Parylene-C microfibrous thin films as phononic crystals
Chandraprakash Chindam, Akhlesh Lakhtakia, Osama Osman Awadelkarim
aa r X i v : . [ phy s i c s . c l a ss - ph ] M a r Parylene-C microfibrous thin films as phononic crystals
Chandraprakash Chindam, Akhlesh Lakhtakia ∗ , and Osama O. Awadelkarim Department of Engineering Science and Mechanics, Pennsylvania State University,University Park, PA 16802, USA ∗ To whom correspondence should be addressed; E-mail: [email protected]
Abstract
Phononic bandgaps of Parylene-C microfibrous thin films ( µ FTF s) were computation-ally determined by treating them as phononic crystals comprising identical microfibersarranged either on a square or a hexagonal lattice. The microfibers could be columnar,chevronic, or helical in shape, and the host medium could be either water or air. Allbandgaps were observed to lie in the 0.01–162.9-MHz regime, for microfibers of re-alistically chosen dimensions. The upper limit of the frequency of bandgaps was thehighest for the columnar µ FTF and the lowest for the chiral µ FTF . More bandgapsexist when the host medium is water than air. Complete bandgaps were observed for thecolumnar µ FTF with microfibers arranged on a hexagonal lattice in air, the chevronic µ FTF with microfibers arranged on a square lattice in water, and the chiral µ FTF with microfibers arranged on a hexagonal lattice in either air or water. The softness ofthe Parylene-C µ FTF s makes them mechanically tunable, and their bandgaps can beexploited in multiband ultrasonic filters.
Keywords: bandgap, chevron, chirality, columns, helix, microfibrous film, ParyleneC, phononic crystal
1. Introduction
Products are generally designed to accomplish only one specific function [1]. But,multifunctionality — the ability to perform multiple functions — is currently emerg-ing as a technoscientific paradigm inspired by nature [2, 3, 4]. Two prime examplesof natural multifunctional entities [5] are the leaves of plants and the skins of animals.
Preprint submitted to arXiv.org July 16, 2018 eaves are designed to transport nutrients across the neighboring cells, release car-bon dioxide, and absorb sunlight, and some are capable of self-cleaning [6]. Skins ofanimals, including humans, hold the body parts in place and maintain the shape of thebody, facilitate perspiration, hold hair for thermal insulation, and also provide the senseof touch. For a sustainable future [7] we need to decrease the use of single-function de-vices by devising and popularizing multifunctional devices. Some engineered productsperform multiple functions; for example, printers copy, print, staple, email, and fax.Although such products usually are conglomerations of many single-function devices,a few dual-function devices and structures have been reported. Examples include light-emitting diodes that also function as photodetectors [8], and photonic-cum-phononiccrystals [9].In the current age of miniaturization, many devices contain thin films for which fab-rication techniques are well-known [10, 12, 11, 13]. Thin films can be so highly denseas to be considered homogeneous [10, 13] or can be porous with engineered morphol-ogy [11, 13]. Microfibrous thin films ( µ FTFs) of the polymer Parylene C are attractiveas multifunctional materials [4]. The Parylene-C µ FTFs are fabricated by a straightfor-ward modification [15, 16] of the industrially used Gorham process [14] to coat variousstructures conformally with dense Parylene-C films [18, 19, 20]. Thus far, Parylene-C µ FTF s comprising parallel and identical microfibers of upright circular-cylindrical,slanted-circular cylindrical, chevronic, and helical shapes have been fabricated [17].The water-wettability [21], crystallinity [21], ability to store electric charge [22], glass-transition temperature [23], and biocompatibility[17, 24] of Parylene-C µ FTF s com-prising slanted-circular cylindrical microfibers have been experimentally investigated.If the identical microfibers are deposited on a topographic substrate decorated witha regular lattice [25, 26], the Parylene-C µ FTF could function as a phononic crystal[27, 28]. Phononic crystals are being investigated as acoustic sensors [29, 30], isolators[31], filters [32, 33], waveguides [33], and concentrators [34]. Most studies consideran array of solid/fluid scatterers in a fluid/solid medium, the scatterers being of simpleshapes such as infinitely long cylinders [27] and spheres [35]. Also in these studies,the scatterers are arranged on either a square or a hexagonal lattice in two-dimensionalspace, or in either a face- or a body-centered cubic lattice in three-dimensional space.2s we were interested in investigating the µ FTF s of Parylene C as ultrasonicfilters, we determined their phononic-bandgap characteristics using the commercialfinite-element-method (FEM) software COMSOL Multiphysics R (cid:13) (version 5.1) [36].We consider all microfibers in a µ FTF to be identical and parallel. µ FTF s compris-ing circular-cylindrical microfibers are referred to as columnar µ FTF , and those withchevronic and helical microfibers as chevronic and chiral µ FTF s, respectively. AsParylene-C µ FTF s are easily removable from topographic substrates [16], we con-sider the free-standing µ FTF as a periodic arrangement of microfibers with the hostmedium as air, i.e., a phononic crystal with host as air. Furthermore, we also examinethe phononic dispersion characteristics of the µ FTF in water keeping biomedical re-search in mind [17, 24]. As Parylene C is a polymer rather than a hard material suchas a metal, the Parylene-C µ FTF s are very different from most phononic crystals re-ported in the literature [27, 28, 29, 30, 31, 32, 33, 34, 35] in that the Parylene-C µ FTF sshall be tunable by the application of pressure [37] in the same way that liquid-crystalelastomers are [38, 39, 40].The plan of this paper is as follows. We describe the geometric details of the chosen µ FTF s in Sec. 2 and theoretical basis (governing constitutive relations and Brillouinzone paths) used for determining the eigenfrequencies in Sec. 3. The procedure toimplement COMSOL Multiphysics R (cid:13) and the validation of that procedure are brieflydescribed in Sec. 4. Finally in Sec. 5 we present the results of the phononic bandgapsof Parylene-C µ FTF s. A time dependence of exp( iωt ) is implicit, with i = √− , t astime, ω = 2 πf as the angular frequency, and f as the linear frequency. The microfibersare arranged on either a square or a hexagonal lattice in the xy plane. Vectors aredenoted in boldface, every unit vector is identified by a caret, and r = x ˆ x + y ˆ y + z ˆ z denotes the position vector.
