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Band gap structure of acoustic phononic crystals with cluster primitives in triangular lattices
Journal of Physics and Chemistry of Solids 136 (2020) 109206
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Band gap structure of acoustic phononic crystals with cluster primitives in triangular lattices Jiaguang Hu School of Information Science, Wenshan University, Wenshan, 663099, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Cluster primitive Energy band structure Phononic crystal Triangular lattice
In this study, a two-dimensional steel/gas acoustic phononic crystal with triangular lattices was considered as the model and single cylinder primitives were substituted with cluster primitives, before investigating the energy band structures of the longitudinal sound wave using the plane wave expansion method. The results indicated that energy band extremes were still present at the high symmetry points or symmetry lines in the Brillouin zone, although the range of the irreducible Brillouin zone increased by two times. Compared with single cylinder primitives, double cylinder primitives could achieve excitation in low frequency broad band gaps at relatively low filling rates. Furthermore, sextuple cylinder primitives achieved primary band gap width increases of 7.3 times. The width of the band gap increased with the filling rates of the cluster primitives. Precise control of the band gap location and width at low filling rates can be achieved by varying the primitive orientation or intercylinder distance in the primitives.
1. Introduction Recently, inspired by photonic crystals, researchers have intensively investigated phononic crystals because of their unique properties in terms of producing an elastic wave band gap. Phononic crystals are composite functional materials with a periodic arrangement of different elastic materials. It has been demonstrated both theoretically and experimentally that the transmission of elastic waves in phononic crystals may lead to band gap structures similar to those in photonic crystals under appropriate conditions [1–6]. The unique band gap properties of phononic crystals mean that they may have applications in various fields, including precise vibration absorption, negative refrac tion, ultrasonic imaging, acoustic cloaking, detectors, directional transmission, and auto-collimation [7–11]. Unlike photonic crystals, the transmission of elastic waves in media with different properties lead to variations in the polarization state of the phononic crystals, thereby complicating the identification and calculation of the band gaps. The mechanisms related to forbidden bands in phononic crystals mainly comprise the Bragg scattering mechanism [12], local resonance mech anism [13], and inertial amplification mechanism [14]. The physical properties of forbidden bands in phononic crystals are dominated by the structural shape of primitive [15–17] characteristics such as the elastic parameters, and spatial distribution properties such as the lattice structure distance, symmetry, and lattice configuration [16–18].
The applications of phononic crystals depend on the generation and regulation of band gaps. In particular, two-dimensional (2D) solid/gas phononic crystals based on Bragg scattering have attracted much attention because of their simplicity in terms of experimental prepara tion, band gap generation, and the establishment of an equivalent calculation model. In Bragg phononic crystals, ultra-low frequency band gaps are usually generated in plate structures [19–21]. Composite structures [22–27], plate structures [28], fractal structures [29], and quasi-Sierpinski carpet unit cells [30] have been introduced into pho nonic crystals to allow excellent band gap tunability in ordinary struc tures. In addition, polyhedrons [31], square cylinders [32], elliptic cylinders [33], hexagonal cylinders, and triangular cylinders [34] have been inserted to reduce the symmetry of the crystal structures in lattices. However, the design of novel structures has focused mainly on square and honeycomb lattices, whereas cluster inclusions have not been considered. In conventional phononic crystals, broad band gaps are readily generated in crystals with triangular lattices, but their filling rates are extremely high [35]. The generation of relatively broad band gaps with enhanced tunability at low filling rates can reduce the cost and facilitate various applications of phononic crystals. In this study, we designed a novel phononic crystal structure with cluster primitives based on composite structures and under the requirement for facile experimental preparation. The variations in the first irreducible Bril louin zone were investigated and we identified the locations of extreme
E-mail address: [email protected]. https://doi.org/10.1016/j.jpcs.2019.109206 Received 4 March 2019; Received in revised form 15 September 2019; Accepted 16 September 2019 Available online 17 September 2019 0022-3697/© 2019 Elsevier Ltd. All rights reserved.
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Journal of Physics and Chemistry of Solids 136 (2020) 109206
energy bands in order to investigate the effects of the shape of the cluster primitives, filling rate, arrangement orientation, and location on the band gaps and the band gap broadening mechanism. It should be noted that the band structures were calculated in this study by only consid ering the propagating modes. The evanescent modes [36] will be calculated in a future study. In addition, we focused in acoustic pho nonic crystals and a similar study of fully elastic continuum structures might obtain different results because of the more complicated band structure. 2. Study model Fig. 1(a) shows conventional simple lattice phononic crystals comprising solid steel cylinders arranged in a triangular lattice in the air with an infinite periodic structure. R and a denote the cross-section radius of the cylinder and the lattice constant, respectively. Fig. 1(b) shows the first irreducible Brillouin zone. As shown in Fig. 2, the single cylinder primitives are substituted with double, triple, sextuple, and septuple cylinder primitives to obtain new phononic crystals. The degree of rotational symmetry for the double, triple, and sextuple cylinder primitives increases successively, and the degree of symmetry for the septuple cylinder primitive is the same as that for the sextuple cylinder primitive. The adjacent cylinders in a primitive are tangent to each other and the lattice point is at the center of each primitive. The tangent of the cylinders in the primitive is the only critical condition for the spatial structure and the adjacent cylinders exhibit no deformation. The tangent of the cylinders is similar to the critical condition where the scatterers simply contact each other when the filling ratio in the crystal is maxi mized. In the present study, we only considered the band gaps below 40 kHz, and thus the wavelength range of the acoustic waves is much larger than the inter-atomic distance. Therefore, a linear elastic model is suitable for this situation. All of the cylinders are identical in size. This composite primitive structure is regarded as a cluster primitive. The tangent locations for the cylinders in the primitive or the inter-cylinder distance are adjustable. These phononic crystals are easy to synthesize and it is convenient to control the band gaps due to the increased quantity of structural parameters involved. In solid/gas systems, we can impose the condition of elastic rigidity on the solid rods because of the high density contrast between the solid and air. Under these conditions, the solid is effectively treated as a fluid and only the longitudinal wave needs to be considered [37]. Vasseur et al. reported that the plane wave expansion method agreed well with the finite-difference time-domain method calculations and the experi mental results obtained for two-dimensional solid/air systems [38]. In the calculation using the plane wave method, the following eigenvalue equation should be solved [39]. � �� �i Xh 0 ω2 λg 1g0 ρg 1g0 k þ g ⋅ k þ g Pg0 ¼ 0; (1) g
Fig. 2. Structure of triangular lattice with cluster primitives: (a) double cyl inder primitives; (b) triple cylinder primitives; (c) sextuple cylinder primitives; and (d) septuple cylinder primitives.
