# Phononic Crystal Plate with Hollow Pillars Actively Controlled by Fluid Filling

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Results and Discussion

#### 2.1. Whispering-Gallery Modes

_{i}and outer radius r. The entire structure is made of cubic silicon, with the elastic constants C

_{11}= 166 GPa, C

_{12}= 64 GPa, C

_{44}= 79.6 GPa, and the mass density being ρ = 2330 kg·m

^{−3}. The crystallographic axis [100] and [010] have been chosen respectively parallel to the phononic crystal axis x and y. In the (x, y) plane, periodic boundary conditions are applied on each side of the unit cell. Dispersion and transmission curves are calculated by the finite element code COMSOL Multiphysics

^{®}(COMSOL Inc., Stockholm, Sweden) and presented as a function of the reduced frequency wa/2πv

_{t}, where v

_{t}= Sqrt (0.5*(C

_{11}− C

_{12})/ρ) is the transverse velocity of sound in silicon along the [110] direction in the (001) plane.

_{i}/a = 0.145, r/a = 0.4, h/a = 0.45, e/a = 0.1, l/a = 0.2. With this set of parameters, the Bragg and low frequency band gaps still appear along ΓX direction while only a Bragg band gap remains along the ΓM direction. In the Bragg band gap, two branches of WGMs occur, marked as branch ‘1’ and ‘2’. Two transmission spectra along each direction are associated with two different incident waves, namely the fundamental anti-symmetric A

_{0}Lamb (blue curve) and the symmetric S

_{0}(red curve) Lamb waves. Although some mode conversion can occur at the exit of PnC, the transmitted wave keeps essentially its original character. Only WGM1 gives rise to a narrow transmitted pass band in both the ΓX and ΓM directions, more significantly with anti-symmetric Lamb wave excitation, marked as peak ‘A’ and ‘B’. In the right panel of Figure 2, we show the displacement fields of the dominant Uz component (displacement along z-axis) for peak ‘A’ and ‘B’. The excitation inside the PnC is symmetric with respect to an xz plane, perpendicular to the pillars and parallel to the propagation direction. This is in accordance with the symmetry of the incident wave (either A

_{0}or S

_{0}) with respect to such a plane. In contrast, to obtain a transmission at the frequency of WGM2, it would be necessary to have an incident wave which has an antisymmetric profile with respect to this plane, which means −/+ force in the unit cell along the y-direction. To explain the higher transmission of WGM1 with the A

_{0}rather than S

_{0}incident wave, it should be noticed that the x and y components of its displacement field are mainly localized in the upper part of the pillar, around the hollow part, while the z component extends down to the bottom of the pillar and is therefore sensitive to a vertical motion in the membrane. However, in some other frequency ranges such as [0.2; 0.4], the transmission is much higher with S

_{0}rather than A

_{0}excitation.

_{i}) 2 of the shell around the hollow increases the acoustic path along the perimeter 2π<r> of the cylinder. As a result, when r

_{i}/a = 0.145 (resp. 0.35), the WGM ‘1’ and ‘2’ fall in the middle of the Bragg (resp. low frequency) gap [22]. We also calculate the corresponding quality factor for the WGM 1, Q = f/Δf, where f is the central frequency of the pass band and Δf is the full width at half maximum of the transmission peak. The right panel of Figure 3 shows a significant increase in the quality factor when increasing the reduced height l/a of the pillar basis. For l/a = 0.35, the quality factor is Q = 280, which is more than 10 times the value obtained without the additional cylinder, paving the way to a high resolved narrow pass band device for filtering applications [22].

_{i}/a = 0.12 and waveguide j with inner radius r

_{j}/a = 0.11. The transmission peaks for waveguide i and j are located at reduced frequency 0.654 and 0.678, respectively. In addition, an efficient subwavelength waveguide is also demonstrated, as the WGMs can be tuned in the low frequency band gap [22].