2. Geometric Preliminaries
In a preceding study [41], we investigated the scattering of an acoustic plane waveby a single finite-sized microfiber of Parylene C in water. The microfiber could beupright circular-cylindrical, slanted-circular cylindrical, chevronic, or helical in shape.However, as a phononic crystal is considered to occupy all space for the purpose of3etermining its phononic dispersion characteristics, only a simple rotation of the co-ordinate system is needed to see that a µ FTF comprising slanted circular-cylindricalmicrofibers is identical to a µ FTF comprising upright circular-cylindrical microfibers.Hence, Parylene C µ FTF s comprising upright circular-cylindrical, chevronic, and heli-cal microfibers, shown in Fig. 1(a), were chosen for investigation as phononic crystals.
Figure 1: (Color online) (a) Microfibers of upright circular-cylindrical, chevronic, and structurally right-handed helical shapes with their dimensions chosen for numerical results presented here. (b) Square andhexagonal lattices used for the columnar µ FTF . Unit cells of (c) square and (d) hexagonal lattices usedfor the chevronic and chiral µ FTF s, with lattice dimensions a and b , the microfiber diameter denoted by d . In (d), the angle β = 2 π/ for chevronic µ FTF s and β = π/ for chiral µ FTF s. In (b)–(d), theunit cell is denoted by Ω , the shaded region A is completely occupied by the Parylene-C microfiber, and theunshaded region Ω −A by either air or water. For the square lattice, the chosen µ FTF s with upright circular-cylindrical, chevronic, and helical microfibers have filling fractions of 0.20, 0.16, and 0.15, respectively; forthe hexagonal lattice, the corresponding filling fractions are 0.23, 0.18, and 0.11, respectively.
The upright circular-cylindrical microfiber is parallel to the z axis and of infinitelength, so that the lattice is really two-dimensional, as shown in Fig. 1(b). This latticecan be either square or hexagonal, the lattice dimension being denoted by a . Thechevronic and chiral microfibers are periodic along the z axis with period b , the three-4imensional unit cells drawn on square and hexagonal lattices of dimension a , as shownin Figs. 1(c,d). The chevron underlying the chevronic microfiber lies in the xz plane.The helix underlying the chiral microfiber is curled about the z axis, the helix beingeither left- or right-handed. We took the chiral microfibers to be right-handed for allcalculations reported here. The microfiber diameter is denoted by d in Figs. 1(b)–(d).The microfiber dimensions shown in Fig. 1(a) were chosen as representative values,based on several scanning-electron micrographs of Parylene-C µ FTF s [17, 21, 24, 41].The distance in the xy plane between a microfiber and its nearest neighbor distance canbe chosen above a minimum value by topographically patterning the substrate [16, 25],but for all calculations reported here the lattice parameters were fixed as follows. Thesquare lattice was chosen of side a = 10 µ m and the hexagonal lattice of side a =10 µ m in Figs. 1, when the microfibers are either columnar or chevronic. The squarelattice was chosen of side a = 12 µ m and the hexagonal lattice of side a = 15 µ m,when the microfibers are helical. As appropriate, the dimension b = 10 µ m was fixed.Since the the inter-microfiber space in as-fabricated µ FTF s is occupied by air, westudied the phononic crystals with air as the host medium. Keeping their biomedicalapplications in mind [17, 24], we also studied the phononic crystals with water as thehost medium.
