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Восточно-Европейский журнал передовым технологий ISSN 1729-3774
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Показано можлив^ть створення сенсора на акустичних хвилях iз застосуванням нано-стрижтв ZnO. Отримано частотне рiвняння, що дозволить визначати швидтсть поширен-ня поверхневог хвилi при заданих геометрич-них параметрах стрижневих наноструктур, вiдомiй густит гх розмiщення на поверхт, а також керувати щею швидтстю, змшюючи указан параметри. Побудована модель спро-щуе процес розробки сенсорiв та аналiз отри-маних за допомогою сенсорiв на акустичних хвилях результатiв
Ключовi слова: сенсор, акустична хвиля, нанострижш ZnO, гидротермальний метод,
пружний пiвпростiр
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Показана возможность создания сенсора на акустических волнах с использованием наностержней ZnO. Получено частотное уравнение, которое позволит определять скорость распространения поверхностной волны при заданных геометрических параметрах стержневых наноструктур, известной плотности их размещения на поверхности, а также управлять этой скоростью, изменяя указанные параметры. Построенная модель упрощает процесс разработки сенсоров и анализ полученных с помощью сенсоров на акустических волнах результатов
Ключевые слова: сенсор, акустическая волна, наностержни ZnO, гидротермальный
метод, упругое полупространство -□ □-
1. Introduction
Nowadays acoustic wave devices are widely applied in many industrial and scientific fields ranging from mobile devices and wireless communication, pressure and viscosity sensors to novel biosensors for DNA detection. Major advantages of acoustic wave sensors include: single sided planar structure, the ability to interact directly with the sensing medium, high sensitivity, low hysteresis, small size, direct frequency output signal and low power consumption. Acoustic wave sensors have great potential for further application in the systems of analysis and quality control of chemical and biological agents. The investigation of novel sensitive materials, their synthesis methods is an actual problem and continuously carried out. For further quality and accuracy enhancement, simplification of manufacturing and improvement of applying the simulation of novel acoustic wave sensors operation should be performed.
At the same time ZnO became a promising component in a wide range of nanoscale devices for future application. It is an attractive material for electronics, photonics and sensing due to having exotic and versatile properties such as mechanical, piezoelectric, optical and electrical properties, biocompatibility, nontoxicity, chemical and photochemical stability, high specific surface area, optical transparency, electrochemical activities and so on [1]. On other hand,
УДК 53.082.4; 534.222.1
ZnO NANOSTRUCTURES AS SENSING ELEMENT OF ACOUSTIC WAVE
SENSOR
O. Bogdan
Deputy director of Scientific and Research Institute of
Applied Electronics** Contact tel.: (044) 236-96-76 E-mail: bogdan@ee.ntu-kpi.kiev.ua A. Orlov PhD, Associate Professor* Contact tel.: (044) 599-18-12 E-mail: a.orlov@kpi.ua O. Petrischev DPhil, Associate Professor Department of Acoustics and Acoustoelectronics** Contact tel.: (039) 493-66-11 E-mail: om.petrischev@aae.kpi.ua V. U l i a n ova Postgraduate student* Contact tel.: 093-764-95-73 E-mail: v.ulianova@gmail.com *Department of Microelectronics** **National Technical University of Ukraine "Kyiv
Polytechnic Institute" 37, Prospect Peremogy, Kiev, Ukraine,03056
ZnO is attracting considerable attention due to its unique ability to form a variety of nanostructures such as nanowires, nanoribbons/nanobels, nanocombs, nanorings, nanocages, nanocastle, nanofibers etc. ZnO nanorods/nan-owires have been employed as bio- or gas sensitive element of acoustic wave sensors [2]. ZnO nanorods provide giant effective surface area and strong bonding sites and this way allow more precision managing of their properties and characteristics.
It is known [3] that a propagation velocity of ultrasonic waves in solids is determined by the shape of the surface along which the elastic disturbances propagate, by resiliency and inertia (density) of the continuum. Rayleigh surface waves or, generally speaking, surface acoustic waves obtain in a narrow not more than two wavelengths near-surface layer of a solid. It is suggested that the propagation velocity of Rayleigh (surface) waves can be controlled by changing of inertial properties of the material particles, which are located near the surface of an elastic half-space.
