title: Band-gap tuning in 2D spatiotemporal phononic crystals
authors: D. Psiachos M. M. Sigalas
text: 
  
  
  
  
  dpsiachos@gmail.com
  
  sigalas@upatras.gr
  Department of Materials Science, University of Patras, 26504, Rio, Greece
  
  
  
  We investigate the effect of small spatiotemporal modulations in subwavelength-dimensioned phononic
  crystals with large band gaps, on the frequency
  spectrum for elastic waves polarized in the plane of periodicity. When the
  radius of cylinders periodically placed inside a matrix of highly-contrasting elastic properties
  	is time-varying,
  	we find that due to 
  	the appearance of frequency harmonics throughout the spectrum, the notion of a
  band gap is destroyed in general, although with the appropriate tuning of parameters, in particular the modulation
  frequency, it is possible that some band-gap region is retained, making such systems possible candidates for
  tunable bandpass filters or phononic isolators, accordingly, and for sensor applications.
  
  
  Band-gap tuning in 2D spatiotemporal phononic crystals
      M. M. Sigalas
      
  ======================================================
  
  
  
  
  
  § INTRODUCTION
  
  
  Phononic lattices
  are necessarily engineered structures made up of composite materials (metamaterials). The 
  tunability of their design can permit novel behaviour not found in conventional materials,
  such as wide, complete band gaps and nonreciprocal propagation <cit.>.
  In the present study, we investigate the effect of spatiotemporal modulation of 
  material properties on the band-gap properties for a phononic-lattice
  prototype: cylinders embedded in a matrix of significantly
  different elastic properties in the subwavelength regime. This type of 
  structure is well-known for yielding the large spectral
  gaps <cit.> desirable in phononic lattices, 
  enabling the transmission of one type of polarization or none at all. Recent work 
  on two-dimensional colloidal phononic crystals has shown that while 
  altering the periodicity of the system results in Bragg gap tuning, 
  the appearance as well as the characteristics of band gaps 
  above and below the Bragg gap is controllable by altering the
  mechanical eigenmodes of the nanoparticles and nanoparticle-membrane
  adhesion respectively <cit.>.
  
  Reciprocal propagation of sound or elastic waves in media <cit.>, is a general principle 
  which assures the interchangeability of 
  source and observer: that waves originating from one of these points
  propagate in exactly the same manner as waves originating from the other point. It 
  is only broken in special situations such as when nonlinearity is present,
  when time-reversal symmetry is broken such as by the inclusion of
  materials with gain or loss, chirality, angular-momentum bias, 
  or in moving systems <cit.>. It may also be broken when the system itself
  is not moving in time but the material parameters are varying in space and time,
  manifesting itself as asymmetric band gaps leading to directional 
  propagation <cit.>.
  
  The rectification of the mechanical 
  properties of phononic materials has been demonstrated in several experimental
  or numerical studies, based on phenomena such as electromagnetism <cit.>,
  piezoelectricity <cit.>, or
  magnetorheological polymers subjected to magnetic fields <cit.>. For
  nonreciprocal propagation to be observed, apart from time-varying material properties, there
  must also be spatial variation present.
  
  Some purely mechanical implementations of spatiotemporally-varying material properties 
  include a time-varying effective acoustic capacitance achieved in simulations 
  involving air-filled waveguides by varying the heights of the attached 
  resonators <cit.>, a metabeam comprised of multiple resonators with different
  reltaive orientations where the resulting stiffness changes in response to 
  rotation <cit.>. Frequency conversion without nonlinearity present can occur
  in media with spatiotemporally-modulated
  material properties. Depending on the direction of the incident wave in a beam with reflective ends
  or a beam containing
  an interface between a homogeneous material and one with time-space varying properties, the frequency
  can be either up or down-converted, a clear indicator of nonreciprocal 
  propagation <cit.>. The production of harmonics in time-space
  modulated media <cit.> may be exploited for uses
  such as unidirectional acoustic isolation <cit.>, or parametric 
  amplification as a possible implementation of a gain medium <cit.>. In 
  an air-filled waveguide system with a vibrating membrane, 
  time-varying the membrane tension leads to phenomena such as frequency
  conversion; adding spatial variation 
  via the addition of a second membrane with a phase 
  difference, is what leads to non-reciprocal propagation <cit.>. Alternatively,
  it has been shown that spatial variation leads to the formation of band gaps while the addition of time-variation
  is what causes the band structure to display a directionality <cit.> via a “tilting"
  of ω-k space, a phenomenon long-known for the case of electromagnetic waves
  propagating on a two-dimensional surface with spatiotemporally-modulated properties <cit.>.
  