3. Theoretical Preliminaries
The unit cell Ω is two-dimensional for columnar µ FTF s (Fig. 1(b)), but three-dimensional for chevronic µ FTF s and chiral µ FTF s (Figs. 1(c–d)). The unit cell isalso the domain of all calculations for phononic crystals. The displacement phasor u ( r ) in an isotropic material is governed by the Navier equation [42] [ λ ( r ) + µ ( r )] ∇ [ ∇ • u ( r )] + µ ( r ) ∇ u ( r ) + ρ ( r ) ω u ( r ) = , (1)5here the Lam´e parameters λ ( r ) and µ ( r ) as well as the mass density ρ ( r ) are piece-wise continuous as follows: λ ( r ) = λ s λ h , µ ( r ) = µ s µ h ,ρ ( r ) = ρ s ρ h , r ∈ A Ω − A . (2)Both traction and displacement were set to be continuous across every interface.The Floquet–Bloch periodicity condition [43] was imposed, as appropriate, on theboundaries of the unit cell. With k = k x ˆ x + k y ˆ y + k z ˆ z denoting the wave vector, thedisplacement, the mass density, and the Lam´e parameters are expanded as the Fourierseries u ( r ) = exp( − i k • r ) P G [ u G exp ( − i π G • r )] ρ ( r ) = P G [ ρ G exp( − i π G • r )] λ ( r ) = P G [ λ G exp( − i π G • r )] µ ( r ) = P G [ µ G exp( − i π G • r )] , (3)where the Fourier coefficients ρ G , λ G , and µ G are known, but u G is unknown for all G = n b + n b + n b . Here, { b , b , b } is the triad of basis vectors of thereciprocal lattice space, whereas the integers n ℓ ∈ ( −∞ , ∞ ) for all ℓ ∈ { , , } . If { a , a , a } is the triad of basis vectors of the lattice, then the vectors b = ( a × a ) /V Ω , b = ( a × a ) /V Ω , and b = ( a × a ) /V Ω , whereas V Ω = a • ( a × a ) is thevolume of Ω . The Fourier coefficients γ G of γ ∈ { ρ, λ, µ } are given by [44] γ G = 1 V Ω Z Ω γ ( r ) exp( i π G • r ) d r . (4)After noting that Z Ω exp[ i π ( G − G ′ ) • r ] d r = V Ω δ ( G − G ′ ) , (5)where δ ( G − G ′ ) is the Dirac delta, we get γ G = γ s p + γ h (1 − p ) , G = ( γ s − γ h ) P ( G ) , G = (6)6here the filling fraction p is the ratio of the volume V A of A to V Ω and the structurefunction P ( G ) = 1 V Ω Z A exp( i π G • r ) d r . (7)In the planewave expansion method (PWEM) [35], substitution of Eqs. (3) in Eq. (1),followed by restricting n ℓ ∈ [ − N, N ] for all ℓ ∈ { , , } with a sufficiently large nat-ural number N for computational tractability, yields a set of linear algebraic equationsfor u G . This set can be be cast as a matrix eigenvalue problem [35]. The eigenvalues(i.e., the eigenfrequencies) can be evaluated using MATLAB R (cid:13) [45] (version R2012a)when both the microfiber (or any other scatterer) and the host medium are solids. How-ever, if either the scatterer medium or the host medium is a fluid, then either µ s = 0 or µ h = 0 , resulting in a large sparse matrix which is ill conditioned [46]. As µ h = 0 for our problem, as noted in Sec. 2, the PWEM could not be used, and we used thecommercial FEM software COMSOL Multiphysics R (cid:13) . Figure 2: (Color online) Irreducible brillouin zones for the chevronic microfiber arranged in (a) square and(b) hexagonal lattices [47]. (c) and (d) are the same as (a) and (b) but for a helical microfiber [48]. [By kindpermission of Dr. Mois Ilia Aroyo on behalf of the Bilbao Crystallographic Server.]
For the square lattice regardless of the microfiber shape, a = a ˆ x , a = a ˆ y , and a = b ˆ z ; for the hexagonal lattice for columnar and chiral µ FTF s, a = a (cid:0) ˆ x + √ y (cid:1) / ,7 = a (ˆ x − √ y ) / , and a = b ˆ z ; and for the hexagonal lattice for chevronic µ FTF s with the microfibers oriented to lie in the xz plane, a = a ( √ x + ˆ y ) / , a = a ( √ x − ˆ y ) / , and a = b ˆ z .The irreducible Brillouin zone (IBZ) and the coordinates of the corners for the threerepeating units of microfibers arranged as square and hexagonal lattice [47, 48] weretaken from the Bilbao Crystallographic Server [49, 50]. The corresponding IBZs areshown in Fig. 2. Eigenfrequencies were estimated for each path of the IBZ separatelyby setting the values of the components of the wave vector as different lengths of thepaths of IBZ. These eigenfrequencies were assembled in MATLAB R (cid:13) to get the com-plete band diagrams.
4. Methodology µ FTF s The domain and boundary conditions mentioned in Sec. 3 were implemented inCOMSOL Multiphysics R (cid:13) , for the unit cells shown in Fig. 1. We imposed the Floquet–Bloch periodic conditions by setting k = k F , where k F = 2 π ( f b + f b + f b ) ,with { f , f } ∈ [ − / , / and f ∈ [0 , / . Figure 3: (Color online) Meshed unit cells.