#### 2.2. Active Control of the WGMs and New Localized Modes by Filling the Hollows with a Liquid

_{i}. For water, the mass density is ρ = 998 kg·m

^{−3}and speed of sound is c = 1490 ms

^{−1}. Let us start with the inner radius r

_{i}/a = 0.17. The modes labeled 1 and 2 are the quadrupolar WGMs discussed in the previous section. However, the filling of the holes with water has the effect of giving rise to two new sets of localized modes in the band gap. One set, labeled M

_{c1}and M

_{c2}, correspond to compressional vibrations inside the liquid column almost independently of the solid; they will be discussed in detail in Section 2.3. The other set called M

_{liq}is a doubly degenerate new mode that is essentially associated to the presence of the liquid and appears in the band gap under some conditions on the geometrical parameters. The strongest vibration of this mode occurs in the liquid where the displacement field is one order of magnitude higher than in the solid part. When decreasing the inner radius r

_{i}/a of the pillars from 0.17 to 0.11 (Figure 5), the WGM1,2, as well as M

_{liq}, see their frequency increasing and going outside the band gap, while the frequencies of M

_{c1}and M

_{c2}remain unchanged because they are dictated by the height of the fluid.

_{c1,2}and M

_{liq}, behave when changing either the inner radius of the hollow pillars or the height h

_{w}of the fluid filling the hollow part of the pillar of total height h. In the upper-left panel, one can see, in accordance with Figure 5, that with an increasing inner radius, WGM1,2 and M

_{liq}decrease to lower frequencies, passing out of the Bragg band gap of the full PnC (h/a = 0.45, r/a = 0.4, l/a = 0). On the other hand, M

_{c2}is practically insensitive to the inner radii, as the compressional mode in the liquid is only related to the height of the liquid (see discussion in Section 2.3.). In the upper-right panel of Figure 6, we present the evolution of those localized modes as a function of the height h

_{w}of water filling the pillar of total height h when the inner radius is r

_{i}/a = 0.19. The liquid compressional modes M

_{c1,2}are very sensitive to h

_{w}, whereas the M

_{liq}decreases through the Bragg band gap when h

_{w}/h increases from 0.3 to 1. The latter mode can be then a good candidate to be tuned gradually by changing the height of water. The vibration of this mode is presented in the lower panel of Figure 6, both in the solid and liquid part. One can see that the elastic and acoustic fields are mainly oriented along the diagonal direction of the square lattice. The vibration in the solid is mostly localized at the top of the pillar although there are still some displacements left in the plate. In the liquid part, the pressure field behaves like a dipolar motion, with −max and +max along the same diagonal direction.

#### 2.3. Compressional Modes along the Height of the Liquid

_{c}in the previous section, which are associated to vertical motion inside the fluid. Due to the high impedance mismatch between most of the liquids and a hard solid, these modes are mainly localized inside the fluid. We give two illustrations about the sensitivity of these modes to the physical properties of the liquid and the variation of its parameters as a function of temperature.

_{liq}with rigid lateral boundaries, rigid bottom boundary and a free upper boundary. The expressions of the frequencies are then f

_{n}= (2n + 1)c/4 h

_{liq}, where n is the resonance number (0, 1, 2, 3, ...), and c is the speed of sound in the fluid; this means that the height can accommodate stationary waves at λ/4, 3λ/4,…

_{i}/a = 0.1, h

_{w}/h = 1, and l/a = 0.1. The first and the second compressional liquid frequencies are f

_{c1}= 0.197 and f

_{c2}= 0.587, corresponding to a wavelength λ

_{c1}/h = 4 and λ

_{c2}/h = 4/3, respectively, as shown clearly in the pressure distributions in the left panel. The set of geometric parameters are chosen to move the WGMs to higher frequencies outside of the Bragg band gap. From the right panel, the first compressional liquid mode is at the edge of the low frequency band gap while the second one is in the middle of the Bragg band gap. We focus on the second compressional liquid mode, as the Bragg band gap is broader and has more potential in the applications.

