Temperature biosensor based on triangular lattice phononic crystals

The technological progress of biosensors for various utilizations has been observed in many fields: petrochemicals, pharmaceutical production, and further water and air pollutant detection.^1–4 Recently, the ability to realize a sensor based on its acoustic properties attracts attention. A wide set of acoustic bio-chemical sensors has been proposed to sense very small changes in the acoustic properties of the fluid.^5,6 To realize such acoustic sensors, various types of structures and designs were proposed. For example, lab-on-chip, biochemical, or acoustic sensors were introduced, dealing with either surface acoustic wave,⁷ Lamb waves in thin films,⁸ or shear acoustic waves.⁹ In addition, smart acoustic sensors based on phononic crystals (PnCs) have newly been investigated as an efficient tool for fluid detection and characterization. PnC biosensors are introduced in relation to the sound velocity and/or the density of a fluid mixture. To be efficient, the PnC biosensor’s transmission or reflection spectra may vary distinctly when varying the nature of the fluidic component.¹ For this, it is worth considering a defect, such as a cavity containing the fluid, within a conventional platform. The introduction of the cavity inside the perfect PnC then leads to the localization of the acoustic energy inside the fluid at the frequency of the resonant mode.² The excitation of the cavity eigenmode thus gives rise to sharp intensity peaks in the transmission or reflection spectra of the PnC, sensitive to the acoustic properties of the fluid. In the hypersonic frequency range, Schneider et al. demonstrated the existence of a localized mode inside the bandgap of a hybrid superlattice structure that could offer sensing opportunities.¹⁰ Different 2D and 3D PnC biosensors were proposed theoretically and fabricated experimentally for various sensing purposes. As such, Oseev et al. presented experimentally a 2D PnC biosensor consisting in a steel plate including a cavity in the center of the structure for measuring the physical properties of gasoline and determining octane numbers of gasoline blends.¹¹ Mukhin et al. introduced a narrow solid/liquid slotted PnC sensor design for the purposes of liquid sensing.¹² In the acoustic metamaterial topic, Jin et al. reported on a phononic crystal plate with hollow pillars filled with a liquid and discussed the behaviors of whispering gallery modes for the purpose of temperature sensing.^13,14 In addition, Amoudache et al.^15,16 presented theoretically the dual photonic/phononic or phoxonic platform containing a fluid for sensing the optical and acoustic properties of biochemical liquids.

The Methyl Nonafluorobutyl Ether (MNE) is an organic mild solvent solution extensively used in cosmetic and beauty product ingredients.¹⁷ Besides, the low surface tension and high boiling point of MNE make it a good candidate as a cleaning solvent and lubricant substance. MNE presents famous properties such as non-flammable, colorless, non-toxic, relatively low vapor pressure, low-odor, non-corrosive, and clear liquid, allowing its use in various pharmaceutical and industrial applications.¹⁸ Previous works reported that the composition of MNE is highly affected by the temperature. Since then, it has been shown that the rate of production of fluoride increases with increasing temperature. Despite its wide use, only few works were reported in this regard, i.e., showing the variation of the chemical and physical properties of MNE as a function of the temperature.^19,20 Among them, Piñeiro et al.¹⁹ demonstrated that the density and acoustic properties of MNE are greatly altered with the temperatures.

The aim of this paper is to design a phononic crystal sensor able to detect the smallest changes in the acoustic properties and temperature of MNE based on the transmission of highly confined modes that relate to the acoustic properties of MNE. We focus on a 2D PnC biosensor made of an epoxy matrix including tungsten (W) cylinders arranged following a triangular array. The triangular lattice presents the advantage to open large phononic bandgaps compared to other structures.²¹ In most of the previous studies reporting on sensing purposes, the insertion of a defect has been done by considering different kinds of finite size cavities. We propose here to introduce a linear waveguide inserted in the perfect phononic crystal. The waveguide is constituted by hollow cylinders that will be filled with MNE. Based on this phononic platform, we aim to demonstrate the efficiency of the temperature detection of the viscous liquid MNE, dealing with guiding modes of low group velocities.

To achieve a wide bandgap in a phononic crystal of triangular symmetry, it is known that we need to start with a soft host matrix containing scatterers cores of stiff materials.²² Therefore, the matrix and scatterers should present high contrast in their respective elastic parameters. To satisfy this condition, we used epoxy and tungsten (W) as the host matrix and scatterers, respectively. The elastic parameters are given in Table I.