Fig. 3 shows images of all unit cells meshed for implementing the finite-elementmethod on the chosen µ FTF s. The following meshing procedures were used to gen-erate the meshes. For the two-dimensional lattice used for the columnar µ FTF , a8 redefined: Fine mesh mode was set in both A and Ω − A , with the maximum andminimum element sizes set as . µ m and . µ m, respectively, along with a ‘max-imum element growth rate’ of 1.3. One pair of adjacent edges of the square weremeshed and copied to the opposite edges to mesh the boundaries. Following that step,a ‘Free Triangular’ mesh was set in both A and Ω − A .For the unit cells of the chevronic and chiral µ FTF s, the ‘Mapped mesh’ was set ontwo faces whose normals are a × a and a × a , and a ‘distribution’ of 12 elementswas set on all the edges of these two faces. Next, the mapped mesh on these faceswas copied to the opposite faces. A ‘Free Quad’ mesh was set on the faces parallelto which a × a is normal, on both the microfiber and host. All these meshed faceswere ‘Converted’ by the ‘Element Split method’ via the Insert diagonal edges option.Thereafter, the ‘quad mesh’ on the bottom face was copied to the opposite face anda ‘Free Tetrahedral’ mesh was set within the entire domain. For all unit cells, initialchecks were made with every microfiber to verify the sufficiency of
Predefined: Fine mesh over
Predefined: Finer mesh. With either kind of mesh, the convergence valuewas less than − . As the computation time was faster for a
Predefined: Fine meshthan for the
Predefined: Finer mesh, we chose the former for all calculations.The ‘parametric sweep’ option was used to determine the set of eigenfrequenciesof every unit cell for different choices of k F . For every k F , 20 eigenfrequencies weredetermined with a ‘search method around shift’ as larger real part around Hz.Based on a perusal of the literature, this constraint is in accord with the conjecturethat any eigenfrequency in a bandgap is likely to fall in the range [0 , c h /a ] ,where c h ≡ p ( λ h + 2 µ h ) /ρ h is the longitudinal-wave speed in Ω − A . This constraint alsodetermined the highest frequency for each of the band diagrams presented in this paper.In the ‘solver configuration’ option, the ‘search method around shift’ was set as‘closest in absolute value’ option with the ‘transformation value’ as . The MUlti-frontal Massively Parallel Sparse (MUMPS) [51] direct solver provided in COMSOLwas used with a memory allocation factor of . .9 igure 4: (Color online) Band diagram calculated using the commercial FEM software COMSOLMultiphysics R (cid:13) for a hexagonal array ( a = 4 . mm) of infinitely long circular cylinders (radius R =1 . mm) of steel in water. This diagram is identical to a published result [52, Fig. 2(b)]. To validate the numerical procedure implemented in COMSOL Multiphysics R (cid:13) ,the two check cases were chosen. The first check case is an array of infinitely long, par-allel, solid circular cylinders of radius R = 1 . mm arranged on a two-dimensionalhexagonal lattice ( a = 4 . mm) in a fluid host medium [52]. Each cylinder was takento be made of steel ( λ s = 148 . GPa, µ s = 73 . GPa, ρ s = 7890 kg m − ) andhost medium was water ( λ h = 2 . GPa, µ h = 0 , ρ h = 1000 kg m − ). With a = a (cid:0) ˆ x + √ y (cid:1) / and a = a (cid:0) ˆ x − √ y (cid:1) / , we obtained b = (2 / a )(2 a + a ) and b = (2 / a )( a + 2 a ) . The IBZ is the path connecting the points Γ (i.e., G = ), M (i.e., G = b / ), and X (i.e., G = b / b / ). The mesh for this unit cell waschosen following the procedure described for columnar µ FTF s in Sec. 4.1, and theeigenfrequencies of the unit cell were determined. Figure 4 shows the band diagramcalculated by us in this way, every band represented by a dotted line that spans a spe-cific path of the IBZ. The band diagram in Fig. 4 is identical to a published result [52,Fig. 2(b)].For the second check case, solid spheres ( λ s = 101 . GPa, µ s = 80 . GPa, ρ s =7850 kg m − ) of radius R = 3 µ m arranged in a simple cubic lattice of side a = 10 µ min a solid host medium ( λ h = 458 . GPa, µ h = 147 . GPa, ρ h = 1142 kg m − )were considered. Since a = a ˆ x , a = a ˆ y , and a = a ˆ z for the simple cubic lattice,10e get b = a − ˆ x , b = a − ˆ y , b = a − ˆ z , p = (4 π/ R/a ) , and P ( G ) = 3 p ( GR ) [sin( GR ) − GR cos( GR )] , (8)where G = | G | . The corresponding IBZ is the path connecting Γ (i.e., G = ), X (i.e., G = b / ), M (i.e., G = b / b / ), and R (i.e., G = b / b / b / ).As the sphere in the unit cell does not touch the boundary of the unit cell, the follow-ing meshing procedure was used for FEM implemented in COMSOL Multiphysics R (cid:13) .A ‘user-controlled mesh’ of fine quality was set for the entire geometry. A ‘Mappedmesh’ was set on three faces whose normals are a , a , and a and a ‘distribution’ of12 elements on all the edges of these three faces. All these meshed faces were ‘Con-verted’ by the ‘Element Split method’ via ‘Insert diagonal edges,’a ‘Free Tetrahe-dral’ mesh was set in Ω , and the eigenfrequencies were evaluated over the IBZ. Also,for comparison, the PWEM [35], described in Sec. 3, was implemented using MATLAB R (cid:13) to determine the eigenfrequencies of the unit cell. Figs. 5(a) and (b), respectively, showthe band diagrams obtained using the FEM and PWEM. Despite some differences, wefound very good agreement between the two band diagrams, which thus provided con-fidence in our COMSOL Multiphysics R (cid:13) implementation.