are the Fourier expansion coefficients for the pressure P(r), mass density, and bulk modulus, respectively, and g is the 2D reciprocal lattice vector, which can be defined as [40]: � � 2π ð m þ 2nÞ pffiffi g¼ ey ; (2) mex þ a 3 where a is the lattice constant, m and n are integers, and ex and ey are unit vectors of the x axis and y axis, respectively. 1 Using fg for Pg’, ρ-1 g , and λg , fg can be calculated as follows: � fg ¼ fA þ ð1 þ FÞfB ; g¼0 0 ; (3) fg ¼ ðfA fB ÞI ðgÞ; g 6¼ 0 where A and B denote the cylinders and background, respectively. The filling rate is defined as F ¼ S0 /S, where S and S0 are the total area of the cell cross-section and the area of cluster primitive cross-section, respectively. The maximum values of F for the five structures are 0.9069 for single cylinder primitives, 0.6046 for double and sextuple cylinder primitives, 0.6802 for triple cylinder primitives, and 0.7054 for septuple cylinder primitives. Based on the characteristics of the Fourier transform, the structure functions of the cluster primitives can be obtained as follows: X 0 I ðgÞ ¼ eig⋅rn IðgÞ; (4)
0
where ω is the angular frequency, k is the wave vector, Pg’, ρ-g1, and λ-g1
rn
where rn is the position vector (x, y) for the center of each cylinder in a cluster primitive and the origin of the coordinates is at the center of the primitive. In the single cylinder primitive, rn is (0, 0). In the double cylinder primitives, rn are (0, R) and (0, –R). In the triple cylinder pffiffiffi pffiffiffi pffiffiffi primitives, rn are ð0; 2R= 3 Þ, ðR; 3 R=3Þ, and ð R; 3 R=3Þ. In pffiffiffi pffiffiffi the sextuple cylinder primitives, rn are ð R; 3 RÞ, ðR; 3 RÞ, ðR; pffiffiffi pffiffiffi 3 RÞ, ð R; 3 RÞ, (–2R, 0), and (2R, 0). In the septuple cylinder pffiffiffi pffiffiffi pffiffiffi primitives, rn are (0, 0), ð R; 3 RÞ, ðR; 3 RÞ, ðR; 3 RÞ, ð R; pffiffiffi 3 RÞ, (–2R, 0), and (2R, 0). I(g) is the structure function for each cyl inder and it can be calculated as [41]. Fig. 1. Structure of simple triangular lattice (a) and the first irreducible Bril louin zone (b). 2
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IðgÞ ¼
Journal of Physics and Chemistry of Solids 136 (2020) 109206
1 S
,
Z e
ig⋅r
dr ¼ 2FJ1 ðjgjRÞ
jgjR
(5)
A
where A is the cross-section of a cylinder and J1 is the first-order Bessel function. Based on Bloch theory, the pressure P(r) in a structure can be written as follows [39]. X PðrÞ ¼ eik⋅r Pg eig⋅r (6) g
By solving equation (1), we can obtain ω and the relationship be tween the energy band and structure (k–Hz). The distributions of the acoustic pressure field under a certain point and a certain value of ω can be determined based on equations (1) and (6). The physical parameters for the steel/air system are: ρ ¼ 7800/1.21 kg/m3, longitudinal wave speed Cl ¼ 6100/343 m/s, λ ¼ ρC2l , and lattice constant a ¼ 2 cm. In total, 625 plane wave numbers were considered to guarantee the convergence and MATLAB software was used for the calculations. 3. Results and discussion 3.1. Extreme locations of energy bands In order to obtain the energy band structure, the natural frequencies of all the wave vectors k in the irreducible Brillouin zone must be considered (shadow in Fig. 1(b)). The natural frequency is defined as extreme because the normal component of the group velocity at high symmetry points along the symmetry plane is zero. The energy band structure of the simple lattices can be obtained by assigning all of the waves to wave vectors k on the boundary of the Brillouin zone (Γ-Х-YΓ). For phononic crystals with cluster primitives, wave scattering varies in all directions due to the varying symmetry of the crystal structures, which may affect the shape of the irreducible Brillouin zone and the extreme location of the energy band. Thus, we mapped the frequencies of the low frequency and high frequency energy bands in the primary Brillouin zone using the selected structural parameters, as shown in Fig. 3, where the dark color denotes low frequencies, and the black and white dots denote the locations of maximum and minimum energy bands, respectively. The results showed that the symmetry of the Bril louin zone varied significantly, especially for crystals with double cyl inder primitives. The irreducible Brillouin zone had to be extended to the ΓMXYΓ zone whereas the symmetry of the Brillouin zone for crystals with the sextuple cylinder primitives remained unchanged. The extreme energy bands for crystals with double cylinder primitives were located on the high symmetry lines and the extreme energy bands of the other crystals were still located at the high symmetry points in the Brillouin zone. Therefore, the values of the wave vectors were still assigned on the boundary of the Brillouin zone (Γ-Х-Y-Γ) to calculate the energy band. In this study, we considered crystals with no cylinders. Thus, the values of the wave vectors were assigned in the entire primary Brillouin zone in order to obtain an accurate energy band structure, which significantly increased the calculations required. Indeed, the uniformity of wave scattering in phononic crystals will affect the symmetry and locations of the extreme energy bands in the Brillouin zone. Reducing the symmetry of the primitives or lattice may eliminate the reflection surface of the unit cell, non-uniform wave scattering, and the reconstructed in teractions of waves and lattice, thereby resulting in variations in the energy band structures [42]. However, the high symmetry of a single cylinder in the cluster primitives can partially counteract the effects of reducing the symmetry of the simple lattice on wave scattering.
Fig. 3. Mapping of the energy band frequency in the primary Brillouin zone. F ¼ 0.5. Lighter colors indicate higher acoustic frequencies. Black and white dots denote the locations of the maximum and minimum energy bands, respectively. (a) Double cylinder primitives, tertiary energy band; (b) triple cylinder primitives, tertiary energy band; (c) sextuple cylinder primitives, ter tiary energy band; (d) septuple cylinder primitives, tertiary energy band; (e) double cylinder primitives, septuple energy band; and (f) double cylinder primitives, octuple energy band.
3.2. Effects of filling rate on the band gap width Fig. 4. Energy band and band gap for a simple triangular lattice. The shadow denotes the band gap. (a) Energy band structure, F ¼ 0.5. (b) Band gap width as a function of F.
