New Torsional Surface Elastic Waves in Cylindrical Metamaterial Waveguides for Sensing Applications

2. Physical Model

2.1. Geometry and Material Parameters of the Waveguide

The geometry of the waveguide supporting the new torsional elastic surface waves is presented in Figure 1. The waveguide consists of a metamaterial elastic cylindrical rod $(0<r\le a)$ embedded in a conventional elastic medium $(r>a$). As it will be shown explicitly in the next Section 2.2 the elastic compliance of the metamaterial cylindrical rod $(0<r\le \mathrm{a})$ can exhibit negative values $s_{44}^{\left(1\right)}\left(\omega \right)<0$ in the frequency range adjacent to zero frequency $\omega =0$. The remaining material parameters of the waveguide, i.e., the density in the metamaterial rod ${\rho }_1$ and in a conventional elastic surrounding medium ${\rho }_2$ as well as its elastic compliance $s_{44}^{\left(2\right)}>0$, are all positive.

2.2. Elastic Compliance $s_{44}^{\left(1\right)}\left(\omega \right)$ of the Metamaterial Elastic Cylinder $0<r\le a$

It is assumed throughout this paper that the elastic compliance $s_{44}^{\left(1\right)}\left(\omega \right)$ of the metamaterial cylinder (rod), as a function of angular frequency $\omega$, changes analogously to the dielectric function $\epsilon \left(\omega \right)$ in Drude’s model of metals [16], namely

$$s_{44}^{\left(1\right)}(\omega )=s_0\left(1-{\displaystyle \frac{{\omega }_p^2}{{\omega }^2}}\right)$$

(1)

where: ${\omega }_p$ is the angular frequency of the local mechanical resonators in the metamaterial and $s_0$ is its reference elastic compliance for $\omega \to \infty$.

In the Drude’s model of metals the angular frequency ${\omega }_p=2\pi f_p$ is called the angular frequency of bulk plasmon resonance [17]. The adjacent medium ($r>a$) is a conventional elastic material with a positive compliance $s_{44}^{\left(2\right)}>0$ and density ${\rho }_2>0$ that are both frequency independent.

It should be stressed that the according to Equation (1) the elastic compliance $s_{44}^{\left(1\right)}\left(\omega \right)$ of the metamaterial rod is negative in the frequency range $0<\omega <{\omega }_p$. How to realize the elastic metamaterial with a negative elastic compliance $s_{44}^{\left(1\right)}\left(\omega \right)<0$ was shown in the recent paper of Kiełczyński [2].

3. Mathematical Model

An unique feature of the new torsional elastic surface waves is the fact that they possess only one component of the mechanical displacement $u_{\theta }$ that is polarized along the angular coordinate $\theta$ which is tangential to the circumference of the cylinder (see Figure 1).

The mechanical displacement of the new torsional elastic surface wave decays rapidly with the distance from the surface of the cylinder ($r=a$) in both directions, i.e., into the metamaterial elastic cylinder ($0<r\le a$) and into the adjacent elastic medium ($r>a$).

3.1. Mechanical Displacement and Shear Stress

The new torsional elastic surface wave propagates along the axis of the cylinder $z$, The mechanical displacement $u_{\theta }^{\left(1\right)}(r,z,t)$ of the new torsional elastic surface wave in the metamaterial elastic cylinder $0<r\le a$ can be expressed in the following generic form:

$$u_{\theta }^{\left(1\right)}(r,z,t)=A·f\left(r\right)·\mathrm{e}\mathrm{x}\mathrm{p}\left[j\left(kz-\omega t\right)\right]$$

(2)

where: the function $f\left(r\right)$ depends only on the radial distance $r$, $j=\sqrt{-1}$ stands for the imaginary unit, $k$ is the wavenumber of the new torsional elastic surface wave, $\omega$ is its angular frequency, $A$ is an arbitrary real constant and $t$stands for time.