5. Results and Discussion
Let us now present band diagrams for the Parylene-C µ FTF s of each of the threemorphologies in Fig. 1 calculated using the following constitutive parameters: λ = .
943 GPa2 . , µ = .
986 GPa00 ,ρ = − − .
225 kg m − , Parylene Cwaterair , (9)The viscoelastic parameters of Parylene C were ignored, as they are not known at highfrequencies [41]. Apart from a trivial eigenfrequency of 0 Hz (at Γ in band diagrams),all eigenfrequencies found exceeded 0.01 MHz.11 igure 5: (Color online) Comparison of band diagrams, for solid spheres of radius R = 3 µ m arranged ona cubic lattice ( a = 10 µ m) in a solid host medium, obtained using the (a) FEM and (b) PWEM. Note that λ s = 101 . GPa, µ s = 80 . GPa, ρ s = 7850 kg m − , λ h = 458 . GPa, µ h = 147 . GPa, and ρ h = 1142 kg m − . .1. Columnar µ FTF
The IBZ of a 2D unit cell for columnar µ FTF s, with the upright circular-cylindricalmicrofibers of Parylene C arranged on either a square or a hexagonal lattice has atotal of three paths. The band diagrams for the chosen columnar µ FTF with the hostmedium being either water or air are shown in Figs. 6(a)–(d).When the host medium is water, we found 6 partial bandgaps for the square lat-tice in the [0 . , . MHz range and 6 partial bandgaps for the hexagonal latticein the [1 . , . MHz range, but no complete bandgap is evident in Figs. 6(a,b).When the host medium is air, we found 3 partial bandgaps for the square lattice in the [0 . , . MHz range and 7 partial bandgaps for the hexagonal lattice in the [0 . , . MHz range, in addition to a complete bandgap (35.8 to 38.2 MHz) for the hexagonallattice, in Figs. 6(c,d). The complete bandgap comprises three partial bandgaps. Therange of bandgaps of the columnar µ FTF immersed in water is about twice that ofthe columnar µ FTF with air as the host medium. Regardless of the host medium,the columnar µ FTF with the hexagonal lattice appears suitable as a multiple-bandstopfilter for the path KM of the IBZ.A bandgap comprises those frequencies for which a plane wave cannot pass throughthe medium of interest. Since the eigenfrequency calculations were made by assum-ing that all mediums in the phononic crystal are nondissipative, one can expect maxi-mums in the back-scattering efficiencies Q b of a solitary unit cell in the bandgaps. In apredecessor study on the scattering characteristics of an individual circular-cylindricalmicrofiber of Parylene C (of the same dimensions as in a unit cell) and immersed inwater [41, Fig. 8(a)], we observed peaks in the spectrum of Q b at 106, 116, 133, 146,165, 175, 184, and 197 MHz, when the incident plane wave propagates normally tothe cylindrical axis. The peak of Q b at 116 MHz can be correlated to the bandgap6 for path XM in Fig. 6(a). Likewise, the peaks at 106, 146, and 165 MHz canbe respectively correlated to the bandgaps 1 , 4 , and 6 in Fig. 6(b). However,we were unable to correlate all bandgaps in Figs. 6(a,b) to the peaks in the spectrumof Q b , which indicates that the scattering response of a solitary unit cell is of limitedusefulness in explaining the bandgaps of a phononic crystal.In the four band diagrams shown in Fig. 6, we observe that the group speed v g ≡ ω/dk F ∼ on certain bands. A zero group speed indicates that there is no energyflow. For a specific path in the IBZ, such eigenfrequencies often fall on a horizontal ora quasi-horizontal band. A flat band (for which v g ≃ ) arises due to a local resonance,i.e., the resonance of an isolated unit cell [53, 54]. We observe that some, but not all,bandgaps lie immediately above and/or below a flat band in Figs. 6(a)–(d). Only abandgap that lies above and/or below a flat band can be correlated to a local resonance.That is the reason why the bandgap 6 in Fig. 6(a) and the bandgaps 1 , 4 , and 6in Fig. 6(b) could be correlated to the peaks of Q b identified in the predecessor study[41, Fig. 8(a)], but the bandgaps 1 – 5 in Fig. 6(a) and the bandgaps 2 , 3 , and5 in Fig. 6(b) could not be similarly correlated.Let us also note that the local resonances underlying the flat bands cannot be dueto absorption. This is because all materials have been taken to be nondissipative, as isclear from the first paragraph of Sec. 5. 14 igure 6: (Color online) Band diagrams for the chosen columnar µ FTF , with the upright circular-cylindricalmicrofibers of Parylene C arranged on either (a,c) a square or (b,d) a hexagonal lattice, the host mediumbeing either (a,b) water or (c,d) air. Bandgaps are shown unshaded and each is identified by a number insidea circle. igure 7: (Color online) Band diagrams for the chosen chevronic µ FTF , with the chevronic microfibers ofParylene C arranged on either (a,c) a square or (b,d) a hexagonal lattice, the host medium being either (a,b)water or (c,d) air. Bandgaps are shown unshaded. The boxed number above each path of the IBZ is thenumber of bandgaps observed on that path. .2. Chevronic µ FTF