At low filling rates, e.g., F ¼ 0.5, low frequency broad band gaps were unlikely to be observed in crystals with a simple triangular lattice due to 3
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The enhancement of the tunability of the phononic crystal band gap by using cluster primitives can be attributed to changes in the in teractions among the acoustic fields in adjacent primitive cells due to variations in the direction of wave scattering in crystals caused by the presence of cluster primitives. In order to study this problem from a microscopic viewpoint, we determined the distributions of acoustic pressure fields at point X in the primary energy bands of crystals with single and double cylinder primitives, as shown in Fig. 6. In Fig. 6, lighter colors indicate greater acoustic pressure. In the phononic crystals with single cylinder primitives, the distribution of the acoustic pressure fields followed the shape of an oblique band and the acoustic fields interacted mainly with their neighbors in adjacent primitive cells at the four boundaries with identical intensities. The acoustic waves could propagate freely between adjacent cells and this was a major cause of energy band degeneracy. In phononic crystals with double cylinder primitives, the distribution of the acoustic pressure field followed the shape of the cluster and the acoustic fields interacted mainly with their neighbors in adjacent primitive cells at two symmetric boundaries. Hence, the majority of the acoustic wave energy was confined in the primitive cell. In phononic crystals with sextuple cylinder primitives, the cluster primitives exhibited a high capacity for capturing acoustic fields, which may be attributed to the strong standing waves induced by the strong resonance of acoustic waves in the central cavity. The symmetry of the acoustic fields in crystals with triple and septuple cylinder prim itives was lower than that in crystals with double and sextuple cylinder primitives. Fig. 7 shows the distributions of the acoustic pressure fields at point Γ in the tertiary energy band of the crystals. In phononic crystals with single cylinder primitives, the distribution of the acoustic pressure was similar to that shown in Fig. 6(b), but more concentrated. In other phononic crystals, six concentration areas were observed for the
the presence of energy band degeneracy (points X and Γ) at the high symmetry points and significant variations in the frequency of one specific energy band (see Fig. 4(a)). It should be noted that only the real part of the Bloch wave vector was considered on the x axis. The shadow denotes the band gap in Fig. 4(a). Increasing the filling rate to about 0.6 did not significantly change the width of the band gap (see Fig. 4(b)). Therefore, the degeneracy had to be eliminated at the high symmetry points to reduce the frequency variations for one specific energy band, which was effectively achieved using cluster primitives. As shown in Fig. 5, the tunability of the band gap for the four cluster primitives, especially the double cylinder primitives and sextuple cylinder primi tives, was superior to that with a simple triangular lattice. In crystals with double cylinder primitives, the primary band gap was present be tween the primary and secondary energy bands, and the degeneracy of the energy band at point X was eliminated. The central frequency of the band gap in crystals with double cylinder primitives was approximately 9 kHz, and the central frequency of the primary band gap in crystals with simple lattices was approximately 17 kHz, thereby indicating that low frequency broad band gaps could be generated in crystals with double cylinder primitives at relatively low filling rates. At F ¼ 0.6, the width of the primary band gap for crystals with double cylinder primitives was 4.4 times higher than that for crystals with simple lattices. In crystals with sextuple cylinder primitives, the secondary band gap located be tween the tertiary and quadruple energy bands had the maximum width. The central frequencies of the band gap for crystals with sextuple cyl inder primitives, secondary band gap for crystals with double cylinder primitives, and central frequency of the primary band gap for crystals with simple lattice were consistent. The degeneracy of the energy band at point Γ was eliminated and the energy band curves were very smooth. At F ¼ 0.6, the width of the secondary band gap for crystals with sextuple cylinder primitives was 7.3 times higher than the primary band gap for crystals with a simple lattice, and three times higher than the secondary band gap width for crystals with double cylinder primitives. In addition, three distribution areas with relatively broad band gaps were observed in the high frequency zone. Therefore, broad band gaps were most readily generated in crystals with sextuple cylinder primitives among all of the samples. The tunability of the band gap for crystals with triple and septuple cylinder primitives was inferior compared with those containing double and sextuple cylinder primitives, but superior compared with those containing simple lattices.
Fig. 6. Distribution of acoustic pressure field for the primary energy band at point X, F ¼ 0.5. Lighter colors indicate greater acoustic pressure. (a) Single cylinder primitives; (b) double cylinder primitives; (c) triple cylinder primi tives; (d) sextuple cylinder primitives; and (e) septuple cylinder primitives.
Fig. 5. Band gap width as a function of F. The shadow denotes the band gap. (a) Double cylinder primitives; (b) triple cylinder primitives; (c) sextuple cylinder primitives; and (d) septuple cylinder primitives. 4
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facilitate the design of new acoustic wave energy collectors. 3.3. Effects of primitive orientation angle on band gap width To further understand the effects of the inter-cylinder structure in the cluster primitives on the band gap, all of the cylinders in each primitive were exposed to synchronous counter-clockwise rotation by θ� around the central axis of the primitive. The position vector of (x,y) represented by rn in equation (4) was changed to (x0 , y0 ) and thus [43]: � 0� � �� � cos θ sin θ x x ¼ : (7) 0 sin θ cos θ y y We found that the extremes of the energy band were still present at the high symmetry points or symmetry lines in the Brillouin zone because the high symmetry of a single cylinder in the cluster primitives could partially counteract the effects of reducing the symmetry of the simple lattice on wave scattering as the reflection plane of the crystal remained the same. Fig. 8 shows the changes in the band gaps. Changes in θ had significant effects on the band gaps in the four phononic crys tals. In particular, the changes in θ could generate new broad band gaps, especially in the crystals with double cylinder primitives. The changes in θ could adjust (expand or reduce) the band gap width and allow precise control of the band gap width at a constant crystal size. The band gap was absent from crystals with simple lattices at these filling rates. The excitation and compression of the band gap could be controlled over a relatively broad range by tuning θ for the cluster primitive, which may greatly facilitate the applications of phononic crystals. The mechanism that allows θ to affect the band gap is explained as follows. Changes in θ reduce the symmetry of the crystal structure and affect the in-phase stacking of reflected waves between adjacent primitive cells, thereby reconstructing the energy of the acoustic waves in primitive cells, as also shown in previous studies of the acoustic pressure field [33]. In some cases, the degenerate states of the energy bands at high symmetry points were broken or strengthened. However, it is not possible to conclude that increasing or decreasing the symmetry of the crystal structure is favorable for generating band gaps because the effects of structural symmetry on the band gap will vary with the frequency zone.
