that join the repeat units into chains, and any covalent bonds that may join adjacent chains together (crosslinkages). In addition, the polymer chains are held together by a variety of in-

4.2 Characterization of Polymers

155

termolecular forces, including hydrogen bonding, dipole-dipole interactions, and London dispersion forces resulting from the synchronization of electron motion in the interacting atoms (see Chapter 5 for a discussion of chemical interactions). The physical and chemical properties of the polymer depend on the types and relative strengths of these inter- and intra-molecular interactions. The sheer volume of polymeric material produced has increased dramatically in the last decade and, insofar as the chemical and physical properties of these materials can be modified, the number of applications for polymers has expanded [ 1,2,5]. In general, the polymer properties of interest can be categorized as diffusion/permeation properties or as mechanical (e.g., viscoelastic) properties. The measurement of diffusion/permeation properties is straightforward when diffusion of a species into a polymer film produces a simple mass-loading effect. Experimental determination of these properties using AW devices will be discussed in Section 4.2.2. In addition to the mass-loading effect, the presence of dispersed molecules in a polymer has a plasticizing effect, inducing changes in viscoelastic properties, as described in Section 4.2.1.3. Measurement of these viscoelastic properties is more complex. There are a number of texts that provide an excellent discussion of the viscoelastic behavior of polymers, including theoretical models to explain such behavior [42-44]. While an in-depth discussion of these models and their ramifications is beyond the scope of this work, a brief summary of viscoelastic behavior is supplied below.

4.2.1

VISCOELASTIC PROPERTIES

The viscoelastic properties of a polymer can be described in terms of how the polymer deforms in response to an applied stress. Elasticity refers to the ability of a material to return to its original shape after it has been stressed. Elastic behavior implies a linear relationship between stress, T, and strain, S, (T oc S). Viscosity is a measure of the flow resistance of the polymer or polymer solution. Viscous behavior implies a linear relationship between shear stress and the rate of strain (T oc OS/Ot). Rigid materials tend to display elastic behavior, whereas fluid or soft materials display viscous behavior. In many polymers, a combination of elastic and viscous responses arises as a direct consequence of the chain structure, hence the term "viscoelastic" properties. The concepts of stress, strain and displacement have already been introduced in Chapter 2 in describing the propagation of acoustic waves in an elastic medium, and in Chapter 3 in describing the various sensing mechanisms. The two deformation modes of interest are elongation and shear deformation. Elongation refers to the change in length

156

4. Materials Characterization

(in a given direction) of a polymer sample upon application of a longitudinal unit stress (i.e., stretching or compressing). Shear deformation refers to the deformation behavior of the sample under the application of a lateral force on one surface. How a polymer behaves under the force of an applied stress depends on a number of variables, including temperature, pressure, and the time frame (i.e., frequency) and nature (i.e., shear vs elongation) of the stress. As described in more detail in Section 3.1.8, the viscoelasticity of a polymer can be described by a complex modulus. The modulus is defined as the stress associated with a unit strain, and has units of force/unit area (dynes/cm2). It can be thought of as the stiffness or rigidity of the polymer, and is related to the inter- and intra-molecular forces at work within the polymer. In general, polymer film/acoustic wave interactions are dominated by the shear component of displacement (see Chapter 3). Thus, it is the shear modulus which can be effectively probed with AW devices. This shear modulus can be represented by G = G' + jG" where G', the storage modulus, is associated with energy storage and release during the periodic deformation associated with the oscillating stress, and G", the loss modulus, is associated with the dissipation of energy, usually as heat. The modulus depends on the molecular structure of the polymer, the average molecular weight, the temperature, and, in general, the rate (frequency) of applied shear stress. The interchangeability between temperature and strain rate in determining the modulus was first described by Williams, Landel and Ferry [45] and became the basis of the so-called "time-temperature superposition principle." This dependence can be explained in terms of the molecular motions in the polymer chain by examining the mechanism by which a polymer reacts to an applied stress. When the polymer is deformed on a time scale, Ts, that does not allow significant thermal motion of polymer chains with respect to each other (i.e., rotational freedom of the polymer chains is limited), the polymer behaves as a rigid or "glassy" material. The glassy state is characterized by large shear moduli, on the order of 101~dynes/cm 2. As temperature increases, thermal energy in the system becomes sufficient to overcome the molecular forces, permitting free rotation around the bonds of the polymer chain. This additional rotational freedom is manifested as a softening or "relaxation" of the polymer, and the polymer is described as an "elastomer." Modulus values of elastomers are on the order of 107 dynes/cm 2. The temperature at which the transition from the glassy to the elastomeric state occurs is called the glass transition temperature Tg. Another way to look at this is to consider that the polymer exhibits a characteristic relaxation time, ~'. If the stress is applied for a time period Ts that is much

4.2 Characterization of Polymers

157

shorter than the relaxation time (Ts < < ~'), polymer chains do not have time to move with respect to each other and the polymer behaves as an elastic solid characterized by a stiffness/x. As temperature increases, z decreases until Ts > > ~', at which point thermal motion allows (uncrosslinked) chains to move with respect to each other and the polymer behaves as a viscous liquid characterized by a viscosity r/. Tg can be defined as the temperature where Ts ~ I", at which point the polymer deforms both elastically and viscously, giving rise to viscoelastic behavior. It should be noted that Tg for an amorphous, glassy polymer is not the same as the melting temperature Tm for a semi-crystalline polymer. Both glassy and semi-crystalline materials are characterized by high modulus values, yet the two transition temperatures are associated with distinctly different phenomena. The former (Tg) is a relaxation, or second-order transition, and exhibits the time (frequency) dependence discussed above. In addition, this Tg transition generally occurs over a significant temperature range (i.e., is not abrupt) due to heterogeneities in the polymer and the fact that chain motion is an activated process. The latter (Tin) arises because of a chemical phase change, or first-order transition, and is independent of frequency. Melting transitions typically occur only in polymers having chains sufficiently linear to allow "packing" in a regular crystalline-like manner. Just like other melting transitions (e.g., ice to water), the temperature at which the transition occurs can depend on whether the temperature is being raised or lowered to induce the transition. This is due to the fact that nucleation of the crystalline phase during cooling does not occur until a lower temperature (i.e., supercooling) as a result of the high curvature of a newly nucleated phase [46]. In polymers, these melting transitions may not always occur at a single temperature. Instead, the presence of different molecular chain structures can result in multiple transitions, often denoted by Greek letters [45]. Even though these transitions are different in many ways, as demonstrated below, the way in which acoustic energy interacts with polymeric materials permits us to use AW devices to probe changes in polymer film viscoelastic properties associated with these transitions. It should be emphasized up front, however, that evaluating the viscoelastic properties (e.g., modulus values) requires an ability to effectively model the film displacement profiles in the viscoelastic layer. As described in Section 3.1.8, the film displacement effects are dictated by the phase shift, ~b, across the film. Since ~b depends on film thickness, perturbations in acoustic wave properties due to changes in viscoelastic properties (e.g., during polymer transitions) do not typically depend simply on the intrinsic polymer properties. This can lead to erroneous predictions if the film

158

4. Materials Characterization

dynamics are not taken into account. However, if these dynamics are effectively modeled, the AW device response can be used to quantitatively evaluate the shear modulus values (see Section 4.2.1.2).

4.2.1.1

Determination of Transition Temperatures

The attenuation and velocity of acoustic energy in polymers are very different from those in other materials due to their unique viscoelastic properties. The use of ultrasonic techniques, such as acoustic spectroscopy, for the characterization of polymers has been demonstrated [47,48]. For AW devices, the propagation of an acoustic wave in a substrate causes an oscillating displacement of particles on the substrate surface. For a medium in intimate contact with the substrate, the horizontal component of this motion produces a shearing force. In such cases, there can be sufficient interaction between the acoustic wave and the adjacent medium to perturb the properties of the wave. For polymeric materials, attenuation and velocity of the acoustic wave will be affected by changes in the viscoelastic behavior of the polymer. Because of the oscillatory nature of the acoustic wave, probing of polymer viscoelastic properties using AW devices is analogous to the high rate/short time scale probing of polymers mentioned previously. The wave period, which is the inverse of the AW frequency, determines the time scale of the applied strain. Wave attenuation and velocity, or resonant amplitude and frequency, can be monitored at a relatively fixed frequency (rate) while scanning the temperature. The use of SAW devices to identify Ts and Tm for a variety of polymers has been reported. Both attenuation (output amplitude) [49,50] and velocity (frequency) [51,52] changes have been monitored. In two of these studies, relatively thick sample films were tested [50,51 ], and the results were consistent with Ts and Tm values from other experimental methods, such as differential scanning calorimetry (DSC). (It should be noted that the slow processes (large Ts) used in techniques such as DSC result in these techniques probing the static or low-frequency Tg values.) An example of this type of trend is shown in Figure 4.1 for a film being pressed onto the surface of a SAW device using a clamping system. No increase in T8 was observed, indicating that the measured attenuation/velocity changes were the result of changes in the AW/polymer coupling due to increased adhesion of the polymer to the AW device surface. This transition from a poorly coupled film, which has a correspondingly low perturbation in wave amplitude, to a film coupled to the acoustic wave, resulting in significant atten-

4.2 Characterization of Polymers

159

1.0

>

O O

E

V

uJ Q

0.8

l" ..J 11. ram=

'< ,.J

0.6

O

.< Z

0 m

U)

0.4

::) n_ I::)

0

Tg=75* 0.2

O 0.0

!~_ 30

I 60

C~o6itie~J 90

, I ...... 120

130

TEMPERATURE (oc) Figure 4.1 Glass transition detection using a polyethylene terephthalate film clamped o n t o the surface of a SAW device. (Reprinted with permission. See Ref. [50]. Copyright 9 1979 American Chemical Society.)

uation of the wave, occured when the polymer became softer as the temperature is raised above the static (low-frequency) Tg. Another technique for evaluating the static Tg uses an indirect approach that probes relatively slow processes. King [53] described how changes in diffusion rates (as indicated by the time to sorb 90% of the final sorption value) and solubility values could be used to probe the change from a glassy (slow diffusion) to a rubbery (diffusion several orders of magnitude faster) state. Using polystyrene on TSM devices, King showed that Tg values in agreement with those

160

4. Materials Characterization

obtained by other techniques could be determined, as well as showing that the transition occurred over a temperature range of about 20~ (interpreted as being due to the sample having a distribution of molecular weights). TSM-determined partition and diffusion coefficients vs temperature have also been used to probe transition temperatures in synthetic lipid multibilayer films [54]. Other studies have demonstrated the utility of FPW devices to identify both the static and dynamic Ts of polymer films simultaneously [55-57]. As shown in Figure 4.2, the static (low-frequency) Tg Was observed as a change in the slope of the acoustic velocity vs temperature curve; the change in slope was interpreted as a change in the rate of polymer expansion at the polymer static (low-frequency) Tg. The dynamic (or frequency-dependent) Ts was identified as a minimum in a plot of the acoustic wave amplitude vs temperature (indicative of a maximum in the loss modulus G"). These basic trends are consistent with results using bulk transducers to generate longitudinal waves at 2.5 MHz in polymer disks combined with a technique for measuring the thickness of the polymer disk with temperature [47]. For the one polymer (poly(vinylacetate)) where both the static and dynamic transitions were observed, the static transition was found to be about

4780

%

4760 A

N "I"

>.

4740

tO Z uJ =)

o

uJ

4720

u..

4700

4680

........ 0

1 10

I 20

.. I 30

..

TEMPERATURE

I 40

.....

I .. 50

I 60

.

.

.

.

.

.

(~

Figure 4.2 Frequency vs temperature for a poly(t-butyl acrylate)-coated FPW device

showing a slope change at the static (low-frequency) Ts. 9 1992 American Chemical Society.)

(Reprinted with permission. See Ref. [56].

4.2 Characterization of Polymers

161

60~ lower than the dynamic transition probed by the 5 MHz FPW device. Previous SAW studies have also reported detecting the dynamic Tg using thin films sprayed or cast on the SAW device surface. The observed Tg values (indicated by trends in the frequency response) were reported to be increased by ~50~ compared to DSC or other low-frequency techniques [50,52]. These results, indicating Tg values at AW frequencies significantly higher than the static Tg values, are consistent with the time-temperature superposition principle. To enable probing of the frequency dependence, one SAW study used a multi-frequency SAW device (i.e., a single ST-quartz substrate bearing five different SAW delay lines) to probe the temperature-dependent behavior of polymer films [58]. Multifrequency probing of viscoelastic properties has also been performed using TSM devices probed over many harmonics using a network analyzer [59,60]. The minimum amplitude (maximum attenuation) reported in the FPW study has also been observed during temperature ramps of polymer-coated SAW devices [61--65]. Examples of data for both velocity and attenuation are shown in Figure 4.3. In this study, it was determined that the observed trends were due to film resonance conditions (see Sections 3.1.8 and 3.2.7). This was strongly indicated by the fact that the temperature of the maximum attenuation decreased with increasing film thickness h; in fact, a more-than-60~ in the temperature of the maximum attenuation is demonstrated for only a three-fold increase in film thickness (0.44 to 1.37/zm). These results highlight the importance of considering film dynamics when investigating viscoelastic properties and transitions using AW devices [61,63,64]. Regarding the FPW work described above, it is important to consider whether this amplitude minimum is due to film resonance or if the films were thin enough (h about 0.5 to 1 /zm) that the phase shift ~b is much less than Ir/2 at the frequency of the FPW device (5 MHz). This latter condition would indicate that the observed amplitude minimum would represent the maximum in G" that occurs at the glass transition. FPW devices have the advantage for this application of high sensitivity at lower frequencies (smaller th values and an ability to stay in the acoustically thin realm). Thus, it appears that the film was acoustically thin for these FPW tests and that the responses are tracking changes in the film properties (i.e., h, G', and G"). This same question regarding film resonance is even more relevant for the earlier higher frequency SAW work [50,52], since film resonance results in frequency trends similar to those reported as being due to the glass transition. If film resonance effects are occurring in these studies, the reported Tg values would still be close to the actual Tg since it is the dramatic change in modulus values during the glass transition that would result in significant changes in the phase shift and the onset of film resonance. However, the

162

4. Materials Characterization 3.5

.....

~

. . I . . . . . . I. . . . . . . I . . . . . . I. . . . . . . I . . . . . .

[ PIB Film (~tm): 3.0 t_[_ , 0.44 ] 9 0.68

:

2.5

~,.,

9

4k)

2.0

1.o

..-'.--m

**

I ~/r

_ . f " _4)0 ,,=

/

&

"~

,'"

,*

-

9

AA A 9 9 0 oeo

an n

9r

aa""

.af

*

0.5 I

_ 9 .,,**,

,nul

t

/

4

~'*

I:]7

1.5[

I

9

'

/

.

AAAA***Q

]

~'Jbtm ' m - a L b ~ P ~ b qk 9

0.0

0

-1 9

,~

*~,

-2

I

. AA~ A

m

- - ~ & & A "

I ~0

~k%, A a.

"ml"

-3

"4 I

-4

"

9 4

.5

I

-20

0

,,

I

20

......

40

I

I

40

60

......

I

80

,

I

I

I00

120

Temperature (C) Figure 4.3 Attenuationand frequency vs temperature for 97-MHz SAW devices with various film thickness of polysobutylene (PIB). The maximum in attenuation and the sigmoidal frequency excursion are due to the onset of film resonance as the polymer softens with temperature. The temperature at which these AW trends occur depends on the thickness of the coating (thicker coatings yield lower temperatures). (Reprintedwith permission. See Ref. [61]. 9 1994 American Chemical Society.)

4.2 Characterization of Polymers

163

specific Tg value, and the interpretation that the AW trends are directly indicating changes in G' and G", would not be accurate if film resonance is causing the observed trends. This is clearly shown by the results in Figure 4.3.