  Given that phononic materials are a very technologically-promising area of research, our 
  aim in this study is to examine whether time variation of the phononic crystals' already
  spatially-varying material
  properties improves or degrades the 
  performance of these systems for various objectives such as acoustic isolators or frequency sensors.
  
  
  
  § METHODS
  
  We consider a composite composed of two homogeneous 
  structures: a solid cylinder inside a background matrix of a different material, 
  together comprising a square unit cell as shown in 
  Fig. <ref>. Time-dependent 
  material properties in this model are imposed implicitly, via a variation of the radius 
  r=r(t)=r_0+A(t) of the cylinder, resulting, due to the nature of the model chosen, in a modification 
  of its density as well as the density of the background matrix, as explained further below. The radius
  of the cylinder r_0 corresponds to a filling factor on the order of 50%. Such large filling factors in
  periodic structures ideally lead to large band gaps <cit.>. The 
  function A(t)=Asin(2π f t) is a slowly-varying sinusoid with
  small modulation amplitude A (up to 2% of r_0), and modulation frequency f. This 
  constitutes a simplified model of a more realistic implementation of
  a system time-varying material properties, such as, for example, a split-ring
  structure where the two semicircular components have a different radius, where one is mechanically 
  modulated in time inside the other. 
  
  
  
  
  
  
   §.§ Formalism
  
  The elastic wave equation in 
  the resulting inhomogeneous elastic medium is
  derived from the following relations <cit.>:
  
  a) Mass conservation,
  In our model, the cylinder expands and contracts according to
  a sinusoidal function, but no mass is actually being added or removed
  from the system. Thus, the mass conservation relation
  
      ∂ρ/∂ t+∇· (ρ𝐯)=0
  
  where 𝐯= ∂𝐮/∂ t is the velocity in terms
  of the displacement vector 𝐮, and ρ is the density, holds
  for all points in space.
  
  b) Euler's Equation,
  
      ρ ∂ v_i/∂ t+ρ∑_j v_j∂/∂ x_j v_i=∑_j ∂ T_ij/∂ x_j
  
  where the i, j are the components in space, specifically x,y,z, of the various 
  quantities, x_i are the spatial coordinates, v_i are the components of the velocity, i.e. 
  v_i= ∂ u_i/∂ t, for displacements u_i, T_ij are the components
  of the stress tensor (defined below).
  
  Equations  <ref> and  <ref> in conjunction yield
  
      ∂(ρ v_i)/∂ t=∑_j ∂ T_ij/∂ x_j
  
  
  after ignoring a term second-order in 𝐯 since we aim to work in the regime of small displacements.
  
  c) Hooke's law,
  
      T_ij=λ u_ijδ_ij+2μ u_ij
  
  where δ_ij is the Kronecker delta function and
  
      u_ij=1/2(∂ u_i/∂ x_j+∂ u_j/∂ x_i)
  
  are the components of the strain tensor u in terms of the displacement
  vector 𝐮 and the spatial coordinates 𝐱, 
  and λ=ρ(C_l^2-2C_s^2) and μ=ρ  C_s^2 are the Lamé coefficients in
  terms of the longitudinal and shear sound velocities C_l and C_s respectively, and where 
  𝐮 as well as the variables ρ, λ, μ have 
  implicit spatial and time dependence governed by the material parameters as determined by
  r(t).
  