Fig. 7. Distribution of acoustic pressure field for the tertiary energy band at point Γ, F ¼ 0.5. Lighter colors indicate greater acoustic pressure. (a) Single cylinder primitives; (b) double cylinder primitives; (c) triple cylinder primi tives; (d) sextuple cylinder primitives; and (e) septuple cylinder primitives.
acoustic pressure. The energy of the acoustic field was completely confined in the primitive cell and the interactions between acoustic fields and their neighbors in adjacent primitive cells were weak. In the crystals with triple and septuple cylinder primitives, the acoustic fields were also confined in the primitive cells, although the concentration ratios of the acoustic fields in these crystals were lower than those in the crystals with double and sextuple cylinder primitives. Hence, the pres ence of cluster primitives affected the symmetry of the crystal structures. Reconstructing the acoustic fields led to degenerate separation states for the energy band at high symmetry points and the energy band curves became smooth, thereby increasing the widths of the band gaps. Ac cording to these results, in order to allow the degenerate states of the energy band to be separated, part of the acoustic pressure must be completely localized in the cell and the acoustic wave cannot propagate freely in the adjacent cells. Under these conditions, the symmetry of the acoustic pressure fields and the concentration of the acoustic pressure energy inside the crystal are the main factors that affect the width of the band gap. Acoustic pressure fields with both high symmetry and a high energy concentration inside the crystal are beneficial for separating the degenerate states of the energy band. The influence of the symmetry of the acoustic pressure field on the width of the band gap is weaker than that of the energy concentration. However, both factors are not related directly to the symmetry of the cluster primitives, thereby indicating that the generation of the band gap is the result of a combination of various factors. Symmetry is only one of the important factors that affect the band gap. The number of single cylinders has no clear regular effect on the distribution of the acoustic pressure field. Our analysis of sextuple cylinder primitives suggests that multi-column primitive structures with large cavities may be more favorable for the excitation of wide band gaps because large cavities can be beneficial for capturing the acoustic waves. Therefore, the acoustic wave energy can be concentrated at different locations in the primitive cell in various forms, and this may
3.4. Effects of inter-cylinder distance in primitives on the band gap width In cluster primitives, the composite structure of the primitives can be
Fig. 8. Effects of the primitive orientation angle θ on the band gap width, F ¼ 0.4. (a) Double cylinder primitives; (b) triple cylinder primitives; (c) sextuple cylinder primitives; and (d) septuple cylinder primitives. 5
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Journal of Physics and Chemistry of Solids 136 (2020) 109206
Fig. 9. Effects of inter-cylinder distance in primitives on the band gap width, F ¼ 0.3, θ ¼ 0. (a) Double cylinder primitives; (b) triple cylinder primitives; (c) sextuple cylinder primitives; and (d) septuple cylinder primitives.
adjusted by tuning the inter-cylinder distance (h). We define h ¼ lR, where l is a positive real number that is not less than 2. Fig. 9 shows the band gaps determined for phononic crystals with double and sextuple cylinder primitives as a function of l. Considering geometric constraints, the filling rate F was set to 0.3 and l ¼ [2,3]. The width of the primary band gap increased significantly with l except in the crystal containing triple cylinder primitives. According to the theory of acoustic waves [44], changing the inter-cylinder distance in the cluster primitives affected the mean free path for acoustic waves in the crystals, thereby changing the phase difference for the coherent acoustic waves, but without directly affecting the symmetry of the crystal structures. As a result, the distribution patterns of the acoustic fields in the crystals were affected. As shown in Fig. 10, as l increased, except for the crystal with triple cylinder primitives, the acoustic field energy tended to converge to each of the original acoustic pressure centers and the interactions be tween the acoustic fields in adjacent primitive cells were reduced, thereby resulting in degenerate state separations in the energy band. Consequently, the energy band curves became smooth and the band gaps were broadened.
Fig. 10. Distribution of the acoustic pressure field, F ¼ 0.3, θ ¼ 0. Lighter colors indicate greater acoustic pressure. (a) Double cylinder primitives, primary en ergy band, point X, l ¼ 2.2. (b) Double cylinder primitives, primary energy band, point X, l ¼ 2.8. (c) Triple cylinder primitives, tertiary energy band, point Γ, l ¼ 2.2. (d) Triple cylinder primitives, tertiary energy band, point Γ, l ¼ 2.8. (e) Sextuple cylinder primitives, tertiary energy band, point Γ, l ¼ 2.2. (f) Sextuple cylinder primitives, tertiary energy band, point Γ, l ¼ 2.8. (g) Septuple cylinder primitives, tertiary energy band, point Γ, l ¼ 2.2. (h) Septuple cylinder primitives, tertiary energy band, point Γ, l ¼ 2.8.
4. Conclusions In this study, we investigated the band structures of triangular acoustic phononic crystals with various cylinder primitive cell clusters using the plane wave expansion method. Compared with crystals con taining single cylinder primitives, the ranges of the irreducible Brillouin zones varied for the proposed crystals, but the extreme energy bands were still present at high symmetry points or on the symmetry lines of the Brillouin zone. Our results indicate that the band gap width is positively related to the filling rate. The width of the low frequency band gaps in crystals with double cylinder primitives was 4.4 times higher than that in crystals with single cylinder primitives, and the width of the secondary band gap for crystals containing sextuple cylinder primitives was 7.3 times higher than the primary band gap for crystals with a simple lattice. Tuning the primitive orientation angle or inter-cylinder distance in the primitive allowed precise control of the location and width of the band gap at low filling rates. The changes in the band gap structures induced by introducing cluster primitives could be attributed to changes in the symmetry of the crystal structure. In particular, changing the symmetry of the crystal structure reconstructed the energy distributions of the acoustic fields and broke the degenerate states in the
energy band, thereby making the energy band curves smoother. The presence of cluster primitives led to the generation of low frequency broad band gaps at low filling rates and enhanced the tunability of the band gaps. Studying cluster primitives can provide insights into the physical properties of phononic crystals. Our findings may facilitate further applications of phononic crystals. It should be noted that the low-frequency band gap investigated in this study is based on the Bragg scattering mechanism and it does not reach the sub-wavelength range. Conflicts of interest We would like to submit the enclosed manuscript entitled “Band Gap Structure of Acoustic Phononic Crystals with Cluster Primitives in Triangular Lattices”, which we wish to be considered for publication in “Journal of Physics and Chemistry of Solids”. We declare that we have no financial and personal relationships with other people or 6
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organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled.