4.2.1.2

Extraction of Storage and Loss Moduli

In this section we will describe how a proper accounting for film dynamics, based on a model of the thin-film/acoustic-wave interactions, can be used to quantitatively evaluate the shear modulus values as a function of temperature. As described in Section 3.1, an equivalent-circuit model can be used to relate the measured TSM electrical characteristics to the elastic properties, density, and thickness of a polymer film coating the device. Consequently, measurements made with polymer-coated TSM devices can be used to extract the shear elastic properties of the film. In order to separate properties of the film from those of the crystal, admittance-vs-frequency (Y-vs-f) measurements are made on the TSM resonator before and after deposition of a film. Fitting the equivalent-circuit model to measurements made on the uncoated device is crucial, allowing extraction of all of the circuit elements except Ze ~ the impedance element arising from the film (Figure 3.7). Once the uncoated resonator has been characterized, the impedance element Ze arising from a film coating. If measurements at only a single harmonic are used, film thickness and density must be known to extract G' and G". Admittance-vs-frequency measurements made at several temperatures on a polyisobutylene-coated TSM resonator were fit to the equivalent-circuit model of Sections 3.1.3 and 3.1.9 to determine values of G' and G" for the film [66]. These extracted values are shown in Figure 4.4, along with 5-MHz values obtained from the literature for polyisobutylene having an average molecular weight of 1.56 • 106 [44]. We note excellent agreement between the extracted and literature values of G' from - 2 0 ~ to 60~ and in G" from - 2 0 ~ to 10~ Above 10~ the extracted G" values are approximately 30% higher than the literature values. These results illustrate how AW devices can be used to quantitatively evaluate the viscoelastic properties of polymer films. Similar models for other AW devices, such as the model for SAW devices coated with viscoelastic layers (Section 3.2.7 and [61 ]), can enable these other devices also to be used to determine modulus values. However, the pure shear motion of the TSM does simplify the model, and the evaluation of the modulus values as compared with the more complex displacements of other AW devices such as the SAW device (a comparison of the models of Section 3.1.9 for the TSM and Section 3.2.7 for the SAW demonstrates this point).

164

4. Materials Characterization

10.0 04

E t~

9.5

c

~

"0 =

t3

i

01 0

S

kAA 9,0

--

8.5

-

'~

~-.

-- ~ --

G

kk

t!

8.0 -20

0

20

40

60

80

Temperature (~ Figure 4.4 Components of the shear elastic modulus extracted from admittance vs frequency measurements using a 15.6 /xm-thick polyisobutylene-coated TSM resonator. Lines are literature values for the polyisobutylene modulus [44] at 5 MHz. (Reprinted with permission. See Ref. [66] @ 1991 IEEE.)

4.2.1.3

Absorption P h e n o m e n a and Plasticization

Absorption of a solute liquid or vapor into a polymer film can profoundly affect the viscoelastic behavior of the polymer. The magnitude of this effect depends on the nature of the solute/polymer interactions and on the amount of solute absorbed. The solute/polymer interactions can range from simple dispersion to hydrogen-bonding and other specific interactions. The extent of absorption can be described by the partition coefficient, K, which quantifies the thermodynamic distribution of the solute between two phases (K = concentration in polymer divided by the concentration in the liquid or vapor phase in contact with the polymer). It has long been known that acoustic wave devices can be used to probe solubility and partition coefficients [53,67]. Due to the relevance of these topics to chemical sensors, more comprehensive discussions of these interaction mechanisms and the significance of the partition coefficient are included in Chapter 5. The major effects of solute absorption by a polymer are swelling (change in

4.2 Characterization of Polymers

165

volume) and plasticization. Both effects are a direct consequence of the solute/polymer interactions. As a solute absorbs into the polymer, it interrupts the intermolecular forces at work between the individual polymer chains, and the polymer swells. For polymers in which these forces are strong, due to a high degree of cross-linking or crystallinity, the swelling will be minimal. Lightly crosslinked or linear polymers can experience significant swelling. A theoretical analysis of the effect of compressive tensions resulting from this swelling is presented by Bartley and Dominguez [68]. The effect of vapor uptake on adhesion of polyimide films, possibly due in part to swelling effects, has been described [69,70]. Grate and coworkers [57,71 ] first proposed and documented, using predicted uptakes from gas chromatograph (GC) retention volumes, how these swelling effects can enhance the sensitivity of SAW chemical sensors over the predicted mass-loading values. This increased sensitivity has been confirmed by separate researchers [72]. Concurrent with the swelling phenomenon, the polymer may undergo significant changes in its viscoelastic properties. The presence of absorbed solute molecules in the regions between the polymer chains can act as a lubricant. Due to the interruption of the polymer intermolecular forces, the individual chains may move more freely and the polymer softens. The net results are a decrease in the Tg of the polymer that is dependent on the concentration of absorbed solute [73,74], and a broadening of the elastomeric region. This effect is called plasticization and has been observed using AW devices [51,61-65]. Mass changes associated with solute absorption will produce a change in the AW velocity without significant attenuation of the wave. Modulus changes associated with the glass transition will produce both velocity and attenuation changes. Examples of experimental results for solvent plasticization are shown in Figure 4.5. This plot is a parametric representation of data similar to that shown in Figure 4.3 for a temperature ramp, except the parameter being changed to move along a given curve is the concentration of the absorbing species in the vapor phase contacting the device [ 10,62,75]. As expected, significant velocity and attenuation changes are observed. In addition, the trends with different chemical species can be used to understand the plasticizing action. Since attenuation does not depend on the mass loading, a position on the curve at a given attenuation (e.g., the point of maximum attenuation) can be used as an indicator of the viscoelastic transition. If the velocity shift at the point of peak attenuation is plotted against the liquid density of the absorbing species, a linear relationship is observed [62]. Extrapolating the line to a density of zero should give the value of velocity shift due to changes in the viscoelastic properties. This is verified by the agreement of this extrapolated velocity shift with the value obtained in an ex-

166

4. Materials Characterization

1.5 PENTANE

1.0

: iN

0.5

"i g i

I

m

0.0

1.5 V VV

METHYLENE CHLORIDE

1.0 A

0.5

I

0 >r

"~

0.0

1.5 1.0

0.5 TRIC HL OROETHYLEN E

0.0

I

2.0

. ~ I DIBROMOMETHANE

1.5

I I

1.0 0.5 0.0

I

'

-1.5-1.0-0.5

A V/Vo

0.0

0.5

1.0

1.5

2.0

( x 10 "3)

Figure 4.5 Normalized attenuation-vs-velocity changes for a polymer-coated SAW device as vapor partial pressures are varied from 0% (at dashed line) to 80% of saturation. The polymer, Kraton D1102, is an ABA triblock copolymer, where A is polystyrene (approximately 28% by weight) and B is polybutadiene. (Reprinted with permission. See Ref. [62].)

4.2 Characterization of Polymers

167

periment where temperature changes were used to induce the viscoelastic transition. These trends are observed even though the maximum attenuations are not due directly to a maximum in the loss modulus (as stated in the original article [62]), but rather to film resonance effects that depend on the changes in the polymer modulus. The correlation with density is consistent with the plasticizing action depending only on the volume of chemical absorbed. This type of plasticizing action would be expected if no specific chemical interactions occurred between the absorbing species and the polymer. In contrast, results with a polyimide film and water, methanol, and ethanol vapors yielded trends which depended on the molecular weight of the absorbing species [76]. These trends indicate that the plasticizing action depends on the number absorbed, possibly indicating that the plasticizing is mainly due to the single hydroxyl group found for each species. Again, extrapolation to a molecular weight of zero can be used to extract the responses due to changes in the polymer properties. These results show that changes in viscoelastic properties with chemical uptake can result in significant AW responses, making these property changes important in developing and optimizing chemical sensors using polymer films (see Chapter 5) [57,61-64,71,72,76-79]. For example, the unique curves generated in a plot of attenuation vs velocity for different chemical species (see Figure 4.5) can be used to discriminate between chemical species, increasing the information provided by an AW chemical sensor [63,64,76,80,81]. These results, combined with those comparing SAW responses to predicted uptakes based on GC retention volumes, also indicate that the common practice of converting frequency shifts to amount absorbed assuming that the response is only due to mass loading can lead to erroneous results when working with viscoelastic polymers. Finally, they show that dual-response (attenuation and velocity) AW devices are particularly well suited for probing viscoelastic property changes. 4.2.2

4.2.2.1

DIFFUSION AND P E R M E A T I O N

Real-Time Monitoring

The wide variability of absorption and diffusional properties of chemical species in organic polymer films makes them useful as selective or complete permeation barriers (e.g., gas separation membranes and passivating layers [1,5,82]) and selective chemical sensor coatings [83]. For these applications, a method for rapidly and directly evaluating the solubility and diffusional properties in thin films is useful. Diffusional properties can be evaluated by monitoring the transient up-

168

4. Materials Characterization

take of a chemical species as it diffuses into a polymer sample. AW devices have sufficient sensitivity to monitor this transient uptake in real time in thin polymer films [84-86]. The use of AW-determined diffusion rates vs temperature for probing polymer transition temperatures is discussed in Section 4.2.1.1. A schematic of the device used in this AW technique is shown in Figure 4.6. A thin film of constant thickness h is formed on the impermeable substrate of the AW device. The film, initially in equilibrium with a partial pressure Pl of a gas-phase species, experiences an absorption transient as species diffuse into the film following an increase in the partial pressure to P2. Experimentally, this change in concentration is typically achieved using a gas test system with valves that can be activated to switch from one stream at Pl (typically Pl = 0) to another stream at P2. This absorption transient results in a transient AW frequency response that can be used to characterize the diffusional properties.

4.2.2.2

Fickian Diffusion

Even though diffusion in polymers is generally a complex process, it is possible to find systems that exhibit relatively simple Fickian diffusional behavior. For example, concentration-independent Fickian diffusion has been observed in many polymers when the temperature is far below the polymer's glass transition tem-

Figure 4.6 Schematic representation of a thin polymer film formed on an impermeable AW device substrate. The SAW device probes the concentration profile C(x,t) integrated over the film thickness. (Reprinted with permission. See Ref. [86].)

4.2 C h a r a c t e r i z a t i o n of P o l y m e r s

169

perature and/or diffusant activity is low [87,88]. In these situations, the concentration profile in the film can be determined from Fick's Second Law for a onedimensional system. For a constant diffusion coefficient D, the relevant equation is [89,90]

OC 02C = D~ Ot Ox2 '

(4.1)

where C(x,t) is the concentration of the absorbing species in the polymer, x is the distance from the polymer/substrate interface, and t is time measured from the onset of the change in the partial pressure of the absorbing species. The relevant boundary and initial conditions for this system are: (1) OC/Ox = 0 at x = 0 and all t, (2) C(h,t) = Co(P2) for t -> 0, and (3) C(x,t) = C0(Pl) for t < 0 and 0 -< x -< h, where Co(p) is the concentration in the polymer in equilibrium with a partial pressure p of the absorbing species. Equation 4.1 can be solved under these conditions to yield the following analytical expression [89]" oo

C(x,t) - Co(P2)

--

2AC0 ~

sin(~x/h)e-q'2~

,

(4.2)

n=l

where ~ = "rr(n- 1/2) and ACo = Co(P2)- Co(p1). Equation 4.2 can be integrated over the film thickness to give the following expression for the total moles, M(t), absorbed as a function of time:

M(t) = Mm~ 1 - 2 ~ e-~O~/h2 n=l 1~2 '

(4.3)

where Mmax is the incremental amount of species absorbed in the film after equilibrium is attained (Mmax - hAACo, where A is the area of the film). Equation 4.3 predicts an accumulation of species proportional to N/t until M(t) is approximately 60% of saturation (Mmax); thereafter, the inability of species m penetrate the substratr decreases the net flux into the film. A common technique for evaluating D is to use a gravimetric method to monitor M(t) [87,90] and then extract D and C0(p) by fitting the data to an equation similar to Equation 4.3 (the exact form of this relationship depends on sample geometry). Since equilibration times for Fickian diffusion are proportional to h2/D, the ability to monitor absorption transients in thin films (small h values) directly using AW devices enables a dramatic decrease in the equilibration time as compared to the use of bulk samples with conventional gravimetric techniques. In addition, since diffusional properties of thin films may differ significantly from bulk samples prepared from the same material [91 ], direct evaluation of thin-film properties can be advantageous.

170

4. Materials Characterization

Fickian diffusion was observed using a polyimide-coated SAW device for a wide variety of chemical species [86]. SAW frequency transients obtained for N20 and methanol are shown in Figure 4.7 (pages 172-173); p increased from zero to the indicated values. The expected behavior is observed: an increase in the response upon increasing p that saturates at a new level, indicating that the film has approached equilibrium with the new gas-phase concentration. The expected linear region of the data when plotted vs X/t is observed. These SAW frequency transients were used to determine diffusion coefficients using an alternative "frequency version" of Equation 4.3 where M(t) is replaced with Af(t)/fo and Mmax is replaced by Afmax/fo. The curve through the points represents a nonlinear least-squares fit of the data to this equation. The variable parameters in this fitting routine were: (1) D, (2) Afmax, and (3) to, the starting time for the change in partial pressure. The value of to was allowed to vary in order to account for the time lag between switching the valves and the arrival of the flow to the device. Excellent fits to the data were obtained with an rms error in both cases of less than 1% of Afmax. The D values obtained were 2.3 • 10-ll cm2/sec for methanol and 8.0 • 10-lo cm2/sec for N20. It should be noted that studies with this film at various methanol concentrations indicated that the diffusion coefficient is not constant, but rather increases with increasing concentration [86]. The use of Equation 4.3 is still justified, however, since the concentration steps shown in Figure 4.7 are small enough that the diffusion coefficient does not change significantly. This concentration dependence can be important for chemical sensors, since it requires challenging the sensor at the low concentrations expected in practice in order to evaluate the speed of the sensor response (see Section 5.3.6). Fickian diffusional behavior in polyimide has also been observed by Denton et al. [82] using a capacitive technique and, for the desorption branch only, by Bartley and Dominguez [68] using a SAW device. The absorption transient in the Bartley and Dominguez study exhibited a non-Fickian linearity with time. As described in detail below, non-Fickian behavior was also observed by Brace et al. [92] in their SAW study. This disagreement is not surprising considering that the various polyimide films differ significantly because of the use of different starting solutions and thermal treatments. These differences in the polymer films also show up in differences in the sign of the frequency response. The polyimide used to generate the data in Figure 4.7 exhibits a positive frequency response when challenged with relatively low concentrations ( P/Po < 0.1, where P0 is the saturation vapor pressure) of the various species tested. The other two SAW studies, however, report negative frequency responses to the vapor challenges. The positive response shown in Figure 4.7

4.2 Characterization of Polymers

171

must be due to a combination of a negative mass response and an additional positive response that is large enough to overwhelm the mass-induced response. This additional response is probably due to viscoelastic property changes caused by the plasticizing action of the absorbing species (see Section 4.2.1.3). These viscoelastic effects probably occurred in the other studies but to a smaller extent compared to the mass-loading effect. As described in Section 4.2.1.3, this makes the evaluation of concentrations in the films based on the frequency response questionable for all of these studies. When mass loading is not the dominant sensing mechanism, sensor response may not be linear with concentration in the film. This departure from linearity has been observed with polymer films [61,64,86]. An investigation into the possible effect of this nonlinearity on the evaluation of D values from SAW frequency transients indicated that errors in D values (factor of two error) could be obtained if the nonlinearity of the response is large [86]. However, using small steps in partial pressure, this nonlinearity in the response can be minimized, allowing the effective evaluation of diffusion coefficients based on AW frequency transients. It has been noted by other researchers that the molecular size of the absorbate has a dramatic effect on the diffusion coefficient [93,94]. An exponential relationship is observed between D and the size (represented by the b parameter in the van der Waals equation of state) of the absorbate [94,95]. As shown in Figure 4.8 (page 174), an exponential dependence on the molar volume of the absorbing species was observed with an almost four-order-of-magnitude decrease in D for only a 2.3-times increase in molar volume. The potential for using this variability in D values to advantage in the development of chemical sensors has been discussed [86,96,97]. The basic concept is to use the evaluation of D to determine the chemical species that is providing the sensor response and the magnitude of the response (e.g., Afmax) to evaluate concentration. The results presented above illustrate the utility of using AW frequency transients to evaluate diffusional processes in thin polymer films. The ability to use thin films allows the rapid evaluation of D values from 10 -9 to 10 -15 cm2/sec [86]. The upper limit on D is set by the requirement for multiple data points during the transient response, while the lower limit results from the long times required to approach equilibrium. Thus, thinner films (hundreds of nanometers) are better for probing slower diffusion, while thicker film (micrometers) are better for faster diffusion. An electronics scheme capable of rapid data acquisition [98] would enable larger D values to be quantified based on following the rapid transients. Another way to probe faster diffusion times is to use very thick films. As men-

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v~" (seconds) 1 / 2 Figure 4.7 Frequency shift as a function of V~ during diffusion of (a) methanol, and (b) N20 into a 1.8-pm polyimide film. The lines through the points are fits of Equation 4.3 to the data, giving D values of 2.3 • 10-11 and 8.0 • 10- lo cm2/sec for methanol

and N20, respectively.