Figure 1(a) shows a schematic representation of the 2D phononic crystal made of W cylinders of radius r = 0.4 mm, embedded in epoxy and arranged according to a triangular array of lattice parameter a = 1 mm. Figure 1(a) presents the primitive unit cell of the perfect PnC structure used for the dispersion curve calculations. The Finite Element Method (FEM) has been used to compute both the dispersion curve diagrams and the transmission spectra. To calculate the dispersion curves [Fig. 1(b), left], periodic boundary conditions have been applied on all edges of the elementary unit cell and the Bloch wave number is solved by sweeping the high symmetric directions ΓMJΓ of the irreducible Brillouin zone. With the chosen geometrical parameters, corresponding to the filling factor $ff=πr2/32a2$ = 58%, we obtained a wide phononic bandgap centered around 1 MHz, in which the origin comes from the Bragg scattering. Phononic bandgap properties follow a scale law with the geometrical parameters of the PnC because of the linearity of the equations of motion. Then, depending on the given lattice parameter, one can reach different ranges of frequencies in relation with the desired application domain. In this paper, we focus on the MHz range, corresponding to the millimeter size of the lattice parameter, which is the most often used area in phononic crystal sensors and which also avoids the high GHz frequencies where structural relaxation processes occur in the MNE polymer.

The right panel of Fig. 1(b) represents the calculation of the transmission coefficient vs the frequency. To get the spectrum, an incident plane wave of longitudinal polarization has been launched in the x direction, at the entrance of the phononic crystal. The amplitude of the signal is then recorded at the output and normalized to the amplitude at the input to get the transmission spectrum. Perfect matching layers are applied at both boundaries of the simulation domain, in the direction of propagation x to avoid any undesired reflective waves. Periodic boundary conditions are applied along the direction perpendicular y to build the phononic crystal. A broad phononic bandgap is found around 1 MHz together with a smaller one around 2.4 MHz. It may be mentioned that the forbidden band can be made wider by increasing the filling factor (see Fig. S-1 of the supplementary material), but this would be at the expense of a difficulty in the perspective of practical realization since the inclusions become very close to each other.

We aim to propose here a linear defect constituted by a line of hollow cylinders containing the MNE fluid and look for guiding modes of low group velocity, traveling along the waveguide. Compared to a single cavity embedded in the phononic structure, the waveguide has several advantages, ensuring the control and manipulation of the waves. It allows an easy integration in lab-on-chip, even for complex component associations. The high localization of the field at the exit also makes easy the detection of the traveling modes.

To solve the basic equations with the help of the finite element method (COMSOL Multiphysics®), we consider in the mixed structure, both the pressure for the fluid and the elastic displacement for the solid. The following equations were used for solving the solid–fluid interactions with boundary conditions in the triangular unit cell. The equations of motions in the mixed solid and liquid structure can be written as²³ 

where M_sg and K_sg are the stiffness and mass matrix of the hard solid domain and the liquid (MNE) region, $Pgk$ and $Usk$ are the acoustic pressure and displacements at the nodes of the MNE and solid meshed domains, and k is the wave vector. The boundary conditions on the interface χ between the solid and the liquid can be written as

where $usn$(r) and $uln$(r) are the normal displacements at the solid/liquid interface.

Before introducing the liquid, we start with the study of the linear waveguide, made of empty hollow cylinders of inner radius r_i = 0.22 × a, inserted inside the perfect phononic crystal. To perform the dispersion curve calculation, we designed a supercell containing a hollow cylinder in the middle of a (7 × 1) elementary cell, as depicted Fig. 2(a). We applied the periodic boundary condition (PBC) at all edges of the supercell. By calculating the dispersion curves, one finds two flat branches of low group velocity, at 1 MHz, in the middle of the large phononic bandgap. These almost flat branches give rise to a narrow passing band in the transmission spectrum, named A [Fig. 2(b)]. Figure 2(c) represents the map of the displacement field of the two eigenfrequencies A where one can see the elastic field strongly localized inside the hollow cylinder section constituting the waveguide. The physical mechanism at the origin of branch A can be understood as an excitation mainly in the tungsten ring inside the polymer matrix. The eigenmode is composed of four lobes, thus corresponding to a quadrupolar mode. Due to the anisotropy of the environment close to the ring, the degeneracy of the mode is slightly raised, leading to two branches of slow dispersions, in which frequencies can be adjusted by varying the value of the inner radius. The representation of the frequency eigenvalue of peak A as a function of inner radius is reported in the supplementary material (Fig. S-2). Such a mode with a high Q factor can be used for filtering purposes, for instance, in telecommunication applications.²⁴ 