  
  
   §.§ Finite-Element Time-Domain (FDTD) Calculations
  
  
  After an initial disturbance was set in motion at a point 
  inside the cylinder, the system was allowed to evolve. The magnitude of the disturbance
  is what determines the amplitude of the response and the resulting displacements 
  may be considered to remain within
  the linear-elastic regime as the problem is completely scalable in this sense.
  Using
  the finite-difference time-domain (FDTD) method, where the spatial as well as the time domain is discretized, 
  and applying periodic boundary conditions and Bloch's theorem <cit.>,
  we solve Eqs. <ref>-<ref> for the propagation of elastic waves in 
  inhomogeneous media. In particular, the
  calculational approach is implemented by discretizing space on a square grid, where the displacements 
  u_x, u_y,u_z are defined in the centre of each grid cell and their derivatives are approximated
  by central-difference formulas in both space and time, resulting in second-order accuracy, while the derivatives for
  the material parameters ρ, λ, μ are approximated by 
  finite differences between grid and time points. The displacements at the following time step are obtained in terms
  of their values, as well as those of other parameters, at previous 
  steps, and Fourier-transformed in order to obtain a spectrum
  of resonant peaks corresponding to the bands in k space <cit.>. 
  
  We utilized a square grid of 90 by 90 points for 
  the calculations of the band structure for the static model A(t)=0 and 180 by 180 points or even denser, up to 
  360 by 360,
  for more-detailed calculations of the frequency responses of the k=0 cases
  with and without time dependence because the denser grid let to less abrupt changes in the material parameters 
  as a function of time, or led to the 
  accessing of smaller amplitudes in A(t). 
  
  All of the lengths are expressed in terms of the lattice parameter a. Frequency 
  is expressed as a dimensionless quantity ν̃=ν a/c where c is taken as
  the speed of sound c_b in the background material and the units of ν
  are kHz. The time step in the simulations was taken 
  as Δ t=7.698× 10^-4 a/c_b.
  
  
  
  In Table <ref> we list the densities and sound velocities of the materials we used for
  the composites in our study.
  
  
  
  § RESULTS
  
  In this section we apply the formalism and methods outlined in the previous section to two
  different composite phononic material systems: iron (Fe) cylinders in epoxy, and silicon (Si) cylinders
  in epoxy.
  
  
   §.§ Non-time-varying case
  
  
  
  The calculated band structure for waves polarized in the plane of the unit cell, of 
  mixed transverse-longitudinal character, for 
  Fe cylinders in epoxy without time-variation of material properties is shown in 
  Fig. <ref>a, for a filling factor of 50%. The plot is shown for a dimensionless
  frequency ν̃. The large band gaps are what make these types of materials ideal phononic materials. The
  ratio of the gap-width defined as Δω to the mid-gap frequency defined as ω_g, for 
  the largest gap, Δω/ω_g, is 0.86
  while for a filling factor of 60% for the same materials, the ratio is 0.91. 
  
  Similarly, for Si cylinders in epoxy without time variation we also find a band 
  structure with large gaps, as shown in Fig. <ref>b,
  for a filling factor of 60%. The gap-width to midgap frequency ratio Δω/ω_g 
  for the largest gap is 0.45, much smaller than for Fe in epoxy even at the same
  filling factor, a result explained by the smaller contrast in the densities between the cylinder 
  and matrix for this case, a factor
  of greater significance in determining this ratio than the disparity
  between the elastic constants <cit.>, which in our case, is about the same
  for both composites. 
  
