Proceeding of the First International Conference on Modeling, Simulation and Applied Optimization, Sharjah, U.A.E. February 1-3, 2005

DISPERSION RELATIONS FOR ACOUSTIC WAVES IN HETEROGENEOUSMULTI-LAYERED STRUCTURES CONTACTING WITH FLUIDS

N.D. Botkin, K.-H. Hoffmann, O.A. Pykhteev and V.L. Turova

Center of Advanced European Studies and ResearchLudwig-Erhard-Allee 2, 53175 Bonn, Germany

pykhteev@caesar.de

ABSTRACT

The paper describes a method and a computer application for thecomputation of the velocity of acoustic waves excited in compli-cated multi-layered structures consisting of anisotropic piezoelec-tric and isotropic layers. The structure assumes to be unboundedin the lateral directions. The top and bottom layers are either semi-infinite or they contact with media such as fluids, gases or vacuum.A special homogenization technique enables to account for bristle-like layers contacting with a fluid. The program is supplied with auser friendly graphical interface and can be useful for researchersworking on acoustic sensors.

1. INTRODUCTION

In contrast to acoustic waves in bulk materials, the wave velocityin multi-layered structures depends on the wave frequency becauseof the interaction between the layers with different acoustic prop-erties. Therefore, one can speak about dispersion relations thatexpress the connection between the propagation velocity and thewave frequency.

Though the proposed algorithm and the program can be ap-plied for a wide range of problems, the primarily pursued appli-cation is the modeling of a biosensor that serves for the detec-tion and quantitative measurement of microscopic amounts of bi-ological substances floating in a liquid. The biosensor is a multi-layered structure consisting basically of anisotropic piezoelectricand isotropic layers. The surface of the top layer is covered withtiny receptors (aptamers) which can bind specific proteins from thecontacting fluid. The operating principle is based on the genera-tion and detection of horizontally polarized shear Love waves: asthe proteins adhere to the aptamers a new layer is being formed,which changes the velocity of shear waves propagating along thesensor surface.

The algorithm is based on the construction of traveling wavesolutions to equations describing deformations in the layers. Thewave velocity is computed from the fitting of appropriate mechan-ical and (if applicable) electrical conditions on the interfaces be-tween the layers. The mechanical conditions express the pressureequilibrium and, depending on the nature of the contacting layers,either the continuity of the displacement field or the continuity ofthe velocity field. Electrical relations express the jump conditionsfor the electric field and electric displacement.

A measure of the inconsistence in the interface conditions isexpressed then by means of a non-negative real function (fittingfunction) whose roots are feasible values of the wave velocity.

2. MATHEMATICAL MODEL

2.1. Description of waves in solid multi-layered structures

A solid multi-layered structure consists of a finite number of stackedlayers as it is depicted in Figure 1. The top and bottom layers areeither semi-infinite or their outer surfaces are in contact with somemedia such as liquids, gases or vacuum. All the layers are infinitein x1 andx2 directions.

Quartz crystal substrateSiO2

4nm Aptamer layer

guiding layerGold layer

Fluid

x

xx

1

3

2

Figure 1: A sample structure.

Besides mechanical properties, each layer can also possesspiezoelectric, dielectric or conductivity properties. In this section,we consider only piezoelectric anisotropic layers as the most gen-eral case. The relations for dielectric and conductor layers looksimpler.

The constitutive relations for piezoelectric layers in the caseof small deformations are of the form:

σij = Cijklεkl − ekijEk, (1)

Di = εijEj + eiklεkl. (2)

Here,σij andεkl are the stress and the strain tensors,Di andEi

denote the electric displacement and the electric field;εkl, ekij ,andCijkl denote the material dielectric tensor, the stress piezo-electric tensor, and the elastic stiffness tensor, respectively. Thecorresponding governing equations for the displacements and theelectric potential are:

ρui tt − Cijkl∂2ul

∂xj∂xk+ ekij

∂2φ

∂xk∂xj= 0, (3)

εij∂2φ

∂xi∂xj+ eikl

∂2ul

∂xi∂xk= 0, (4)

whereui, i = 1, 2, 3, are components of the displacement vector,φ is the electric potential such thatEi = ∂φ/∂xi, andρ is thedensity of the piezoelectric.