By definition, the associated shear stress ${\sigma }_{r\theta }^{\left(1\right)}(r,z,t)$ of the new torsional elastic surface wave in the metamaterial elastic cylinder $0<r\le a$ is given by:

$${\sigma }_{r\theta }^{\left(1\right)}(r,z,t)={\displaystyle \frac{1}{s_{44}^{\left(1\right)}\left(\omega \right)}}r{\displaystyle \frac{\partial }{\partial r}}\left({\displaystyle \frac{u_{\theta }^{\left(1\right)}(r,z,t)}{r}}\right)$$

(3)

Analogous expressions can be written for the mechanical displacement $u_{\theta }^{\left(2\right)}(r,z,t)$ and the associated shear stress ${\sigma }_{r\theta }^{\left(2\right)}(r,z,t)$ in the adjacent conventional elastic medium ($r>a$), namely

$$u_{\theta }^{\left(2\right)}(r,z,t)=B·g\left(r\right)·\mathrm{e}\mathrm{x}\mathrm{p}\left[j\left(kz-\omega t\right)\right]$$

(4)

and

$${\sigma }_{r\theta }^{\left(2\right)}(r,z,t)={\displaystyle \frac{1}{s_{44}^{\left(2\right)}}}r{\displaystyle \frac{\partial }{\partial r}}\left({\displaystyle \frac{u_{\theta }^{\left(2\right)}(r,z,t)}{r}}\right)$$

(5)

where: the function $g\left(r\right)$ depends only on the radial distance $r$ and$B$ is an arbitrary real constant.

The functions $f\left(r\right)$ and $g\left(r\right)$ with a radial argument $r$describe the change in the amplitude of the new torsional elastic surface wave inside the cylindrical metamaterial rod and in the surrounding medium, respectively.

The functions $f\left(r\right)$ and $g\left(r\right)$ will be given in a closed analytical form by Equations (10) and (11) and the wavenumber $k$ will be determined from the dispersion relation Equation (14) in Section 3.5 of this paper.

3.2. Equations of Motion

The mechanical displacements of the new torsional elastic surface wave: in the metamaterial elastic rod $u_{\theta }^{\left(1\right)}$ and in the adjacent conventional elastic medium $u_{\theta }^{\left(2\right)}$ satisfy the following equations of motion, written in the cylindrical system of coordinates:

$${\rho }_1{s}_{44}^{\left(1\right)}{\displaystyle \frac{{\partial }^2{u}_{\theta }^{\left(1\right)}}{\partial t^2}}={\displaystyle \frac{1}{r^2}}·{\displaystyle \frac{\partial }{\partial r}}\left(r^3{\displaystyle \frac{1}{s_{44}^{\left(1\right)}}}{\displaystyle \frac{\partial }{\partial r}}\left({\displaystyle \frac{u_{\theta }^{\left(1\right)}}{r}}\right)\right)+{\displaystyle \frac{\partial }{\partial z}}\left({\displaystyle \frac{1}{s_{44}^{\left(1\right)}}}·{\displaystyle \frac{\partial u_{\theta }^{\left(1\right)}}{\partial z}}\right)0<r\le a$$

(6)

and

$${\rho }_2{s}_{44}^{\left(2\right)}{\displaystyle \frac{{\partial }^2{u}_{\theta }^{\left(2\right)}}{\partial t^2}}={\displaystyle \frac{1}{r^2}}·{\displaystyle \frac{\partial }{\partial r}}\left(r^3{\displaystyle \frac{1}{s_{44}^{\left(2\right)}}}{\displaystyle \frac{\partial }{\partial r}}\left({\displaystyle \frac{u_{\theta }^{\left(2\right)}}{r}}\right)\right)+{\displaystyle \frac{\partial }{\partial z}}\left({\displaystyle \frac{1}{s_{44}^{\left(2\right)}}}·{\displaystyle \frac{\partial u_{\theta }^{\left(2\right)}}{\partial z}}\right)r>a$$

(7)

where: for the sake of clarity the arguments in the mechanical displacements $u_{\theta }^{\left(1\right)},u_{\theta }^{\left(2\right)}$ were omitted.