_{c2}in the Bragg band gap is isolated, allowing a phononic sensor application to sense the probed parameters on a sufficiently broad frequency range. The efficiency of the phononic sensor is detected by changing the physical properties of the filled liquid in the hollow pillar. Six kinds of liquids are employed to test the efficiency [9,10]. Figure 8 left panel presents the evolution of the M

_{c2}induced transmission peaks as a function of the acoustic velocity. It is observed that the transmission peaks are very sensitive to the acoustic velocity of the infiltrated liquid, with high quality factors Q = f/Δf larger than 1000. Figure 8 (right panel) shows the relationship between the frequency of transmission peak and the corresponding acoustic velocity. In order to qualify the sensitivity, a common measurement is to calculate the slope of the lines in the right panel, named sensitivity S, as S = Δf/Δc, where Δf is the difference of the reduced frequencies of two infiltrated liquids and Δc = (c

_{liq}

^{i}− c

_{liq}

^{j})/v

_{t}is the difference of the reduced velocities of two infiltrated liquids by dividing the transverse velocity of silicon v

_{t}to have a dimensionless quantity. The average of S is 1.761, with a deviation 2.53% for the minimum and 0.87% for the maximum.

_{i}/a = 0.1, h

_{w}/h = 1, l/a = 0.1. By tuning the temperature of water from 0 °C to 70 °C, the frequency of the M

_{c2}increases with its corresponding quality factor decreasing, as shown in Figure 9. In the range of [0 °C; 50 °C], the frequency moves significantly in the step of 10 °C. Therefore, tuning the temperature of the infiltrated liquid is another way to actively control the M

_{c2}induced transmission peak.

#### 2.4. Influence of Filling the Holes with Mercury on Whispering Gallery Modes

^{−3}and speed of sound is c = 1490 ms

^{−1}. The upper frequency range limited by two horizontal cyan lines is the Bragg band gap of background full PnCs (h/a = 0.45, r/a = 0.4, l/a = 0), and the lower one is the low frequency band gap. The tunable frequency range of the WGM1 increases when the inner radius becomes larger, as the corresponding filling ratio φ = A

_{hole}/A

_{pillar}increases, where A

_{hole}, A

_{pillar}are the area of inner hole and whole pillar, respectively. For a given inner radius, the tunable frequency range for mercury is wider than that for water due to the fact that the impedance Z

_{m}= ρc of mercury is much larger than the impedance of water, even larger than that of silicon. In the Bragg band gap, mercury plays a more important role in actively controlling the WGM1.

_{m}/h > 0.4. The red dots show that the quality factor of WGM1 does not change too much within the range h

_{m}/h = [0; 0.6] and increases from 140 at h

_{m}/h = 0.6 to 210 at h

_{m}/h = 1. The regime where the WGM1 changes significantly with respect to h

_{m}is still located in the Bragg band gap, allowing for the realization of an actively tuned multichannel wavelength multiplexer.

_{m}

^{c}/h = 0.4 and h

_{m}

^{d}/h = 0.9, respectively. The geometric parameters for the two waveguides are h/a = 0.45, r/a = 0.4, l/a = 0.2, r

_{i}/a = 0.11. The background full cylinders have parameters as h/a = 0.45, r/a = 0.4, l/a = 0. The transmission is detected by exciting the anti-symmetric Lamb wave in front of the PnC. In the left panel, two narrow pass bands c and d appear in the Bragg band gap at reduced frequency 0.66 and 0.61, respectively. The higher height of mercury significantly shifts the transmission frequency to a lower value inside the band gap. The corresponding displacement field distributions in the solid silicon are presented in the right panel of Figure 12, showing a multichannel wavelength multiplexer behavior.