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Note that the stress piezoelectric tensorekij vanishes for purelydielectric layers and, therefore, the elasticity equation (3) and equa-tion (4) describing the electric potential are decoupled:

ρui tt − Cijkl∂2ul

∂xj∂xk= 0, (5)

εij∂2φ

∂xi∂xj= 0. (6)

In conductor layers like metals there is no electric field and, there-fore, equation (6) for the electric potential disappears.

We are looking for a solution corresponding to a plain wavepropagating inx1 direction, which means that the displacementfield and the electric potential are of the form:

ui(x1, x3) = ai(x3) cos(κx1−ωt)+bi(x3) sin(κx1−ωt), (7)

φ(x1, x3) = f(x3) cos(κx1 − ωt) + g(x3) sin(κx1 − ωt), (8)

whereκ is the wave number andω is the circular frequency. Sub-stituting (7) and (8) into system (3), (4) and equating the coeffi-cients oncos andsin, we obtain a system of ordinary linear differ-ential equations. Using the state vector

~q := (a1, a2, a3, b1, b2, b3, f, g, a1, a2, a3, b1, b2, b3, f , g), (9)

this system can be rewritten in the normal form as follows:

~q = A ~q, (10)

where A is the corresponding matrix, the dot denotes the differ-entiation with respect to variablex3 = κx3. Let λ1, ..., λ16 and~h1, ...,~h16 be eigenvalues and eigenvectors ofA. Due to the natureof the phenomena, polynomial solutions are not expected, whichmeans that all the eigenvectors corresponding to ak-multiple eigen-value form thek-dimensional eigensubspace. Therefore, the gen-eral solution of (10) is of the form:

~q(x3) =

16∑j=1

Dj~hij e

λijκx3 , (11)

whereDj are arbitrary coefficients. If the layer is semi-infinite inthex3 (resp.−x3) direction, only the terms decreasing towardsx3

(resp.−x3), i.e. the terms with negative (resp. positive)Reλij ,are to be kept.

For each layer, the coefficientsDj are determined from me-chanical and electrical conditions holding on the interfaces be-tween the layers and on outer surfaces. It is assumed that the ver-tical displacements are very small so that the shape of the layers ispreserved.

For every two neighboring layersI andII the following me-chanical conditions expressing the continuity of the displacementfield and the equilibrium of the normal pressures must hold on theinterface between the layers:

uIi = uII

i , (12)

CIi3kl

∂uIl

∂xk− eI

ki3∂φI

∂xk= CII

i3kl∂uII

l

∂xk− eII

ki3∂φII

∂xk. (13)

Electrical interface conditions express the continuity of the normalcomponent of the electric displacement and the continuity of thetangent component of the electric field:

εI3j

∂φI

∂xj+ eI

3kl∂uI

l

∂xk= εII

3j∂φII

∂xj+ eII

3kl∂uII

l

∂xk, (14)

∂φI

∂x1=

∂φII

∂x1. (15)

In the simplest case, the structure does not contact with any-thing. In this case the following relations on the free surface of thetop layer hold:

Ci3kl∂ul

∂xk− eki3

∂φ

∂xk= 0, (16)

ε3j∂φ

∂xj+ e3kl

∂ul

∂xk= 0, (17)

∂φ

∂x1= 0. (18)

Note that the terms of the relations (13) – (18) involving thestress piezoelectric tensor disappear in the case of a purely dielec-tric layer. For a metallic layer, the terms containing the electricfield vanish additionally.

Computingui andφ with the use of (7) – (9), and (11) andsubstituting them into (12) – 18) for all the pairs of neighboringlayers, we obtain a homogeneous system of linear equations forunknown coefficientsDj . Denote byG(ω, κ) the matrix of thissystem. Fixing the circular frequencyω and denoting byV =ω/κ the unknown wave velocity, we can considerG as a func-tion of V . The wave velocity is feasible if and only if the sys-tem has a nontrivial solution, which is equivalent to the condition:

det∣∣∣GT

(V )G(V )∣∣∣ = 0. The last equation can be easily solved

because the computation of the left-hand-side runs very quicklyeven on an ordinary computer. Usually, three roots are being foundwhenever the layers are sufficiently thin, which corresponds tothree types of waves propagating with different velocities.

2.2. Treatment of the contact with a fluid

Assume now that the upper surface of the top layer contacts witha viscose, weakly compressible fluid governed by the followinglinearized Navier-Stokes equations:

%0vi t − ν∆vi − (ζ +ν

3)

∂

∂xidiv~v +

∂

∂xip = 0, (19)

αpt + %0∂

∂xivi = 0. (20)

Here,vi, i = 1, 2, 3, are the components of the velocity field,p isthe static pressure,ν andζ are the dynamic and volume viscositiesof the fluid, respectively,%0 is the initial density of the fluid andα = ∂%/∂p expresses the compressibility of the fluid.