3.3. Explicit Analytical Formulas for the Mechanical Displacements $u_{\theta }^{\left(1\right)}$ and $u_{\theta }^{\left(2\right)}$

Substituting Equation (2) for the mechanical displacement $u_{\theta }^{\left(1\right)}$ in the metamaterial rod into the equation of motion Equation (6) we obtain an ordinary differential equation for the unknown radial function $f\left(r\right)$. It can be shown that the solution for this differential equation takes the following form:

$$f\left(r\right)=I_1\left({\gamma }_{1}r\right)$$

(8)

where: ${\gamma }_1$ is the radial wavenumber ${\gamma }_1={\left[k^2-{\omega }^2{\rho }_1{s}_{44}^{\left(1\right)}\left(\omega \right)\right]}^{1/2}$ and $I_1$ stands for the modified Bessel function of the first kind of order 1.

Similarly, substituting Equation (4) for the mechanical displacement $u_{\theta }^{\left(2\right)}$ in the adjacent medium into the equation of motion Equation (7) we obtain an ordinary differential equation for the unknown radial function $g\left(r\right)$, whose solution reads

$$g\left(r\right)=K_1\left({\gamma }_{2}r\right)$$

(9)

where: ${\gamma }_2$ is the radial wavenumber ${\gamma }_2={\left(k^2-{\omega }^2{\rho }_2{s}_{44}^{\left(2\right)}\right)}^{1/2}$ and $K_1$ stands for the modified Bessel function of the second kind of order 1.

Finally, substituting Equation (8) into Equation (2) and Equation (9) into Equation (4) one obtains

$$u_{\theta }^{\left(1\right)}(r,z,t)=A·I_1\left({\gamma }_{1}r\right)·\mathrm{e}\mathrm{x}\mathrm{p}\left[j\left(kz-\omega t\right)\right]0<r\le a$$

(10)

$$u_{\theta }^{\left(2\right)}(r,z,t)=B·K_1\left({\gamma }_{2}r\right)·\mathrm{e}\mathrm{x}\mathrm{p}\left[j\left(kz-\omega t\right)\right]r>a$$

(11)

3.4. Boundary Conditions

The mechanical displacement $u_{\theta }$ and shear stress ${\sigma }_{r\theta }$ of the new torsional elastic surface wave must be continuous across the surface of the cylindrical surface $r=a$ of the metamaterial rod, i.e.,

$${\left.u_{\theta }^{\left(1\right)}\right|}_{r=a}={\left.u_{\theta }^{\left(2\right)}\right|}_{r=a}$$

(12)

$${\left.{\sigma }_{r\theta }^{\left(1\right)}\right|}_{r=a}={\left.{\sigma }_{r\theta }^{\left(2\right)}\right|}_{r=a}$$

(13)

3.5. Dispersion Equation

In the first step in determination of the dispersion equation for the new torsional elastic surface waves we will substitute Equations (2)–(5) into the boundary conditions Equations (12) and (13). As a result, we will obtain a system of two linear homogeneous algebraic equations for the unknown constants $A$ and $B$. Equating the determinant of this system of equations to zero we obtain the following dispersion equation of the new torsional elastic surface waves:

$${\displaystyle \frac{K_2\left({\gamma }_{2}a\right)}{K_1\left({\gamma }_{2}a\right)}}+{\displaystyle \frac{s_{44}^{\left(2\right)}}{s_{44}^{\left(1\right)}\left(\omega \right)}}{\displaystyle \frac{{\gamma }_1}{{\gamma }_2}}{\displaystyle \frac{I_2\left({\gamma }_{1}a\right)}{I_1\left({\gamma }_{1}a\right)}}=F\left(\omega ,k\right)=0$$

(14)

where: $I_1$ and $I_2$ are the modified Bessel functions of the first kind of order 1 and 2, and similarly, $K_1$ and $K_2$ are the modified Bessel functions of the second kind of order 1 and 2.

