Exploring Periodic Bicontinuous Cubic Network Structures with

Article pubs.acs.org/JPCC

Exploring Periodic Bicontinuous Cubic Network Structures with Complete Phononic Bandgaps Kahyun Hur,†,‡,§ Richard G. Hennig,‡,∥ and Ulrich Wiesner*,‡ †

Center for Computational Science, Korea Institute of Science and Technology, Seoul 136-791, Republic of Korea Department of Materials Science and Engineering, Cornell University, Ithaca, New York 14853, United States § Nanomaterials Science and Engineering, Korea University of Science and Technology, Daejon 305-333, Republic of Korea ∥ Department of Materials Science and Engineering, University of Florida, Gainesville, Florida 32611, United States ‡

W Web-Enhanced Feature * S Supporting Information *

ABSTRACT: Controlling the phononic properties of materials provides opportunities for better thermal insulation, reduction of sound noise, and conversion of wasted heat into electricity. Phononic crystals are periodically structured media composed of two or more dissimilar materials offering a unique pathway to control the transmission of phonons, responsible for sound and heat transport. In particular, phononic crystals with cubic network structure possessing complete phononic bandgaps are highly desirable for energy applications but have not been thoroughly investigated, hampering progress in this field. Here we computationally obtained phononic band structures of 16 cubic network structures that could be made by fabrication techniques including block copolymer self-assembly and identified six structures that exhibit complete phononic bandgaps. The champion phononic bandgap structure is the so-called I-WP structure with a bandgap width of 0.41. On the basis of simulation results, design rules to tailor network structures for larger phononic bandgaps are elucidated. We expect that our results will provide guidance to develop novel materials for sonic and thermal devices.

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properties.11 Considering the diversity of known cubic structures, critical questions arise, for example, about the optimal cubic network structure and how to tailor it to obtain increasingly larger phononic bandgaps. These questions motivated our methodical study to discover periodic cubic network structures with complete phononic bandgaps. The existence and characteristics of phononic bandgaps depend on material properties such as density, speed of sound, and component volume fractions as well as structural characteristics such as symmetry, connectivity, and geometrical shape. These parameters should be incorporated into the study, thus requiring a search over a high-dimensional parameter space. In order to accelerate the search, we developed a finite difference frequency domain simulator that efficiently calculates phononic bands of inhomogeneous periodic media and identifies phononic bandgap structures.

INTRODUCTION The promise of phononic crystals has attracted considerable attention toward the search for periodically structured media that can block phonons, are responsible for sound and heat transport, and can be readily fabricated using current techniques.1 Phononic crystals are periodically structured media composed of two or more dissimilar materials offering a unique pathway to control phonon transport, e.g., for improved thermal insulation, reduction of sound noise, and conversion of wasted heat into electricity.1−5 Among various periodic structures, three-dimensionally (3D) periodic network structures that exhibit complete phononic bandgaps while providing electron transport through networked semiconducting materials are particularly desirable. The reduction of the thermal conductivity independent of the electric conductivity provides potential access to better thermal-to-electric energy conversion efficiency.6,7 Recent advances in fabrication techniques including bottom-up type block copolymer selfassembly enable facile fabrication of such functional networked nanomaterials.8−10 Cubic network structures are particularly attractive due to their 3D isotropic nature of transport © XXXX American Chemical Society

Received: July 23, 2017 Revised: September 17, 2017 Published: September 19, 2017 A

DOI: 10.1021/acs.jpcc.7b07267 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C

Figure 1. Identification of periodic bicontinuous cubic network structures with phononic bandgaps. (a) Schematic of inhomogeneous bicontinuous cubic network material fabrication using the plumber’s nightmare structure, previously obtained by block copolymer self-assembly.12 (b) Phononic band structures were calculated for the 16 bicontinuous cubic network structures shown here, from which 6 phononic bandgap structures were identified (upper left box). The material for the minority networks is assumed to be bismuth telluride (ρ = 7.642 g/cm3, ct = 2027 m/s, cl = 3120 m/ s), and the majority host material, not shown for better display of the minority network structures, is assumed to be epoxy (ρ = 1.180 g/cm3, ct = 1161 m/s, cl = 2540 m/s). (c) Phononic bandgap maps as a function of volume fraction. The structures change from discontinuous (dotted lines) to bicontinuous network structures (solid lines) with increasing volume fraction. a is the cubic lattice dimension and ct is the transverse speed of sound for bismuth telluride.