In this paper we suggest the simulation of the plane harmonic surface wave propagation process in an isotropic elastic half-space with rod nanostructures on surface for determining of the propagation velocity of surface waves at given geometric parameters of the rod nanostructures, at certain density of their placement on the surface of the half-space and attached masses. This model will simplify the
© O. Bogdan, A. Orlov, O. Fetrisohev, V. Ulianova, 2012
development of ZnO nanorods SAW sensors and analysis of the results.
2. SAW sensors with ZnO nanorods and their simulation.
Problem definition
To obtain nanorods on surfaces of different materials such methods as vapor phase synthesis, laser ablation, electrochemical method, biotemplating method, chemical method are applied [4]. In particular, hydrothermal method allows synthesis of ordered rod structures which satisfy the requirements for acoustic wave sensors design at the same time being a rather simple and inexpensive.
Fig. 1 shows schematically the structure of the acoustic wave sensor using ZnO nanorods. Thus, two interdigitated transducers are formed on the piezoelectric substrate by photolithography, between them ZnO nanorods are grown by hydrothermal method.
Fig. 1. The structure of the acoustic wave sensor: 1, 2 — interdigitated transducers, 3 - piezoelectric substrate, 4- array of ZnO nanostructures
Due to the fact that propagation velocity of ultrasonic waves in solids is determined by the shape of the surface along which the elastic disturbances propagate, by resiliency and inertia (density) of the continuum, propagation velocity of Rayleigh (surface) waves can be controlled by changing of inertial properties of the material particles, which are located near the surface of an elastic half-space. Thus, for the application of shown sensor design it is necessary to carry out preliminary consideration of harmonic surface waves propagation in an elastic half-space on which surface rod nanostructures are located. It is necessary to build a model by which the assessment of propagation velocity of surface waves at given geometric parameters of the rod-shaped nan-ostructures, at certain density values of their placement on the surface of half-space and attached masses will be feasible. Based on this model it will be possible to control propagation velocity of surface waves by changing these parameters.
The elastic half-space (Fig. 2) is considered. Rod nanos-tructures are located on its surface. Rod nanostructures are absolutely rigidly connected to the surface of the half-space at the base, i. e. in the plane x3 = 0 . The rods are arranged equidistantly along the axes Ox1 and Ox2 with the step A , so that the areal density on the surface of the half-space is N = 1/A2 . The point masses M0 are located at the free ends of the rod nanostructures. To fix the idea, we assume that all the nanostructures have the same geometric parameters, i. e. length L and unvarying along the length of the rod cross-
section, which has the shape of a ring with the radii R1 and R2 (R2 > Rj) , (Fig. 2) or a circle with the radius R2. The rod nanostructures don't interact with each other, i. e. an energy transfer over the surface of the half- space is absent.
Suppose, a plane harmonic wave exists in the volume of the elastic half-space (x3 < 0), its kinematic characteristics are determined by the displacement vector of material particles u(x2,x3)e™', here u(x2,x3) is the spatially developed amplitude of the displacement vector of material particles; i = V-i - the imaginary unit; ra - is known angular frequency of the sign change of displacement; t - is time. We assume that the elastic wave propagates from left to right along axis Ox2 (Fig. 2). In this case the components of the vector u(x2,x3) must be determined as follows:
ui (x2,x3 )= ^ u2 (x2,x3 ) = u2 (x3 K u3 (X2,X3 ) = u3 (X3 )e
-'1X2
(1)
here y is the wave number of the plane surface wave - the value to be determined in the course of the problem solving.
Fig. 2. The analytical model of the problem
Propagating surface wave imparts following motion to ends x3 = 0 of rod nanostructures:
- vertical displacement u3 (0)e-iY% ;
- horizontal displacement u2 (0)e-r|% ;
- rotations in x3Ox2 plane by y = e23 (x2,0),
where e23(x2,x3) = (9u2/9x3 + 9u3/9x2)/2 is the elastic strain tensor components.