(Reprinted with permission. See Ref. [86].)

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4.2 Characterization of Polymers

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4. Materials Characterization

- 9 ! . - ' ~ N20

1.8 pm POLYIMIDE FILM 25C

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tioned in Chapter 3, FPW devices can operate when coated with a thick gel having a solids concentration belc, w about 5%. The device behaves as if it were simply in contact with the liquid solvent for the gel, and no significant change in FPW device velocity or attenuation occurs as the gel sets because dilute gels have very low shear moduli [99]. The semi-logarithmic plot of Figure 4.9 illustrates the use of the FPW device to follow diffusion in a gel. Here, a 500-/xmthick, 2% wt./vol, agar gel was made on a FPW device, with deionized water as solvent. After the gel had set, it was exposed to a 0.1 M NaCI solution; the ions diffused into the gel and finally reached the mass-sensitive region within the evanescent decay length of the membrane, a distance of 16/zm in this case. From the observed mass loading, one can determine the diffusion constant of ions in the gel to be 9.8 • 10 - 6 cm2/s, two orders of magnitude higher than could be probed with thinner films on SAW devices [99]. A similar test was made with whole human blood; in this case, the gel acted as a filter that allowed only the smallest molecules to diffuse toward the membrane and be detected, while holding back blood cells and other large molecules.

4.2 Characterization of Polymers 10000 -

175

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TIME (seconds) Figure 4.9 Response of 5.6-MHz FPW device coated with 500-/zm-thick agar gel upon immersion in 0.1 M NaCI, showing gravimetric detection of ions that diffuse to within an evanescent decay length of the sensor membrane. (Data provided by Amy Wang and Ben Costello, U.C. Berkeley and Berkeley Microlnstruments, Inc., respectively.)

4.2.2.3

Non-Fickian Diffusion

The Fickian diffusion described above is relatively easy to analyze, and demonstrates the capabilities of AW devices for monitoring transient uptakes. However, Fickian diffusion in polymers is the exception rather than the rule. A wide variety of transient responses have been observed, often due to the long time constants required for relaxation of the polymer chains upon absorption of species into the film [93,95]. A detailed discussion of these trends is beyond the scope of this book, and the reader is referred to the polymer literature for these details [93,95]. Brace et al. [92] investigated polymer/water interactions using SAW devices coated with either polyimide or cellulose acetate butyrate (CAB). In this study thermodynamic parameters were evaluated from the absorption isotherms, and transient responses to step changes in concentration were monitored. The transient responses observed were not consistent with Fickian diffusion, but could be described using a generalized relaxation equation containing two additive terms. Results under various conditions indicated that relaxation in the polymer system is much slower than diffusion of water.

176

4. Materials Characterization

Laatikainen and Lindstr/Sm [ 100] used TSM devices to investigate absorption in cellulose acetate and poly-(hexamethylene adipamide). In addition to measuring absorption isotherms and partition coefficients, they reported on transient responses to changes in methanol concentration for a cellulose-acetate-coated TSM device (Figure 4.10). At low concentrations, the linear response with X/t is consistent with Fickian behavior, and diffusion coefficients can be evaluated (D = 4.8 X 10 -l~ and 1.6 x 10 -9 cm2/sec for steps 1 and 2, respectively). It is seen that the initial diffusion rate increases with concentration in the polymer (based on the initial slope of the curves), until, at higher concentrations, a two-stage absorption transient occurs. This behavior, which is typical of glassy polymers, is due to the fact that diffusion begins to become faster than the polymer relaxations [95]. Recent work investigating gas sensor applications using TSM devices coated with the conductive polymer poly(pyrrole) revealed in some interesting diffusional properties. In one study on absorption of various alcohols [ 101 ], methanol was found to show Fickian behavior (D = 2.2 • 10 -12 cm2/s), while larger alcohols were found to have slower diffusion rates (D = 1.3 • 10 -12, 6.4 X 10 -13, and 2.4 • 10 -13 cm2/s for ethanol, n-propanol, and n-butanol, respectively) and trends indicative of non-Fickian diffusion. In another study that used a TSM device combined with measurements of film conductivity [102], the trends were consistent with Fickian diffusion except for the TSM frequency response, which demonstrated non-Fickian trends for methanol. These observations were interpreted as indicating that the conductivity changes to methanol were due solely to one stage of the two-stage sorption observed with the TSM. This may be due to the conductivity only probing the swelling of the polymer and not any subsequent sorption. In this study, the TSM measurements helped in determining the mechanism of conductivity changes in poly(pyrrole)films. In a final study investigating dichloromethane absorption from aqueous solutions [103] into poly(N-methylpyrrole) and poly(N-methylpyrrole/polystyrenesulfonate), the sorption rate was found to be independent of film thickness. This was interpreted as being due to rapid diffusion through pores in the polymer, followed by slow diffusion into the bulk of the polymer. The effect of oxidation state on sorption rates was also investigated. The preceding results show that the ability of AW devices to follow the transient uptake of a species into a thin film allows these devices to be used to probe a wide variety of diffusional processes. As described for Fickian diffusion, a significant advantage of the AW technique is the ability to use thin films, which results in the rapid evaluation Of the diffusional properties even in polymers that exhibit very slow transient uptake.

4.2 Characterization of Polymers

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(seconds) Figure 4.10 Interval absorption of methanol in cellulose acetate measured using a TSM device. Each curve represents the transient response in going from an initial (tk~ to a final (tk*) volume fraction of methanol in the polymer. Fickian diffusion at low volume fractions turns into two-stage absorption at higher volume fractions. (Reprintedwith permission. S e e R e f . [ 1 0 0 ] . )

178

4. Materials Characterization

4.2.2.4

Permeation Rate Evaluation

There are some situations where permeation through a film is the property of direct interest. For example, for a film being proposed as a gas-separation membrane or a permeation barrier, the permeation rate through the film is the critical parameter. The diffusion coefficient can be misleading, since it measures the time required for absorption of the species into the polymer but does not effectively probe small holes (which can allow species to permeate rapidly through the film even though the calculated diffusion coefficient is low). This permeation through the film can be evaluated using a multilayer coating. For example, SAW responses for a polysiloxane film over a porous silicate coating upon exposure to methanol are shown in Figure 4.11. For comparison, the SAW responses due to the polysiloxane film alone are also shown. In each case, a rapid frequency decrease is observed upon exposure to the methanol. The small size of the response for polysiloxane alone could be due to the mass increase from methanol

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TIME (min) Figure 4.11 Experimental SAW frequency transients during exposure to methanol (P/Po = 0.05) for 360 nm-thick polysiloxane film deposited on bare quartz (m) and on a 150 nm-thick porous silicate film on the quartz substrate (e). The larger initial frequency drop with the porous film is due to permeation of methanol through the polysiloxane film and adsorption onto the surfaces of the underlying porous film. Data points are overlapping at early times. (Reprintedwith permission.See Ref. [86].)

4.3 Surface Adsorption and Characterization of Porous Films

179

adsorption on the interfaces of the film. However, the larger response for the device with the porous film underneath, due to adsorption on the high-surface-area silicate following rapid permeation of the methanol through the siloxane overcoat, verifies that this fast response is due to rapid diffusion in the polymer. Further results with this type of polysiloxane using a wide variety of species support this interpretation [80]. This type of multilayer coating is useful for effectively probing permeation rates.

4.3

Surface Adsorption and Characterization of Porous Films

4.3.1

PHYSISORPTION: CHARACTERIZING SURFACES AND PORES

Adsorption on a solid surface is the process of a species present in a gas or liquid phase "adhering" to the surface of the solid [46,104,105]. This adsorption occurs due to molecular interactions between the adsorbing species and the solid. If adsorption is characterized by relatively weak interactions, such as those typical of van der Waals forces, the process is called physisorption. Because such weak forces occur between all molecules, physisorption is typically reversible and will occur at any surface when the normalized concentration of the adsorbing species is sufficiently high. For a gas-phase species, the normalized concentration is equal to P/Po where p is the partial pressure of the species and P0 is its saturation vapor pressure. The endpoint for physisorption occurs when the concentration of the adsorbing species reaches its saturation value. For a gaseous contacting phase, condensation occurs at this point, (i.e., when p = P0). 4.3.1.1

Characterizing Surfaces and Adsorbed Layers

Physisorption on a surface can be used as a method of probing the molecular interactions occurring at a solid surface. Since the nature of the adsorbing species can be known, this provides a technique for probing the properties of the solid [106]. These measurements often utilize noble gases to provide an adsorbing species with minimal potential for specific chemical interactions. An example from one of these studies by Krim is shown in Figure 4.12 (page 180) for krypton on a gold surface. The adsorption isotherm labeled A was for adsorption onto a gold surface that was briefly exposed to air prior to being placed into the vacuum system where the measurements were taken. Due to surface contamination on the gold surface, this shows a relatively smooth curve without any specific adsorption steps. In contrast, the data for Curve B were obtained after heating

180

4. Materials Characterization

u.I r LIJ

~' O rO

2-

0.0

A

0.2

0.4

0.6

0.8

1.0

P/Po Figure 4.12 Adsorption isotherms for krypton on gold obtained using TSM devices in a vacuum chamber. Curve A was at 90~ before baking while Curve B was at 96~ after baking the gold film at 350~ for 12 hours. (Reprinted with permission. See Ref. [106].)

the gold surface to 350~ for 12 hours under vacuum prior to measurement. This isotherm clearly shows stepwise adsorption, with the first monolayer adsorbing at very low partial pressures and the adsorption, of a second monolayer occurring at a relative saturation of about 0.5. This stepwise adsorption was stated to indicate that the uniformity of the gold surface was sufficient to result in layerby-layer condensation of atomic layers [ 106]. The ability to monitor submonolayer mass changes on well-characterized surfaces using AW devices has resulted in their use in a variety of these studies [106-108], allowing interesting behavior including wetting transitions and step-wise isotherms to be observed. Comparison of the results to theories of gas-surface interactions enables these theories to be verified and optimized [107,108].

4.3 Surface Adsorption and Characterization of Porous Films

181

AW devices can also be used to probe the orientation and extent of adsorption. For example, using a gold-coated TSM device, Tsionsky and Gileadi [ 109] investigated the adsorption of various vapors. Based on the amount of adsorption, they were able to show that benzene adsorbed flat on the surface while pyridine attached in an upright orientation due to the nitrogen atom. In addition, they found that water, methanol, and 1-propanol occupied the same number of surface sites per molecule on the surface. This study also demonstrated the importance of considering the small but detectable effects of the pressure and the viscosity and density of the gas in contact with the surface when investigating the small responses due to monolayer adsorption onto the surface of an AW device. They proposed that performing the experiments by changing the partial pressure of the adsorbing species in an inert atmosphere maintained at a constant total pressure (common technique used in most AW adsorption experiments) minimized the effects of these additional interactions compared with varying the total pressure to change the partial pressure of the adsorbing species. The "nanotribology" of adsorbed layers has been investigated by Krim and coworkers using TSM devices. They observed changes in the Q (eletrical quality factor): of gold- and silver-coated TSM sensors upon adsorption of Kr, Xe, and N2 at cryogenic temperatures, and for water and cyclohexane at room temperature [ 110,111]. These changes were attributed to slippage of the adsorbed layers at the surfaces due to the large velocities imposed by the high frequency of the TSM device [ll0,111]. Models to account for these results in terms of film-substrate interfacial viscosity [ 112] and in terms of a characteristic slip time [112,113] have been developed. This information is relevant to topics such as dendritic film growth, fluid flow and slip near boundaries, phase transitions in adsorbed films, and adsorbed monolayers on liquid surfaces [ 112]. These considerations may also explain the nonmonotonic frequency responses with vapor partial pressure observed for adsorption of non-polar molecules onto the quartz surface of a SAW sensor [114]. Since it is well known that surface coverage increases monotonically with partial pressure, these trends are inconsistent with a simple mass-loading interpretation. In this paper, the authors proposed a coverage-dependent SAW-adsorbate interaction to explain the anomalous results for the weaker binding non-polar species (no anomalous trends were observed with polar adsorbates). 4.3.1.2

Characterizing Film Surface Area and Pore Size Distribution

There are applications of thin films where it is important to obtain information regarding properties relating to the microstructure of the coating. Some exam-

182

4. Materials Characterization

pies are: (1) gas separation membranes where the size of the pores and their accessibility from the gas phase are critical; (2) chemical sensor coatings where a high surface area is important for films being used to selectively adsorb species [ 115]; and (3) passivating and protective coatings where a lack of porosity is necessary. A standard technique for characterizing the microstructure of bulk samples is the use of adsorption isotherms [ 116-118]. These measurements can provide information regarding sample surface areas, pore size distributions, and total pore volumes. Obtaining an adsorption isotherm involves measuring the extent of uptake of a chemical species (typically called an adsorbate) by a sample as a function of the partial pressure (p) of the adsorbate in the gas phase over the sample. In general, the adsorbate partial pressure is increased from zero to a value near the saturation vapor pressure (Po) and then back to zero. Uptake of the adsorbate occurs due to adsorption onto the accessible surfaces of the sample, as well as condensation into the accessible sample pores. The most commonly used adsorbate is nitrogen at its boiling point (77~ [ 116]. A variety of commercial instruments are available for performing nitrogen adsorption measurements on bulk samples. These instruments measure the extent of uptake using volumetric, gravimetric, or flow-through methods [ 117]. Sample surface area is determined from a nitrogen adsorption isotherm using the well-established BET model developed by Brunauer, Emmett, and Teller [116,119]. This model accounts for adsorption onto the surfaces of the sample using two binding energies: one between the surface and the adsorbate for the first monolayer, and a second binding energy between the adsorbate molecules for adsorption of subsequent monolayers. The isotherm described by this model is given by [116,119] n nm

=

c(p/po)

(4.4)

(1 - P/Po)[ 1 + (c - 1)( p/po)]

where n is the number of adsorbed molecules, nm is the number of adsorbed molecules in a single monolayer on the available sample surfaces, and c is a constant that depends on the two binding energies. For a molecule like N2, which exhibits high-affinity adsorption (large c value), this model predicts a Type II [ 120] isotherm shape (see Figure 4.13 for an example, or Figure 5.7 for schematic examples of Type I-V isotherms) consisting of: enhanced adsorption at low P/Po values due to adsorption of the first monolayer on the surfaces, a knee in the isotherm indicative of the completion of the adsorption of this initial monolayer, and slow uptake at intermediate p values until multilayer adsorption begins to

4.3 Surface Adsorption and Characterization of Porous Films

183

become significant as p approaches the value for condensation (P/Po = 1). For ease of analysis, Equation 4.4 is typically rearranged to the form: fl _

P/Po

n(1 - P/Po)

=

1

+

nmC

l) P_ nmC Po

(c -

(4.5)

where/3 is defined here for simplification. If the BET model is applicable, a plot of/3 vs P/Po yields a straight line. In practice, due to surface heterogeneities altering the adsorption energies at low P/Po values and the onset of capillary condensation at higher P/Po values, this plot is typically linear only in the P/Po range from about 0.1 to 0.3 [ 116,117]. The slope and intercept of this linear region can be used to evaluate c and nm" c-" 1 + slope/intercept and nm "- 1/(slope + intercept). Using the well-characterized area of an adsorbed N2 molecule (am = 0.162 nm 2 [116]), the sample surface area, As, can be calculated using As = nmam. This BET analysis can be effectively applied to samples that exhibit Type II or IV (described in the next section) isotherms. Pore size distributions are determined based on the fact that condensation in small pores occurs at a lower partial pressure than is required to obtain condensation on a flat surface. The relationship quantifying this effect is the Kelvin equation which, for a hemispherical meniscus of radius rc, has the form [ 116]: In P'__.L= 2trVm Po - rcRT

(4.6)

where p,. is the partial pressure at which capillary condensation occurs, tr is surface tension, Vm is molar volume, R is the gas constant, and T is the absolute temperature. The expression 2trVm/RT is equal to 0.96 nm for nitrogen. This capillary condensation results in enhanced uptake over what would be expected for adsorption onto the surfaces of the film. As the pores fill, however, the surface area decreases until finally, at high P/Po values, the pores are completely filled and N2 uptake ceases. The value of adsorption at this plateau can be used to determine the total volume of pores using the liquid density of the adsorbate (0.808 g/cm 3 for nitrogen at 77~ If the total volume of the sample is known by some independent means, this pore volume can be used to determine the fractional porosity of the sample. Due in large part to geometric constraints, the partial pressure at which pore filling and pore evacuation occur are not identical. This results in hysteresis in the adsorption branch (p increasing) falling beneath the desorption branch (p decreasing). These trends result in what is called a Type IV isotherm shape [120] (see Figure 4.14 for an example). To extract a pore size distribution (volume of pores as a function of pore size)

184

4. Materials Characterization

from an adsorption isotherm, one uses a reference point at high partial pressures (P/Po near 1.0) where the isotherm is fiat. In the first step, the change in amount of N2 uptake by the sample as P/Po is decreased is used to indicate the amount of capillary condensation that occurred over this P/Postep. Using the Kelvin equation to determine the size of the radius of the evacuated area and accounting for the thickness of the adsorbed layer at this P/Po value, the size and the volume of pores of this size can be determined. Before proceeding, the surface areapresent in these evacuated pores is calculated so that in subsequent steps, changes in N2 uptake due to changes in the thickness of the adsorbed layer on these surfaces can be accounted for in the calculations. By continuing to lower P/Po values in this way, a pore size distribution can be determined. Pores from 3 to 50 nm in diameter, typically called mesopores, can be characterized in this manner. Further details on these calculation procedures are presented elsewhere [116,117,121]. Samples with pores smaller than this range (i.e., pore diameters < 2 nm), typically called micropores, provide the final type of commonly observed nitrogenadsorption isotherm. The interaction of the nitrogen molecules with a microporous solid is enhanced due to the close proximity of the surfaces of a pore which has a diameter that may be only a few times the diameter of the nitrogen molecules (approximately 0.4 nm). This enhanced interaction is seen in large amounts of adsorption and pore filling at low P/Po values. This type of adsorption isotherm is called Type I. This different mechanism for pore filling results in an overestimation of the surface area using the BET analysis, and an inability to determine a pore size distribution using the Kelvin equation. Because of these limitations in applying standard calculation procedures, research on new techniques for the quantitative evaluation of microporosity is ongoing.