The IBZ of a 3D unit cell for chevronic µ FTF s has a total of 12 paths, whetherthe lattice is square or hexagonal. The band diagrams for the chosen chevronic µ FTF with the host medium being either water or air are shown in Figs. 7(a)–(d).When the host medium is water there is a complete bandgap (13.4–29.7 MHz) forthe square lattice; see Fig. 7(b). This complete bandgap is made of 12 partial bandgaps.In other words, the chevronic µ FTF is the ultimate bandstop filter, because it does notallow transmission for any incidence direction in that spectral regime, provided that itis sufficiently large in all directions. For many, but not all, paths, the complete bandgapwidens to 10.1–34.7 MHz. No other complete bandgap is evident in Fig. 7.The host medium has two distinct effects on the bandgaps, as can be gleaned bycomparing Figs. 7(a,b) with Figs. 7(c,d). The first effect is on the extent of the spectralregime containing the bandgaps. When the host medium is water, bandgaps are foundin the [0 . , . -MHz spectral regime, whether the lattice is square or hexagonal.That regime narrows down to [0 . , . MHz and [0 . , . MHz for the squareand the hexagonal lattices, respectively, when the host medium is air. The secondeffect is on the number of partial bandgaps: 58 and 67 partial bandgaps are present forthe square and hexagonal lattices, respectively, when the host is water, whereas only 24and 39 partial bandgaps were identified for the same lattices when the host is air. Thus,for the chevronic µ FTF , both the number of partial bandgaps and the extent of thespectral regime containing the bandgaps are halved when the host medium is changedfrom water to air.Regardless of the host medium, the chevronic µ FTF has many more partial bandgapsfor many symmetric paths of the IBZ than the columnar µ FTF . Thus, the former ismuch more suitable than the latter as a multiple-bandstop filter.In a predecessor study on the planewave-scattering characteristics of a single chev-ron of Parylene C in water [41, Fig. 8(c)], several peaks in the spectrum of Q b werefound. The spectral locations of these peaks depend on the direction of propagation ofthe incident plane wave. We found that a partial bandgap (77.2–85.7 MHz) for the path Γ X in Fig. 7(a) can be correlated to a Q b -peak at 82 MHz, a partial bandgap (76.9–85.8 MHz) for the path Γ Y in Fig. 7(a) can be correlated to a Q b -peak at 82 MHz, and17 partial bandgap (115.7–119.8 MHz) for the path Γ Y in Fig. 7(a) can be correlatedto a Q b -peak at 118 MHz, but no other correlation was found for the square lattice.A similar exercise for the hexagonal lattice was infructuous, thereby reaffirming thatthe resonances of a solitary unit cell can explain some but not all of the bandgaps of aphononic crystal.Let us also note that the band diagrams in Fig. 7 do not change if the chevronic µ FTF is rotated about the z axis by ◦ , i.e., when the chevrons lie in the yz planeinstead of the xz plane. µ FTF
The IBZ of a 3D unit cell for chiral µ FTF s has a total of 9 paths, whether the latticeis square or hexagonal. The band diagrams for the chosen chiral µ FTF with the hostmedium being either water or air are shown in Figs. 8(a)–(d).When the host medium is water, there is a complete bandgap (3.1–4.6 MHz) forthe hexagonal lattice, as is evident from Fig. 8(b). This bandgap comprises 9 partialbangaps. For four paths of the IBZ, this bandgap widens to 0.01–7.1 MHz. Whenthe host medium is air, there is a complete bandgap (3.7–6.0 MHz) for the hexagonallattice, as is evident from Fig. 8(d). This bandgap comprises 9 partial bangaps. For fourpaths of the IBZ, this bandgap widens to 0.01–8.5 MHz. No other complete bandgapis evident in Fig. 8.Just as for the chevronic µ FTF in Sec. 5.2, the choice of the host medium hastwo distinct effects on the bandgaps of the chosen chiral µ FTF , as becomes clearon comparing Figs. 8(a,b) with Figs. 8(c,d). First, when the host medium is water,bandgaps are found in the 0.01–78.6-MHz and 0.01–77.6-MHz spectral regimes forthe square lattice and the hexagonal lattice, respectively. The two regimes narrowdown to 0.01–39.3 MHz and 0.01–41.3 MHz, respectively, when the host mediumis air. Second, 40 and 63 partial bandgaps are present for the square and hexagonallattices, respectively, when the host is water, whereas only 22 and 43 partial bandgapswere identified for the same lattices when the host is air. Thus, for the chiral µ FTF —just as for the chevronic µ FTF —both the number of partial bandgaps and the extentof the spectral regime containing the bandgaps are halved when the host medium is18 igure 8: (Color online) Band diagrams for the chosen chiral µ FTF , with the upright helical microfibersof Parylene C arranged on either (a,c) a square or (b,d) a hexagonal lattice, the host medium being either(a,b) water or (c,d) air. Bandgaps are shown unshaded. The boxed number above each path of the IBZ is thenumber of bandgaps observed on that path. µ FTF offers many more partial bandgaps for many than the columnar µ FTF of Sec. 5.1. Thus, both the chiral and the chevronic µ FTF s are more suitablethan columnar µ FTF as multiple-bandstop filters.In Fig. 8, some but not all bandgaps lie immediately above and/or below a flatband and can therefore be ascribed to the resonances of a solitary unit cell. ComparingFigs. 