## 3. Conclusions

_{liq}in the band gap with a dipolar shape displacement field, which can be a good candidate for tuning as a function of the liquid height. The phononic crystal plate with fluid filled in the hollow pillars can be a good candidate for wireless communication and sensing applications with the possibility of active control.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Kushwaha, M.S.; Halevi, P.; Dobrzynski, L.; Djafari-Rouhani, B. Acoustic band structure of periodic elastic composites. Phys. Rev. Lett.
**1993**, 71, 2022–2025. [Google Scholar] [CrossRef] [PubMed] - Sigalas, M.; Economou, E.N. Band structure of elastic waves in two dimensional systems. Solid State Commun.
**1993**, 86, 141–143. [Google Scholar] [CrossRef] - Pennec, Y.; Vasseur, J.O.; Djafari-Rouhani, B.; Dobrzynski, L.; Deymier, P.A. Two-Dimensional phononic crystals: Examples and applications. Surf. Sci. Rep.
**2010**, 65, 229–291. [Google Scholar] [CrossRef] - Hussein, M.I.; Leamy, M.J.; Ruzzene, M. Dynamics of phononic materials and structures: Historical origins, recent progress and future outlook. Appl. Mech. Rev.
**2014**, 66, 040802. [Google Scholar] [CrossRef] - Khelif, A.; Choujaa, A.; Benchabane, S.; Djafari-Rouhani, B.; Laude, V. Guiding and bending of acoustic waves in highly confined phononic crystal waveguides. Appl. Phys. Lett.
**2004**, 84, 4400. [Google Scholar] [CrossRef] - Sun, J.H.; Wu, T.T. Propagation of acoustic waves in Phononic-Crystal plates and waveguides using a finite-difference Time-Domain method. Phys. Rev. B
**2007**, 76, 104304. [Google Scholar] [CrossRef] - Rostami-Dogolsara, B.; Moravvej-Farshi, M.K.; Nazari, F. Acoustic add-drop filters based on phononic crystal ring resonants. Phys. Rev. B
**2016**, 93, 014304. [Google Scholar] [CrossRef] - Pennec, Y.; Djafari-Rouhani, B.; Vasseur, J.O.; Khelif, A.; Deymier, P.A. Tunable filtering and demultiplexing in phononic crystals with hollow cylinders. Phys. Rev. E
**2004**, 69, 046608. [Google Scholar] [CrossRef] [PubMed] - Lucklum, R.; Li, J. Phononic crystals for liquid sensor applications. Acoustic band structure of periodic elastic composites. Meas. Sci. Technol.
**2009**, 20, 124014. [Google Scholar] [CrossRef] - Amoudache, S.; Pennec, Y.; Djafari-Rouhani, B.; Khater, A.; Lucklum, R.; Tigrine, R. Simultaneous sensing of light and sound velocities of fluids in a two-dimensional phoXonic crystal with defects. J. Appl. Phys.
**2014**, 115, 134503. [Google Scholar] [CrossRef] - Yang, S.; Page, J.H.; Liu, Z.; Cowan, M.L.; Chan, C.T.; Sheng, P. Focusing of sound in a 3D phononic crystal. Phys. Rev. Lett.
**2004**, 93, 024301. [Google Scholar] [CrossRef] [PubMed] - Jin, Y.; Torrent, D.; Pennec, Y.; Pan, Y.; Djafari-Rouhani, B. Simultaneous control of the S0 and A0 Lamb modes by graded phononic crystal plates. J. Appl. Phys.
**2015**, 117, 244904. [Google Scholar] [CrossRef] - Jin, Y.; Torrent, D.; Pennec, Y.; Pan, Y.; Djafari-Rouhani, B. Gradient index devices for the full control of elastic waves in plates. Sci. Rep.