4.3.1.3

AW Device Results with Porous Films

Nitrogen adsorption isotherms obtained for two films using an AW technique are shown in Figures 4.13 and 4.14 (page 186). These films were both prepared by dip-coating SAW devices using solutions containing oxide precursors prepared by sol-gel processing, a technique that can be used to tailor the microstructure of oxide coatings [3,122,123]. The measurements were made by monitoring SAW frequency changes as the concentration of nitrogen flowing over the device was varied from zero to about 95% and back to zero, using mass-flow controllers to control the dilution of nitrogen with helium, a nonadsorbing species at 77~ [ 117]. The device temperature was maintained at 77~ by immersing the test fix-

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I. . . . . .

I

....

'

0.4

,

'

0.6

'

I,

0.8

, ,,

'

' -

O

1.0

P/Po Figure 4.14 Nitrogen adsorption isotherm for a SAW device dip coated using a sol containing four oxide precursors ("four component"). This coating was prepared from a solution aged for two weeks at 50~ and pH 3. The large amount of adsorption relative to the film thickness and the Type IV isotherm shape indicate that this film is highly porous. (Reprinted with permission [10] by courtesy of Marcel Dekker, Inc.)

r "l

mj ..~ N ..~ g~

4.3 Surface Adsorption and Characterization of Porous Films

187

ture containing the device in liquid nitrogen. An alternative method for varying the nitrogen partial pressure is to monitor pressure as the test fixture containing the AW device is first evacuated, then refilled by slowly bleeding in nitrogen, and finally reevacuated. This technique has been demonstrated using TSM devices in a vacuum chamber [ 124] and for SAW devices by fabricating a test fixture that attaches to a Micromeritics A S A P - 2 0 0 0 nitrogen adsorption instrument [125]. Using the ASAP-2000, the wide pressure range available with this commercial system allows an effective probing of microporosity (P/Po from 10 -6 to 0.1 range) and of larger mesopores (P/Po values very close to 1.0). The data in Figure 4.13 were obtained for a film formed from an acid-catalyzed sol-gel solution ("A2") that is known to produce relatively dense coatings [ 126]. The isotherm shape resembles a Type II isotherm indicative of a sample with no porosity in the mesoporous range. BET plots (/3 vs. P/Po) are linear for both branches of the isotherm, and the calculated BET surface area is 0.95 cm2/(cm 2 of film), very close to the value of 1.0 cm2/(cm 2 of film) expected for a flat, nonporous film on the device surface (BET values are typically thought to be accurate to within 20% [ 116,117]). This data set is useful for demonstrating the extreme sensitivity of SAW devices, since it verifies that submonolayer mass changes can be detected even on a flat surface (full scale is only 120 ng/cm2). For comparison, a single monolayer of nitrogen on a flat substrate contributes a mass loading of 29 ng/cm 2. Analysis of a bulk sample prepared from this same A2 solution using a conventional instrument (Micromeritics DigiSorb 2600) showed a large uptake of nitrogen [123,127] with a Type I isotherm shape, indicative of microporosity. Thus, these data also demonstrate the importance of being able to directly characterize thin film samples rather than risking the erroneous predictions that can occur by using the common procedure of characterizing bulk samples and attributing their properties to thin films. The data in Figure 4.14 were obtained from a film formed from a sol-gel system in which the number and the size of pores can be varied by altering the reaction time prior to film preparation [ 128]. Adsorption isotherms for films formed from the as-prepared solution resemble the data in Figure 4.13, and have low calculated BET surface areas (1.3 cm2/cm 2) [ 129]. This solution contains small precursor polymers (hydrodynamic radius --3 nm) that collapse under the large capillary pressures that occur during film drying [123,128]. However, aging the solution at pH 3 and 50~ results in an increase in the polymer size and, due to the inability of these polymers to interpenetrate [122,123], an increase in the amount and size of porosity in the film. The data from Figure 4.14 were obtained from a film prepared using a solution aged two weeks. The total amount of ad-

188

4. Materials Characterization

sorption is much larger than that observed with the acid-catalyzed A2 film (see Figure 4.13). The BET plot of these data is shown in Figure 4.15. The data are linear in the range expected, and the BET surface area (based on a linear leastsquares fit to the data for P/Po from 0.1 to 0.35) is 46 cm2/cm 2, significantly larger than the value obtained with the A2 film. This value can be converted into the more conventional units of m2/g by calculating the film mass/area (mr): mf = h(l -fp)psk, where h is film thickness, fp is the fractional porosity, and Psk is the skeletal density of the oxide matrix. Based on ellipsometric analysis, this film is 165 nm thick and has a refractive index of 1.21. Using the Lorentz-Lorenz relationship [ 130] and a skeletal refractive index of 1.45, a fractional porosity of 0.50 is estimated. However, from the total amount of adsorption in the plateau region at high P/Po values, a fractional porosity of 0.33 is determined. Since nitrogen adsorption only probes the open porosity, the refractive index estimate, which is sensitive to all porosity in the film, may be more accurate for determining my. Using this porosity value and a skeletal density of 2.1 g/cm 3 yields mf = 1.7 • 10 -5 g/cm 2 and a surface area of 270 m2/g. This relatively large surface area and the fact that the isotherm is of Type IV indicate that this film has porosity in the mesoporous range. A pore size distribution using the desorption branch of the isotherm is shown in Figure 4.16 (page 190). A fairly unimodal distribution is obtained with a median pore diameter of 3.8 nm. Results obtained with samples at aging times up to three weeks indicate that the microstructural properties can be varied over a wide range based solely on the sol-gel aging time; percent porosities ranged from 0 to 52%, median pore diameters ranged from <0.2 to 6.1 nm, and surface areas ranged from 1.5 to 270 m2/g [84,123]. Microporous samples have also been analyzed using AW devices. For example, Bein and coworkers have measured adsorption isotherms for N2 at 77~ and for vapors at 25~ using TSM devices coated with zeolites [131-133] and with layered silicate materials that have been ion exchanged with alkylammonium cations [134]. As expected for these types of materials, Type I isotherms indicative of microporosity were observed in many cases. In addition, Hietela and coworkers [125] used the ASAP-2000 system with a SAW device to look at the A2 sol-gel film described above. In agreement with the results shown in Figure 4.13, the BET surface area calculated from the N2 adsorption isotherm gave a value of 1 cm2/(cm 2 of film). However, tests using CO2 at 196 and 273 ~ gave large uptakes and a Type I isotherm shape, indicative of a microporous sample. The surface area (calculated using the Dubinin-Radushkevich equation useful for microporous samples) was 55 cm2/(cm 2 of film). They interpreted the lack of porosity to N2 to be due to activated diffusion limitation for entrance of the N2

, ,

0.4

,,

,~

9 ADSORPTION II DESORPTION ,

,

i

1

Ilimlll

I

~, 0.3 01 @

E

O

@

c~ 0 . 2 m.

r

r

O.1

m.

N

i

O.0 O.O

@ =3 @ o @ =

O.1

0.2

0.3

0.4

0.5

P/Po Figure 4.15 BET plot of 13 (see Equation 4.5) vs P/Po for the film formed from a two-week-aged, four-component sol (see Figure 4.14). Based on a linear least-squares fit to the data from P/Po = 0.1 to 0.35, a BET surface area of 46 cm2/(cm 2 of film) or 270 m2/g is obtained. (Reprintedwith permission [10] by courtesy of Marcel Dekker, Inc.)

mo

1.o ! ........... i ..........

~

.........I. . . . .

I

W

I

...... I "

I

FOUR COMPONENT TWO WEEK AGE DESORPTION BRANCH

~Z 0.8 ,,,,J

tm~ t~ m/ mm. t~ m

0 >

I,,IJ

~r

0.6

"I

O

tm~ eD m. N

..J

mt @

0.4

z

I,U

13: I,,U IJ.

r

0.2

0

4

8

PORE

12

DIAMETER

16

20

(nm)

Figure 4.16 Pore size distribution obtained from the desorption branch of the nitrogen adsorption isotherm for the film formed from the two-week-aged, four-component sol (see Figure 4.14). Median pore diameter is 3.8 nm. (Reprinted with permission

[ I 0 ] by c o u r t e s y o f M a r c e l D e k k e r , Inc.)

4.3 Surface Adsorption and Characterization of Porous Films

191

molecules into the micropores at 77~ This is commonly observed for coals but was unexpected with the thin film sample used with the SAW experiment. The results described above demonstrate the utility of this AW technique [ 135] for characterizing thin film microstructure. This technique uses the wellestablished nitrogen adsorption isotherm technique commonly used to evaluate porous bulk samples. By increasing the sensitivity to the amount of adsorbed Ng, this AW technique allows thin films to be characterized directly. 4.3.2 4.3.2.1

CHEMISORPTION Overview of Chemisorption

Chemisorption occurs when strong interactions, including hydrogen bonding and covalent and ionic bond formation, occur between the adsorbate and the solid surface. Chemisorption typically occurs even at very low concentrations, and the chemisorbed species are often "irreversibly" bound to the surface, i.e., they will not readily desorb under ambient temperature conditions. The endpoint for chemisorption is when all the active sites on the solid surface are occupied by chemisorbed molecules. The specific interactions that occur during chemisorption raise a broader range of questions to answer in order to characterize the solid/adsorbate interactions. AW measurements of equilibrium adsorption, as well as adsorption and desorption kinetics, can be exceptional tools for evaluating such parameters as activation energies and heats of adsorption, particularly for the study of the submonolayer regime. Czandema [136] has reviewed many of these types of studies using very sensitive conventional microbalances along with the types of information that they can provide. The simplicity of using acoustic wave devices, as compared with sophisticated balance systems, to obtain the required sensitivity for these studies on low surface area samples should be a significant advantage. Studies of corrosion processes, detailed in Section 4.4.3, have demonstrated the capability of SAW devices to monitor relatively low rates of chemisorption, including the conversion of a thin copper film to Cu2S at an initial rate of 4% of one molecular monolayer/day. The use of SAW devices to monitor the real-time desorption of species from a metal film in response to a temperature ramp has been shown to yield information about both the energy and extent of chemisorption [I 14]. TSM studies of chemisorption of O2 and CO on very thin Ti films were used to determine that the oxide being formed is Ti203 and that the oxidation depth is approximately one nm [ 137]. For further discussion and additional examples of chemisorption, the reader is referred to Section 5.4.4.3, where these

192

4. Materials Characterization

interactions are described in some detail because of their relevance to the development of highly sensitive chemical sensors. 4.3.2.2

Self-Assembling Films

Self-assembling films rely on a unique combination of three interactions to produce highly ordered, durable films ranging in thickness from one to tens of molecular layers [138]. The process is best illustrated using an example, the self assembly of a hexadecanethiol (CH3(CH2)IsSH) monolayer on a gold substrate. When the gold surface is exposed to a low concentration of this molecule in either the gas or liquid phase, the SH "head" group of each thiol molecule chemisorbs onto the gold surface, creating a thiolate bond. The creation of a thiolate bond results in the sulfur molecule being bound to a three-fold site on the gold (11 l) crystal lattice accompanied by the loss of hydrogen [ 139]. This head group binding of the thiol, the strongest of the three interactions, ensures that all the chemisorbed molecules are oriented in roughly the same direction. The second interaction involves van der Waals forces, which are relatively weak on a "per bond" basis (measured to be 3.3 kJ/mole per methylene unit using temperature-programmed desorption (TPD) [140]). However, these forces can be cumulatively quite strong (16 C-C van der Waals bonds form between two properly aligned hexadecanethiol chains), bringing all the chains into alignment. Thus, the sulfur atom determines the spacing on the gold lattice while the van der Waals interactions between adjacent chains determine the orientation. Another weaker interaction that should be considered involves the nature of the tail group, in our example, a methyl group. As the monolayer forms, the solvent is excluded from the film as the surface free energy of the film is maximized at the solvent-monolayer interface favoring a more highly ordered structure. For this reason, the tail group along with the solvent properties influence monolayer formation, especially for the last 5% of the monolayer's formation. For tail groups such as carboxylic acid groups (-COOH), the interactions between the tail groups in the monolayer can also be important. The sum of the headgroup chemisorption energy, the van der Waals interactions for the long hydrocarbon tails, and the properties of the tail group/solvent interactions results in a relatively robust, anchored film. These materials show promise in applications including electronic and optical devices, chemical sensors, lubricants, model biological membranes, permeation barriers, and catalysis [141,142]. Work using SAW devices has focused on the gas-phase self-assembly process [143,144]. Figure 4.17 (page 194) shows the utility of a SAW device in monitoring the assembly of a hexadecanethiol monolayer on a polycrystalline gold

4.3 Surface Adsorption and Characterization of Porous Films

193

film on the device surface [143]. Not only is the SAW device's mass resolution sufficient to show that the equilibrium coverage exceeds that calculated for a single crystalline gold surface by about 15% (a result of the polycrystalline film having a roughness factor exceeding unity), but the kinetics of this chemisorption/assembly process are readily discerned in real time. Analysis of the kinetic adsorption data reveals a good fit to a simple, first-order Langmuir rate law, meaning that the rate of adsorption is proportional to the product of the partial pressure of the thiol and the unoccupied concentration of gold adsorption sites on the surface. Significantly, this result indicates that the process of chemisorption of the thiol head group, as opposed to the process of alignment of the adsorbed chains through van der Waals forces, limits the overall rate of film assembly [143]. Chemical reactions between a surface-bound self-assembled monolayer (SAM) and a second species that reacts with one-to-one stoichiometry and modifies the chemical properties of the surface layer, have also been studied with submonolayer accuracy [144]. In solution-phase work, Buttry and coworkers have shown that the TSM oscillator can monitor the process of electrochemically inducing changes in the adsorption of redox centers attached to long alkyl chains [145,146]. In particular, changing the oxidation state of the redox center results in the rapid desorption of the adsorbed layer, a process that is readily followed using the TSM oscillator's mass sensitivity. For sufficiently long alkyl chains, the process slows enough that the kinetics of desorption can be followed in real time. When a thiol or disulfide moiety is attached to the redox center, the resulting chain will chemisorb strongly on silver or gold electrodes on the surface of the TSM oscillator, allowing the oxidation state of the redox center to be changed without desorption occurring. In this case, it is possible to monitor the flux of charge-compensating ions into and out of the film during the electrochemical process [147,148]. Karpovich and Blanchard [149] also looked at adsorption kinetics of SAMs using TSM devices. They found that the adsorption was rapid and was described by the Langmuir adsorption isotherm. Based on a concentration dependence of the monolayer formation rate, they concluded an equilibrium exists for the adsorption/desorption process.