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Acknowledgements This study was supported by the National Natural Science Founda tion of China (Grant No. 11664044), Department of Science and Tech nology of Yunnan Province (Grant No. 2016FC001), Applied Basic Research Programs of Yunnan Province (Grant No. 2014FD055), Sci entific Research Fund of Education Department of Yunnan Province (Grant No. 2019J0911), and Joint Special Fund Project for Basic Research of Local Universities in Yunnan Province (Research topic: Structural Design and Bandgap Characteristics of Primitive Mixed Pho nonic Crystals). References [1] M.S. Kushwaha, P. Halevi, L. Dobrzynski, B. Djafari-Rouhani, Phys. Rev. Lett. 71 (1993) 2022–2025. [2] M. Sigalas, E.N. Economou, Solid State Commun. 86 (1993) 141–143. [3] M.H. Lu, L. Feng, Y.F. Cheng, 12 (2009) 34–42. [4] Y. Dong, H. Yao, J. Du, J. Zhao, C. Ding, Phys. Lett. A 383 (2019) 283–288. [5] A. Srivastava, Int. J. Smart Nano Materials 6 (2015) 41–60. [6] E. Coffy, S. Euphrasie, M. Addouche, P. Vairac, A. Khelif, Ultrasonics 78 (2017) 51–56. [7] Y. Pennec, J.O. Vasseur, B. Djafari-Rouhani, L. Dobrzy� nski, P.A. Deymier, Surf. Sci. Rep. 65 (2010) 229–291. [8] A. Mehaney, Ultrasonics 93 (2019) 37–42. [9] Y. Pennec, Y. Jin, B.D. Rouhani, Adv. Appl. Mech. 93 (2019) 37–42. [10] A. Khelif, A. Adibi. Phononic Crystals: Fundamentals and Applications, SpringerVerlag Press, New York, 2016. [11] E. Soliveres, I. Perez-Arjona, R. Pico, V. Espinosa, V.J. Sanchez-Morcillo, K. Staliunas, Appl. Phys. Lett. 99 (2011) 151905. [12] D. Sievenpiper, L.J. Zhang, F. Romulo, B. Jimenez, N.G. Alexopolous, E. Yablonovitch, IEEE Trans. Microwave Theory Tech. 47 (1999) 2059–2074.
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Band gap structure of acoustic phononic crystals with cluster primitives in triangular lattices Jiaguang Hu School of Information Science, Wenshan University, Wenshan, 663099, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Cluster primitive Energy band structure Phononic crystal Triangular lattice
In this study, a two-dimensional steel/gas acoustic phononic crystal with triangular lattices was considered as the model and single cylinder primitives were substituted with cluster primitives, before investigating the energy band structures of the longitudinal sound wave using the plane wave expansion method. The results indicated that energy band extremes were still present at the high symmetry points or symmetry lines in the Brillouin zone, although the range of the irreducible Brillouin zone increased by two times. Compared with single cylinder primitives, double cylinder primitives could achieve excitation in low frequency broad band gaps at relatively low filling rates. Furthermore, sextuple cylinder primitives achieved primary band gap width increases of 7.3 times. The width of the band gap increased with the filling rates of the cluster primitives. Precise control of the band gap location and width at low filling rates can be achieved by varying the primitive orientation or intercylinder distance in the primitives.
1. Introduction Recently, inspired by photonic crystals, researchers have intensively investigated phononic crystals because of their unique properties in terms of producing an elastic wave band gap. Phononic crystals are composite functional materials with a periodic arrangement of different elastic materials. It has been demonstrated both theoretically and experimentally that the transmission of elastic waves in phononic crystals may lead to band gap structures similar to those in photonic crystals under appropriate conditions [1–6]. The unique band gap properties of phononic crystals mean that they may have applications in various fields, including precise vibration absorption, negative refrac tion, ultrasonic imaging, acoustic cloaking, detectors, directional transmission, and auto-collimation [7–11]. Unlike photonic crystals, the transmission of elastic waves in media with different properties lead to variations in the polarization state of the phononic crystals, thereby complicating the identification and calculation of the band gaps. The mechanisms related to forbidden bands in phononic crystals mainly comprise the Bragg scattering mechanism [12], local resonance mech anism [13], and inertial amplification mechanism [14]. The physical properties of forbidden bands in phononic crystals are dominated by the structural shape of primitive [15–17] characteristics such as the elastic parameters, and spatial distribution properties such as the lattice structure distance, symmetry, and lattice configuration [16–18].
The applications of phononic crystals depend on the generation and regulation of band gaps. In particular, two-dimensional (2D) solid/gas phononic crystals based on Bragg scattering have attracted much attention because of their simplicity in terms of experimental prepara tion, band gap generation, and the establishment of an equivalent calculation model. In Bragg phononic crystals, ultra-low frequency band gaps are usually generated in plate structures [19–21]. Composite structures [22–27], plate structures [28], fractal structures [29], and quasi-Sierpinski carpet unit cells [30] have been introduced into pho nonic crystals to allow excellent band gap tunability in ordinary struc tures. In addition, polyhedrons [31], square cylinders [32], elliptic cylinders [33], hexagonal cylinders, and triangular cylinders [34] have been inserted to reduce the symmetry of the crystal structures in lattices. However, the design of novel structures has focused mainly on square and honeycomb lattices, whereas cluster inclusions have not been considered. In conventional phononic crystals, broad band gaps are readily generated in crystals with triangular lattices, but their filling rates are extremely high [35]. The generation of relatively broad band gaps with enhanced tunability at low filling rates can reduce the cost and facilitate various applications of phononic crystals. In this study, we designed a novel phononic crystal structure with cluster primitives based on composite structures and under the requirement for facile experimental preparation. The variations in the first irreducible Bril louin zone were investigated and we identified the locations of extreme
E-mail address: [email protected]. https://doi.org/10.1016/j.jpcs.2019.109206 Received 4 March 2019; Received in revised form 15 September 2019; Accepted 16 September 2019 Available online 17 September 2019 0022-3697/© 2019 Elsevier Ltd. All rights reserved.
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energy bands in order to investigate the effects of the shape of the cluster primitives, filling rate, arrangement orientation, and location on the band gaps and the band gap broadening mechanism. It should be noted that the band structures were calculated in this study by only consid ering the propagating modes. The evanescent modes [36] will be calculated in a future study. In addition, we focused in acoustic pho nonic crystals and a similar study of fully elastic continuum structures might obtain different results because of the more complicated band structure. 2. Study model Fig. 1(a) shows conventional simple lattice phononic crystals comprising solid steel cylinders arranged in a triangular lattice in the air with an infinite periodic structure. R and a denote the cross-section radius of the cylinder and the lattice constant, respectively. Fig. 1(b) shows the first irreducible Brillouin zone. As shown in Fig. 2, the single cylinder primitives are substituted with double, triple, sextuple, and septuple cylinder primitives to obtain new phononic crystals. The degree of rotational symmetry for the double, triple, and sextuple cylinder primitives increases successively, and the degree of symmetry for the septuple cylinder primitive is the same as that for the sextuple cylinder primitive. The adjacent cylinders in a primitive are tangent to each other and the lattice point is at the center of each primitive. The tangent of the cylinders in the primitive is the only critical condition for the spatial structure and the adjacent cylinders exhibit no deformation. The tangent of the cylinders is similar to the critical condition where the scatterers simply contact each other when the filling ratio in the crystal is maxi mized. In the present study, we only considered the band gaps below 40 kHz, and thus the wavelength range of the acoustic waves is much larger than the inter-atomic distance. Therefore, a linear elastic model is suitable for this situation. All of the cylinders are identical in size. This composite primitive structure is regarded as a cluster primitive. The tangent locations for the cylinders in the primitive or the inter-cylinder distance are adjustable. These phononic crystals are easy to synthesize and it is convenient to control the band gaps due to the increased quantity of structural parameters involved. In solid/gas systems, we can impose the condition of elastic rigidity on the solid rods because of the high density contrast between the solid and air. Under these conditions, the solid is effectively treated as a fluid and only the longitudinal wave needs to be considered [37]. Vasseur et al. reported that the plane wave expansion method agreed well with the finite-difference time-domain method calculations and the experi mental results obtained for two-dimensional solid/air systems [38]. In the calculation using the plane wave method, the following eigenvalue equation should be solved [39]. � �� �i Xh 0 ω2 λg 1g0 ρg 1g0 k þ g ⋅ k þ g Pg0 ¼ 0; (1) g
Fig. 2. Structure of triangular lattice with cluster primitives: (a) double cyl inder primitives; (b) triple cylinder primitives; (c) sextuple cylinder primitives; and (d) septuple cylinder primitives.