Obviously, that y = e23 (0) e-iY% ,
where £23 (0) = (-iYu3 (0) + du2 (X3 )/dx3 = 0)/2 .
It is clear that the above written expressions for the displacements of the ends of the rod nanostructures are accurate only in the case when the strong inequality 2R2 << XR holds, i. e. when the maximum size of the cross-sectional diameter of the nanotube or nanorod substantially smaller than the Rayleigh wave XR. Otherwise, it's needed to use averaged
over the cross sectional area of the rod nanostructures kinematic characteristics. In the following exposition we assume that the inequality 2R2 << XR holds.
In this case, the vertical movements u3 (0)e-iYt2 initiate longitudinal oscillations in the nanorod, which, as a first approximation, can be considered as one-dimensional. In this case the vertical displacements w3 (x3 )eimt of material particles of the rod satisfy Newton's second law in differential form, which is written in the form:
The last relations completely determine the sequence of steps for solving the problem of controlling the speed of propagation of surface acoustic waves. First, the density n!j and n^ of tangential and normal loads is evaluated, and then the components of the displacement vector u2 (x3 )e-iY% and u3 (x3 )e-iYX2 which satisfy the boundary conditions (5) and (6) are determined. Desired wave number y and phase velocity vf = ra/Y of surface waves are determined when these conditions are completed.
a2w3 (x3 )
э x2
+ k2w3 (x3 ) = 0 V x3 e[0,L],
(2)
3. The response simulation of the nanostructured
elements
where k = ra/vcT is the wave number of longitudinal waves in a rod; vcT = E/p0 is the rod velocity; E and p0 are Young's modulus and the density of the rod material, respectively.
Horizontal displacements and rotations in the cross sect-ionx3 = 0 cause harmonic oscillations of the lateral bending w3 (x3 ) = Acoskx3 + Bsinkx3,
in the rod nanostructure. At the same time the longitudinal displacements w2 (x3) eirat of material particles of the rod satisfy the standard equation of the lateral bending, which is written in the form
We first define the response n3 of the array of nanostructures.
The solution of equation (2) is obvious:
(7)
d"W2 (X3 )
Э x4
-X4w2 (x3) = 0Vx3 e[0,L],
(3)
where X is the wave number of the harmonic oscillations of the lateral bending, and X4 = ra2p0S/(EJ4 ); S = n(R2 - Rj2) is the cross-sectional area of the rod; J4 is the moment of inertia of the cross-section of the rod about Ox4 axis. The geometric characteristic J4 of the rod cross-section is calculated as follows [5]:
2n R2
J = Jx2dS = J J P3 sin2 фdфdp = n(R2 - R?).
(4)
where A and B are constants to be determined. On the surface x3 = 0 the displacement of the end x3 = 0 oftherod w3 (0) = A havetobeequaltothedisplacement u3 (0)e-iYX2 of element of the half-space. So A = u3 (0) e-iYX2. According
to Newton's third law x3 = L :
ES
dw3 (x3 )
Эх,
= -ra2M„w3 (x3 )| x
(8)
Substituting in (8) the expression (7) after obvious transformations, we obtain the following result:
B = _ a(kUcoskL - sinkL) = _u3^ (kL), (9)
(kL£,sinkL + coskL) 31 ; w
From (4) it follows that for the solid rod (R1 = 0) geometric characteristic is J1 = rcR^.
Normal stresses o^3 (0) or normal forces N3 (0) = o33 (0)S appear as a result of compression - tension oscillations of a nanorod in its cross section x3 = 0. The surface density of the normal forces is n^ = N3 (0) N = o^3 (0)SN .
Oscillations of the lateral bending form the moment of flection M8 and the lateral force Q (0) in the cross-section x3 = 0 of the fixed nanorod. The surface density of lateral forces is n = Q (0) N .