given in Table S1 in the Supporting Information (SI). The equation F(x,y,z) − t = 0 defines the interface, the so-called intermaterial dividing surface16 that separates regions of dissimilar materials and the variable t determining the volume fraction of each domain. In the present calculations, we take F(x,y,z) − t > 0 as the region filled with bismuth telluride17 and F(x,y,z) − t < 0 as the region consisting of epoxy.18 Bismuth telluride is one of the most efficient thermoelectric materials, while epoxy is assumed as a polymer template. Computational Methods. The general elastic wave equation for a spatially inhomogeneous linear medium composed of two or more dissimilar materials is given by ρü = Cε,18 where ρ is the material density, u the displacement vector, C the elastic tensor, and ε the strain tensor. The strain tensor is defined by ε = [∇u + (∇u)T]/2. If all of the materials constituting the medium are isotropic, the equation simplifies to

Our strategy is to structure common materials at the mesoscale in order to mediate destructive interference of phonons, leading to phononic bandgap formation. A schematic of the fabrication process is depicted in Figure 1a. Resulting materials are amorphous or polycrystalline at the atomic level but crystalline at the mesoscale ranging from tens to hundreds of nanometers. Such mesoscale crystals have successfully been fabricatede.g., using block copolymer structure-directing oxides,12 semiconductors,9 and metals.13 Our specific focus is on bicontinuous cubic network structures that can be generated by current techniques including bottom-up type block copolymer self-assembly. Block copolymer self-assembly results in the formation of minimal surfaces between dissimilar polymeric blocks minimizing their repulsive interactions and enables large-scale and low-cost materials fabrication.14

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THEORETICAL METHODS Structure Preparation. Cubic network structures belong to one of the 36 groups ranging from space group 195 to 230.15 Candidate structures that meet the symmetry requirements can be generated utilizing a level set approach16 leading to a function F(x,y,z) with the desired cubic space group symmetry.15 The symmetrized function, F, is a linear combination of structure factor terms that are invariant under the symmetry operations of that space group. The level set functions for 16 bicontinuous cubic network structures are

ρ

∂ 2ui ∂t

2

⎛ ∂u ⎞ = ∇·(ρct 2∇ui) + ∇·⎜ρct 2 ⎟ ∂xi ⎠ ⎝ ∂ + [(ρc l 2 − 2ρct 2)∇·u] ∂xi

where ct is the transverse speed of sound. cl is the longitudinal speed of sound for the materials, and ui is the i axis component of u. The displacement vector is replaced by ui(r,t) = B

DOI: 10.1021/acs.jpcc.7b07267 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C ψi(r) exp[i(k·r − ωt)], where ψi(r) is a periodic function and ω is the frequency of phonons with a fixed momentum, k, leading to an eigenvalue equation 1 1 ⎛ ∂u ⎞ ω 2ui = − ∇·(ρct 2∇ui) − ∇·⎜ρct 2 ⎟ ρ ρ ⎝ ∂xi ⎠ −

1 ∂ [(ρc l 2 − 2ρct 2)∇·u] ρ ∂xi

We developed a finite difference frequency domain simulator utilizing parallel ARPACK19 in order to quickly solve the equation in the frequency domain, resulting in eigenvectors u(r,t) with an eigenvalue ω at a k point. Our simulator was benchmarked with previous simulation results for bodycentered cubic spheres18 and obtained consistent results (see Figure S1b in the SI). Simulation parameters ct and cl for silicon,20 bismuth telluride,17 steel,18 and epoxy18 were obtained from elastic tensor components ct = C44 /ρ and

Figure 2. Strut thickness effect on phononic bandgap formation in plumber’s nightmare structures. The level set function F(x,y,z;0 ≤ s ≤ 1) = s(c1xc1y + c1y c1z + c1z c1x) + (1 − s)(c2x + c2y + c2z ) is used for changing the strut thickness by varying the parameter s, where cαn denotes cos(nπα/ a). Networks with the thickest/intermediate/thinnest struts are obtained with s = 4/7 (red)/s = 2/3 (green)/s = 4/5 (blue), respectively (green results are from Figure 1c). All of the structures shown in the insets have the same network volume fraction, f = 0.45, for comparison. The structures change from discontinuous (dotted lines) to continuous (network) structures (solid lines) with increasing volume fraction. a and ct are the same as those in Figure 1.