We now turn our interest to the behavior of the dispersion and transmission curves when filling the hollow cylinders with Methyl Nonafluorobutyl Ether (MNE). In the first step, we do not take account of the liquid viscosity. Figure 3(a) represents the dispersion curves for MNE at 10 °C, i.e., with a mass density of ρ_MNE = 1554 kg/m³ and a sound velocity of v_MNE = 650.5 m/s. When filling the hollow cylinders, two additional double flat branches (B and C) occur in the wide bandgap, respectively, below and above the branches A. To understand the origin of these modes, we proceed to the eigenmode calculation at the relevant frequencies B and C. The acoustic displacement field modulus in the fluid has been deduced from the pressure according to

where u is the displacement field vector in the liquid, ρ_MNE and ω represent the mass density of the liquid and the angular frequency, respectively, and p is the acoustic pressure. In Fig. 3(b), the left and right bars correspond to the amplitude of the displacement in the solid and in the liquid, respectively. One can see that, at the resonance, the amplitude of the acoustic field in the liquid is 100 times higher than the amplitude in the solid. In the transmission calculation, the excitation of the modes will depend on the symmetry and the polarization of the incident field.

As mentioned earlier, to define a sensor, the frequency of the resonant mode has to be very sensitive to any change in the acoustic properties of the material, which fill the hollow cylinders. It is worth mentioning that MNE is considered as a temperature-sensitive material.^16,17 For that reason, we now study the resonant properties and wave confinement of modes A, B, and C at different MNE temperatures. The acoustic properties of MNE are reported in Table II as a function of temperature. One can see that both the mass density ρ_MNE and the longitudinal velocity c_MNE decrease when the temperature grows from 10 to 40 °C, following the respective slopes $ΔρMNEΔT=−2.65kg−1m−3K−1$ and $ΔcMNEΔT=−3.29m−1s−1K−1$⁠.

The more direct way to estimate the frequency shift of the modes is to calculate the transmission spectrum, by recording the output signal compared to the input. Figure 4(a) reports the transmission when filling the hollow cylinders with MNE and then changing the temperature from 10 to 40 °C in a 5 °C step. The black curve represents the 10 °C case where we find three narrow peaks inside the bandgap, which correspond to the three flat bands A, B, and C in the dispersion curves. When changing the temperature, one can see that only the modes B and C shift in frequency, while mode A is almost unchanged. This is due to the high localization of the acoustic field in the liquid for modes B and C, in contrast to mode A, which remains mainly confined in the solid. However, we note that the frequency offset of mode C is lower than mode B. Furthermore, the narrow bands between two steps of temperature are clearly separated, away from any overlap, which might appear for frequency shifts lower than 3 kHz, which would correspond to a variation of temperature less than 1 °C. Figure 4(b) represents an illustration of the acoustic and elastic field modulus at the resonant frequency f_C = 878 kHz, obtained for T = 10 °C. The transmission through the waveguide is clearly observed. Nevertheless, the high resonances in the MNE (right bar) are coupled together through evanescent fields present in the solid (left bar). Physically, it means that the narrow band comes from the coupling between the MNE resonance modes inside the hollow cylinders, leading to a narrow-guided band of low group velocity. Figure 4(c), left panel, represents the transmission peak C at 10 °C, magnified in the frequency range [870, 890] kHz. We can see that C is not exactly a resonance peak, but rather a narrow bandwidth, resulting in an average quality factor of Q ∼ 400.

The effect of the viscosity is introduced via the general description of the acoustic wave’s propagation in a compressible Newtonian fluid, based on Navier–Stokes equations and boundary conditions,²⁶ expressed following the liquid waves’ velocity v, without considering the thermal effects,

with $v=jωρliq∇p$⁠.

The shear viscosity for MNE at T = 10 °C is μ = 8.8610⁻⁴ Pa s.²⁵ In the absence of data for μ_B, we took the water value, 2.480 × 10⁻³ Pa s at T = 25 °C, associated with a shear viscosity of μ_water = 8.88 × 10⁻⁴ Pa s, very close to the one in MNE.

The middle panel of Fig. 4(c) has been obtained considering only the effect of the longitudinal viscosity on the resonant peak C at T = 10 °C. We found that the amplitude of the narrow branch decreased by 15%. When including, in addition, the shear viscosity [Fig. 4(c), right panel], the amplitude of the peak drops by two orders of magnitude, reaching a quite negligible value. It means that the losses due to the shear viscosity have killed the resonant peak C. The shear displacement at the solid–liquid interface leads to an important energy dissipation due to shear viscosity in the thin interfacial liquid/solid layer. To recover the resonant peak, one way has been to shorten the length of the waveguide. Figure 5 shows the transmitted signal when considering a waveguide of three units’ cells long, calculated for a non-viscous (solid line) and a viscous (dashed lines) liquid. In this case, the transmitted peak is divided by three from the non-viscous to the viscous case, but the decrease is only about 50% as compared to the non-viscous result of Fig. 4, so it still remains reachable. It is also worth noting that the quality factor is not much affected because despite the decrease of the amplitude, the width of the peak is not altered.