**2016**, 6, 24437. [Google Scholar] [CrossRef] [PubMed] - Pennec, Y.; Djafari-Rouhani, B.; Larabi, H.; Vasseur, J.O.; Hladky-Henion, A.C. Low-Frequency gaps in a phononic crystal constituted of cylindrical dots deposited on a thin homogeneous plate. Phys. Rev. B.
**2008**, 78, 104105. [Google Scholar] [CrossRef] - Wu, T.T.; Huang, Z.G.; Tsai, T.C.; Wu, T.C. Evidence of complete band gap and resonances in a plate with periodic stubbed surface. Appl. Phys. Lett.
**2008**, 93, 111902. [Google Scholar] [CrossRef] - Pennec, Y.; Djafari-Rouhani, B.; Larabi, H.; Akjouj, A.; Gillet, J.N.; Vasseur, J.O.; Thabet, G. Phonon transport and waveguiding in a phononic crystal made up of cylindrical dots on a thin homogeneous plate. Phys. Rev. B
**2009**, 80, 144302. [Google Scholar] [CrossRef] - Oudich, M.; Li, Y.; Assouar, M.B.; Hou, Z. A sonic band gap based on the locally resonant phononic plates with stubs. New J. Phys.
**2010**, 12, 083049. [Google Scholar] [CrossRef] - Assouar, M.B.; Oudich, M. Enlargement of a locally resonant sonic band gap by using double-Sides stubbed phononic plates. Appl. Phys. Lett.
**2012**, 100, 123506. [Google Scholar] [CrossRef] - Coffy, E.; Lavergne, T.; Addouche, M.; Euphrasie, S.; Vairac, P.; Khelif, A. Ultra-Wide acoustic band gaps in pillar-Based phononic crystal strips. J. Appl. Phys.
**2015**, 118, 214902. [Google Scholar] [CrossRef] - Midtvedt, D.; Isacsson, A.; Croy, A. Nonlinear phononics using atomically thin membrances. Nat. Commun.
**2014**, 5, 4838. [Google Scholar] [CrossRef] [PubMed] - Graczykowshi, B.; Sledzinska, M.; Alzina, F.; Gomis-Bresco, J.; Reparaz, J.S.; Wagner, M.R.; Sotomayor Torres, C.M. Phonon dispersion in hypersonic two-dimensional phononic crystal membranes. Phys. Rev. B
**2015**, 91, 075414. [Google Scholar] [CrossRef] - Jin, Y.; Fernez, N.; Pennec, Y.; Bonello, B.; Moiseyenko, R.P.; Hemon, S.; Pan, Y.; Djafari-Rouhani, B. Tunable waveguide and cavity in a phononic crystal plate by controlling whispering-Gallery modes in hollow pillars. Phys. Rev. B
**2016**, 93, 054109. [Google Scholar] [CrossRef] - Jin, Y.; Bonello, B.; Pan, Y. Acoustic metamaterials with piezoelectric resonant structures. J. Phys. D Appl. Phys.
**2014**, 47, 245301. [Google Scholar] [CrossRef] - Popa, B.T.; Cummer, S.A. Non-Reciprocal and highly nonlinear active acoustic metamaterials. Nat. Commun.
**2014**, 5, 3398. [Google Scholar] [CrossRef] [PubMed] - Wang, P.; Casadei, F.; Shan, S.; Weaver, J.C.; Bertoldi, K. Harnessing buckling to design tunable locally resonant acoustic metamaterials. Phys. Rev. Lett.
**2014**, 113, 014301. [Google Scholar] [CrossRef] [PubMed] - Sato, A.; Pennec, Y.; Shingne, N.; Thurn-Albrecht, T.; Knoll, W.; Steinhart, M.; Djafari-Rouhani, B.; Fytas, G. Tuning and switching the hypersonic phononic properties of elastic impedance contrast nanocomposites. ACS Nano
**2010**, 4, 3471. [Google Scholar] [CrossRef] [PubMed] - Demirci, U.; Montesano, G. Single cell epitaxy by acoustic picolitre droplets. Lab Chip
**2007**, 7, 1139. [Google Scholar] [CrossRef] [PubMed] - Tilli, M.; Motooka, T.; Airaksinen, V.; Franssila, S.; Paulasto-Krockel, M.; Lindroos, V. Handbook of Silicon Based MEMs Materials and Technologies, 2nd ed.; William Andrew: London, UK, 2015. [Google Scholar]

