3.1. Premises
The theoretical model is based on the analysis of dispersion curves of guided surface acoustic waves. Resonance peaks in the frequency spectra of the acquired signals (e.g.,
Figure 4a) are dots on the wavelength-frequency plane
(e.g.,
Figure 4b). They indicate that at the corresponding wavelength-frequency pairs (
), the laser beam excites SAWs. To use this information for the probing (i.e., quantitative evaluation) of the effective material constants of HSQ-NW composite medium, one needs, first, a mathematical and computer model allowing adequate prediction of the SAW characteristics for a given set of input sample’s parameters (geometry, elastic moduli, and density). Second, it is necessary to specify an objective function which determines the deviation of the calculated SAW dispersion characteristics from the experimentally obtained ones. The minimization of this function in the space of input parameters of the specimen should result in approaching of the calculated characteristics to the experimental data. Additionally, the set of variable parameters providing the global minimum can be treated as the effective sample’s parameters within the framework of the model used.
The analysis of the scanning electronic microscope pictures of cleaved specimens (e.g.,
Figure 5a) tells us that, in general, the samples can be considered as three-layer elastic waveguides (
Figure 5b). Counting top-down, the HSQ-NW composite is the core (second) layer covered by an electrical-contact coating (first layer) and underlain by a thick silicon substrate which may be simulated by an elastic half-space (third domain). It is expected that the total thickness of the first two layers deposited on the substrate is about 1 μm while the coating takes about one-tenth.
It is obvious that, due to a complex microstructure of the nanowired-based material, its elastic properties can be simulated neither by the properties of separate constituents (HSQ polymer or NW material) nor by their jointly averaged characteristics. Vertical elongation of the nano-inclusions results in a strong anisotropy of elastic properties that manifests itself in a large difference between vertical and horizontal deformations. Consequently, within the mathematical model, its stress-strain relations should be described by a transversely isotropic elastic medium with the horizontal plane of isotropy
(
Figure 5b). Such materials are specified by five independent elastic moduli
and density
.
The material parameters of the CrPt alloy in the coating and Si[111] in the substrate are generally known and may be taken from literature. The underlying silicon half-space is of cubic symmetry specified by three independent elastic constants. The full set of Si[111] parameters used in the calculations is presented in
Table 1.
The first layer (coating) is isotropic (two independent elastic constants that are expected to be known, in such cases the values from
Table 1 are used as input). However, due to the roughness of the top surface, the averaged elastic moduli of the top layer tend to be smaller than those of the pure CrPt alloy. Therefore, the effective material constants of the first layer and its thickness
are also generally unknown and, in many cases, it is determined simultaneously with the HSQ-NW parameters.
The engineering constants are related to the elastic moduli through the connection between the
stiffness matrix
C, which elements
are written in the Voigt notation, and the compliance matrix
S:
. In the case of transversely isotropic medium with the vertical axis of symmetry
, the compliance matrix
S written in terms of the Young modulus
and
, shear modulus
, and Poisson’s ratios
and
takes the form
A three-layer elastic anisotropic half-space is a basic model for the SAW simulation. However, since some samples present straight and homogeneously grown NWs (e.g.,
Figure 1), in other samples, the NWs are of “
pyramidal” shape (
Figure 2) in the sense where the straight NWs present a large base resulting from the parasitic layer grown at their bottom due to the lower temperature. It means that the elastic properties and density of the HSQ-NW layer vary with the depth, and it should be modeled by a vertically inhomogeneous (layered or functionally gradient) elastic medium. Nevertheless, at the first stage, such a core layer can also be roughly modeled by a homogeneous transversely isotropic material. And the numerical analysis shows that such an approximation provides quite reasonable results. However, judging by the pictures of “
pyramidal” NWs, it is more precise to divide such a layer in two sublayers with different material properties (e.g., sublayers 2.1 and 2.2 separated in
Figure 5b by the horizontal dashed line).
Thus, the wave processes in the samples under study are simulated on the basis of solutions to the elastodynamic boundary value problems (BVP) for multilayered anisotropic half-spaces.