Thus, at the propagation of a plane surface wave in the half-space, on which surface x3 = 0 nanostructured elements are located, the kinematic characteristics of elastic waves must satisfy the following boundary conditions:
where = M0/(p0LS) - is the relative mass of a point on the end of the rod x3 = L (Fig. 2); F3 (kL) - is a frequency-dependent function.
A normal stress o^3 (x3) in a random cross-section of the rod is defined as follows:
О33 (x3 )= E
dw3 (x3 )
Эхч
(10)
= kEu3 (0) e-'"1"2 [-sin kx3 - F3 (kL) coskx3 ]. From (10) o;3 (0) = - kEF3 (kL)u3 (0)e-1Y"2.
ES
n33 = - N—F^kL) u3 (0) e-
(11)
du(X2,X3 )
Эх,
2c
du (X2,X3 )
Эх,
= n3, k,l = 2,3,
= n2, k,l = 2,3,
(5)
(6)
where c33kl and c4
are the matrix elements of coefficie-
nts of elasticity of the half-space material; in (1.5) and (1.6) summation over repeated indices k and l is implied.
where F3*(kL) = kLF3 (kL).
Fig. 3 shows graphs of the frequency dependence of the functions F3 (kL) that were calculated for different values of £, and the fixed quality factor QCT = 100 of material of the nanorod.
Fig. 3, b shows shifts large-scaled resonance frequencies of longitudinal oscillations of rods at small values of the relative mass of a point on the end of the rod x3 = L . The increase in the mass M0 of a point is accompanied by an increase in the resonant frequency of the oscillating system.
L
x, = L
c
x, = 0
x, = 0
CeHcopHi HanienpceigHUKOBi npucTpoi
It is obvious that the equation Do (XL,%) = 0,
Fig. 3. Graphs of the frequency dependence of the functions
Fj(kL)
The solution of the differential equation (3) has the form:
has infinite set of roots XnL (n = 1,2,... - number of the root), which correspond to an infinite set of resonant frequencies raj^ of oscillating system.
Because XL = gVkL is the dimensionless wave-number, where g - dimensionless geometric parameter g = ^SLTj! (for a given cross-sectional shape of the nanot-ube g = ^L2/(R2 + R2) ), obviously XnL = g^i^L.
For nanotubes with dimensions L = 10-6 m, R2 = 50 nm , R1 = 25 nm ( g = 4,2295 ) dimensionless frequency k1L depend on parameter % like that:
% = 0, k1L = 1,231766; % = 0,0001, k1L = 1,231519; % = 0,001, k1L = 1,229317; % = 0,01, k1L = 1,208592; % = 0,1, k1L = 1,082022.
It is clearly seen that the increase in mass M0 of a point is accompanied by a decrease of the first resonance frequency of lateral oscillations of a nanorod. It can be noted that the value of the first resonance frequency of lateral oscillations are more sensitive to changes in parameter % than the value of the first resonance frequency of longitudinal oscillations of a rod.
The lateral force Q(x3) = EJ193w2(x3)/9x3 atx3 = 0, i.e., in the plane of the fixing of the nanotube on the surface of the elastic half-space defined by the following equation:
w2 (x3) = C1 sinXx3 + D1cosXx3 + C2shXx3 + D2chXx3, (12) Q(x3)| = EJ1X3 (-C1 + C2),
(19)
where C1, D1, C2 and D2 - are constants to be determined.
The solution of (12) should implement the following conditions.
First of all, the conditions of the kinematic coupling of the rod cross-section x3 = 0 with an element of the surface of the elastic half-space must be performed. The longitudinal displacement and rotation angles must be equal. The dynamic conditions at the end x3 = L must be performed too. There are vanishing of moment of flection and equation of the lateral force to the force of inertia of a point mass (Newton's third law). These speculations lead to the following system of equations for desired constants C1, D1, C2 and D2.