c l = C11/ρ . Polarization of Phonons. In order to identify the polarization characteristics of the phonon waves, we decomposed the displacement vector, u, into its transverse, u⊥, and longitudinal, u||, components as u=u⊥ + u||, where u|| is parallel and u⊥ is perpendicular to the momentum vector k. The transverse fraction, f⊥, is given by f⊥ = ∫ dr3 |u⊥(r)|2/∫ dr3 | u⊥(r)|2 + ∫ dr3 |u∥(r)|2 and measures the polarization. As expected in materials with cubic symmetry,21 two degenerate transverse bands and one longitudinal band are observed for the three lowest frequency bands. The mechanical Poynting vector S, i.e., the energy flux vector mediated by phonons, is v *·T given by S = − 2 ,22 where v* is the complex conjugate of the velocity vector v and T is the stress tensor.

structures vary with network volume fraction, we explored candidate structures with varying volume fractions, as visualized by the phononic bandgap maps shown in Figure 1c. The volume fraction variation leads to a structural change from discontinuous (micellar) to continuous structures, which are denoted by dotted and solid lines, respectively, in Figure 1c. The champion phononic bandgap structure among the continuous structures is I-WP, which has the largest bandgap width, Wg = 0.41, at a network volume fraction of 0.39, where ω −ω Wg = (ω max+ ω min) / 2 . As revealed by the bandgap maps in Figure

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RESULTS AND DISCUSSION We investigated 16 bicontinuous cubic network structures, of which 6 were identified to have a complete phononic bandgap,

max

Table 1. Level Set Functions for Bicontinuous Cubic Network Structures with a Complete Phononic Bandgapa level set F(x,y,z)

structure

space group

F23 double diamond I-WP plumber’s nightmare K

196 (F23) 224 (Pn3m ̅ )

4c1xc1y c1z − 4s1xs1y s1z − c2xc2y c2z s1xs1y + s1y s1z + s1z s1x + c1xc1y c1z

229 (Im3̅m) 229 (Im3m ̅ )

c1xc1y + c1y c1z + c1z c1x 2(c1xc1y + c1y c1z + c1z c1x) + c2x + c2y + c2z

221 (Pm3̅m)

Neovius

195 (P23)

−2(c1x + c1y + c1z ) − 2(c2x + c2y + c2z ) − (c1xc1y + c1y c1z + c1z c1x) −3(c1x + c1y + c1z ) − 4c1xc1y c1z )

a

(

2π

)

Here, we simplify trigonometric functions as cαn = cos n a α and

sαn

= sin

2π naα

(

min

1c, significantly different bandgap frequency ranges and widths are observed for different structures. Such a large variation of phononic bandgaps implies that the choice of network structure provides an important control parameter in addition to the constituting materials and the lattice dimension. The space group is the primary factor that governs structural characteristics. For a given space group, a variety of network structures exist. The present calculations compare phononic band structures for three different structures with the same space group 229 (Im3̅m): body-centered cubic (BCC) spheres (Figure S1c), I-WP (Figure S3c), and plumber’s nightmare (Figure S3e). They exhibit similar band structures but display different bandgap maps, e.g., compare results for I-WP and plumber’s nightmare structures in Figure 1c. The similarity in phononic band structures has its origin in the same space group. The most distinguishing feature between these three structures is the connectivity between their structural components. The BCC sphere structure has no connectivity, resulting in the widest bandgap range among the three structures. This is expected as network struts that bridge between unit cells mediate phonon transport and thus lead to bandgap narrowing. In order to clearly demonstrate such effects on phononic bandgap formation, we further studied three different plumber’s nightmare structures with varying strut thickness. Using a linear combination of structure factor terms in the level set function controlled by a factor s leads to a change in the strut thickness (see Figure 2). Considering only continuous structures denoted

), where a is the cubic lattice dimension.