Instead of the condition (16) holding on the free surface ofthe top layer, we consider now the following matching conditionsexpressing the continuity of the velocities and the equilibrium ofthe normal pressures:

∂ui

∂t= vi, (21)

Ci3kl∂ul

∂xk− eki3

∂φ

∂xk=

− pδi3 + ν

(∂vi

∂x3+

∂v3

∂xi

)+ (ζ − 2

3ν)δi3 div~v.

(22)

Look for the velocity in the form similar to (7) and for thepressure in the form like (8). Substitute this ansatz into equations(19) and (20) to arrive at a linear system like (9), (10). Solutionsof such a system are of the form (11). Substitution them in theinterface conditions (21) and (22) results in linear equations withrespect to the arbitrary coefficients (compare withDj from sec-tion 2.1), which contributes to the matrixG(V ) (see section 2.1).

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2.3. Treatment of the interface with a surrounding dielectricmedium

The conditions (17) and (18) are not quite precise because theyassume the absence of the electric field outside the solid structure,which is not true as a rule. To account for dielectric propertiesof the contacting medium, the conditions (17) and (18) must bereplaced by:

ε3j∂φ

∂xj+ e3kl

∂ul

∂xk= ε0

∂φ

∂xj, (23)

∂φ

∂x1=

∂φ

∂x1. (24)

whereε0 is the dielectric permittivity of the surrounding mediumandφ is the electric potential for this medium satisfying the equa-tion:

ε0∂2φ

∂xi∂xj= 0. (25)

2.4. Treatment of the interface between a bristle-like structureand a fluid

In many applications, the surface of the top layer contacting withthe fluid is covered with a very thin periodic bristle-like structure(see Figure 2) which plays an essential role for the functionality.An example of such a structure is an aptamer-protein layer men-tioned in section 1.

Figure 2: A bristle-like solid-fluid interface.

The height of the bristles is very small and their density isvery high, which makes impossible a direct modeling of such astructure using fluid-solid interface conditions. Instead of that, thebristle-fluid structure is replaced by an averaged material whoseproperties are derived as the number of bristles goes to infinitywhereas their thickness goes to zero, thereby the height of the bris-tles remains constant. Thus, we end up with a new layer whosethickness is equal to the height of the bristles and whose behav-ior is described by limiting equations derived using the followingtechnique.

Let Ω ⊂ R3 be the total domain of the bristle-fluid structureconsisting of the domainΩS occupied by bristles andΩF occupiedby a fluid. The interface separating these continua is denoted byΓ.The structure is assumed to be periodic inx1 andx2 directions andindependent ofx3. The basic governing equations read as follows:

ρF ~vt − divP~v~x +∇p = 0 in ΩF , (26)

αpt + ρF div~v = 0 in ΩF , (27)

ρS ~utt − div C~u~x = 0 in ΩS . (28)

Note that the equations describing the fluid part are the same as(19) and (20) but rewritten in a tensor form. The boundary andinitial conditions are assumed to be homogenous for simplicity.The no-slip and pressure equilibrium conditions on the fluid-solidinterface read

~ut = ~v on Γ, (29)

C~u~x · ~n =(− p I + P~v~x

)· ~n on Γ. (30)

Here,ρF andρS are the densities of the fluid and the solid parts,respectively;~v is the velocity field in the fluid,p is the pressurein the fluid, ~u is the displacement field for the solid part. Thecoefficientα characterizes the compressibility of the fluid. Thefourth-rank tensorsP andC express the viscosity of the fluid andthe elastic stiffness of the solid, respectively.

To overcome the difficulty with the non-slip condition (29),the approach proposed by J.-L. Lions in [1] is used. The basicidea is to use the velocity instead of the displacement in equation(28) as the state variable, which is done by means of the followingintegral operator:

Jt~v =

∫ t

0

~v(s)ds.

The equation (28) assumes then the form

ρS~vt − div CJt~v~x = 0, (31)

where~v = ~ut. The pressurep in (27) is expressed through thevelocity~v as follows

p = −ρF

αdivJt~v. (32)

Assume that the (x1, x2)-projection of the base cell of thebristle structure is a squareΣ containing just one bristle (see Fig-ure 3). LetΣS be the corresponding projection of the bristle andΣF = Σ \ cl (ΣS ).