Where constants C1 and C2, defined from (13) - (16)
C1 =
e->Y%
—-.-7TI u2 (0)(cos XLshXL + 2XL% cos XLchXL +
D0 (XL,%)
^23 (0),
D1 + D2 = u2 (0) e-iYX2, 1
C + C2 =jX £23 (0) e-'1"2 ,
-C1 sin XL - D1 cos XL + C2shXL + D2chXL = 0 ,
C1 (-cos XL + XL%sin XL) + D1 (sin XL + XL% cos XL) +
+ C2 (chXL + XL%shXL) + D2 (shXL + XL%chXL) = 0
(13)
(14)
(15)
, (16)
where % = M0/(p0LS) - is the relative mass of a point on the end of the rod x3 = L .
The equation determinant D0(XL,%) of system (13) - (16) is written as follows:
D0 (XL, %)= (17)
= 2 [1 + cos XLchXL + XL% (shXL cos XL - chXL sin XL)].
+ sin XLchXL) + 23 v ' (1+cos XLchXL + 2XL%sin XLshXL) X
C2 D0 (XL, %)X
x [-u2 (0)(cos XLshXL + 2XL% cos XLchXL + sin XLchXL) +
+ £23 (0)x
+ X X (20)
x(1 - sin XLshXL - 2XL% sin XLshXL + cos XLchXL)].
Substituting (20) in the definition (16) of the lateral force we define:
Q(x3)|^ =o = -2ra2mCTe-iYt2 [u2 (0)f (XL) + Le23 (0)f2 (XL)] , (21)
where mcT = p0LS - is a mass of a nanorod; 1
f1 (XL)=XLD0 (XL, %) + sin XLchXL),
1
(cos XLshXL + 2XL% cos XLchXL +
f2 (XL) =
(XL)2 D0 (XL, %)
(sin XLshXL + 2XL% sin XLchXL). (22)
The tangential load or density n2 of lateral forces on the density of the surface of an elastic half-space x3 = 0 is proportional to the surface density N of sources of lateral forces, i.e.:
П = NQ (x3 )| = о =
= -2ra2NmCTe-r|X2 [u2(0)f1 (XL) + Le23(0)f2(XL)].
(23)
Thus, the reactions n2 and n3 of nanostructured elements on the surface of an elastic half-space are defined. Now we need to determine the wave number у of surface waves, which create and balance these reactions.
4. The evaluation of kinematic characteristics of Relay waves in the half-space with nanostructures elements on its surface
Amplitude values uk(xk) (k = 1,2,3) of the components of the displacement vector u(xk)eimt are determined by Newton's second law in differential form, which for an isotropic to elastic properties of solid in the absence of bulk forces is written as follows:
(ц + 2G)graddivU(xk) - GrotrotU(xk) + + P0™2U (Xk ) = О V Xk eV,
(24)
where and G - are Lame elastic constants of an isotropic solid; p0 - is a density of a material; V - is a half-space volume.
The displacement vector u(xk) of material particles, as, indeed, the vector of any physical nature, can be written in the form of Helmholtz representation [3], i.e.:
U(Xk) = gradФ^) + rot^(Xk),
(25)
where 0(xk) and ^(xk) - are scalar and vector potentials of the wave field of elastic displacements of material particles of solid.
If div^(xk) = 0 in any point of solid, then, substituting the Helmholtz representation (25) into the steady harmonic oscillations equation (24), we can get two of the Laplace equation of the form:
V2Hl (k, ) = 0,
V2Hl (ks ) = 0,
Where h (k) = V^(xk) + k2Ф(xk),
(26) (27)
the harmonic functions S4 (k,) and S2 (ks) are constants, and are not equal. If the domain of existence of solutions of the Laplace equation is unbounded (space, half-space, infinite layer, and infinite rod), these constants are zero (otherwise it violates the principle of physical feasibility of the process) and the potentials 0(xk) and ^(xk) satisfy the Helmholtz equation, i. e.:
V^(xk ) + k2Ф(xk ) = 0, - rotrot Ф (xk ) + k2xP (xk ) = 0.