F23, double diamond,23 I-WP,24 plumber’s nightmare,12 K surface,25 and Neovius,26 where F23 stands for a cubic network structure that we found from structure factor terms invariant under the symmetry operations of space group 196 (F23) (see Table 1 and Figure 1b). All of the structures have a 6−7 gap, except the double diamond structure, which has a 12−13 gap. The two numbers denote the nth lowest frequency bands that bracket the phononic bandgap. Because phononic band C

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Figure 3. Material effect on phononic bands of plumber’s nightmare structure. (a−c) Phononic band structures of the plumber’s nightmare structure with a minority network volume fraction of 0.3 and varying minority network material: (a) silicon (ρ = 2.320 g/cm3, ct = 5710 m/s, cl = 9090 m/s); (b) bismuth telluride (ρ = 7.642 g/cm3, ct = 2027, cl=3120 m/s); and (c) steel (ρ = 7.780 g/cm3, ct = 3227 m/s, cl = 5825 m/s). The host material is epoxy, as in Figures 1 and 2. The phonon frequencies are in units of 2πct/a, where a is the cubic lattice parameter and ct the transverse speed of sound for the respective network materials. The transverse fractions, f⊥, for all of the phononic bands are represented in colors, representing values as encoded in (d). (e) Brillouin zone and k point path for the BCC primitive unit cell. (f) Displacement vectors, u (blue arrows), and phononic energy flux vectors, S (red arrows), for the four eigenstates indicated in (b) (i−iv) at k = 2π/a[0,0.2,0]. D

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predominantly localized in the low-density material, i.e., the host material. Such distinguishing local distributions are more pronounced in discontinuous structures like BCC spheres. On the basis of characteristics of the energy fluxes in the network structures, we found that if the phonons are more localized in the low-density material the seventh lowest band frequency increases. This is because the effective material density for the band decreases and thus the phonon frequency increases as ω ≈ 1/ ρeff , where ρeff is the effective material density. Therefore, a smaller density contrast of materials leads to a smaller frequency difference between the sixth and seventh lowest bands, resulting in the bandgap closing in the case of silicon. Therefore, one can engineer larger phononic bandgaps utilizing materials with a higher density contrast.

by solid lines in Figures 1c and 2, the thinnest strut structure with s = 4/5, with a network volume fraction of 0.43, has the largest phononic bandgap, Wg = 0.45, which is even larger than the value for the I-WP structure described earlier. Therefore, strut thickness is a critical parameter to control the bandgap range, and thinner strut structures are advantageous for obtaining larger bandgaps. Following the study of structural effects on phononic bandgap formation, we moved on to explore the effects of different materials constituting the networks. To that end, we studied plumber’s nightmare networks composed of silicon20 and steel.18 Compared with bismuth telluride, a surprisingly different phononic band structure was obtained for silicon, while for steel results were quite similar. Despite considerable differences in the speeds of sound, the networks with bismuth telluride and steel show similar phononic band structures and bandgap widths, as demonstrated in Figure 3b,c (note that ρ = 7.642 g/cm3, ct = 2027 m/s, and cl = 3120 m/s for bismuth telluride vs ρ = 7.780, ct = 3227 m/s, and cl = 5825 m/s for steel). In contrast, light materials like silicon (ρ = 2.320 g/cm3) do not exhibit any bandgaps, as shown in Figure 3a. Therefore, material density seems to be another dominant factor that dictates phononic band structure. Material density primarily determines phonon frequency, i.e., phonon energy, E = ℏω. Heavy materials for networks provide large density contrast against the light material used as templates and are preferred to obtain larger phononic bandgaps. Our in-depth analyses provide rich information on phononic band characteristics. Phononic bands are usually classified as transverse and longitudinal bands. The displacement vectors, u, of transverse waves are perpendicular to the phonon momentum vector, k, while those of longitudinal waves are parallel to k. The polarization of phonons is a particularly important indicator for sound waves because transverse waves are usually inactive for acoustic applications. However, most of the phononic bands in 3D structured materials cannot be clearly classified because they usually have both transverse and longitudinal components. Thus, we determine the transverse fraction of phonons, f⊥, to characterize the phonon waves (see the Computational Methods section). Figure 3a−c shows the polarization of the phonon band structures. We found that the two lowest degenerate bands (i) are transverse and the third lowest band (ii) is longitudinal, which is also reflected in the displacement vectors, u (blue arrows) in Figure 3f. Many higher frequency bands have transverse and longitudinal components, i.e., are quasi-transverse or quasi-longitudinal bands. Despite the mixed characteristics, one can see that at high frequencies the transverse component dominates. Figure 3f shows the displacement vectors, u, and energy flux vectors, S (red arrows), of phonon modes to reveal the underlying physics of the phononic bandgap formation (see the Computational Methods section). The six lowest frequency bands are acoustic because the displacement vectors are inphase along the k direction, as shown in Figure 3f(i−iii) and Movies S1, S2, and S3 (note that the two acoustic-transverse branches and one acoustic-longitudinal branch are all doubled by the Brillouin zone folding resulting in the six lowest frequency bands). Meanwhile, higher frequency bands are optical (see Movie S4). Thus, the 6−7 gap originates from the frequency difference between the acoustic and optical modes. Low frequency bands (i−iii) consist of phonons that are mainly localized in the high-density material, i.e., the network material. Meanwhile, the phonons of the high frequency bands (iv) are