ΣS

ΣF

Figure 3: Periodic cellΣ.

Let χ(x1, x2) be theΣ-periodic extension of the characteristicfunction of the domainΣF to all R2. We define then the character-istic function of the domain occupied by fluid as follows:

χε(~x) = χ(x1

ε,x2

ε), (33)

whereε is a refinement parameter. The valueε = 1 correspondsto the original structure, the bristles become finer and their densitygrows wheneverε → 0. Using (33), we can rewrite equations (26)– (28) as one equation with discontinues coefficients in the wholedomainΩ:

ρε ~v εt − div ~Mε~v ε

~x = 0, (34)

where

ρε = ρF χε + ρS (1− χε),

~Mε = χεP +(χε ρF

αI ⊗ I + (1− χε)C

)Jt.

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Figure 4: Arrangement of a structure corresponding to a biosensor.

The interface condition (29) is equivalent to the continuity of~v ε

onΓ but the condition (30) assumes now the form

CJt~vε~x · ~n =

(ρF

αdivJt~v

ε · I + P~v ε~x

)· ~n onΓε, (35)

when accounting for (32).As it is shown in [2] the sequence~v ε of solutions of (34)

and (35) converges in the two-scale sense (see [3]) to a limitingfunction~v which satisfies the following limiting equation

ρθ~vt − divJtC~v~x − div P~v~x − div

∫ t

0

ω(t− s)~v~x(s) ds = 0.

(36)The computation of the tensorsP , C, andω(τ) is based on ananalytical representation of solutions of the so-called cell equationwhich arises in homogenization theory. The integral term in theleft-hand-side of (36) is considerably smaller than the other terms.Further, it is omitted for simplicity. Substituting~u = Jt~v resultsin the final equation for bristle-like layers:

ρui tt − Cijkl∂2ul

∂xj∂xk− Pijkl

∂2utl

∂xj∂xk= 0. (37)

The conditions at the interface with the underlying solid layer andthe overlying fluid are similar to those described in sections 2.1 and2.2, respectively, if we observe that the components of the pressureon the surface with the normal(0, 0, 1) are expressed as:

pi = Ci3kl∂ul

∂xk+ Pi3kl

∂utl

∂xk, (38)

wherei = 1, 2, 3. The form of (37) and (38) allows us to embedsuch structures into the general scheme described in section 2.1quite easily.

3. PROGRAM DESCRIPTION

The program has a standard windows users friendly interface writ-ten in Visual C++.

The usage of the program includes the following steps. Firstof all one composes a multi-layered structure and specifies its pa-rameters and the excitation frequencyω (see Figure 4). The struc-ture can contain arbitrary number of layers of diverse types: solid,fluid, medium, or bristle. If a layer is anisotropic, the orientationof its material is described in terms of successive rotations of thereference system. Since some materials require a lot of parametersto be specified (up to 46 independent parameters for a piezoelectricanisotropic materials) an option supporting the import of materialparameters from existing models is implemented. For bristle-likelayers the geometry of the basic (structural) cell and the elastic pa-rameters of the solid material are specified. The coefficients of thelimiting equation (37) are calculated then automatically by meansof an embedded finite element subsystem.

After composing the structure and specifying the wave fre-quency, the fitting function is computed and graphically presented(see Figure 5). Now, the roots of the fitting function can be local-ized and found precisely. As one of the feasible wave velocitiesis found, the corresponding wave is defined and can be observed(see Figure 6). Moreover, the whole dispersion curve for the se-lected wave type (dependency between the wave velocity and thefrequency) can be automatically computed and drawn. Such fea-tures make the program useful for researchers working on acousticsensors. Note that the processing of bristle-like structures is a quitenew option of our development. As far as we know, other programs(see e.g. [4]) do not provide such a capability.

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Proceeding of the First International Conference on Modeling, Simulation and Applied Optimization, Sharjah, U.A.E. February 1-3, 2005

Figure 6: Profiles of the electric potential, components of the displacement and the electric field. The case of a shear wave (onlyu2 6= 0).

4. EXAMPLES

Figure 7 shows a dispersion curve generated by the program for asimple structure consisting of a half-space isotropic substrate cov-ered with an isotropic layer. For such a simple structure, an an-alytical solution exists (see [5]). The squares mark points foundanalytically.