**Figure 1.**(

**a**) schematic view of the PnC unit cell in the square array consisting of hollow pillars deposited on a thin homogeneous plate with an additional cylinder of height l at the basis to improve the confinement of the modes in the hollow pillars. a is the lattice constant, e is the thickness of plate, h is the height of hollow pillar, r

_{i}and r are the inner and outer radius of the hollow pillar, respectively; (

**b**) the irreducible first Brillouin zone of the square lattice.

**Figure 2.**(

**Left panel**): Dispersion curves of the confined hollow pillars on a thin silicon plate in the ΓX and ΓM directions of the first irreducible Brillouin zone in the reduced frequency range [0; 0.75]. On each side of the dispersion curves, we give the corresponding transmission spectra in blue and red respectively for which the incident wave is either anti-symmetric A

_{0}or symmetric S

_{0}. The Bragg and low frequency band gaps are marked as red and blue rectangular hatched regions, respectively. The geometric parameters are chosen as r

_{i}/a = 0.145, r/a = 0.4, h/a = 0.45, e/a = 0.1, l/a = 0.2; (

**Right panel**): Uz component of the displacement fields with the anti-symmetric A

_{0}Lamb wave excitation at transmission peak A along the ΓX direction and peak B along the ΓM direction; Uz component of the displacement fields of WGM1 and WGM2 at Γ point.

**Figure 3.**(

**Left panel**): the frequency evolution of the WGM 1 and 2 as a function of the inner radius of the hollow pillar. The upper frequency range limited by two horizontal cyan lines is the Bragg band gap of background full PnCs (h/a = 0.45, r/a = 0.4, l/a = 0) and the lower one is the low frequency band gap. (

**Right panel**): The quality factor of the WGM 1 grows when increasing the confinement height l/a, r

_{i}/a = 0.145.

**Figure 4.**The multichannel wavelength multiplexer: (

**Left panel**): Transmission spectrum of the antisymmetric Lamb wave when the inner radius inside waveguides i and j are r

_{i}/a = 0.12 and r

_{j}/a = 0.11; (

**Right panel**): Displacement field distributions at the frequency of the two narrow pass bands i and j. The geometric parameters of the multiplexer are h/a = 0.45, r/a = 0.4, l/a = 0.2.

**Figure 5.**Dispersion curves of the hollow pillars on a thin silicon plate in the ΓX direction with different inner radii (

**left**) r

_{i}/a = 0.11, (

**middle**) r

_{i}/a = 0.14, (

**right**) r

_{i}/a = 0.17. The other geometric parameters are h/a = 0.45, r/a = 0.4, l/a = 0.2.

**Figure 6.**(

**Upper-left panel**): evolution of WGM1 (

**black dotted line**), WGM2 (

**red dotted line**), M

_{c1}(

**cyan dotted line**), M

_{c2}(

**blue dotted line**), M

_{liq}(

**green dotted line**) as a function of inner radius when the hollow pillars are fully filled with water (h

_{w}/h = 1). The two horizontal pink dotted lines are the limits of Bragg band gap of the full PnC; (

**Upper-right panel**): evolution of WGM1 (

**black dotted line**), WGM2 (

**red dotted line**), M

_{c1}(

**cyan dotted line**), M

_{c2}(

**blue dotted line**), M

_{liq}(

**green dotted line**) as a function of the height of filling water h

_{w}/h when the inner radius is r

_{i}/a = 0.19. (

**Lower panel**): representation of the acoustic (pressure) and elastic (displacement) field of the mode M

_{liq}respectively in the fluid and solid part for h

_{w}/h = 1 (left) and h

_{w}/h = 0.5 (right) when r

_{i}/a = 0.19. The other geometric parameters are h/a = 0.45, r/a = 0.4, l/a = 0.2.

**Figure 7.**(

**Left panel**): 3D-schematic view of the pressure fields in water of the M

_{c1}(

**left-lower**) and M

_{c2}(

**left-upper**) compressional modes; (

**Right panel**): Dispersion curves of the PnC with geometric parameters h/a = 0.4, r/a = 0.39, r

_{i}/a = 0.1, h

_{w}/h = 1, l/a = 0.1 along the ΓX direction.