are the Fourier expansion coefficients for the pressure P(r), mass density, and bulk modulus, respectively, and g is the 2D reciprocal lattice vector, which can be defined as [40]: � � 2π ð m þ 2nÞ pffiffi g¼ ey ; (2) mex þ a 3 where a is the lattice constant, m and n are integers, and ex and ey are unit vectors of the x axis and y axis, respectively. 1 Using fg for Pg’, ρ-1 g , and λg , fg can be calculated as follows: � fg ¼ fA þ ð1 þ FÞfB ; g¼0 0 ; (3) fg ¼ ðfA fB ÞI ðgÞ; g 6¼ 0 where A and B denote the cylinders and background, respectively. The filling rate is defined as F ¼ S0 /S, where S and S0 are the total area of the cell cross-section and the area of cluster primitive cross-section, respectively. The maximum values of F for the five structures are 0.9069 for single cylinder primitives, 0.6046 for double and sextuple cylinder primitives, 0.6802 for triple cylinder primitives, and 0.7054 for septuple cylinder primitives. Based on the characteristics of the Fourier transform, the structure functions of the cluster primitives can be obtained as follows: X 0 I ðgÞ ¼ eig⋅rn IðgÞ; (4)
0
where ω is the angular frequency, k is the wave vector, Pg’, ρ-g1, and λ-g1
rn
where rn is the position vector (x, y) for the center of each cylinder in a cluster primitive and the origin of the coordinates is at the center of the primitive. In the single cylinder primitive, rn is (0, 0). In the double cylinder primitives, rn are (0, R) and (0, –R). In the triple cylinder pffiffiffi pffiffiffi pffiffiffi primitives, rn are ð0; 2R= 3 Þ, ðR; 3 R=3Þ, and ð R; 3 R=3Þ. In pffiffiffi pffiffiffi the sextuple cylinder primitives, rn are ð R; 3 RÞ, ðR; 3 RÞ, ðR; pffiffiffi pffiffiffi 3 RÞ, ð R; 3 RÞ, (–2R, 0), and (2R, 0). In the septuple cylinder pffiffiffi pffiffiffi pffiffiffi primitives, rn are (0, 0), ð R; 3 RÞ, ðR; 3 RÞ, ðR; 3 RÞ, ð R; pffiffiffi 3 RÞ, (–2R, 0), and (2R, 0). I(g) is the structure function for each cyl inder and it can be calculated as [41]. Fig. 1. Structure of simple triangular lattice (a) and the first irreducible Bril louin zone (b). 2
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IðgÞ ¼
Journal of Physics and Chemistry of Solids 136 (2020) 109206
1 S
,
Z e
ig⋅r
dr ¼ 2FJ1 ðjgjRÞ
jgjR
(5)
A
where A is the cross-section of a cylinder and J1 is the first-order Bessel function. Based on Bloch theory, the pressure P(r) in a structure can be written as follows [39]. X PðrÞ ¼ eik⋅r Pg eig⋅r (6) g
By solving equation (1), we can obtain ω and the relationship be tween the energy band and structure (k–Hz). The distributions of the acoustic pressure field under a certain point and a certain value of ω can be determined based on equations (1) and (6). The physical parameters for the steel/air system are: ρ ¼ 7800/1.21 kg/m3, longitudinal wave speed Cl ¼ 6100/343 m/s, λ ¼ ρC2l , and lattice constant a ¼ 2 cm. In total, 625 plane wave numbers were considered to guarantee the convergence and MATLAB software was used for the calculations. 3. Results and discussion 3.1. Extreme locations of energy bands In order to obtain the energy band structure, the natural frequencies of all the wave vectors k in the irreducible Brillouin zone must be considered (shadow in Fig. 1(b)). The natural frequency is defined as extreme because the normal component of the group velocity at high symmetry points along the symmetry plane is zero. The energy band structure of the simple lattices can be obtained by assigning all of the waves to wave vectors k on the boundary of the Brillouin zone (Γ-Х-YΓ). For phononic crystals with cluster primitives, wave scattering varies in all directions due to the varying symmetry of the crystal structures, which may affect the shape of the irreducible Brillouin zone and the extreme location of the energy band. Thus, we mapped the frequencies of the low frequency and high frequency energy bands in the primary Brillouin zone using the selected structural parameters, as shown in Fig. 3, where the dark color denotes low frequencies, and the black and white dots denote the locations of maximum and minimum energy bands, respectively. The results showed that the symmetry of the Bril louin zone varied significantly, especially for crystals with double cyl inder primitives. The irreducible Brillouin zone had to be extended to the ΓMXYΓ zone whereas the symmetry of the Brillouin zone for crystals with the sextuple cylinder primitives remained unchanged. The extreme energy bands for crystals with double cylinder primitives were located on the high symmetry lines and the extreme energy bands of the other crystals were still located at the high symmetry points in the Brillouin zone. Therefore, the values of the wave vectors were still assigned on the boundary of the Brillouin zone (Γ-Х-Y-Γ) to calculate the energy band. In this study, we considered crystals with no cylinders. Thus, the values of the wave vectors were assigned in the entire primary Brillouin zone in order to obtain an accurate energy band structure, which significantly increased the calculations required. Indeed, the uniformity of wave scattering in phononic crystals will affect the symmetry and locations of the extreme energy bands in the Brillouin zone. Reducing the symmetry of the primitives or lattice may eliminate the reflection surface of the unit cell, non-uniform wave scattering, and the reconstructed in teractions of waves and lattice, thereby resulting in variations in the energy band structures [42]. However, the high symmetry of a single cylinder in the cluster primitives can partially counteract the effects of reducing the symmetry of the simple lattice on wave scattering.