4.3.3

ADSORPTION AT SOLID-LIQUID INTERFACES

Adsorption from liquids can involve both physisorption and chemisorption; however, one needs to consider the molecular interactions between the adsorbing species and the liquid, as well as those with the solid. The relative magnitudes of the liquid/adsorbate and solid/adsorbate interactions will determine the extent

-0

0 - N2 A

E

O H 3 ( 0 H 2 ) 1 5 SH in N 2 - 0.2

-5

r "I m, m

- 0.4 ::= m,mml

O

-I0

~')

- 0.6 -15

O O

- 0.8

O m

Q r

(/)

-20

- 1.0

13" 0 ,-,- -25

- 1.2 I

I

!

I

0

20

40

60

......

!

80

,I

1O0

,!

120

! .....

140

Time (min) Figure 4.17 Frequency shift as a function of time measured for a gold-coated SAW device exposed to an N2 gas stream containing CH3(CH2)IsSH at approximately 25% of its saturation vapor pressure. Leveling of the frequency shift at approximately 1.15 monolayers indicates that the polycrystalline gold film has a roughness factor of 1.15. The kinetic data fit a simple, first-order Langmuir rate law. (Reprinted with permission. See R e f . [ 1 4 3 ] . 9 1991 A m e r i c a n Chemical Society.)

gr

4.3 Surface Adsorption and Characterization of Porous Films

195

of adsorption at a given concentration. An interesting example of the importance of this concept is adsorption of surfactant molecules, which contain a hydrophilic "head" making them soluble in water, but a hydrophobic "tail" that causes ordering of water molecules. The entropy decrease upon removal of this tail from water upon adsorption at a surface results in an additional driving force for adsorption. This interaction is sometimes called hydrophobic bonding [104]. The ability to use TSM, APM, and FPW devices in contact with liquids enables them to monitor adsorption at the solid-liquid interface. This ability is probably most useful in evaluating the adsorption of polymer or surfactant molecules, since this adsorption is used in a variety of applications [46,104,105]: modification of surface wettability, removal of dirt and oils using surfactants (e.g., detergents), and the dispersion of small particles (e.g., pigments in paints). Limited studies on this ability have been performed with AW devices. Figure 4.18 shows data for adsorption of bovine serum albumin (BSA) onto the surface of an FPW device [150]. Similarly, Laatikainen and Lindstr6m [ 151 ] monitored the adsorption of BSA on a polysulfone-coated TSM device. They found the adsorption trends were in agreement with expectations; however, the absolute magnitude of

5.94

-

[i

i

i

El 9

i i ii 9

i

i 9

i

it

ii

9

iiii qr a

i

iiiiii

i

i / Ii

iii 9

5.93 'k ~ 5.9'2 0r

::2

o" 5.91

5.90 ""~0

0.5

1

1.5

2 Time

2.5

3

3.5

4

(minutes)

Figure 4.18 Response of a 5.9-MHz FPW device as bovine serum albumin (BSA) deposits onto the device surface from a phosphate buffer solution. (Dataprovided by Ben Costello, Berkeley Microlnstruments, Inc.)

196

4. Materials Characterization

the adsorbed amount, calculated based on assuming that the frequency shift was due solely to the mass increase due to the adsorbing BSA, was too large. By allowing adsorption to occur from the liquid and then drying the crystal, significantly smaller values (approximately 30% of those from the in-situ measurements) were obtained. They interpreted the larger response for the in.situ measurement as being due to the fact that the TSM was sensitive to the hydrated mass of the adsorbed layer, and thus, was also detecting the mass of the water and electrolytes bound to the BSA layer. Adsorption of other biomolecules has also been monitored with AW devices. For example, TSM devices were monitored with a network analyzer during adsorption and denaturation of proteins [152] and during adsorption, immobilization, and hybridization of DNA [153]. The multidimensional information was used to look at the amount of adsorption, the kinetics, and to probe the conformation of the protein molecules on the surface [152] or the viscoelastic properties of the DNA layers [153]. A Love plate device was used to follow the kinetics and amount of adsorption of human IgG, anti-human IgG, and protein A onto polymethylmethacrylate polymer layers [154]. Similarly, an APM device was used to measure kinetics and amounts of adsorption for antigen-antibody reactions [155]. The adsorption of a variety of surfactants onto the quartz surface of APM and TSM devices has also been monitored [145,156]. Values in reasonable agreement with the literature were obtained in several cases by assuming that the frequency shift was due to the adsorbed mass of the surfactant. However, anomalous results (i.e., frequency increasing with solution concentration) were also observed under some conditions. These studies demonstrate that one of the challenges in using AW devices for monitoring adsorption at the solid-liquid interface is the ability to quantitatively convert the device response to an adsorption level (mass/area or moles/area). Additional responses are the result of the sensitivity of these devices to such effects as the mass of bound or entrained liquid, changes in the liquid properties in the vicinity of the surface due to changes in the electric double layer (the layer of ions formed around a charged surface [104]), or modification of the wettability of the surface (this has been proposed to result in slipping of the liquid layer at the surface [ 157]). Even with these challenges, the simplicity and speed of measurement possible with an AW device as compared with other measurement techniques can make these studies useful even if only to evaluate adsorption trends quickly. The sensitivity of AW devices to these additional effects, which here is put forth as a complication, may be used to advantage to study these effects. For example, the ability to probe a thin liquid layer near a charged surface may permit study of the liquid properties in the electric double layer. This layer is typi-

4.4 Real-Time Monitoring of Chemical and Materials Processes

197

cally less than 10 nm thick [104] and thus is difficult to monitor with conventional techniques. As demonstrated in some of the studies described [152,153], monitoring of more than one sensor response can be helpful in determining the amount of adsorption and the properties of the adsorbed layer.

4.4

Real-Time Monitoring of Chemical and Materials Processes

Because acoustic wave devices are sensitive and respond rapidly, they are ideally suited for real-time monitoring of chemical and physical systems. As discussed in the introduction to this chapter, thin films represent a growing industrial and technological concern for a variety of applications. The use of acoustic devices to characterize the physical properties of these films has been dealt with in the previous sections. Here we describe how these devices can be used to monitor film formation or dissolution processes, or to observe and characterize film properties as a function of time (similar to the monitoring of diffusion in polymers described in Section 4.2.2). 4.4.1

MONITORING FILM AND PARTICLE DEPOSITION AND REMOVAL

Perhaps the simplest application of real-time monitoring is to monitor the deposition or removal of a thin layer. The use of TSM devices for deposition monitoring is well known, and commercial instruments using TSM devices are routinely used for such applications as monitoring the thickness of metal films formed by vacuum evaporation, as shown in Figure 1.5 [158,159]. Other types of deposition can be monitored; for example, the deposition of Langmuir-Blodgett (LB) films on TSM devices was studied by McCaffrey, et al. [160]. They observed a linear decrease in the resonant frequency of the device as multiple layers of a calcium stearate LB film were deposited. From the slope of the resulting plot (Hz/layer), they calculated a surface mass density in the film of 322 ng/cm 2, significantly greater than the theoretically calculated value of 266 ng/cm 2. This result suggested that water was trapped in the film during formation or that the surface topography of the substrate may provide a slightly larger effective surface area for film formation than previously assumed. In addition, the straight-line plot had a negative intercept, from which they concluded that the first layer of film was less dense than subsequently deposited layers. Wohltjen, et al. [161 ] also measured SAW frequency shifts as a function of the num-

198

4. Materials Characterization

ber of LB layers. Using devices operating at three different frequencies (31, 52, and 112 MHz), they were able to verify the dependence of SAW response on the square of the operating frequency (see Section 3.2.4). However, the absolute magnitudes of the responses were significantly lower (about 50%) than predicted based on mass loading. This discrepancy was interpreted as being due to elastic effects in the LB film. In contrast, Grate and Klusty [ 162] found excellent agreement with predictions based on mass loading using 2t)0-400 MHz SAW resonators and LB layers of soft organic polymer materials. In two final LB studies, Hanley, et al. [163] used TSM devices to estimate the molecular area and percent ionization of LB layers composed of fatty acids, while Ariga and Okahata [164] used TSMs to determine the amount of water incorporated into LB films as a function of parameters such as deposition rate and surface roughness. The use of SAW and APM devices for real-time monitoring of film deposition processes is demonstrated by the results shown in Figures 3.22, 3.26, and 3.34 for the vacuum evaporation of thin metal films. In these examples, the film thickness and deposition rate was monitored with submonolayer accuracy. In addition, for the deposition shown in Figure 3.26, the conductivity of the film could be probed along with the film thickness by monitoring both the velocity and attenuation of the wave, and relating these total responses to the combined effects of the mass-loading and the acoustoelectric interactions. For monitoring deposition of viscoelastic polymers, it should be possible to probe the viscoelastic properties of the film being deposited based on the relative magnitudes of velocity and attenuation (or frequency and admittance) values. This ability to probe film properties while monitoring the deposition rate makes AW devices uniquely suited for studying film deposition processes. Conversely, AW devices can effectively track the loss of film material as the result of some chemical or physical process. For example, Joshi utilized a dual delay-line SAW oscillator to monitor the plasma-etching of a photoresist film [165]. One delay line was coated with the photoresist film, while the other delay line was left uncoated to act as a reference. Because plasma-etching produces significant heating of the substrate, the dual-device approach was used to correct for temperature-induced frequency changes: changes in frequency difference between the two delay-lines were attributed to film loss from the device surface. The TSM has also been used to study sputtering yields [ 166] (i.e., loss of material from a surface under ion bombardment) by monitoring frequency changes due to mass loss as a function of ion dose. While these studies have usually been performed using metal or silicon films [ 167], recent studies have investigated the sputtering process associated with ion bombardment of organic films. UUevig and Evans monitored the frequency of TSM devices coated with polystyrene (PS)

4.4 Real.Time Monitoring of Chemical and Materials Processes

199

and polymethyl methacrylate (PMMA) films during bombardment with 1 keV argon ions [168]. In addition, TSM devices have also been used for laser ablation studies. In one study, the TSM sensor served as a precise probe of etch rates in the region of the ablation threshold intensity [ 169]. In another study, the TSM was used to measure variations in the amount of sample ablated for normalizing results of laser ablation-inductively coupled plasma atomic emission spectrometry [ 170]. AW devices have also been demonstrated for real-time monitoring of attachment of particles. For example, early work looked at the ability of TSM devices to measure particles and found that detection efficiency decreased with particle size above 2/zm on uncoated devices but could be improved for these larger particles using viscous coatings [171,172]. The use of an electrostatic precipitator was demonstrated for determining mass concentrations of suspended particles [ 171,173], while a portable instrument was demonstrated for monitoring changes in particle concentrations due to short-term changes such as starting a machine or the presence of a smoker [171,174]. In more recent work, SAW devices have been used to increase the sensitivity for mass detection [ 175]. In addition, TSM devices have been combined with a cascade impactor to provide data on size and amount of particles. This particle-sizing capability has been used to measure changes in stratospheric particle concentrations due to volcanic eruptions [ 176-178]. In a serendipitous experiment designed to measure removal of SiO2 layers by cosmic radiation, a TSM device on Saliut-6, a Soviet-Romanian joint space flight, was able to quantitatively measure accumulation of cosmic dust after a solar explosion [28]. The delay time from the explosion to the start of deposition enabled an estimate of the solar wind at 697.7 km/hr. The lack of a frequency increase indicative of mass loss prior to the arrival of the cosmic dust also demonstrated the good stability of SiO2 in space. Real-time monitoring of deposition or dissolution from liquids can be performed using TSM, APM, or FPW devices. For example, Kanazawa and Doss [ 179] demonstrated the use of TSM devices as real-time rate monitors of electroless nickel deposition, while Ricco and Martin [ 180] demonstrated a similar use of APM devices for electroless deposition of copper directly onto the quartz substrate. TSM devices have also been used to monitor surface accumulation of thermal degradation products from jet fuels [181]. The ability to rapidly quantify the rate of accumulation can be used in designing jet fuels with improved thermal stability [182,183]. Turning to material removal, monitoring the dissolution of metals in aqueous environments has also been demonstrated [ 184]. In addition, by intentionally contaminating a TSM with a surface layer similar to the anticipated contamination on a component to be cleaned, a real-time moni-

200

4. Materials Characterization

tor of contaminant removal has been demonstrated [ 185]. This capability could be used for optimizing cleaning processes, as an in-line process monitor, or as a probe to determine cleaning efficiency in order to extend the life of cleaning baths. An important parameter for many photoresist materials is the rate of dissolution in the developer as a function of light exposure. Hinsberg et al. [ 186] utilized TSM devices in a special cell that enabled flow of developer across a photoresist layer on one side of the device. They demonstrated the ability to monitor the removal of the photoresist material in real time as a function of the exposure. Rapid dissolution rates could be monitored (e.g., total removal of the layer in less than 30 seconds), as well as verification of the lack of removal of the unexposed resist. Using a previously formed conductive layer, electroplating processes can also be monitored. Of the AW devices discussed in this book, TSM devices have an advantage for this application since the electrode metalization (used for AW transduction) can also be used as an electrode for the electroplating process. Commercial instruments using TSM devices are currently available for monitoring the thickness of electrodeposits (see Appendix D). Electrodeposition has been used for calibrating FPW sensors while they are in contact with a liquid, since liquid loading affects their gravimetric sensitivity [ 187] (see Section 3.4.2.3). Using APM devices, the ability to use both velocity and attenuation to simultaneously measure film thickness and probe deposit roughness has been demonstrated [ 188]. The film roughness is detected by an increase in the attenuation observed at higher electroplating currents, interpreted as being due to the generation of compressional waves from the roughened surface. The ability to probe surface roughness with TSM devices by monitoring the admittance magnitude has also been demonstrated (see Section 3.1.6). AW devices have also been used in solution to measure the attachment or binding of various biomaterials. For example, TSM devices have been used to measure attachment of cells to the surface of the TSM [ 189-191 ]. As might be expected, the cells were found to behave viscoelastically [ 189]. In addition, the removal of cells by a virus infection has also been monitored [ 190]. By improving stability to enable long-term monitoring, the slow attachment and growth of biofilms was monitored in one study [192]. 4.4.2

POLYMERIZATION REACTIONS

Reaction rate and time to completion are two important yet difficult-to-measure parameters in many polymerization processes. The mass changes that" occur during polymerization provide a velocity or resonant frequency response for a

4.4 Real-Time Monitoring of Chemical and Materials Processes

201

polymer-coated AW device. In addition, the formation of cross-links affects the elastic properties of a polymer. The free movement of one polymer chain relative to its neighbors, which is possible in the uncrosslinked polymer, is severely inhibited by the formation of crosslinks between chains. As a result, crosslinking significantly increases the film's shear modulus while simultaneously decreasing viscous damping in the polymer [193]. Thus, the shear modulus of the film gives an indication of the crosslink density and the extent to which a polymerization reaction has proceeded. The combination of these elastic property changes with the mass changes that occur during polymerization enables the use of acoustic wave devices to monitor polymer film crosslinking reactions in situ and in real time. For example, Hager et al. [ 194] demonstrated the potential of a TSM for characterizing the curing process for a gelatin-based photoresist film. By monitoring amplitude and frequency of the acoustic wave simultaneously during the photocuring process, they were able to differentiate between solvent evaporation and photo-hardening processes occurring in films of varying thickness. The early stages of film drying were characterized by concurrent increases in both amplitude and frequency, consistent with solvent evaporation. The loss of solvent mass produces an increase in frequency, while the effective increase in viscosity and density of the film as the film "dries" was thought to result in a decrease in energy dissipation in the film (i.e., output amplitude increases). By contrast, illumination of a well-dried film produces further increases in frequency at essentially constant amplitude. This was interpreted as an effective mass-increase due to greater coupling efficiency for photo-crosslinked films. Martin and Ricco [ 195] monitored photo-polymerization in negative photoresist films (Waycoat HR-100 [196]) using APM devices. Shown in Figure 4.19 (page 202) are changes in APM propagation velocity and attenuation measured as the film crosslinks upon exposure to 380 nm illumination. This wavelength is near the peak spectral sensitivity reported [ 196] for photo-polymerization of HR100. During the reaction, APM velocity increased while attenuation decreased. After an exposure time of 140 minutes, velocity and attenuation reached stable values, indicating the completion of the cross-linking reaction. Martin and Ricco state that each cross-link formed in the HR-100 film has two effects that can perturb the APM propagation velocity: an increase in the elastic storage modulus (G') and a decrease in the surface mass density (ps) through the liberation of two N2 molecules. Both of these effects should result in an increase in APM velocity, consistent with the positive velocity shift observed during cross-linking. In addition, cross-linking typically decreases the loss modulus (G"), a result of restricting dissipative processes [193]. This is consis-

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EXPOSURE ENERGY ( 3 / c m 2 ) Figure 4.19 Fractional velocity shift and change in attenuation as a function of time, measured during cross-linking of HR100 negative photoresist film. (Reprinted with permission. See Ref. [ 1951.)