In many cases reported in the literature, the introduction of the liquid inside the phononic structure for sensing applications has been done by considering the insertion of a single cavity. Figure 6(b) represents the case of a cavity consisting of a hollow cylinder, filled with MNE, and inserted inside the phononic crystal of five units’ cell long. The resonant mode C inside the MNE appears now in the transmission curve as a very narrow peak at the frequency of 878.30 kHz, with a frequency resolution of 0.01 kHz. In this case, mode C cannot escape from the cavity into the crystal because of the bandgap and the field is highly confined [see Fig. 6(b)], giving rise to a very high-quality factor, above 10⁵. Such a structure would theoretically be useful to sense with high efficiency small variations of temperature, less than 10⁻² °C, corresponding to a frequency shift of 0.1 kHz. However, the peak will shrink even in the presence of a very small damping in MNE,²⁷ as mentioned in Fig. 4(c) for the waveguide. This will yield to experimental difficulties in the detection of such a narrow peak, not necessarily useful for practical investigations. Therefore, the waveguide represents a good alternative, for which the experimental measurement can be done, and its geometry adapted to the background noise or the viscosity of the liquid component as proposed above.

To estimate the performance of the phononic sensor, we first present the evolution of the frequencies of modes A, B, and C inside the bandgap as a function of MNE temperature [Fig. 7(a)]. The behavior is linear and can be quantified considering the sensitivity, defined as S = Δf/ΔT. We found, for the two modes B and C, S_B = −7.1 and S_C = −4.4 kHz/K, while the slope for mode A is almost zero. To go further and clarify independently the physical effects of the velocity and the density of the liquid, we have changed each physical parameter by 10%, keeping constant the other one. It appears that most of the frequency shift comes from the velocity of the fluid rather than the mass density.

The sensitivity by itself is not sufficient to fully characterize the performance of the sensor. The figure of merit (FoM) given by FoM = S · Q/f_R allows a full description of the efficiency of the sensor, where Q = f_R/Δf_1/2 is the quality factor, f_R is the resonant frequency, and Δf_1/2 is the frequency width at the half maximum, FWHM. The FoM calculations have been performed in the non-viscous MNE. The evolution of the FoM is given for modes B and C in Fig. 7(b). The non-regular variation of the FoM vs the temperature (especially for mode B) results in the Q factor variations. Indeed, the Q factor strongly depends on the frequency position of the flat branch inside the gap, which can interact differently with the bandgap edges. Consequently, the quality of the confinement is then affected in a different manner depending on the frequency position of the mode, thus leading to a non-regular variation of the Q factor as a function of temperature.

We developed a two-dimensional phononic crystal sensor to detect the sensitivity of the Methyl Nonafluorobutyl Ether (MNE) acoustic properties with the temperature. The MNE is a non-toxic, non-flammable, and colorless organic mild solvent solution widely used in cosmetic and beauty products. Previous works reported that the composition of MNE is highly affected with the temperature. The phononic crystal sensor proposed here consists of a hexagonal lattice of tungsten cylinders embedded in an epoxy background in which a line of hollow cylinders is introduced as a waveguide. We show that when filling the hollow cylinders with the MNE, two guided modes of low group velocity appear in the large bandgap. We demonstrated that the sharp peaks associated with these modes in the transmission spectrum were sensitive to the variation of temperature of the MNE. Moreover, we found that among the two acoustic parameters ρ_MNE and c_MNE defining the liquid, the longitudinal velocity was the one, which produces the highest contribution. We also introduced the effect of damping and found that, compared to the longitudinal, the shear viscosity induces the more important losses. The adaptation of the length of the waveguide has represented an efficient solution to recover the resonant peak in the presence of viscosities. Finally, compared to the excitation of a waveguide, the cavity resonant modes offer peaks with higher quality factor, which could be, however, very sensitive to losses. Therefore, for current detections of frequency with shifts larger than 3 kHz, the waveguide platform represents a good compromise in terms of FoM, offering an experimental way easier to implement and also technically adapted to strong liquid viscosity.

See the supplementary material for the evolution of the width of the bandgap as a function of the radius of the tungsten cylinders embedded in epoxy and the evolution of the guided mode of type A, which can be used as a filtering device; Figure S-1: Evolution of the bandgap width (BGW) as a function of the radius of the cylinders (the value r = 0.40 mm is the one chosen in the paper) and Figure S-2: Evolution of the frequency of mode A as a function of the inner radius r_i of the hollow cylinders inside the absolute bandgap limited by the red hatched areas.

Y.P., B.D.R., A.G., and R.L. thank the national French agency Grant No. ANR-18-CE92-0023 and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) for supporting this work in the frame of the project “Tubular Bell,” under Grant Nos. VE 483/2-1 and LU 605/22-1.

The data that support the findings of this study are available from the corresponding authors upon reasonable request.