Fig. 3. Mapping of the energy band frequency in the primary Brillouin zone. F ¼ 0.5. Lighter colors indicate higher acoustic frequencies. Black and white dots denote the locations of the maximum and minimum energy bands, respectively. (a) Double cylinder primitives, tertiary energy band; (b) triple cylinder primitives, tertiary energy band; (c) sextuple cylinder primitives, ter tiary energy band; (d) septuple cylinder primitives, tertiary energy band; (e) double cylinder primitives, septuple energy band; and (f) double cylinder primitives, octuple energy band.
3.2. Effects of filling rate on the band gap width Fig. 4. Energy band and band gap for a simple triangular lattice. The shadow denotes the band gap. (a) Energy band structure, F ¼ 0.5. (b) Band gap width as a function of F.
At low filling rates, e.g., F ¼ 0.5, low frequency broad band gaps were unlikely to be observed in crystals with a simple triangular lattice due to 3
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The enhancement of the tunability of the phononic crystal band gap by using cluster primitives can be attributed to changes in the in teractions among the acoustic fields in adjacent primitive cells due to variations in the direction of wave scattering in crystals caused by the presence of cluster primitives. In order to study this problem from a microscopic viewpoint, we determined the distributions of acoustic pressure fields at point X in the primary energy bands of crystals with single and double cylinder primitives, as shown in Fig. 6. In Fig. 6, lighter colors indicate greater acoustic pressure. In the phononic crystals with single cylinder primitives, the distribution of the acoustic pressure fields followed the shape of an oblique band and the acoustic fields interacted mainly with their neighbors in adjacent primitive cells at the four boundaries with identical intensities. The acoustic waves could propagate freely between adjacent cells and this was a major cause of energy band degeneracy. In phononic crystals with double cylinder primitives, the distribution of the acoustic pressure field followed the shape of the cluster and the acoustic fields interacted mainly with their neighbors in adjacent primitive cells at two symmetric boundaries. Hence, the majority of the acoustic wave energy was confined in the primitive cell. In phononic crystals with sextuple cylinder primitives, the cluster primitives exhibited a high capacity for capturing acoustic fields, which may be attributed to the strong standing waves induced by the strong resonance of acoustic waves in the central cavity. The symmetry of the acoustic fields in crystals with triple and septuple cylinder prim itives was lower than that in crystals with double and sextuple cylinder primitives. Fig. 7 shows the distributions of the acoustic pressure fields at point Γ in the tertiary energy band of the crystals. In phononic crystals with single cylinder primitives, the distribution of the acoustic pressure was similar to that shown in Fig. 6(b), but more concentrated. In other phononic crystals, six concentration areas were observed for the
the presence of energy band degeneracy (points X and Γ) at the high symmetry points and significant variations in the frequency of one specific energy band (see Fig. 4(a)). It should be noted that only the real part of the Bloch wave vector was considered on the x axis. The shadow denotes the band gap in Fig. 4(a). Increasing the filling rate to about 0.6 did not significantly change the width of the band gap (see Fig. 4(b)). Therefore, the degeneracy had to be eliminated at the high symmetry points to reduce the frequency variations for one specific energy band, which was effectively achieved using cluster primitives. As shown in Fig. 5, the tunability of the band gap for the four cluster primitives, especially the double cylinder primitives and sextuple cylinder primi tives, was superior to that with a simple triangular lattice. In crystals with double cylinder primitives, the primary band gap was present be tween the primary and secondary energy bands, and the degeneracy of the energy band at point X was eliminated. The central frequency of the band gap in crystals with double cylinder primitives was approximately 9 kHz, and the central frequency of the primary band gap in crystals with simple lattices was approximately 17 kHz, thereby indicating that low frequency broad band gaps could be generated in crystals with double cylinder primitives at relatively low filling rates. At F ¼ 0.6, the width of the primary band gap for crystals with double cylinder primitives was 4.4 times higher than that for crystals with simple lattices. In crystals with sextuple cylinder primitives, the secondary band gap located be tween the tertiary and quadruple energy bands had the maximum width. The central frequencies of the band gap for crystals with sextuple cyl inder primitives, secondary band gap for crystals with double cylinder primitives, and central frequency of the primary band gap for crystals with simple lattice were consistent. The degeneracy of the energy band at point Γ was eliminated and the energy band curves were very smooth. At F ¼ 0.6, the width of the secondary band gap for crystals with sextuple cylinder primitives was 7.3 times higher than the primary band gap for crystals with a simple lattice, and three times higher than the secondary band gap width for crystals with double cylinder primitives. In addition, three distribution areas with relatively broad band gaps were observed in the high frequency zone. Therefore, broad band gaps were most readily generated in crystals with sextuple cylinder primitives among all of the samples. The tunability of the band gap for crystals with triple and septuple cylinder primitives was inferior compared with those containing double and sextuple cylinder primitives, but superior compared with those containing simple lattices.
Fig. 6. Distribution of acoustic pressure field for the primary energy band at point X, F ¼ 0.5. Lighter colors indicate greater acoustic pressure. (a) Single cylinder primitives; (b) double cylinder primitives; (c) triple cylinder primi tives; (d) sextuple cylinder primitives; and (e) septuple cylinder primitives.
Fig. 5. Band gap width as a function of F. The shadow denotes the band gap. (a) Double cylinder primitives; (b) triple cylinder primitives; (c) sextuple cylinder primitives; and (d) septuple cylinder primitives. 4
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facilitate the design of new acoustic wave energy collectors. 3.3. Effects of primitive orientation angle on band gap width To further understand the effects of the inter-cylinder structure in the cluster primitives on the band gap, all of the cylinders in each primitive were exposed to synchronous counter-clockwise rotation by θ� around the central axis of the primitive. The position vector of (x,y) represented by rn in equation (4) was changed to (x0 , y0 ) and thus [43]: � 0� � �� � cos θ sin θ x x ¼ : (7) 0 sin θ cos θ y y We found that the extremes of the energy band were still present at the high symmetry points or symmetry lines in the Brillouin zone because the high symmetry of a single cylinder in the cluster primitives could partially counteract the effects of reducing the symmetry of the simple lattice on wave scattering as the reflection plane of the crystal remained the same. Fig. 8 shows the changes in the band gaps. Changes in θ had significant effects on the band gaps in the four phononic crys tals. In particular, the changes in θ could generate new broad band gaps, especially in the crystals with double cylinder primitives. The changes in θ could adjust (expand or reduce) the band gap width and allow precise control of the band gap width at a constant crystal size. The band gap was absent from crystals with simple lattices at these filling rates. The excitation and compression of the band gap could be controlled over a relatively broad range by tuning θ for the cluster primitive, which may greatly facilitate the applications of phononic crystals. The mechanism that allows θ to affect the band gap is explained as follows. Changes in θ reduce the symmetry of the crystal structure and affect the in-phase stacking of reflected waves between adjacent primitive cells, thereby reconstructing the energy of the acoustic waves in primitive cells, as also shown in previous studies of the acoustic pressure field [33]. In some cases, the degenerate states of the energy bands at high symmetry points were broken or strengthened. However, it is not possible to conclude that increasing or decreasing the symmetry of the crystal structure is favorable for generating band gaps because the effects of structural symmetry on the band gap will vary with the frequency zone.