4.4 Real-Time Monitoring of Chemical and Materials Processes

203

tent with the decrease in attenuation observed during cross-linking. A plot of attenuation vs velocity change [ 195] is linear during this cross-linking process, indicating a linear relationship between incremental changes in the storage and loss moduli. More work is needed to develop a realistic model to relate cross-link formation to changes in viscoelastic parameters. This model would permit the evaluation of film properties as a function of cross-linking. The wavelength dependence of the photosensitivity of the resist film was determined by measuring the rate at which APM velocity changed in response to cross-linking as a function of incident optical wavelength A0. The rate of change of APM velocity at each wavelength is defined by" R(h~o) =

1 dv Vo dt

hO"

(4.7)

Wavelength was stepped from 200 to 650 nm in 10 nm increments. The 15second dwell time at each wavelength was sufficient for a change in film properties to be measured without appreciably cross-linking the film: the entire spectral response experiment cross-linked the film by less than 20%. The relative spectral sensitivity for film cross-linking is R/I, where l(h0) is the spectral density of the source. A plot of R/I vs optical wavelength is shown in Figure 4.20 (page 204) for an HR-100 photoresist film. The peak in spectral sensitivity is at approximately 370 nm, in reasonable agreement with the sensitivity peak of 360 nm cited by the manufacturer [ 196]. Also shown in Figure 4.20 is a UV-visible absorbance spectrum recorded for a similarly prepared HR-100 film on a quartz optical flat. The absorbance spectrum has a peak at 355 nm, corresponding to the peak observed in the acoustic measurement of R/I. The film also exhibits very high absorbance at wavelengths below 300 nm, a result not observed in the acoustic measurement. The spectral response measured using an acoustic device is an action spectrum in that it registers only optical absorptions that lead to a chemical reaction and thereby contribute to changes in the elasticity or density of the film. Not all absorbed photons lead to chemical reaction. The peak in the absorbance spectrum (Figure 4.20) at 355 nm is due to photoinitiator absorption, which leads to the formation of cross-links in the polymer and a corresponding peak in the acoustic R/I response at 370 nm. The strong absorption at wavelengths less than 300 nm, however, is likely due to the polymer backbone (cyclized polyisoprene), which is not photoactive; thus, its optical absorption affects neither mass density nor the elastic properties of the film. This method of obtaining cross-linking photoaction spectra in real time using acoustic wave devices is a unique analytical technique for characterizing photoactive films.

9

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WAVELENGTH (nm) Figure 4.20 Acoustically probed photoaction specmnn of 1.I/zm thick HR-100 photoresist (left axis); rate of change of APM velocity R(Ao), normalized to spectral density l(Ao), indicates rate of film cross-linking as a function of optical wavelength Ao. Optical absorption spectrum of identical film on quartz flat (right axis); spectrum peaks at 355 nm due to absorption by photoinitiator, while strong absorption below 300 nm is due to the polymer backbone. (Reprintedwith permission.

See Ref. [195].)

mmm

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4.4 Real-Time Monitoring of Chemical and Materials Processes 4.4.3

205

CORROSION MONITORING

The ability to use SAW devices for real-time monitoring of corrosion of thin metal films in gaseous environments has been demonstrated. Metal films, ranging in thickness from a few Angstroms to a micron, were deposited on the propagation path between transducers. Since the acoustic-wave propagation velocity depends upon the mass and mechanical properties of the film, any change in these properties due to corrosion alters the velocity. Results with a copper film that was vacuum-deposited between the transducers on a 100-MHz SAW device upon exposure to 5 ppm of H2S in dry nitrogen are shown in Figure 4.21 (page 206). During this initial sulfidation process, an overall frequency shift of - 2 3 ppm occurred. If elastic effects are negligible, this would correspond to a mass change of 180 ng/cm 2. The presence of a minimum in the response may be due to the presence of adsorbed H20 that is formed by the reaction of H2S with a surface layer of copper oxide. On this time scale, the reaction appears to stop after only a few minutes. However, experiments for longer times indicate that after this relatively fast reaction and apparent stabilization upon initial contact with H2S, the frequency again begins to decrease [ 184]. For example, in another experiment with a copper film challenged by 100 ppb H2S, 150 ppb NO2, and 60% relative humidity at 40 ~ a second frequency decrease began after about 10 hours of exposure. The rate of corrosion, assuming that mass loading dominates, ranged from 22 ng/cm2-hr at the early stages of this linear decrease to 15 ng/cm2-hr at the end of this eightday experiment. In addition, a copper film passivated by a corrosion inhibitor (benzotriazole) showed a much slower rate of corrosion; frequency shift (after the initial rapid decrease) is approximately 7% of that observed with the untreated copper film. Hager et al. used TSMs as electrode substrates to monitor corrosion processes occurring at the surface of an iron-coated electrode [ 197]. The frequency response of a TSM in solution depends on the combined effects of mass gain, film stress, hydrodynamic coupling, and electrokinetic coupling. Since the dependence of the electrokinetic coupling on solution conditions is not presently well-modeled, interpretation of data is problematic for systems where electrokinetic effects may be significant. For system responses dominated by mass or hydrodynamic coupling effects, however, the TSM can be used to study corrosion processes quantitatively. These results demonstrate that acoustic wave sensors can be used as real-time corrosion monitors. Their small size enables them to be used in situ in corrosive process environments or as a component to warn of the impending failure of a

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TIME ( s e c o n d s ) Figure 4.21 Monitoring of the corrosion of a copper film by a dilute H2S stream. The corrosion reaction can be seen in real time using the SAW device as a sensitive probe. ( R e p r i n t e d w i t h p e r m i s s i o n [10} b y c o u r t e s y o f M a r c e l D e k k e r , Inc.)

4.4 Real-Time Monitoring of Chemical and Materials Processes

207

circuit or structure. In addition, they should prove useful in research by providing real-time kinetic data on corrosion rates. Extreme sensitivity allows rapid evaluation of even relatively slow corrosion processes, as well as testing of corrosion inhibitors. 4.4.4

ELECTROCHEMICAL STUDIES

There is a growing body of literature on the use of acoustic sensors for in-situ experiments, particularly in conjunction with electrochemical measurements [198]. These studies are most commonly performed using a quartz resonator (TSM) device that has been designed or adapted for use in solution. When used in this manner, these acoustic devices are commonly referred to as electrochemical cluartz crystal _microbalances (EQCM). To be consistent with the terminology used in this book, these devices will be referred to here as electrochemical TSMs or ETSMs. In a typical experiment, one of the electrodes used to excite the acoustic wave mode in the substrate is used concurrently as the working electrode in an electrochemical cell. This configuration allows for the simultaneous monitoring of electrochemically-induced mass changes at the electrode surface and the consumption of charge (electron flow) associated with the electrode process/reaction. An example of some typical results is represented in Figure 4.22 (page 208). The ETSM provides information on changes in surface mass (film or solution property changes can also be monitored) concurrent with the measurement of the current-voltage properties. This ETSM information is useful in the characterization of charge-transfer electrode processes, including the influence of the solution/electrode interface (e.g., depletion layer effects [199]), solvent effects (e.g., differences in frequency shifts for H20 and D20 have been used to probe hydration of polymer layers [200,201]), and kinetics [202-204]. The diversity of ETSM studies is evident in several recent reviews [ 198,205,206] and references contained therein. Some examples of these applications are briefly discussed in the following paragraphs. The high mass sensitivity of ETSM sensors renders them particularly suited for the analysis of monolayer and submonolayer films. In fact, the earliest applications of the ETSM involved studying the electrochemical deposition of monolayers, including the formation of metal oxides [207], electrosorption of halides [208], and the underpotential deposition of metal atoms [209-213]. In some cases, the electrovalency (i.e., the ratio of moles of electrons transferred at the electrode to moles of adsorbate deposited) was found to vary with adsorbing species; the adsorption of iodide onto gold, for example, occurs with complete charge transfer from the halide to the electrode, whereas the adsorption of bro-

208

4. Materials

Characterization

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Figure 4.22 Simultaneous idealized representation of ETSM processes. Curve a (solid line) represents the current flow in the cell for a redox cycle, while curve b (dotted line) represents the concurrent frequency shift associated with the adsorption (forward arrow) and desorption (reverse arrow) of mass at the electrode surface during the redox electrode reaction. The reference electrode is a saturated calomel electrode (SCE). mide on gold occurs with only partial electron transfer [208]. Studies of reductive desorption of self-assembled monolayers also provided key information on processes occurring at the electrode surface [214]. The ETSM has been used to study the adsorption characteristics of surfactants containing electroactive groups, and how the adsorption and micellization of these species could be altered by electrochemical reactions at the electrode [ 145]. Bubble evolution during electrochemical processes has also been investigated [215]. In addition, these devices have been used to study thick (multilayer) films of electroactive materials, including charge-transfer salts [216], electrochromic (colored) materials [217,218], and a variety of oxide films [219-221]. The use of the ETSM to study polymeric systems, especially redox and conducting polymers, is a rapidly growing area of research. It has been used to elucidate the mechanisms of film formation, ion and solvent transport phenomena, and compositional changes that occur in these films upon redox cycling. Among the redox polymer systems that have been studied are poly(vinylferrocene)

4.4 Real.Time Monitoring of Chemical and Materials Processes

209

[222-224], nickel ferrocyanide [201,225,226], and ferrocene-siloxane copolymers [227]. Conducting polymer systems that have been investigated include poly(pyrrole) [228-230], poly(aniline) [203,231], poly(bithiophene) [232,233], and a variety of co-polymer and composite polymer films [231,234]. Because viscoelastic effects may come into play for polymer films, special care must be taken in the interpretation of frequency data for these materials [227,233]. 4.4.5

MISCELLANEOUS M A T E R I A L S / PROCESS CHARA CTERIZA TION

The use of AW devices to monitor relaxation and phase transitions in polymeric materials was discussed previously. These devices can also be used to monitor phase transitions in other materials. Changes in the general fluidity or viscoelasticity upon transition from solid to liquid crystalline (LC) states can be detected as frequency changes in TSMs coated with these materials. Okahata and Ebato [235] studied the behavior of smectic and nematic LC materials, as well as lipid bilayers, as the ambient temperature was increased above the phase transition temperature (Tc). For the smectic LC phases, they observed a dramatic increase in frequency upon transition to the LC state, whereas for the nematic phase, there was minimal frequency change at To. From these results, they concluded that the smectic materials form multilayers that are oriented perpendicular to the crystal surface, while the nematic materials do not form a similarly layered structure. Below To, the solid coating vibrates with the crystal. Above To, the multilayered smectic LC coatings become fluid and slippage occurs between layers, the effective mass coupled to the surface decreases, and the AW-device frequency increases. This behavior is observed for smectic LC materials in both air and in water, and is readily reversible. Similar behavior is observed for lipid bilayers in contact with aqueous solution; for films in contact with air, however, no frequency change at Tc is observed. Miramatsu and Kimura [236] also used a TSM to study transitions in Langmuir-Blodgett films with two phase transitions, and their results suggested that the viscosity and mass increased at the first phase transition, while the elasticity decreased and viscosity increased at the second transition. In addition to the characterization of polymers and liquid crystals, AW devices have been used in the characterization of layered metal films, especially superlattices [237-239]. Metal superlattice structures are of interest because they display interesting characteristics including, in some cases, superconductive properties. Superlattice metal films can exhibit a "supermodulus effect" (i.e., an increase in the Young's and torsional modulus) or an anomalous softening (---35%)

210

4. Materials Characterization

in the elastic shear modulus compared to single-metal films. The correlation of anomalous elastic behavior (i.e., modulus changes) with the layer dimensions and/or composition of these films can provide insight into the superconductivity effect. The moduli of these films are usually determined by Brillouin scattering, an optical technique requiring an extremely high-quality surface. The SAW technique permits the nondestructive evaluation of thin superlattice films to obtain moduli data that are comparable to Brillouin scattering results. The SAW has also been used to monitor the effects of contaminants or dopants on the elastic behavior of superlattice structures [240]. Acoustic sensors have also been reported as alternatives to existing pharmacochemical animal test procedures. Okahata and Ebato [241] report on the use of lipid-coated TSM devices as an alternative to the Draize test, which involves using animals to predict eye-irritant potential. As shown in Figure 4.23, they found excellent agreement between the partition coefficient obtained from the TSM response and the Draize score for a variety of surfactants. These results indicate the potential for using AW devices as an inexpensive, fast, and humane alternative to the controversial Draize test. Kurosawa et al. [242] used actomyosin from fresh carp and monitored the changes when adenosine 5'-triphosphate (ATP) was added, and noted changes that were not observed with aged actomyosin, potentially making this useful for a meat-freshness sensor. Some final notes refer to studies showing real-time monitoring of chemical modifications of films. For example, acoustoelectric interactions with metalcoated SAW devices have been shown to be useful for probing chemisorptioninduced mass and conductivity changes [243]. TSM devices have been used to monitor DNA hybridization [244]. CdS particle formation in cadmium arachidate Langmuir-Blodgett films upon exposure to hydrogen sulfide [245]. SAW devices have been used to determine rate laws and activation energies for the reaction of styrene vapor with an organoplatinum complex in a polymeric matrix [246]. Finally, using TSM devices to monitor uptake of water and calcium ions in phospholipid LB films as a function of temperature, significant differences in uptake above and below the phase transition temperatures have been observed [247,248].

4.5

Summary

For some film-characterization techniques, the sensitivity of AW devices to film mass density allows these devices to be used as sensitive microbalances (nanogram mass changes can be effectively quantified). This thin-film mass balance can be used to monitor absorption into polymers and adsorption onto sur-

4.5 Summary

211

50 A

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Ik==

O O

30

O Ik==

O O

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13 Ik== E3

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O

0 2.0

2.5

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log 10 P Figure 4.23 Comparison for the same surfactants of the partition coefficient (P) obtained from TSM measurements with the Draize score (cornea) obtained from animal e x p e r i m e n t s . (Reprinted with permission. See Ref. [241]. 9 1991 American Chemical Society.)

faces. By measuring nitrogen adsorption isotherms, thin-film surface area and microstructure can be evaluated. AW devices can also be used for real-time monitoring of: (1) diffusional transients (allows the rapid evaluation of diffusion coefficients as low as 10 -15 cm2/sec), (2) film deposition processes, and (3) corrosion and etching processes. For other film characterization techniques, the sensitivity of the AW device to the mechanical or electrical properties of a film is used. The presence of two sensor responses (i.e., velocity and attenuation or, equivalently, the frequency and admittance magnitude) has the significant advantage that the amount of in-

212

4. Materials Characterization

formation about the film is increased. This additional information helps in discriminating between sensing mechanisms and applying models for the interactions that allow the evaluation of thin-film properties. For example, these two responses enable one to determine thin-film viscoelastic properties and glass transition temperatures. In addition, they facilitate the understanding of dynamic processes such as polymer crosslinking and corrosion reactions. The results presented here demonstrate that thin films can be characterized based on acoustical monitoring of changes in film mass density, conductivity, and viscoelasticity. Additional sensing mechanisms are available to probe film properties. Some examples are thin-film dielectric constant, stress, and structure (e.g., roughness). Some of these sensing mechanisms will be hard to quantify since they involve a complex interaction (e.g., wave attenuation based on wave scattering due to film roughness); however, they may still be useful to provide a qualitative monitor based on empirical data. In conclusion, the results presented in this chapter demonstrate the extreme versatility of AW devices for the characterization of materials. The inherent sensitivity of AW properties to the mechanical and electrical properties of thin films can be used to advantage to directly monitor a wide variety of film properties. Since the properties and behavior of thin-film materials can be very different from those of similar bulk materials, this ability to directly measure thin film properties can be a significant advantage in materials research and development. The ability to use thin films instead of bulk samples has the added advantage that the time required to perform an evaluation of dynamic processes such as diffusion and corrosion can be greatly decreased. The number of applications of AW devices to thin-film characterization continues to increase, and is limited only by the ingenuity of AW device researchers and developers.