3.2. Simulation
To simulate the frequency spectra of the laser-generated SAWs, the BVP is formulated with respect to the complex displacement amplitude
of the time-harmonic oscillation
;
is angular frequency,
f is frequency,
is a point in the Cartesian coordinate system; plane
and axis
z are aligned with the sample’s surface and outward normal (
Figure 5b).
The components of the displacement vector
obey the elastodynamics equations
The elastic stiffness tensor
and the density
are piecewise constant functions of the transverse coordinate
z, thereby keeping constant values within the sublayers of thickness
,
. The bottom substrate is a half-space (
). The outer surface
is stress free, except within the loading region
D:
and the sublayers are perfectly bonded with each other. Here
is a tension vector at a horizontal surface area
,
is a given load simulating the action of the laser beam generating SAWs.
The solution to this BVP is obtained in terms of the Green’s matrix
and the vector of load
applied to a surface area
D, or equivalently, via their Fourier symbols
and
[
16,
17,
20]:
Here is the Fourier transform with respect to x and y variables; the Fourier parameters and play the role of wavenumbers for the waves propagating along the surface. The integration paths and go along the real axes, deviating from them into the complex planes and for rounding the real poles of the matrix K elements.
The frequency domain Green’s matrix
is formed from the solution vectors
corresponding to the point loads applied along the basic coordinate vectors
. In the case under consideration,
, where
is the solution to BVP (
1) and (
2) with
in the boundary conditions at the surface
;
is Dirac’s delta-function. The exhaustive description of the algorithms of matrix
K calculation can be found in Refs. [
16,
20]. Ibid, as well as in Reference [
17], the derivation of asymptotic representations for SAWs generated by the surface load
is described in details.
The SAWs are extracted from the path integrals of Equation (
3) as the residues from the real and nearly real poles
. Since the action of the laser beam on the sample’s surface can be simulated by a vertical load
, only the third column of matrix
K is involved in the representation of the laser-generated wave field
. Moreover, since only the third (vertical) component
of the excited displacement
is laser acquired as the sample’s response, only one element
of the matrix
K is needed to simulate the experimental measurements. Under these conditions, the SAW asymptotics of Reference [
16] is reduced to the form
Here
N is the number of real and closest to the real axis complex poles
held in the asymptotics,
is a characteristic wavenumber; for definiteness, the amplitude factors are shown for the SAWs propagating in the
x-direction (
,
,
). Each term of expansion (
4) describes the guided wave, for which the pole
is the wavenumber. With real
, they are traveling waves propagating with the phase and group velocities
and
or slownesses
. The imaginary part
of a complex
results in the exponential attenuation
as
, and the corresponding waves are leaky or pseudo-surface acoustic waves (PSAW).
The poles
are the roots of the
denominator, therefore, the characteristic equation that relates the wavelength
with frequency
f can be written in the form
where
is treated as a function of
and
f at
and
. With a fixed wavelength
, the roots of this equation
are the frequencies of SAW/PSAW modes, which should coincide with the TGM measured frequencies of the sample’s resonance response, e.g., peak frequencies in
Figure 4a. And vice versa, with a fixed frequency
f, the roots
yield the SAW wavenumbers
.
3.3. Validation
The magnitude of
reaches the maximal values at the sets
specified by the roots of characteristic equation (
5); theoretically it is infinite with real
. Therefore, the ridges in the level-line plots depicting
as a function of
and
f visually show the SAW/PSAW dispersion curves in the plane
. To confirm the ability of the developed Green’s matrix based model to correctly simulate SAWs generated in the samples, we had to make sure that the ridges pass through the experimentally obtained points
indicating the peaks of the frequency spectrum for a sample with known material parameters.
First of all, such a validating comparison has been performed for a simple two-layer specimen Ni/Si. The material properties of the isotropic nickel coating were taken from the handbooks (
GPa,
GPa, and
kg/m
3) while the silicon substrate is the same in all experiments (
Table 1). In this sample, the thickness of the coating (
nm) was very small as compared with the range of SAW wavelengths (2–15 μm) in the measurements. Therefore, its influence was insignificant and the only Rayleigh-like SAW was excited, and, as expected, the only ridge of the calculated surface
passed through the experimental points (
Figure 6). This has confirmed both the measurement and simulation accuracy.