(28)
(29)
It is easy to show that the displacement field of the material particles of an elastic half-space, where components of the displacement vector are given by (1), is completely determined by the following potentials:
0(x2,x3) = 0(x3)e-iYx2, ^ (x2,x3) = ^fo)e-iYx2 , (30)
where the components of the vector potential (x2 ,x3) and (x2,x3) are equal to zero. Substituting the assumed solutions (3.7) in the Helmholtz equation (28) and (29), we obtain:
a2o(x3) , , , a2T, (x3) , , ,
-«2°(x3) = 0, -P2^1 (x3) = 0, (31)
where a = Jy2 - k2 and P = ^/y2 - k2 - are lagging in phase by the angle rc/ 2 components Ox3 of the wave vector
kt and ks.
The solutions of equation (3.8), which do not contradict the physical sense of the problem, are as follows:
Ф^3) = AeaX3 , ^ (x3) = BeßX3 ,
(32)
where A and B - are constants, which are determined from the conditions of implementation of Newton's third law to the x3 = 0 surface of an elastic half-space, i.e., from the boundary conditions (1.5) and (1.6). In respect to an isotropic solid, moduli of elasticity are determined by the Lame constants and G moreover cijkl = ^8ij8kl + G(8ik8jl + 8il8jk), where 8ij,..,, 8 jk are Kronecker symbols, the left part of these conditions are written as follows:
°33 (X2,X3 ) = 2Ge 33 (X2,X3 ) + ц divU (X2,X3 ) = = 2Ge 33 (X2,X3 )- k2ф(X2,X3 ), O32 (x2,x3 )= 2Ge 32 (X2,X3 ) ,
(33)
(34)
»2 (k, ) = - rotrot Y(Xk ) + k2^(Xk )
are functions; kt and ks - are wave numbers of longitudinal (index , ) and shift (index s ) of noninteracting harmonic waves, and kt = ra/vt and ks = ra/vs , where vt and vs - are velocities of longitudinal and shift waves, respectively, with values v =7^ + 2Gyp0 and vs = VGPo .
Since the solutions of the Laplace equation (3.3) and (3.4) are harmonic functions [7] certainly bounded above,
where £33(x2,x3) = du (x2,x3)/9x3 - is compression-stretching deformation along the axis Ox3 ;
£ 32 (x2,x3 )=[du2 (x2,x3 )/dx3 +du3 (x2,x3 )/dx2 ]/2 - is
shearing strain in the plane x3Ox2.
Substituting the Helmholtz representation (25) solutions (32), we obtain u4(x2,x3) = 0 the other two components of the amplitude values of the displacement vector of the material particles are defined by the following expressions:
u2 (x2,x3 ) = (-iyAeax3 + ßBeßx3 )e-1YX2
(35)
u3 (x2,x3 ) = (aAeaX3 + iyBePx3 )e-,YX2
(36)
Ar = a41 (ra)a22 (ra) - a12 (ra)a21 (rn) = 0 .
(43)
Defined by (3.12) and (3.13) displacement vector components correspond to the following values of the amplitudes of spatially developed the strain tensor components:
£33(x2,x3) = (a2AeaX3 + iyPBe1^ )e-'1"2 ,
(37)
£32("2,x3)= -[-2iyaAe"X3 + (y2 + p2)BePx3]e-^2. (38)
Substituting (3.14) and (3.15) in the determination of mechanical stresses (3.10) and (3.11), we obtain:
o33("2,x3) = G[(y2 +p2)Ae""3 + 2iYPBep"3]e-^2 , (39)
o32(x2,x3) = G[-2iYaAeKX3 + (y2 + p2)Bep"3]e-rYX2 . (40)
From (3.13) h (3.15) u2 (x2,0) = (-iyA + PB)e-rFt2 ,
u3 (x2,0) = (aA + ipB) e-iYX2
and £32 (0) = [-2iyA + (y2 + P2 )b]/2 .
Substituting these values into the definition of reactions n2 and n3, and substituting results into the boundary conditions (1.5) and (1.6), in the left sides of which values of o32(x2,0) and o33(x2,0) are written, after obvious transformations we obtain:
a11 (ra)A + ia12(ra)B = 0 , -ia21 (ra) A + a22 (ra)B = 0,
(41)
(42)
where:
an (ra) = y2 + P2 + NS—aF3*(kL) ;
a12 (ra) = 2yP + NS— YF3*(kL);
a21 (ra) = 2Ya + 2NS[^Lf (XL) + Ya(ksL)2f2 (XL)] ,
a22 (ra) = Y 2 + P2 +
+ NS [2pk2Lf1 (XL) + (Y 2 + P2 )(ksL)2 f2 (XL)].