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Chapter 5

Chemical and Biological Sensors

5.1 Introduction Preceding chapters have described the detailed operating principles of acoustic wave (AW) devices and how these devices can function as sensors of various physicochemical phenomena in surrounding media. This chapter describes the extension of these capabilities to the detection and quantitation of chemical and biochemical species. An introduction to the fundamental background of various important physical and chemical interactions is presented for those not especially familiar with these topics. Coated bulk-wave oscillators operating in the thickness shear mode (TSM quartz devices, commonly referred to as quartz crystal microbalances--QCMs) were first demonstrated as viable organic vapor sensors by King in 1964 [ 1]. The subsequent proliferation of reports on this class of TSM sensors for the measurement of airborne and dissolved species has been reviewed by several authors [2-8]. The advent of the interdigital transducer provided the capability for generating surface waves in piezoelectric media [9,10], and the development of chemical sensors based on surface acoustic-wave (SAW) technology soon followed [11,12]. Compared to TSM chemical sensors, coated-SAW sensors are relative newcomers to the chemical sensor arena. The advantage of higher sensitivity with SAW sensors, coupled with the ability for mass production using planar microfabrication methods, are key factors responsible for the increasing interest in this class of chemical sensors, which have also recently been reviewed [13-15]. Shearhorizontal acoustic plate mode (SH-APM) sensors [16], and flexural acoustic plate wave (FPW) sensors [ 17-20] can be considered close relatives of SAW de-

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Copyright 9 1997 by Academic Press All rights of reproduction in any form reserved. ISBN 0-12-077460-7

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vices wherein a different type of wave or mode is employed. They are the most recently developed AW sensors and their capabilities for chemical sensing have only recently been characterized. With a few notable exceptions, obtaining adequate sensitivity and selectivity for the measurement of a given analyte requires a chemical or biochemical interface, henceforth referred to as "the coating." The coating, which should be physically or chemically bound to the sensor surface, may consist of a solid adsorbent, a chemical reagent, or a sorptive liquid or polymer. The coating acts as a chemically sensitive and selective element that immobilizes a finite mass of some chemical species from the environment. Resultant changes in physical and/or chemical properties of the coating, in turn, perturb the underlying AW device. Perturbations of AWs resulting from interactions of the coating with one or more analytes constitute the basis for detection and quantitation. The ultimate performance of the sensor depends on both the device configuration (substrate material, acoustic mode, operating frequency) and the nature and extent of coating-analyte interactions. Perturbation mechanisms for the various acoustic devices were discussed in general terms in Chapter 3. In this chapter, these mechanisms are reviewed specifically in the context of chemical and biochemical analysis. Performance criteria are discussed, and the fundamental coating-analyte interactions giving rise to sensor responses are presented as a basis for classification. Relevant physical and chemical models of these interactions are described, and examples of analytical applications employing each type of interaction are given to illustrate their advantages and limitations. While references have been included to illustrate specific points, this chapter is not intended to comprise an exhaustive review of the literature, particularly for TSM resonators, for which the number of references is far too great to be fully reviewed here. For more detailed information on the diversity of sensor applications, the reader is referred to the many review articles that have been published on these topics [2-8,13-15].

5.2

Detection Mechanisms

The detection of chemical analytes can be based on changes in one or more of the physical characteristics of a thin film or layer in contact with the device surface. Some of the intrinsic film properties that can be utilized for detection include mass/area, elastic stiffness (modulus), viscoelasticity, viscosity, electrical conductivity, and permittivity. In addition, changes in extrinsic variables such as

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5. Chemical and Biological Sensors

temperature and pressure can also produce a sensor response, affecting the AW either directly or via changes in the film's intrinsic properties. Not all detection mechanisms are of practical significance for all types of sensors, and several mechanisms can operate simultaneously (synergistically or antagonistically) to affect a response. The range of detection mechanisms is thus a double-edged sword that leads to a high degree of versatility and a broad range of potential analytes, but that can also make interpretation of sensor responses difficult in certain cases. The development of equivalent circuit models or network analysis has recently been used to assist in the evaluation of sensor response behavior, and can provide insight into the relative importance of a given transduction mechanism for a specific sensor application. As with the materials characterization applications described in Chapter 4, the transduction mechanisms employed in analytical applications involve changes in the velocity and/or the amplitude of the acoustic waves. Although wave velocity is generally the preferred measurement, wave attenuation sometimes provides an additional indicator of a particular interaction and, importantly, aids in distinguishing one sort of perturbation from another [21,22]. In many cases, the relevant perturbation mechanisms can depend on the nature of the environment contacting an AW sensor. For example, all the AW devices discussed in this book have been examined in detail in vapor-phase environments and, under typical laboratory conditions, the density and viscosity of the contacting gas(es) are of little consequence. In contrast, the effect of contact by a liquid phase is such that only TSM, SH-APM, and FPW sensors operate effectively in such an environment ~ a consequence of the nature of the waves they propagate. For SAW devices, propagating waves are excessively attenuated by liquids, leading to inefficient operation [16,23]. SH-APM and FPW devices have the added advantage that acoustic energy is present on the surface opposite where the electrodes are placed, permitting isolation of the electrical contacts from a liquid-phase environment. The nature of the sensitive coating ~ both its inherent physicochemical properties and the physical particulars (thickness, uniformity, etc.) of a specific layer deposited on a specific device ~ often influences the detection process. Thickness, uniformity, and other characteristics are affected by the method of deposition, be it painting, dipping, solvent casting, spraying (air-brushing), spin-casting, or subliming. These methods are detailed in Chapter 6. LangmuirBlodgett and self-assembling monolayer films represent a special case wherein molecularly ordered film layers are deposited [24-26]. Coating uniformity over the device surface can vary greatly with the deposition method. For situations

5.2 Detection Mechanisms

225

involving equilibrium sorption, only the average thickness is important, so film nonuniformities can be tolerated within limits. Particularly for liquid-phase analysis, adhesion of films to the sensor surface can be problematic, requiring chemical immobilization (i.e., chemical bonding) of the coating on the sensor surface. Such methods of attachment are fairly well developed [ 16,27-29], although some limit the coating thickness that can be deposited, thereby limiting analytical sensitivity. The selection and/or design of a coating appropriate for a given sensing application requires consideration of the predominant detection mechanism(s) to be utilized. These are discussed in the following section, along with the relevant coating properties that affect chemical sensor response. 5.2.1

MASS LOADING

Of all the detection mechanisms listed previously, changes in the wave velocity (or, equivalently, oscillation frequency) by the addition or removal of mass at the surface of the sensor is the most commonly used and easily interpreted. Simple mass loading perturbs the wave velocity without producing attenuation effects, a fact that distinguishes it from the other modes of detection discussed in the following. Furthermore, this effect is accessible to all acoustic-wave modes. As a result, the vastmajority of analytical applications have sought to utilize changes in mass loading. TSM and SAW sensors have often been referred to in the past as "gravimetric sensors" because of the supposed predominance of mass changes in causing velocity (frequency) shifts. It is now clear that other factors, including changes in mechanical and electrical properties of the coating, can affect the sensor response, making such a generalization inappropriate in many cases (vide infra). Surface mass changes can result from sorptive interactions (i.e., adsorption or absorption) or chemical reactions between analyte and coating, and can be used for sensing applications in both liquid and gas phases. Although the absolute mass sensitivity of the uncoated sensor depends on the nature of the piezoelectric substrate, device dimensions, frequency of operation, and the acoustic mode that is utilized, a linear dependence is predicted in all cases. This allows a very general description of the working relationship between mass-loading and frequency shift, Afm, for AW devices to be written: Afro = - KSmAmA,

(5.1)

in which Sm is a device-specific constant that depends upon the factors mentioned above, K is a geometric factor for the fraction of the active device area being per-

226

5. Chemical and Biological Sensors

turbed, and AmA is the change in mass/area on the device surface. (Mathematical expressions that give explicit details of the relationship between mass changes and frequency/velocity shifts can be found in Chapter 3.) Note that r can be as simple as the fraction of the center-to-center spacing between IDTs that is coated with a uniform chemically sensitive film in the case of delay lines, or it can be a complex function that relates some nonuniform coating geometry with the (nonuniform) distribution of acoustic energy for resonators. As defined in Equation 5.1, Sm includes dependencies on frequency and substrate thickness of various orders for the different categories of AW devices. Table 5.1 summarizes these dependencies and also gives expressions for Sm that allow the operating frequency or substrate thickness to be accounted for, yielding a numerical value for Sin. The relationship among sensitivity, operating frequency, and substrate thickness was previously a limiting factor in the applicability of TSM sensors. As the plate thickness decreases, the operating frequency and mass sensitivity increase proportionally. The mechanical stability of the substrate, however, constrained the TSM configuration to an upper limit operating frequency of about l0 MHz. Recent evaluation of chemically milled substrates has demonstrated that stable operation at frequencies of 30 MHz can be achieved in liquids [32], and that even higher frequencies may be possible. The last column of Table 5.1 must not be misinterpreted: although a larger S,, value is indicative of a greater inherent mass sensitivity, the minimum detectable mass (MDM) change is limited by the noise level of the AW device-containing circuit, which can vary by orders of magnitude from one device/circuit combination to the next. As an example, the AIN thin-film resonators have the highest Sm value of any of the examples in Table 5.1, a result of their being extremely thin and thus operating at relatively high frequency. But initial results with these devices give noise estimates of nearly 1000 Hz, leading to a MDM of about 1.7 ng/cm 2 [33]. In contrast, the 97-MHz SAW delay lines of Table 5.1 have a shortterm noise level of about 1 Hz, yielding a MDM of 80 pg/cm 2, better by a factor of greater than 20. Noise levels are highly dependent upon the design of the circuit and the RF fixturing (see Chapter 6), as well as the method used to calculate the noise. Comparison of noise levels is, therefore, of limited utility; data on noise levels for these AW devices have been purposely omitted. The operation of AW devices in liquids has been reported and models have been developed to interpret observed behavior for the TSM [37-41 ], FPW [20] and SH-APM devices [ 16]. For sensing in liquids, the effective surface mass depends on the thickness of the liquid/coating layer that is "coupled" to the propagating AW. The thickness of this layer depends on the density and viscosity of the contacting liquid as well as operating frequency. For thin-film, acoustically

Table 5.1 Typical Mass Sensitivities of Acoustic Wave Devices i

Device Description AT-cut quartz T S M resonator

i

i

Examples

Refs.

Theoretical Mass Sensitivity (Sin)a (Hz "cm2/ttg)

Freq. Range (MHz)

Frequency (MHz)

[30]

2.3 .fo ~

5-30 b

6

0.26

[31]

5.7/t 2

30

0.055

[32]

Substrate Thickness (ram)

1111

(Hz-cm2/ttg) 84 2170

Sputtered A1N thin-film

-,i

longitudinal-mode resonator

[33]

0.59 .fo E 16.3/t 2

900-1000

957

0.0055

X-propagating S A W delay lines ST-cut quartz Y-cut quartz

[34] [35]

1.32. fo 2 1.35 9f02 e

25-500

97 112

N/A N/A

12,200 16,900

ST-quartz S A W resonator

[36]

1.26.fo 2

ZnO/AI/Si3N4 F P W delay line

[7]

0.019/t 2 d 0.022/t 2 d

200 4.7

N/A 0.003 c

50,400 2,110

2.6

0.003 e

2,470

ST-quartz S H - A P M delay line i,i

[16]

80/t 2 f

[16]

400It 2 g

25-500 25-500 1-10 25-200

i i

539,000

104

0.20

2,000

158

0.19

11,000

i

ii

,

value tO be used with Equation 5.1. Unless otherwise noted, these values have been experimentaUy confirmed within 5% or better. Multiply by (J'o in M H z ) 2 or divide by (t in mm) 2, as indicated, to obtain working value of S,,,. bConventional wisdom usually places the upper limit for the fundamental mode of quartz TSM devices between 10 and 15 MHz. Alternative device structures, however, can achieve a fundamental frequency significantly greater than this, with operating frequencies of up to 100 MHz. CNot experimentally confirmed to better than 50%, but believed accurate. aExpedmentally confu'med within 7%. eBecause the membranes through which the FPW propagates are three-layer, microfabricated composites, the thickness of each layer, as well as effects of residual strain, must be accounted for to give accurate Sm values. 9 or n -> 1 modes; sensitivity is one-half as great for n = 0 mode (see Chapter 3 and ref. 15). gFor unresolved modes. Value is experimental only, as calculation is not possible for mixed modes (see ref. 25). N/A = not applicable. aS m

228

5. Chemical and Biological Sensors

rigid coatings that are strongly bound to the surface, mass accumulation in or on the coating yields a proportional frequency decrease as in Equation 5.1. Most physical and chemical interactions between analytes and sensor coatings lead to changes of mass. Thus, this sensing mechanism offers the greatest latitude in the selection of sorptive or reactive coating materials, including a wide variety of organic polymers [42]. In addition, the performance of a given coating can sometimes be predicted a priori through knowledge of chemical reactions or by reference to solubility theory and/or appropriate models, as described in Section 5.4. The key challenge to implementing this detection mechanism in a useful sensor is imparting selectivity. Strategies for accomplishing this are discussed in later sections.

5.2.2

MECHANICAL PROPERTIES

Interactions with an analyte can cause changes in the mechanical properties of the coating. An increase in mass loading alone produces a decrease in frequency without affecting attenuation. In contrast, changes in mechanical properties of the coating can produce changes in both the frequency and the attenuation of the AW, as described in Chapters 3 and 4. Furthermore, these changes can either increase or decrease either or both of the two AW propagation parameters, depending on the details of the relationship between film thickness, acoustic wavelength, and the complex modulus of the film at the frequency and temperature of operation. The mechanical properties of a thin film can be generally classified as either elastic or viscous in nature. In many cases, these two properties are so interdependent that treating one without the other is neither practical nor realistic, so they are considered together as viscoelastic properties. The case of purely viscous interactions was treated in Chapter 3 for the contact of liquids with AW devices. A few liquid absorbent films and liquid-like, low molecular weight polymer films (which might be adequately treated as liquids in terms of their physical interactions with acoustic waves), have been examined [43-47]. Similarly, there are some cases where mechanical effects are (almost) purely elastic; this case is discussed next. In general, however, many thin-film materials, including most polymers, must be treated as viscoelastic materials to fully account for their interactions with acoustic waves. Investigation of viscoelastic effects on acoustic wave sensor response represents a particularly active area of research.

5.2 Detection Mechanisms

229

In the instance that mechanical effects can be adequately treated as purely elastic, the frequency of a SAW device is perturbed by modulus changes according to A f e = S ef 2oA - 7

A + 21a,

'

(5.2)

where S e is a constant that depends on the substrate material, h is the coating film thickness, v is the surface-wave velocity, and/x and A are the shear modulus and Lam6 constant (bulk modulus) of the coating, respectively. Note that the presence of the "A" outside the term in parentheses indicates that the change in the entire term is utilized to compute the elastic perturbation. This form is convenient because it applies to either the deposition of an elastic material (in which case h changes from zero to the thickness of the deposited film), or changes in the elastic moduli and/or thickness of a film already present on the device surface. Organic polymers comprise the most common type of coating used with AW sensors due to their capability to reversibly sorb vapors and liquids. For those polymers whose interactions with AWs can be treated as perfectly elastic, the fact that h is invariably larger than/z means that the value of the term (A + /z)/(A + 2/x) is constrained between 0.67 and 1; thus, this ratio can be approximated using a value of 0.84. The magnitude of a purely elastic perturbation is then proportional to the product of shear modulus and thickness, with no more than a 16% error. In much of the work published on the use of polymer coatings for SAW vapor sensing, the polymer's elastic modulus has been considered small enough for modulus effects to be neglected; most of these studies did not, however, consider viscoelastic effects at all. Furthermore, the modulus values assumed in such cases have been based on static or low-frequency determinations. This is a likely source of additional error, because the effective modulus of a polymer increases with the frequency of applied stress. In fact, for (nonacoustic) measurements performed in the range of 1-30 MHz (the highest frequencies reported), shear moduli in the range of 108-109 N/m E are found for many organic polymers that have low-frequency/z values of 106-107 N/m 2 [48-51 ]. Shear-modulus values in this range are sufficiently large so that they must be accounted for if the effect of adding a polymer film to an AW device surface is to be properly modeled. The frequency shift obtained upon deposition of such a coating is smaller ( ~ 10%) than that predicted from mass effects alone, because the sign of the righthand side of Equation 5.2 is opposite to that of Equation 5.1.