The dispersion relation Equation (14) is a transcendental nonlinear algebraic equation for the wavenumber $k$ at a fixed angular frequency $\omega$ which can be solved numerically using appropriate numerical procedures, such as e.g., the iterative Newton-Raphson method.

3.6. Group Velocity

Group velocity $v_{gr}\left(\omega \right)$ of the new torsional elastic surface waves was evaluated analytically using the following formula: $v_{gr}\left(\omega \right)=-{\displaystyle \frac{\partial F\left(\omega ,k\right)/\partial k}{\partial F\left(\omega ,k\right)/\partial \omega }}$, where the function$F\left(\omega ,k\right)$ represents the dispersion equation (see Equation (14)) which is obviously an implicit function of the angular frequency $\omega$ and wavenumber $k$. Thus, after lengthy but quite elementary algebra we obtain:

$$v_{gr}=-{\displaystyle \frac{\begin{array}{c}\\ \left[s_{44}^{\left(1\right)}\left(\omega \right)\frac{\partial {\gamma }_2}{\partial k}+s_{44}^{\left(1\right)}\left(\omega \right)\left(\frac{\partial {\gamma }_1}{\partial k}\frac{{\gamma }_2}{{\gamma }_1}-2\frac{\partial {\gamma }_2}{\partial k}\right)\right]\frac{K_2\left({\gamma }_{2}a\right)}{K_1\left({\gamma }_{2}a\right)}+\left[s_{44}^{\left(2\right)}\frac{\partial {\gamma }_1}{\partial k}+s_{44}^{\left(2\right)}\left(\frac{\partial {\gamma }_2}{\partial k}\frac{{\gamma }_1}{{\gamma }_2}-2\frac{\partial {\gamma }_1}{\partial k}\right)\right]\frac{I_2\left({\gamma }_{1}a\right)}{I_1\left({\gamma }_{1}a\right)}+\left[s_{44}^{\left(1\right)}\left(\omega \right){\gamma }_{2}a\frac{\partial {\gamma }_1}{\partial k}-s_{44}^{\left(2\right)}{\gamma }_{1}a\frac{\partial {\gamma }_2}{\partial k}\right]\frac{I_2\left({\gamma }_{1}a\right)}{I_1\left({\gamma }_{1}a\right)}\frac{K_2\left({\gamma }_{2}a\right)}{K_1\left({\gamma }_{2}a\right)}-\left[s_{44}^{\left(1\right)}\left(\omega \right){\gamma }_{2}a\frac{\partial {\gamma }_2}{\partial k}-s_{44}^{\left(2\right)}{\gamma }_{1}a\frac{\partial {\gamma }_1}{\partial k}\right]\end{array}}{\left[{\frac{\partial s_{44}^{\left(1\right)}\left(\omega \right)}{\partial \omega }{\gamma }_2+s}_{44}^{\left(1\right)}\left(\omega \right)\frac{\partial {\gamma }_2}{\partial \omega }+s_{44}^{\left(1\right)}\left(\omega \right)\left(\frac{\partial {\gamma }_1}{\partial \omega }\frac{{\gamma }_2}{{\gamma }_1}-2\frac{\partial {\gamma }_2}{\partial \omega }\right)\right]\frac{K_2\left({\gamma }_{2}a\right)}{K_1\left({\gamma }_{2}a\right)}+\left[s_{44}^{\left(2\right)}\frac{\partial {\gamma }_1}{\partial \omega }+s_{44}^{\left(2\right)}\left(\frac{\partial {\gamma }_2}{\partial \omega }\frac{{\gamma }_1}{{\gamma }_2}-2\frac{\partial {\gamma }_1}{\partial \omega }\right)\right]\frac{I_2\left({\gamma }_{1}a\right)}{I_1\left({\gamma }_{1}a\right)}+\left[s_{44}^{\left(1\right)}\left(\omega \right){\gamma }_{2}a\frac{\partial {\gamma }_1}{\partial \omega }-s_{44}^{\left(2\right)}{\gamma }_{1}a\frac{\partial {\gamma }_2}{\partial \omega }\right]\frac{I_2\left({\gamma }_{1}a\right)}{I_1\left({\gamma }_{1}a\right)}\frac{K_2\left({\gamma }_{2}a\right)}{K_1\left({\gamma }_{2}a\right)}-\left[s_{44}^{\left(1\right)}\left(\omega \right){\gamma }_{2}a\frac{\partial {\gamma }_2}{\partial \omega }-s_{44}^{\left(2\right)}{\gamma }_{1}a\frac{\partial {\gamma }_1}{\partial \omega }\right]}}$$