  
  
   §.§ Time-varying material properties
  
  
  As we time-vary the cylinder radius as outlined in the previous section, for an Fe or a Si 
  cylinder in epoxy, we obtain the spectrum shown in 
  Fig. <ref> for various maximum amplitude variations of a sinusoid and fixed modulation frequency, scaled
  respectively to be dimensionless as A^'=A/a and f^'=fa/c, at
  k=0. The vertical axis depicts the amplitude of the frequency spectrum  
  √(u_x(ν̃)^2+u_y(ν̃)^2), obtained after Fourier-transforming
  the value of the displacement at a fixed point inside the cylinder, near its centre. The units
  for the displacement are arbitrary as mentioned in Sec. <ref> owing to the scalability
  of the problem as far as the amplitude of the initial pulse is concerned. The 
  horizontal arrows denote
  the location of the absolute band gaps in the non-time-varying case (viz. 
  Fig. <ref>). Most notable are the blueshifting of the main resonance peaks, as well 
  as the appearance of accompanying sidebands upon adding time-modulation, offset from 
  the main resonances by the amount of the modulation
  frequency, and the existence higher harmonics fading with distance in intensity. The sidebands do not necessarily
  have a smaller amplitude than the main resonance peak. As the modulation amplitude is increased, the
  resonance peaks become increasingly blueshifted and the sidebands cover a wider
  frequency range, as can be seen clearly by the insets for the region around the resonance peaks. In 
  the case of Fe in epoxy (Fig. <ref>a), the two lowest peaks in the non-time-varying case 
  corresponding to a frequency of 0.0679 and 0.1760 are shifted towards the values of 0.0687 and
  0.1768 respectively for A^' being 0.011 and 0.022 respectively, as shown
  in the insets, while the respective sidebands are separated from the respective peaks 
  by multiples of the value f^'=0.00157, as shown, to within an excellent approximation. Further
  details, involving smaller amplitude modulations and regarding the amplitudes of the displacements
  upon adding time-modulation, calculated for the system of an Fe cylinder in epoxy, for a larger 
  filling factor, 60%, may 
  be found in the Appendix.
  
  In Fig. <ref>b, which depicts the frequency spectrum for an Si cylinder in epoxy
  at 60% filling, the largest A^' depicted is two times larger than that shown in 
  Fig. <ref>a for Fe cylinders in epoxy and all of the band gaps are completely obliterated although
  it is clear that the harmonics destroy the band gaps in all cases - especially as a result of the contribution
  of other wavevectors given that the bands themselves are not flat. However, for small amplitude modulations,
  the harmonics do die off quickly and in principle, some region within the largest band gap would be 
  left intact. We note that the bands for Fe in epoxy are flatter than for Si in epoxy in the non-time
  varying case (viz. Fig. <ref>), something which
  occurs even for the same filling factors and whose origin lies in the greater hybridization of the 
  continuum bands with the localized resonances from rigid-body modes of the cylinder in the case
  of Fe, due to the sharper contrast in the densities <cit.>.
  
  
  
  For fixed amplitude modulation and varying frequency modulation of the cylinder radius, we obtain
  the results in Fig. <ref>, for the two systems, confirming that the sidebands  
  are offset by an amount equal to the modulation frequency f^' and that the blueshift of the main
  resonance peak compared to the non-time-varying case is unaffected by the value of f^'. In the
  case of Fe in epoxy in Fig. <ref>a, the lowest peak in the non-time varying case
  corresponding to a frequency of 0.0679 is shifted towards the value of 0.0683 for both
  f^'=0.00157 and 0.00314 for A^'=0.011 while the sidebands appear at successive separations
  of f^' from the main peak. The results for the frequency variation confirm that upon adding
  time-variation, a blueshift is in fact occurring because, as the resonances for different
  frequency modulation but the same amplitude modulation coincide, there should be a resonance
  from both modulation frequencies immediately to the left
  of the nonmodulated resonance in the inset of Fig. <ref>a if the resonances
  were redshifted instead of blueshifted, 
  as this peak would have corresponded to the main resonance as opposed to a sideband.
  
  
  
  As it constitutes a more realistic implementation of a time-varying radius, we repeated
  some of our calculations on a simplied model of a mechanically-modulated split-ring
  structure, that of a hollow cylinder. The results were similar, with
  large band gaps for a non-time-varying radius and resonances with harmonics when the outer
  radius is periodically modulated in time. The details and results of out calculations are shown in the Appendix.
  