7 and 8, we note that the spectral widths of bandgaps are the same for chiral andchevronic µ FTF s when the host medium is air, but the spectral widths are lower forthe chiral µ FTF than for the chevronic µ FTF when the host medium is water.Several peaks in the spectrum of the back-scattering efficiency Q b of a single-turn helical microfiber of Parylene C in water were found in a predecessor study[41, Fig. 8(d)]. We found for the path M X in Fig. 7(a) a partial bandgap (41.6–52.1 MHz) can be correlated to the Q b -peak at 45 MHz and another partial bandgap(52.1–61.0 MHz) can be correlated to Q b -peaks at 59 MHz and MHz. No othercorrelations between the Q b -peaks and the bandgaps were found for either of the twolattices, confirming that the resonances in the scattering response of a solitary unit cellare of limited usefulness in explaining all of the bandgaps of a phononic crystal.If all structurally right-handed helical microfibers were to be replaced by their struc-turally left-handed counterparts, the Brillouin zone and hence the IBZ would remainunchanged [55, Fig. 1]. Hence, the eigenfrequencies and the bandgaps would be thesame, regardless of the structural handedness of the chiral µ FTF . Several flat bands exist in the band diagrams shown in Figs. 6, 7, and 8, each flatband being indicating of a local resonance [53, 54]. Among the four choices for thecombination of the lattice and the host medium, the phononic crystal comprising mi-crofibers arranged in a hexagonal lattice and immersed in air was found to have manyflat bands. Furthermore, the band diagrams contain many partial and few completebandgaps. Bandgaps are attractive for filtering applications, because of either appre-ciable insensitivity (in the case of a partial bandgap) or total insensitivity (in the caseof a complete bandgap) to the direction of propagation of the plane wave incident on a20 igure 9: (Color online) Band diagrams (a) on path KM for columnar µ FTF s, (b) on path
Y T for chevronic µ FTF s, and (c) on path MK for chiral µ FTF s, when the lattice is hexagonal and the host medium is air.Values of d and p are marked for each band diagram, whereas a , b , and β (as applicable) were left unalteredfrom their values stated in Fig. 1. p on the band diagrams, the host medium being air. For this investigation ona specific band diagram, we selected one path of the IBZ containing a large number offlat bands, and computed the band diagram for the same path by altering the microfiberdiameter d in order to change p , whereas a , b , and β (as applicable) were left unaltered.The calculated band diagrams for p varying by about ± are shown in Fig. 9. Ineach row of this figure, the center panel is the band diagram for the unaltered p (Fig. 1),whereas the left and right panels are band diagrams for lower and higher values of p ,respectively. The eigenfrequencies vary by no more than ± for p varying by about ± . The flat bands remained virtually unchanged, thereby confirming that they aredue to local resonances [53, 54]. Also, the bandgaps change by less than ± forthe p -variations considered. We concluded from these and other numerical results thatmanufacturing variations will not drastically affect the performances of the phononiccrystals under consideration. As compared to scatterers of simple shapes such as circular cylinders [27] andspheres [35], we deduce from Figs. 6 to 8 that the number of partial bandgaps increasesas the complexity of the shape of the scatterer in the unit cell increases. The increasein the number of partial bandgaps is not just due to the increase in the number ofsymmetric paths of the IBZ, but because of increase in the number of partial bandgapsper path—as can be established by dividing the number of partial bandgaps by thenumber of unique paths for the IBZ in a band diagram.From Figs. 6 to 8 we note that chiral µ FTF s have lower eigenfrequencies comparedto columnar and chevronic µ FTF s. Hence, the center-frequencies of the bandgaps arealso lower for the chiral µ FTF s than for the columnar and chevronic µ FTF s. Thecolumnar µ FTF were found to have the highest center-frequencies. These observa-tions are in complete agreement with the eigenfrequencies of the individual microfibers22ound previously by us [41, Table II].Existing bulk-acoustic-wave (BAW) filters