**Figure 8.**(

**Left panel**): Evolution of the second liquid compressional mode induced transmission peak (lines, corresponding to the left y-axis) and quality factor (dots, corresponding to the right y-axis) as a function of the acoustic velocity of filled liquid. The geometric parameters are h/a = 0.4, r/a = 0.39, r

_{i}/a = 0.1, h

_{liq}/h = 1, l/a = 0.1. (

**Right panel**): The frequency of transmission peak corresponds to the acoustic velocity of different fluids.

**Figure 9.**Varied frequencies (

**blue triangular dots**) and corresponding quality factors (

**red circle dots**) of the second liquid compressional mode by tuning the temperature of water in the holes. The geometric parameters are h/a = 0.4, r/a = 0.39, r

_{i}/a = 0.1, h

_{w}/h = 1, l/a = 0.1.

**Figure 10.**Varied range of WGM 1 frequency when the holes are respectively empty or fully filled with the liquid: water (

**blue dotted lines**) and mercury (

**red dotted lines**), corresponding to different inner radii. The other geometric parameters are h/a = 0.45, r/a = 0.4, l/a = 0.2. The upper frequency range limited by two horizontal cyan lines is the Bragg band gap of background full PnCs (h/a = 0.45, r/a = 0.4, l/a = 0) and the lower one is the low frequency band gap.

**Figure 11.**The evolutions of the WGM 1 frequency (

**blue triangle dots**) and its corresponding quality factor (

**red circle dots**) as a function of the filled mercury height. The geometric parameters are h/a = 0.45, r/a = 0.4, l/a = 0.2, r

_{i}/a = 0.11.

**Figure 12.**The multichannel wavelength multiplexer. (

**Left panel**): Transmission spectrum of the antisymmetric Lamb wave when the filled mercury heights inside waveguide c and d are h

_{m}

^{c}/h = 0.4 and h

_{m}

^{d}/h = 0.9; (

**Right panel**): Displacement field distributions at the frequency of the two narrow pass bands c and d. The geometric parameters of the multiplexer are h/a = 0.45, r/a = 0.4, l/a = 0.2, r

_{i}/a = 0.11.

**Table 1.**Density and speed of sound of a mixture of water and 1-propanol at different molar ratio x.

Molar Ratio x | Density (kg·m^{−3}) | Speed of Sound (ms^{−1}) |
---|---|---|

0 (water) | 998 | 1490 |

0.021 | 990 | 1545 |

0.056 | 974 | 1588 |

0.230 | 908 | 1421 |

0.347 | 881 | 1367 |

0.596 | 841 | 1298 |

T (°C) | Mass Density (kg·m^{−3}) | Speed of Sound (ms^{−1}) |
---|---|---|

0 | 999 | 1405 |

10 | 999 | 1447 |

20 | 998 | 1482 |

30 | 997 | 1497 |

40 | 992 | 1529 |

50 | 986 | 1547 |

60 | 983 | 1550 |

70 | 977 | 1554 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Jin, Y.; Pennec, Y.; Pan, Y.; Djafari-Rouhani, B.
Phononic Crystal Plate with Hollow Pillars Actively Controlled by Fluid Filling. *Crystals* **2016**, *6*, 64.
https://doi.org/10.3390/cryst6060064

**AMA Style**

Jin Y, Pennec Y, Pan Y, Djafari-Rouhani B.
Phononic Crystal Plate with Hollow Pillars Actively Controlled by Fluid Filling. *Crystals*. 2016; 6(6):64.
https://doi.org/10.3390/cryst6060064

**Chicago/Turabian Style**

Jin, Yabin, Yan Pennec, Yongdong Pan, and Bahram Djafari-Rouhani.
2016. "Phononic Crystal Plate with Hollow Pillars Actively Controlled by Fluid Filling" *Crystals* 6, no. 6: 64.
https://doi.org/10.3390/cryst6060064