(15)

where: ${\displaystyle \frac{\partial {\gamma }_1}{\partial k}}={\displaystyle \frac{k}{{\gamma }_1}}$; ${\displaystyle \frac{\partial {\gamma }_2}{\partial k}}={\displaystyle \frac{k}{{\gamma }_2}}$; ${\displaystyle \frac{\partial {\gamma }_1}{\partial \omega }}={\displaystyle \frac{-{\rho }_1\left(\omega s_{44}^{\left(1\right)}\left(\omega \right)+s_0{\omega }_p^2/\omega \right)}{{\gamma }_1}}$; ${\displaystyle \frac{\partial {\gamma }_2}{\partial \omega }}={\displaystyle \frac{-\omega {\rho }_2{s}_{44}^{\left(2\right)}}{{\gamma }_2}}$ and ${\displaystyle \frac{\partial s_{44}^{\left(1\right)}\left(\omega \right)}{\partial \omega }}=2s_0{\omega }_p^2/{\omega }^3$.

At first glance Equation (15) looks lengthy and intimidating, with doubtful operational significance, however it can be easily implemented in numerical calculations using standard procedures from software packages, such as Scilab or Matlab.

4. Results of Numerical Calculations

4.1. Material Parameters of the Waveguide

Numerical calculations were performed employing an exemplary waveguide structure consisting of metamaterial cylindrical rod ($0<r\le a$) made of ST-Quartz with embedded local oscillators and PMMA surrounding medium ($r>a$). We assume that, the frequency of the local elementary oscillators equals $f_p=1MHz$. The radius of the metamaterial cylindrical rod is $a=1cm$. Losses in the cylindrical waveguide structure are neglected. The actual values of the material parameters used in the numerical calculations are given in Table 1.

Numerical calculations were performed with the help of the Scilab software packed

4.2. Dispersion Curve

The dispersion curve of the new torsional elastic surface wave was calculated from the solution of the dispersion relation Equation (14) and plotted in Figure 2 as the wave frequency $f$ versus the wave number $k$. The phase velocity of the bulk shear elastic waves in the surrounding conventional elastic medium was denoted in Figure 2 as $v_2={\left({\rho }_2{s}_{44}^{\left(2\right)}\right)}^{1/2}$. Surface resonant frequency $f_{sp}=f_p/\sqrt{1+s_{44}^{\left(2\right)}/s_0}$ is an upper cut-off frequency, since above this frequency the new torsional elastic surface wave cannot propagate.

4.3. Phase Velocity

Using the solution of the dispersion Equation (14), the plot of the phase velocity $v_p\left(\omega \right)=\omega /k$, as a function of the wave frequency$f$was evaluated and presented in Figure 3. In our calculations the surface resonant frequency $f_{sp}$ is equal to $143.569kHz$.

4.4. Group Velocity

Figure 4 shows the plot of the group velocity $v_{gr}\left(\omega \right)$ of the newly discovered torsional elastic surface wave as a function of the wave frequency $f$. The numerical calculations were performed using an analytical formula, Equation (15).

Note that group velocity $v_{gr}$ of the new torsional elastic surface wave is always lower than its phase velocity $v_p$ shown in Figure 3, but has the same cut-off frequencies.