Fig. 7. Distribution of acoustic pressure field for the tertiary energy band at point Γ, F ¼ 0.5. Lighter colors indicate greater acoustic pressure. (a) Single cylinder primitives; (b) double cylinder primitives; (c) triple cylinder primi tives; (d) sextuple cylinder primitives; and (e) septuple cylinder primitives.
acoustic pressure. The energy of the acoustic field was completely confined in the primitive cell and the interactions between acoustic fields and their neighbors in adjacent primitive cells were weak. In the crystals with triple and septuple cylinder primitives, the acoustic fields were also confined in the primitive cells, although the concentration ratios of the acoustic fields in these crystals were lower than those in the crystals with double and sextuple cylinder primitives. Hence, the pres ence of cluster primitives affected the symmetry of the crystal structures. Reconstructing the acoustic fields led to degenerate separation states for the energy band at high symmetry points and the energy band curves became smooth, thereby increasing the widths of the band gaps. Ac cording to these results, in order to allow the degenerate states of the energy band to be separated, part of the acoustic pressure must be completely localized in the cell and the acoustic wave cannot propagate freely in the adjacent cells. Under these conditions, the symmetry of the acoustic pressure fields and the concentration of the acoustic pressure energy inside the crystal are the main factors that affect the width of the band gap. Acoustic pressure fields with both high symmetry and a high energy concentration inside the crystal are beneficial for separating the degenerate states of the energy band. The influence of the symmetry of the acoustic pressure field on the width of the band gap is weaker than that of the energy concentration. However, both factors are not related directly to the symmetry of the cluster primitives, thereby indicating that the generation of the band gap is the result of a combination of various factors. Symmetry is only one of the important factors that affect the band gap. The number of single cylinders has no clear regular effect on the distribution of the acoustic pressure field. Our analysis of sextuple cylinder primitives suggests that multi-column primitive structures with large cavities may be more favorable for the excitation of wide band gaps because large cavities can be beneficial for capturing the acoustic waves. Therefore, the acoustic wave energy can be concentrated at different locations in the primitive cell in various forms, and this may
3.4. Effects of inter-cylinder distance in primitives on the band gap width In cluster primitives, the composite structure of the primitives can be
Fig. 8. Effects of the primitive orientation angle θ on the band gap width, F ¼ 0.4. (a) Double cylinder primitives; (b) triple cylinder primitives; (c) sextuple cylinder primitives; and (d) septuple cylinder primitives. 5
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Journal of Physics and Chemistry of Solids 136 (2020) 109206
Fig. 9. Effects of inter-cylinder distance in primitives on the band gap width, F ¼ 0.3, θ ¼ 0. (a) Double cylinder primitives; (b) triple cylinder primitives; (c) sextuple cylinder primitives; and (d) septuple cylinder primitives.
adjusted by tuning the inter-cylinder distance (h). We define h ¼ lR, where l is a positive real number that is not less than 2. Fig. 9 shows the band gaps determined for phononic crystals with double and sextuple cylinder primitives as a function of l. Considering geometric constraints, the filling rate F was set to 0.3 and l ¼ [2,3]. The width of the primary band gap increased significantly with l except in the crystal containing triple cylinder primitives. According to the theory of acoustic waves [44], changing the inter-cylinder distance in the cluster primitives affected the mean free path for acoustic waves in the crystals, thereby changing the phase difference for the coherent acoustic waves, but without directly affecting the symmetry of the crystal structures. As a result, the distribution patterns of the acoustic fields in the crystals were affected. As shown in Fig. 10, as l increased, except for the crystal with triple cylinder primitives, the acoustic field energy tended to converge to each of the original acoustic pressure centers and the interactions be tween the acoustic fields in adjacent primitive cells were reduced, thereby resulting in degenerate state separations in the energy band. Consequently, the energy band curves became smooth and the band gaps were broadened.
Fig. 10. Distribution of the acoustic pressure field, F ¼ 0.3, θ ¼ 0. Lighter colors indicate greater acoustic pressure. (a) Double cylinder primitives, primary en ergy band, point X, l ¼ 2.2. (b) Double cylinder primitives, primary energy band, point X, l ¼ 2.8. (c) Triple cylinder primitives, tertiary energy band, point Γ, l ¼ 2.2. (d) Triple cylinder primitives, tertiary energy band, point Γ, l ¼ 2.8. (e) Sextuple cylinder primitives, tertiary energy band, point Γ, l ¼ 2.2. (f) Sextuple cylinder primitives, tertiary energy band, point Γ, l ¼ 2.8. (g) Septuple cylinder primitives, tertiary energy band, point Γ, l ¼ 2.2. (h) Septuple cylinder primitives, tertiary energy band, point Γ, l ¼ 2.8.
4. Conclusions In this study, we investigated the band structures of triangular acoustic phononic crystals with various cylinder primitive cell clusters using the plane wave expansion method. Compared with crystals con taining single cylinder primitives, the ranges of the irreducible Brillouin zones varied for the proposed crystals, but the extreme energy bands were still present at high symmetry points or on the symmetry lines of the Brillouin zone. Our results indicate that the band gap width is positively related to the filling rate. The width of the low frequency band gaps in crystals with double cylinder primitives was 4.4 times higher than that in crystals with single cylinder primitives, and the width of the secondary band gap for crystals containing sextuple cylinder primitives was 7.3 times higher than the primary band gap for crystals with a simple lattice. Tuning the primitive orientation angle or inter-cylinder distance in the primitive allowed precise control of the location and width of the band gap at low filling rates. The changes in the band gap structures induced by introducing cluster primitives could be attributed to changes in the symmetry of the crystal structure. In particular, changing the symmetry of the crystal structure reconstructed the energy distributions of the acoustic fields and broke the degenerate states in the
energy band, thereby making the energy band curves smoother. The presence of cluster primitives led to the generation of low frequency broad band gaps at low filling rates and enhanced the tunability of the band gaps. Studying cluster primitives can provide insights into the physical properties of phononic crystals. Our findings may facilitate further applications of phononic crystals. It should be noted that the low-frequency band gap investigated in this study is based on the Bragg scattering mechanism and it does not reach the sub-wavelength range. Conflicts of interest We would like to submit the enclosed manuscript entitled “Band Gap Structure of Acoustic Phononic Crystals with Cluster Primitives in Triangular Lattices”, which we wish to be considered for publication in “Journal of Physics and Chemistry of Solids”. We declare that we have no financial and personal relationships with other people or 6
J. Hu
Journal of Physics and Chemistry of Solids 136 (2020) 109206
organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled.
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