Figure 8 demonstrates the application of our method to the

Figure 5: The main window of the program. Roots of the fittingfunction are the feasible wave velocities for different wave types.

verification of physical experiments related to a biosensor devel-oped atcaesar [6, 7, 8]. A 9 nm copper film is deposited on thetop layer of the biosensor. Curve (a) shows the time performanceof the etching of the cooper film. The water flux is being alter-nated with the flux of an acid solution. The phase shift is beingmeasured. Curve (b) represents the phase shift computed usingdispersion relations. The simulation proves the assumption thatthe jump at the acid-to-water transition is caused by the change ofthe fluid viscosity.

The main application of the developed tools is the estimationof the mass sensitivity of a Love wave sensor (see section 1) whosestructure is shown in Figures 1 and 4. The adhered protein is rep-

Figure 7: Dispersion curve generated by the program and pointsfound analytically.

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Proceeding of the First International Conference on Modeling, Simulation and Applied Optimization, Sharjah, U.A.E. February 1-3, 2005

Figure 8: Verification of physical experiments through numericalsimulations: (a) experiment, (b) simulation.

2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 4 . 5 5 . 0 5 . 5 6 . 01 3 81 4 01 4 21 4 41 4 61 4 81 5 01 5 21 5 4

The s

ensiti

vity

T h e t h i c k n e s s o f t h e g u i d i n g l a y e r

Figure 9: Dependence of the sensitivity of the Love wave sensoron the thickness of the guiding layer.

resented as a bristle-like layer located on the top of the bristle-likeaptamer layer. The sensitivity is expressed by:

ω0

( lV1− l

V0)

ρhτ, (39)

wherel is the travel distance of the wave,τ is the packing densityof the molecules defined as|ΣS |/(|ΣS | + |ΣF |) (see Figure 3),h is the molecular diameter of the protein, i.e. the thickness ofthe protein layer,ρ is the density of the protein,ω is the operationfrequency of the biosensor, andV1 andV0 are the wave velocitiesin the case of the presence and absence of the adhered protein,respectively.

The operation frequency of the biosensor is calculated from aprescribed wave length using the following iterating relation:

ωn+1 =2πV (ωn)

λ, (40)

whereλ = 28µm is the optimal operation wave length determinedfrom some technical reasons. The dependenceV (ω) is calculatedby means of the developed program. Figure 9 shows the calculateddependence of the sensitivity of the sensor on the thickness of theguiding layer.

5. CONCLUSION AND PERSPECTIVE

The methods presented in this paper has already been successfullyapplied to the development of a biosensor at the research centercaesar.

Sometimes, investigated structures contain very much of pe-riodically alternating layers whose thickness is significantly lessthan the wave length. Such a sandwich can be replaced by anaveraged layer whose properties can be derived by means of thehomogenization technique. Note that the homogenization will beperformed in the vertical direction only. Therefore, explicit for-mulas for parameters of the averaged material can be obtain. Thisoption will also be implemented.

6. REFERENCES

[1] Lions, J.L., Quelques methodes de resolution des problemesaux limites non lineaires, Dunod Gauthier-Villars, Paris,1969.

[2] Hoffmann, K.-H., Botkin, N.D., and Starovoitov, V.N., “Ho-mogenization of interfaces between rapidly oscillating fineelastic structures and fluids”, to appear in SIAM J. Appl.Math.

[3] Allaire, G., “Homogenization and two-scale convergence”,SIAM J. Math. Anal., Vol.23(6), 1482–1518, 1992.

[4] Pavlakovic, Lowe, Alleyne, and Cawley, “Disperse: a gen-eral purpose program for creating dispersion curves”, Annualreview of progress in quantitative NDE (ed. Thompson andChimenti), Plenum Press, New York, 155–192, 1997.

[5] Landau, L.D., and Lifschitz, E.M., Elastizitatstheorie,Akademie–Verlag, Berlin, 1975.

[6] Annual Report 2002, Center of Advanced European Studiesand Research, Bonn (http://www.caesar.de).

[7] Botkin, N., Schlensog, M., Tewes, M., and Turova, V.,“A mathematical model of a biosensor”, Proc. MSM 2001,Hilton Head Island, 231–234, 2001.

[8] Botkin, N.D., Hoffmann, K.-H., Starovoitov, V.N., and Tur-ova, V.L., “Modeling the interaction between bristle elasticstructures and fluids”, Proc. MSM 2003, San Francisco, 126–129, 2003.

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