230

5. Chemical and Biological Sensors

Just as in the case of film deposition, exposure of a polymer to an analyte must generally be considered in the context of viscoelastic changes in the film. Again, there are some cases in which the perturbation is largely elastic. In such instances, absorption of the analyte more commonly causes a modulus decrease, thereby enhancing the magnitude of the negative frequency shift attributable to mass loading. Recent reports suggesting a greater role for changes in the stiffness or viscosity of the coating film are preliminary and, in some cases, contradictory. Bartley and Dominguez suggested that internal stress created during vapor sorption in polymers might lead to an increase in frequency [52]. They derived an expression for the sensor response that included terms for mass loading and elastic stiffness changes (as in Equations 5.1 and 5.2), as well as a third term to account for the increase in frequency expected if compressive stresses were created within the coating upon vapor sorption. For typical polymers, however, this term is expected to be about an order of magnitude less than the mass-loading term. Interfacial stress was implicated by Thompson et al. to explain transient frequency increases in liquid-phase bioassays using TSM sensors with immobilized coatings [53,54]. Zellers et al. reported on the relative responses of polymercoated FPW and SAW sensors to changes of mass, density, and elastic stiffness [ 18] using a computer model developed previously. Results using high-frequency modulus values for different devices operating at the same acoustic wavelengths indicated that the fractional change of velocity resulting from changes of elastic stiffness were small relative to thickness- and density-induced changes. In contrast, a follow-up of earlier work [55] by Grate et al. compared partition coefficients, K, determined by gas chromatography to those determined using the same stationary phases as coatings on a SAW device [56]. They found that the SAW-derived values were four to six times greater than chromatographically determined values, with the latter reflecting only mass uptake. Typical comparisons are presented in Figure 5.1 for various polymers. Although differences between the chromatographic support material and the planar SAW device surface might account for part of the disagreement, the discrepancies were largely attributed to changes in the modulus of the coating upon vapor sorption and the effect of such changes on SAW frequency. Several factors hinder a complete analysis of these results, including uncertainties in modulus values at high frequency, as well as the effects of absorbed vapors on the moduli as a function of vapor concentration. The conclusion derived from this work was that responses of polymer-coated SAW vapor sensors might be dominated by modulus effects rather than mass-loading effects.

5 A

L.

0 C

I

4

W m nxm DI23

3 E

I

e

rn x

x rn x

2 io

0 ..J

taalb O

1

2

3 Log K (from G LC)

4

5

Figure 5.1 Comparison of K values (partition coefficient) calculated from SAW response data with K values from GLC retention. The K values are for fluoropolyol (FPOL - 1 ) , poly(epichlorohydrin) (PECH - F]), and polyisbutylene ( P I B - x) exposed to a variety of organic vapors. The solid line is the line of perfect correlation. (Adaptedwith permission. See Ref. [56]. 9 1992 American Chemical Society.)

t~ IT i.mo

I',O

232

5. Chemical and Biological Sensors

Other studies have reported frequency increases for polymer-coated SAW devices upon exposure to water vapor at elevated temperatures (<80~ consistent with a predominance of elastic stiffening of the coating [57]. At lower temperatures (25-35~ the same sensors exhibited frequency decreases upon exposure to water vapor of the same concentration, a response consistent with mass loading. Thus, the dominant mechanism may be determined by a variety of factors, including the type of vapor-coating interactions and the ambient temperature. Note that ambient temperature can have profound effects on film viscoelastic properties (see Chapter 4), which in turn can influence the results of vapor exposure. At least some of the controversy might be resolved by applying models that account for viscoelastic and film resonance effects, as described in Chapters 3 and 4. The apparent discrepancies in the studies may arise from the assumption underlying Equation 5.2, namely that the entire coating moves synchronously with the device surface. This assumption is valid only for rigid films (in the case of polymers, glassy or crystalline) that are acoustically thin: the film thickness should be less than approximately 1% of the acoustic wavelength in thefilm I. For thicker films of viscoelastic materials, the assumption of synchronous motion is not generally valid. Martin and Frye [58] recently derived an equivalent-circuit model relating the near-resonant electrical characteristics of a TSM to the viscoelastic properties of the coating. They demonstrated that the resonant frequency of the TSM can either increase or decrease, depending on the relative acoustic thickness of the film. Evidence of film resonance effects has been reported for polymer-coated SAWs exposed to organic vapors [59a]; and a formal film resonance model has been developed for SAW/polymer interactions [59b]. Lu and Lewis [60] also investigated the response of coated TSM devices with respect to elastic effects. They related the frequency response upon coating deposition, and subsequent interaction with the analyte, to the acoustic impedance of the overlay film. The relative importance of the mass-loading and viscoelastic contributions to the observed acoustic sensor response is an issue that has yet to be resolved. Capitalizing on these effects to improve chemical selectivity and detection sensitivity requires further characterization of sensor response, in terms of both velocity and attenuation changes, in addition to more accurate models describing how coating-analyte interactions affect relevant film properties. ~Note that the acoustic wavelengthin the film depends on the velocityof sound in that material and can therefore be very different (often much smaller) than the wavelengthin the AW device substrate. Consequently,the "1% rule" is an approximation,at best, unless correction is made for the acoustic wavelengthin the film.

5.2 Detection Mechanisms

5.2.3

233

RHEOLOGICAL PROPERTIES

AW propagation velocity is sensitive to changes in the density of any medium in contact with the substrate surface. Inspection of the mass-loading term for AW devices (Chapter 3; Equation 5.1) reveals an implicit dependence of wave velocity on the density of a coating layer. Note that a density change need not occur in a surface-immobilized film in order to be detected: changes in the ambient medium produce an effect as well. TSM [61], SAW [ 11 ], and FPW devices have been utilized to monitor the pressure (via the density) of gases. For liquid-phase sensing, both density and viscosity, as well as the nature of the acoustic mode, play a role in AW perturbations. For TSM and SH-APM devices, in-plane motion of the substrate surface entrains a thin layer of liquid through viscous coupling. Entrainment of a liquid layer by the sensor surface constitutes a mass load proportional to the product of thickness and density of the coupled liquid layer, giving rise to a velocity change. The inelastic nature of the liquid layer also results in attenuation of the AW, the magnitude of which depends upon frequency, density, and viscosity. For highfrequency SAW devices, this attenuation is far more severe than for shear-mode devices: Rayleigh waves have a substantial surface-normal component that generates compressional waves in a contacting liquid, dissipating most of the SAW energy (the extent of the attenuation is about 4 dB/cm-MHz). As a result, liquidphase sensing using SAW devices is limited to frequencies below approximately 10 MHz [23,62]. At first glance, the fact that FPWs also possess a substantial surface-normal component would appear to preempt them from liquid-phase applications as well. But the velocity of FPWs is extremely l o w - lower than the velocity of sound in ordinary l i q u i d s - so dissipation of energy through compressional wave generation is minor [17] and these devices function well in contact with liquids. While rheologically-based detection mechanisms have value for characterization of liquids (see Chapter 3), they are generally impractical for most chemical sensing applications due to limited sensitivity. For most solutes, sizable changes in concentration are required to produce a measurable change in density and/or viscosity, and viscosity is often not a monotonic function of concentration. Furthermore, the sensitivities of both density and viscosity to temperature changes mean that exceptional temperature control and/or compensation are required for such measurements to be meaningful. Two possible exceptions are (1) a few analytes, e.g., sucrose or glycerol, that cause very dramatic changes in viscosity with concentration [63], and (2) chemical reactions in a bulk solution or thin liquid layer that dramatically alter viscosity. Examples of the latter case are the

234

5. Chemical and Biological Sensors

gelation/agglutination reactions involving biochemical species discussed in Section 5.4.7 [64-66].

5.2.4 ELECTRICAL PROPERTIES Because piezoelectric substrates are used for AW chemical sensing applications, an electric field accompanies the propagating wave as it travels through the substrate. This field can interact with mobile charge carriers in a surface layer/ coating and affect both the velocity and amplitude of the wave, as discussed in Section 3.2.5; changes in acoustic velocity and attenuation due to the interaction of charge carriers are given by Equations 3.35a and b. Taking advantage of this detection mechanism in the design of a chemical sensor requires judicious selection of the piezoelectric substrate [34]. Changes in the attenuation and velocity of surface acoustic waves depend on the square of the electromechanical coupling coefficient of the substrate, K 2, which varies by orders of magnitude from one material to another. YZ-lithium niobate (LiNbO3), for example, has a/(2 value of 0.048, which is more than 40 times higher than that of ST-quartz (K2 = 0.0011). Thus, LiNbO3 is a more appropriate substrate to choose if this detection mechanism is to be employed. Appropriate temperature controls will be essential, however, since the temperature coefficient of LiNbO3 SAW devices is over 20 times that of comparable ST-quartz devices. If a coating material is used to concentrate the analyte near the sensor surface, it must be a semiconductive material with a sheet conductivity (the product of film thickness and bulk conductivity) that falls within a "window of sensitivity" determined by the substrate. Using values for LiNbO3 [from ref. 67] maximum sensitivity is obtained in the range of 10-7-10 -5 ~ - l , and the maximum possible frequency shift (Af/fo) is K2/2 = 24,000 ppm. For quartz, the window of sensitivity is an order of magnitude lower (10-8-10 -r l-l-l) and the maximum shift obtainable is only 550 ppm; this is, however, more than adequate for the observation of a significant effect (i.e., sensor response) under the proper conditions [34]. If interaction with an analyte increases the conductivity of the coating film (in the appropriate range), the frequency decreases and the conductivity change enhances frequency shifts due to increased mass or decreased modulus. If it is desired to isolate the conductivity effects from mass and mechanical effects, a reference device can be coated with an initial layer of metal whose sheet conductivity significantly exceeds the window of sensitivity, thereby shorting the electric field (this does not affect sensitivity to mass changes). The chemically sensitive coating material is then applied over the metal film [67]. The signal from this device can be compared with that from a sensing device having no

5.2 Detection Mechanisms

235

metal layer, and the resulting difference in frequency shifts depends only on conductivity changes. The substrate material can also be important in isolating the conductivity effect; LiNbO3 is again the material of choice, with a mass sensitivity less than half that of other common substrates such as quartz and ZnO-onSi [67]. In addition to having sheet conductivity within the aforementioned window of sensitivity, the sheet conductivity of the coating material must be altered as a result of a (bio)chemical stimulus. Much of the earlier applications used films of phthalocyanine (H2Pc) or metal phthalocyanine (MPc, where M = Pb, Cu, Co, Fe, etc.) [24,67-70]. Pc-based films are attractive as coatings because they exhibit suitable conductivity and are thermally stable to 400-500~ although their propensity for sublimation limits extended operating temperatures to about 250~ and below. Films of H2Pc, MPcs, and their derivatives can be deposited by a variety of techniques, including vacuum sublimation [68], Langmuir-Blodgett transfer [24,26], and plasma polymerization [70]. Interaction with electronegative gases, such as NO2, can enhance the p-type conductivity of these materials by several orders of magnitude; this is a result of the creation of mobile holes via electron transfer to the electrophile [71,72]. Changes in conductivity within the range of 10-9-10 -6 l'~-1 have been reported for MPc films upon exposure to 2.8 ppb of NO2 [72]. The conductivity and sensitivity of the Pc film to a given analyte can be modified by varying the central metal atom [70], by using substituted Pc materials, or by adding dopants to the coatings after deposition [71 ]. Ricco and Martin [67] used the metal-coated reference-device scheme described above to confirm their calculations comparing mass and conductivity effects. Their results are presented in Figure 5.2, showing the SAW response to NO2 for a Pc-coated SAW device with and without a conductive metal layer. The calculations indicated that the frequency change due to NO2-induced conductivity variation in PbPc on a LiNbO3 SAW device was orders of magnitude greater than the frequency shift from mass loading for that particular system. 2 More recent studies have focussed on the characterization of the conductive polymer, polypyrrole, and its use as a coating for acoustic wave vapor sensors [74-77]. One advantage of polypyrrole is that it can be generated directly on an 2Nieuwenhuizen et al., performed similar studies using PbPc on quartz SAW devices [73] and concluded that, for quartz substrates, both mass-loading and acoustoelectric effects contributed to the observed response. This is consistent with data provided in a table in reference 67; the greater mass sensitivity and smaller K 2 value for quartz relative to LiNbO3 combine to result in a ratio of acoustoelectric to mass effects of about 10, rather than the value of 1000 calculated for LiNbO3, for the same film thickness and operating frequency. This difference serves to reinforce the point made earlier, that the choice of substrate is crucial when examining such effects.

L~

t,n

rg r

r162

B

am 9

mini

lg

m9

z o

am..

r~ r om/

Figure 5.2 Response of lead phthalocyanine-coated SAW to NO2 with (or decoupled) and without (or coupled) a metal subcoating. The data indicate that the response of the LiNbO3 SAW is due to acoustoelectric effects, rather than mass loading. The "single" designation indicates that the response is from an individual SAW sensor (i.e., no reference sensor was employed). (Reprinted with permission. See Ref. [67]. 9 1985 Elsevier Publishers.)

5.2 Detection Mechanisms

237

electrode surface by electropolymerization, which makes it particularly suited for the TSM (quartz resonator) devices. The conductance of the polypyrrole film changes upon sorption of organic vapors, with nucleophilic vapors producing an increase in film resistance and electrophilic vapors causing a decrease in resistance [77]. The oxidation state of the deposited film also appears to influence its sorptive capabilities [75]. The observed response of polypyrrole-coated TSM sensors, however, is likely dominated by physical effects (e.g., mass loading and/or stiffness changes). The polypyrrole films are particularly well suited for hybrid sensor configurations. For example, the use of a SAW sensor (for mass measurements) in tandem with a chemiresistor (for conductivity measurements) is feasible; such hybrid devices have been previously reported using phthalocyanine coatings [78]. A unique advantage of the TSM devices is the presence of an electrode that can function as a working electrode in an electrochemical cell. Thus, sorption of vapors into the film can produce a mass loading response and, if the vapor is electroactive, the vapor may also be involved in redox reactions (current flow) with the film at the electrode surface [76]. Such a configuration could provide additional means of discriminating between sorbed vapors. An uncoated substrate can also be affected by the interaction between polar analytes and the electric field associated with the propagating acoustic wave. This effect may have contributed to SAW sensor responses to high relative humidity levels by Huang [79]. Using lithium niobate substrates, increased attenuation of the SAW was observed in the RH range of 89% to 98%; minimal frequency shifts were observed during these studies. Increases in attenuation may have occurred as a result of electrical leakage between electrode finger pairs in the IDTs. Other studies have reported liquid sensing/detection of ionic species using uncoated acoustic devices, where the frequency of the device changes in response to the conductance of the solution. Yao et al. reported the use of a TSM device to monitor acid-base [80] and compleximetric titrations [81]. The frequency of the TSM changed with the specific conductance of the solution, which depends strongly on the ion(s) present and their respective concentrations. By plotting frequency shift vs titrant volume, the endpoint was determined as a change in slope. This sensor was subsequently used for the multi-component determination of aspirin and salicylic acid in solution [82]. Nomura and Kanazawa also used an uncoated TSM to analyze for metal ions in solution [83]. They concluded that the negatively charged quartz surface attracted and adsorbed metal ions; the frequency shift observed was likely the result of mass loading. While uncoated devices lack selectivity, the general response to charged and/or polar species may be an advantage for some applications, for example as a detector for ion chromatography.

238

5.2.5

5. Chemical and Biological Sensors

THERMAL EFFECTS

The AW velocity in piezoelectric media is sensitive to changes in temperature. For most practical sensors, this fact necessitates some means of temperature control and/or compensation, as discussed in detail in Chapter 6. For most chemical sensing applications, substrates such as ST-quartz for SAW sensors and ATquartz for TSM sensors, which have small temperature coefficients, are selected. There are cases where the temperature sensitivity of acoustic-wave devices is of potential, but thus far undemonstrated, value. Pyroelectric materials, e.g., ZnO, LiTaO3, and VOx (2 < x < 3), have the property that a temperature gradient causes a potential drop across the substrate [84,85]. This enthalpimetric capability has been used to measure the heats of chemical reactions occurring at a sensor surface, the most common application being the detection of biological species via enzyme-catalyzed reactions [86-88] because of the considerable heats generated (up to 100 kcal/mole [89]). The temperature sensitivity of AW velocity makes similar enthalpimetric measurements feasible without the need for a pyroelectric layer. As in the case of the acoustoelectric effects discussed above, maximum sensitivity would be obtained through judicious choice of the substrate; YZ-LiNbO3