  
  
  
   §.§ Displacement profiles
  
  
  
  
  
  Upon adding time-variation to the material properties, we find that the localization 
  character of the vibrations associated with each resonance remains unchanged;
  even up to the second harmonic, it remains constant with the only difference from the non-time-varying
  case being that the surrounding material is also disturbed. In Fig. <ref> we show
  the displacement profile u_x^2+u_y^2 of the lowest resonance peak at the 
  frequency ν̃=0.0681 in (a) the non-time-varying 
  case (point X1 in Fig. <ref>a), 
  and (b) its counterpart for the case where the maximum amplitude is A^'=0.022 and frequency 
  f^'=0.00157 where the main
  peak is now at a frequency of ν̃=0.0689. We also show, in Fig. <ref>c, 
  the localization of the first sideband on the 
  right having a frequency of ν̃=0.0705. In all of these cases, the vibrations were highly localized
  on the outer edge of the cylinder, albeit blurred in the case of the time-varying case. In all of the 
  higher-frequency resonances, the vibrations were localized outside the cylinder. For example,
  for point X2 in Fig. <ref>a, at a reduced frequency of ν̃=0.2352 without time-variation, we have  
  Fig. <ref>a which is only slightly disturbed in Fig. <ref>b, by adding amplitude and frequency 
  modulation of A^'=0.022 and f^'=0.00157 respectively, whereupon
  the resonance frequency shifts slightly to ν̃=0.2361. In all of the above
  situations, the displacement profile was generated by exciting with a plane wave at the frequency being
  investigated, and examining the square of the displacement in the plane u_x^2+u_y^2, at steady state once the excitation has long died out. During
  the steady state, there are of course vibrations but the localization character of each resonance remains unchanged.
  
  
  
  In Fig. <ref> we plot the displacement profile u_x^2+u_y^2 for a Si cylinder in epoxy 
  for the lowest resonance peak at the frequency 0.1205 in 
  the non-time-varying case (point Y1 in Fig. <ref>b), and in order to compare it 
  with its counterpart at the frequency 0.1217 in the time-varying case
  where A^'=0.044 and f^'=0.00157 (viz. inset in Fig. <ref>b) 
  in Fig. <ref>b. In Fig. <ref>c
  we show the second harmonic of the lowest resonance at ν̃=0.1248. Similar to the case
  of Fe in epoxy, the vibrations are 
  highly localized on the edge of the Si cylinder (outlined in the figures)
  although it is markedly blurry in the time-varying case. There are bulges near the corners of the unit cell
  which are associated with the proximity of the neighbouring cylinders. They are not present in the case
  of the Fe cylinder in epoxy depicted earlier because there the filling factor was 50%, but they are seen
  in that system as well for a filling factor of 60%. As for the case of Fe in epoxy, for the higher
  resonances, the vibrations are localized in the matrix outside the cylinder. In 
  Fig. <ref>d we show the localization
  of the vibrations for the point Y2 in Fig. <ref>b located at ν̃=0.2169. Point Y2
  actually corresponds to two very nearly degenerate frequencies, with virtually-identical localization 
  profiles. Just one of these is shown in Fig. <ref>d, with more details provided
  in the Appendix. Due to the
  close proximity of the cylinders at this large filling factor in neighbouring unit cells, the vibrations
  of this resonance are preferentially localized around the corners of the unit cell, where the
  spacing between neighbouring cylinders is greatest.
  
  
  
  
  § CONCLUSIONS
  
  We have examined the effect of periodic strucural modulations in two phononic materials composed
  of a cylinder inside a matrix of highly-contrasting elastic properties and we have found
  that in general the large band gaps are destroyed by harmonics generated by the temporal
  modulation. However, the resonances' time-harmonics 
  induced by structural modulations may be exploited
  in order to achieve tunable phononic isolation or alternatively, bandpass filtering given that 
  in phononic systems with fairly flat bands, by employing a large-enough modulation frequency, it is feasible that
  small band gaps could be retained for some regions. The harmonics'
  periodicity and wide range could also find implementation in a 
  frequency sensor. Modulating the cylinders' radius may be achieved
  relatively simply: such as by a mechanically-modulated split-ring structure where the two components have
  a different radius.
  