operate in the higher-MHz and the lower-GHz regimes [56] as narrow-bandstop filters. Requiring tedious fabrication procedures[57], BAW filters are used in 0.3–3-MHz regime in aviation and AM radio circuits, inthe 3–30 MHz regime in shortwave radio circuits, and in the 30–300 MHz regime inFM radio circuits. Given their simple fabrication procedures [16, 17, 24], chevronicand chiral µ FTF s of Parylene C can be of use in aviation, AM radio, and shortwaveradio circuits as bandstop filters. Parylene-C columnar µ FTF can be used as bandstopfilters upto 163 MHz in FM radio circuits.Surface-acoustic-wave (SAW) devices often used for mobile and wireless commu-nications [58], operate in the 0.1–1000-MHz regime. These devices are also used inmicrofluidics [59]. The manufacture of SAW devices requires a series of steps in alayer-by-layer procedure, as well as the fabrication of an intricate pattern of metallicinterconnects [60]. Given the large number of bandgaps on various paths of the IBZsin Secs. 5.1–5.3, we propose that Parylene-C µ FTF s fabricated over piezoelectric sub-strate can be used as SAW filters.Based on Figs. 6 to 8, we note that our Parylene microfibrous thin films havebandgaps in the 1–160-MHz regime. Since ultrasonic transducers operate in the 0.5–25-MHz regime [61], Parylene-C µ FTF s can be cascaded onto these transducers tofilter out specific spectral regimes. Thus, Parylene-C µ FTF s can be classified as low-frequency ultrasonic filters and thin-film bulk acoustic resonators. We have numericallyascertained that bandgap engineering is possible by changing the filling fraction p , butthat issue lies outside the scope of this paper.Parylene C is more amorphous in its microfibrous form than in its bulk form [21],indicating that the Parylene-C µ FTF s are softer than the bulk Parylene C. Since the µ FTF s are softer, the distance between adjacent microfibers can easily be adjustedby the application of an external pressure or strain [37, 62], similar to tunable pho-tonic crystals [63] and cholesteric elastomers [38, 39, 40]. As the bandgaps can there-fore be dynamically tuned for every combination of microfiber morphology and lattice,Parylene-C µ FTF s can be called soft tunable phononic crystals .23 . Concluding Remarks
Parylene-C microfibrous thin films were treated as phononic crystals comprisingidentical microfibers arranged either on a square or a hexagonal lattice. The microfiberscould be columnar, chevronic, or helical in shape, and the host medium could be eitherwater or air.For these µ FTF s with microfibers of realistically chosen dimensions [41], all thebandgaps were observed to lie in the 0.01–162.9-MHz regime. Complete bandaps wereobserved for the following µ FTF s: (i) columnar µ FTF with microfibers arranged ona hexagonal lattice in air, (ii) chevronic µ FTF with microfibers arranged on a squarelattice in water, and (iii) chiral µ FTF with microfibers arranged on a hexagonal latticein either water or air. The upper limit of the frequency of bandgaps was the highestfor the columnar µ FTF s and the lowest for the chiral µ FTF s. For all µ FTF s par-tial bandgaps along many symmetric directions were found. The number of partialbandgaps was higher when the host medium is water than air.The obtained bandgaps for all the Parylene-C µ FTF s suggests their possible useas multi-band bulk-acoustic-wave filters. These filters can be used in conjunction withultrasonic transducers as well as surface-acoustic-wave devices. By patterning the sub-strates a higher inter-microfiber distance could be achieved and the desired bandgapsin the lower MHz regime can be obtained. Furthermore, the low elastic modulus ofParylene C also makes the µ FTF s suitable for mechanical tuning. We are currentlyinvestigating the terahertz photonic properties of these engineered micromaterials, andplan to extend our work to other parylenes such as Parylene N [64].
Acknowledgments.
We thank the Research and Cyberinfrastructure Center of Instituteof Cyber Science (ICS) of the Pennsylvania State University for computing resources,Brian Van Leeuwen and Hirofumi Akamatsu of the Department of Materials Scienceand Engineering (Penn State) for help in identifying irreducible Brillouin zones, andAnand Kumar Singh (ICS) and Chien Liu (COMSOL support team) for assistance withmesh-related issues. CC and AL are grateful to the Charles Godfrey Binder Endow-ment at Penn State for financial supporting this research.24 eferencesReferences [1] Otto, K.N., Wood, K.L., 2001. Product Design: Techniques in Reverse Engineer-ing and New Product Development. first ed., Prentice–Hall, Upper Saddle River,NJ, USA.[2] Nicole, L., Laberty-Robert, C., Rozes, L., Sanchez, C., 2014. Hy-brid materials science: a promised land for the integrative de-sign of multifunctional materials. 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