In writing the expressions for calculating the coefficients
«21 (ra)
and a22 (ra)
it was taken into account that mcT/G = p0LS/G = LS/vs2,
where vs = G/p0 - is shear wave velocity.
The homogeneous system of algebraic equations (41) and (42) has nontrivial (non-zero) solutions for the coefficients A and B only if the determinant AR of this system is equal to zero, i. e.:
The relation (43) has a sense of the condition of existence of Rayleigh surface wave at a given frequency ra and can be read as follows: constants A and B are not equal to zero, i. e. in the elastic half-space with a nanostructured surface Rayleigh wave propagates, only in the case when the wave numbers a and y satisfy the equation (43).
It is easy to see that at zero surface density ( N = 0 ) of nanostructures, condition (43) becomes to the standard [3] condition for the existence of classical Rayleigh wave,
Ar|n=0 = (Y2 + p2)2 - 4Y2«P = 0.
If N ^ 0 , as it follows from (43), the wave number y becomes non-linear dependence on frequency, i. e., there is frequency dispersion of the velocity propagation and, consequently, there are differences between the phase and group velocities of propagation of individual spectral components in the case of existence of pulse ultrasonic signals in a half-space. The frequency dispersion particularly appears at the resonant frequencies of nanostructural elements. Because on the resonance frequency the oscillating system (nanostructure element) consumes from the source of oscillations, i. e., from the surface wave, the maximum amount of energy, it can be argued that the velocity of propagation of surface waves at these frequencies will decrease sharply.
The most significant difference from the classical Ra-yleigh wave is in the fact that due to the energy transfers between the material particles of the elastic half-space and nanostructured elements, in which energy is dissipated, the wave number y in equation (43) should be considered as a complex number where imaginary part is dramatically increased at resonance frequencies of nanostructural elements.
5. Conclusion
In this paper the feasibility of the development of surface acoustic wave sensors with ZnO nanostructures as sensing element is shown.
The process of propagation of a plane harmonic surface wave in an isotropic elastic half-space with the rod nanostructures on surface is observed. The frequency equation from which we can determine the velocity of propagation of a surface wave for a given geometrical parameters of the rod nanostructures, the values of the density of their distribution on the surface of the half-space and added masses is obtained. It is clear that variation of these parameters can change (control) the speed of propagation of a surface wave.
Performed simulation extends the range of tools for further development of nanorods-based SAW sensors due to comprehensive math-based explanation of their operating principles.
Further continuation of the simulation and adjustment of hydrothermal method of ZnO nanorods growing will simplify the creation of reliable and sensitive sensors on surface acoustic waves.
,. e.
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Abstract
In spite of the wide application of SAW devices the investigation of novel sensitive materials, their synthesis methods is continuously carried out. For further quality and accuracy enhancement, simplification of manufacturing and improvement of applying the simulation of novel acoustic wave sensors operation should be performed. In ZnO nanorods-based sensors propagating surface wave imparts following motion to the ends of hydrothermally grown between two interdigitated transducers rod nanostructures: vertical displacement, horizontal displacement and rotations in plane. Thus, response of nanostructured elements to that displacements and kinematic characteristics of Relay waves are evaluated. During the simulation the frequency equation from which we can determine the velocity of propagation of a surface wave for a given geometrical parameters of the rod nanostructures, the values of the density of their distribution on the surface of the half-space and added masses is obtained. This model simplifies the development and analysis of the results from ZnO nanorods-based SAW sensors for biological and chemical agents, including molecules, cells and tissues. Performed simulation extends the range of tools for further development of nanorods-based SAW sensors due to comprehensive math-based explanation of their operating principles
Keywords: sensor, acoustic wave, ZnO nanorods, hydrothermal method, elastic half-space