  
  
  § APPENDIX
  
  
      
  
  
   §.§ Fe cylinder in Epoxy, filling factor of 60%
  
  In Fig. <ref> we show  results for the frequency
  spectrum for the lowest resonance at k=0 (analogous to point X1 in Fig. <ref>a in the
  main text) for the system of an Fe cylinder in epoxy for a filling factor of 60% for a
  time-varying cylinder radius for different maximum amplitude and frequency variations.
  The blueshift for the smallest amplitude variation (A^'=0.0055) is not discernible
  for the calculational parameters used but it is evident for the larger A^'. In
  general, the results are similar to the case of 50% filling described in the main text. In
  the case of a time-varying cylinder radius,
  sidebands and higher harmonics appear around the main resonance peak, separated by the value
  of the dimensionless frequency f^'.
  
  
  
  In Fig. <ref> we show, for the same system, for a fixed as well as for a
  time-varying cylinder radius, the variation of the material displacement at a
  point inside the cylinder, as a function of time, where the timestep is
  Δ t=7.698× 10^-4 a/c_b for a being the lattice parameter and c_b
  the speed of a longitudinal sound wave in the background medium (epoxy). The magnitude
  of the displacement is entirely scalable as it varies linearly with the magnitude
  of the initial pulse used to excite a point inside the system. Therefore, the displacements
  may in this manner be constrained to the regime of linear elasticity. As can be seen from the
  figure, the displacements are on the same scale either for a fixed or for a time-varying cylinder
  radius.
  
  
  
  
   §.§ Hollow Cylinder of Fe in Epoxy
  
  
  If, instead of a solid cylinder as in the main text, we have
  a hollow cylinder of Fe in Epoxy with the same outer
  dimension as the cylinder described in the main text (of 50% filling factor) and
  inner dimension as shown by the outlines in Fig. <ref>, we obtain the band structure
  of Fig. <ref> for elastic waves polarized in the plane of the unit cell for a non-time-varying
  system. In general, the band structure is very similar to its counterpart for a solid cylinder
  (Fig. <ref>a in the main text).
  
  
  
  
  
  As we time-vary the outer radius of the cylinder, we obtain the spectrum shown in
  Fig. <ref> for different maximum amplitude variations and modulation frequencies, scaled
  respectively to be dimensionless as A^'=A/a and f^'=fa/c, at
  k=0. The vertical axis depicts the amplitude of the frequency spectrum
  √(u_x(ν̃)^2+u_y(ν̃)^2), obtained after Fourier-transforming
  the value of the displacement at a fixed point inside the cylinder, near its centre. The
  horizontal arrows denote
  the location of the absolute band gaps in the non-time-varying case (viz. Fig. <ref>).
  
  
  
  In Fig. <ref> we show the displacement profiles of the system for the points Z1-Z4 at k=0
  in Fig. <ref>. The
  point Z2 actually corresponds to four nearly-degenerate frequencies. Specifically, we display
  a plane projection of the displacement u_x^2+u_y^2 of a point inside the
  ring in order to visualize the localization of the vibrations.
  
  The results are very similar to the corresponding case of a solid cylinder. The displacement
  profile of point Z1 has the same character as that of point X1 (viz. Fig. <ref>a).
  Point Z4 here corresponds to the displacement profile of point X2 in that case (viz.
  Fig. <ref>a). Point Z2 has a strong lifting of its degeneracy and only two of the four
  closely-spaced modes show the same localization
  as in the solid cylinder case: Z2(2) and Z2(3) which correspond to the two
  nearly-degenerate modes located at the point Y2 in Fig. <ref>d for the Si
  cylinder in epoxy. In Fig. <ref>d we
  showed only one of these (corresponding to Z2(2) here). The
  profiles Z2(1) and Z2(4) show symmetries not seen in the case of the solid cylinder
  showing strong indications of mixing with the higher modes Z3 and Z4.
  
  35
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