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CONCLUSIONS In conclusion, utilizing our finite difference frequency domain simulator, we investigated 16 bicontinuous cubic network structures and identified 6 structures with a complete phononic bandgap. The champion phononic bandgap structure is I-WP with a bandgap width of 0.41. Network structures can be tailored to possess even larger phononic bandgaps by reducing the strut thickness bridging between unit cells. It is important to note that our champion I-WP and plumber’s nightmare base structures have already been realized experimentally via selfassembly approaches.12,27 Materials density is a key parameter that determines phonon frequencies and significantly affects the phononic band structure. A higher density contrast between the material components of the structure leads to a larger phononic bandgap. The results provide design rules for phononic crystals with complete phononic bandgaps and give impetus toward novel applications of sonic and thermal devices.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.7b07267. Benchmark of the simulator, level set functions, models of all of the bicontinuous cubic network structures studied, and phononic band structures (PDF) W Web-Enhanced Features *

Movies S1−S4 corresponding to Figure 3f(i−iv), where the phonon modes are animated for one cycle

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Address: 330 Bard Hall, Ithaca, NY 14853. Phone: (607) 255-3487. Fax: (607) 255-2365. ORCID

Kahyun Hur: 0000-0002-5563-3986 Richard G. Hennig: 0000-0003-4933-7686 Ulrich Wiesner: 0000-0001-6934-3755 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work has been supported by the Korea Institute of Science and Technology (Grant No. 2E26940), the National Research Foundation of Korea (Grant No. NRF-2016M3D1A1021142, NRF-2014M3C1A3054143), and the U.S. National Science Foundation (DMR-1707836). The authors also acknowledge E

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(21) Ashcroft, N. W.; Mermin, N. D. Solid state physics; Holt: New York, 1976; p xxi. (22) Auld, B. A. Acoustic fields and waves in solids; R.E. Krieger, 1990. (23) Thomas, E. L.; Alward, D. B.; Kinning, D. J.; Martin, D. C.; Handlin, D. L.; Fetters, L. J. Ordered bicontinuous double-diamond structure of star block copolymers: a new equilibrium microdomain morphology. Macromolecules 1986, 19 (8), 2197−2202. (24) Schoen, A. H. Infinite periodic minimal surfaces without selfintersections; National Aeronautics and Space Administration: Washington, DC, 1970. (25) Karcher, H. The triply periodic minimal surfaces of Alan Schoen and their constant mean curvature companions. manuscripta mathematica 1989, 64 (3), 291−357. (26) Gòzd̀ z̀, W.; Holyst, R. From the plateau problem to periodic minimal surfaces in lipids, surfactants and diblock copolymers. Macromol. Theory Simul. 1996, 5 (2), 321−332. (27) Stucky, G. Direct imaging of the pores and cages of threedimensional mesoporous materials. Nature 2000, 408 (6811), 449− 453.

support from the Disaster and Safety Management Institute funded by the Ministry of Public Safety and Security of Korean government (Grant No. MPSS-CG-2016-02).

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