Figure 7 presents the results of the theory-to-experiment comparisons with a more complex three-layer specimen Ni/HSQ400/Si. The core layer is a pure HSQ polymer, still without NWs, annealed at the temperature
°C. This material is isotropic, therefore, it was not too difficult to determine its material constants manually. It required just a few turns to fit the
ridges (
Figure 7b) to the experimental points
(
Figure 7a). The fit is achieved with the HSQ parameters
GPa,
GPa, and
kg/m
3.
The real roots of Equation (
5) calculated with these parameters allowed tracing the dispersion curves
(
Figure 7c). Obviously, the curves go along the
ridges but not the whole length. For example, the two upper-right experimental points get on the ridge in
Figure 7b, but in
Figure 7c this area is blank. The reason is that this part of the ridge is already associated with the complex pole
, i.e., the corresponding SAW actually becomes PSAW (leaky wave). The transformation of a traveling surface wave into a leaky wave occurs when its phase velocity
becomes greater than the velocity
of the bulk shear waves in the bottom half-space (i.e., in the silicon substrate in the case). Or equivalently, when its slowness
is less then the substrate’s
S-wave slowness
. Depicting dispersion curves in the frequency-slowness plane is more convenient, since, in contrast with the phase velocity curves that can come down from infinity, the slowness magnitudes vary in a limited range specified by the frequency independent bulk-wave slownesses. In
Figure 7d and similar figures below, the latter are shown by horizontal dashed lines. They are indicated by the symbols
,
,
, and
,
,
, which relate to longitudinal and transversal (
P and
S) bulk waves propagating in the
x-direction in each of the sublayers numbered from top to bottom. One can see that those two points, getting in the blank area in
Figure 7c, lie below the
line in
Figure 7d.
In general,
Figure 7 demonstrates a good match of the SAW dispersion curves calculated for manually selected HSQ parameters with the experimental data. However, the HSQ-NW composite, which are the main goal, do not promise such an easy selection of proper effective parameters. Creating an algorithm of the fitting, i.e., the development of methods for the inverse problem solution, is necessary.
3.4. Inverse Problem
To obtain the effective elastic moduli of the interlayer HSQ-NW, the sample parameters (the matrix of elastic moduli
C, the density
and the thickness
h of each sublayer), which are inputs to calculate
K within the multilayered mathematical model used, vary to match the experimental points. This goal is achieved through the minimization of a certain objective function that can be constructed in various forms. A natural way is to minimize the discrepancy Δ between the measured (
) and calculated (
) SAW dispersion characteristics (e.g., between the SAW wavelengths or group velocities at the measured resonance frequencies
):
To search for the global minimum of such objective functions, genetic algorithms are usually employed [
21]. For example, a similar approach was successfully applied to the evaluation of the effective elastic properties of layered composite fiber-reinforced plastic plates [
22].
However, the calculation of dispersion characteristics at each step is unreasonably time-consuming, because, to obtain each of them, the matrix
K are to be computed hundreds or even thousands of times. Drastic cost reduction is achieved by using the objective function
where (
) are experimentally obtained points in the wavelength-frequency plane. The calculation of this function requires only one call of the procedure
for each experimental point. The appropriateness of function (
7) as an objective function is explained by the fact that it becomes equal to zero when the variable set of sample parameters
C,
, and
h yields, together with the fixed known parameters, the SAW characteristics
corresponding to the measured resonance points
. A similar approach was also recently proposed for real-time assessment of anisotropic plate properties using elastic guided waves [
23]. To obtain the set of parameters providing the minimal value of function
(i.e., to find the effective sample’s parameters), we use the method of coordinate-wise minimization within prescribed ranges of parameter variations. It usually requires no more than 50 iterations, which is computationally cost free even as compared with the plotting of the dispersion curves obtained.