5. Discussion

The characteristic feature of the dispersion curve (see Figure 2) of the newly discovered torsional wave is that as the wave frequency grows and approaches the surface resonance frequency $f_{sp}$, the wavenumber $k$increases significantly. Consequently, the wavelength $\lambda =2\pi /k$ decreases and can reach the subwavelength region. This feature is responsible for the increase in the concentration of wave energy near the surface of the cylinder $r=a$, see Figure 1, therefore for a substantial increase in the mass sensitivity of the new torsional wave.

The quest for sensors with enhanced parameters, such as very high sensitivity or very low detection threshold, is driven by the requirements resulting from numerous applications in medicine, biology, environmental studies, toxicology, etc.. In fact, an early detection of harmful bacteria, viruses or toxins requires development of the appropriate sensors with a very high sensitivity and very low threshold of detection.

These important goals can be achieved using in general two different ways: first, by improvement of the existing sensors and technologies and second, by employment of new concepts, new types of waves or new materials. Evidently, the second approach offers a chance to develop new revolutionary solutions with sensors of extraordinary parameters. However, it requires the navigation on uncharted waters, which may be not only very difficult but very often disappointing.

In this paper we adhere to the second approach. In fact, in order to develop sensors with a very high mass sensitivity we propose to use the new type of elastic torsional surface waves which were discovered recently by the authors. These newly discovered elastic torsional waves propagate in the vicinity of the curved surface of the metamaterial cylindrical rod in which the elastic compliance $s_{44}^{\left(1\right)}\left(\omega \right)$ is analogous to the dielectric function $\epsilon \left(\omega \right)$ in Drude’s model of metals.

Our choice of the new type of torsional elastic surface waves, propagating in metamaterial waveguides, can be justified by their extraordinary properties, which cannot be found in the existing elastic surface waves propagating in conventional pure elastic waveguides. For example, the newly discovered torsional elastic surface waves propagating along metamaterial cylinders (rods) exhibit the following unique properties:

• very high concentration of the wave energy in the vicinity of the cylindrical guiding surface ($r=a$) of the waveguide
• subwavelength penetration depth in both directions from the cylindrical guiding surface ($r=a$)
• very low phase and group velocities (see Figure 3 and Figure 4).

As a matter of fact, all the above characteristics of the new torsional elastic surface waves can be employed in development of ultrasonic sensors with a very high mass sensitivity. In addition, the cylindrical shape of the waveguide supporting the new torsional elastic surface waves can be advantageous in operation in a liquid environment.

6. Conclusions

In this paper we discovered and present new torsional elastic surface waves that propagate along elastic metamaterial rods (cylinders) embedded in a conventional elastic medium. The new torsional elastic surface waves have the following unique properties:

• they constitute an elastic analogue of the Surface Plasmon Polariton (SPP) electromagnetic (optical) waves propagating in layered dielectric-metal cylindrical waveguides,
• new torsional elastic waves can inherit such fascinating properties of SPP optical waves as: a) superlensing, b) superresolution, and c) ability to amplify evanescent waves
• they have only one component of the mechanical displacement polarized along the angular coordinate,
• the energy of the wave is strongly confined in the vicinity of the guiding cylindrical surface ($r=a$) of the metamaterial rod,
• penetration depth of the wave in both direction from the guiding cylindrical surface of the metamaterial rod can be subwavelength,
• their phase and group velocities tend to zero as the wave frequency approaches the upper cut-off frequency.

Consequently, due to their unique properties, presented above, the newly discovered torsional elastic surface waves, analyzed in this paper, have a significant potential for development of a new generation of ultrasonic sensors, biosensors and chemosensors with a very high mass sensitivity for applications in medicine, biology, chemistry and environmental research.

This work has an interdisciplinary character and therefore can be of interest to a wide range of researchers and engineers working in different domains of science and technology, such as: acoustics, ultrasonics, optics, physics, microwaves, elastic metamaterials, ultrasonic sensors, biosensors and chemosensors.