# From Pure Mathematics to Macroscale Applications: The Genesis of Schwarzites

Levi C. Felix  [  Rushikesh Ambekar  [  Raphael M. Tromer  [  Cristiano F.
Woellner  [  Varlei Rodrigues  [  Pulickel M. Ajayan  [  Chandra S. Tiwary  [
Douglas S. Galvao<EMAIL_ADDRESS>[

###### Abstract

Schwarzites are porous (spongy-like) carbon allotropes with negative Gaussian
curvatures. They were proposed by Mackay and Terrones inspired by the works of
the German mathematician Hermann Schwarz on Triply-Periodic Minimal Surfaces
(TPMS). This review presents and discusses the history of schwarzites and
their place among curved carbon nanomaterials. We summarized the main works on
schwarzites available in the literature. We discuss their unique structural,
electronic, thermal, and mechanical properties. Although the synthesis of
carbon-based schwarzites remains elusive, the recent advances in the synthesis
of zeolite-templates nanomaterials bring them closer to reality. Atomic-based
models of schwarzites have been translated into macroscale ones that have been
3D printed. These 3D printed models have been exploited in many real-world
applications, including water remediation and biomedical ones.

State University of Campinas] Applied Physics Department, State University of
Campinas, Campinas, SP, 13083-970, Brazil [State University of Campinas]
Center for Computational Engineering and Sciences, State University of
Campinas, Campinas, SP, 13083-970, Brazil Indian Institute of Technology]
Department of Metallurgical and Materials Engineering, Indian Institute of
Technology Kharagpur, West Bengal, India State University of Campinas] Applied
Physics Department, State University of Campinas, Campinas, SP, 13083-970,
Brazil [State University of Campinas] Center for Computational Engineering and
Sciences, State University of Campinas, Campinas, SP, 13083-970, Brazil
Federal University of Paraná] Physics Department, Federal University of
Paraná, Curitiba, PR, Brazil State University of Campinas] Applied Physics
Department, State University of Campinas, Campinas, SP, 13083-970, Brazil Rice
University] Department of Materials Science and Nanoengineering, Rice
University, Houston, TX, USA Indian Institute of Technology] Department of
Metallurgical and Materials Engineering, Indian Institute of Technology
Kharagpur, West Bengal, India State University of Campinas] Applied Physics
Department, State University of Campinas, Campinas, SP, 13083-970, Brazil
[State University of Campinas] Center for Computational Engineering and
Sciences, State University of Campinas, Campinas, SP, 13083-970, Brazil

##  1 Introduction

The emergence of curved carbon nanostructures in the mid-1980s, such as
fullerenes 1 and nanotubes 2, have motivated the study and the advent of new
ones 3. Inspired by the work of the German mathematician Hermann Schwarz
(1843-1921), who did seminal works on the field of topology and geometry of
periodic minimal surfaces 4, Mackay and Terrones proposed a new class of
structures called schwarzites 5, after Schwarz. Shwarzites are porous (spongy-
like) three-fold coordinated carbon allotrope structures exhibiting negative
Gaussian curvatures and with one-atom-thick pore-walls. They attracted much
attention for their interesting mechanical 6, 7, 8, 9, electronic 10, 11, 12,
13, 14, 15, 16, 17, thermal 18, 19, 20, 21, structural 22, 23, and many others
properties 24, 25, 26.

Due to their complex structural topologies, the synthesis of schwarzites
represents a real challenge. First adopted as models for random graphene foams
27, 28, 29, 30, the possibility of their synthesis through zeolite templating
31, 32, 33, 34 and/or molecular assembly into negative Gaussian-curvature sub-
structures 35, 36, 37, have also been proposed, but their experimental
realization remains elusive. However, recent advances in 3D print
manufacturing technology have allowed the creation of 3D printed macroscale
models of several Schwarzites structures 22. These structures are lightweight
and have high energy absorption capabilities 22, 38. These studies have
inspired other works on 3D printed carbon-based lattices, such as tubulanes
39, pentadiamond 40, and zeolite-templated carbons 41. Further works of 3D
printed schwarzite have emerged in catalysis, wastewater treatment, building
blocks, etc., exploiting engineering the interface with functional
materials/nanomaterials for enhanced mechanical, absorption, and fracture-
resistant properties 42. In particular, using such porous structures in the
remediation of microplastic-polluted waters has also shown promising results
43.

This review aims to contextualize the place of Schwarzites among curved carbon
nanomaterials, summarize the main works on schwarzites available in the
literature, and emphasize their potential in many daily-life applications.
Since the property of negative Gaussian curvature is the one that mainly
distinguishes them from other curved structures, the connection between
topology and geometry of carbon materials will be briefly reviewed in the
first section. Then, the stability of non-carbon schwarzites and their
structural differences will be discussed. Separate sections will be dedicated
to a summary of reported results in the literature on the mechanical, thermal,
and adsorption properties. Its early use as models for graphitic random foams,
alongside possible routes to their synthesis, will also be briefly reviewed. A
summary of applications of 3D-printed macroscopic Schwarzites is also
discussed.

##  2 Topology and Curved Carbon Nanomaterials

Carbon in its sp2-hybridized state is found in graphite, graphene, and other
planar allotropes. Such a well-defined hybridization state is attributed to
the mirror symmetry of all $p_{z}$ orbitals, which does not distinguish where
it is up or down the plane. However, the presence of a curvature breaks such
symmetry by distorting these orbitals. Many curved carbon nanomaterials still
possess three-fold coordination per atom, such as fullerenes 1 and nanotubes
2, being also denominated graphitic structures.

In most common planar allotropes of carbon, such as graphite and graphene,
there are only six-membered carbon rings and no curvature. The existence and
type of curvature can be uniquely determined by the presence of different
types of rings that induce lattice distortions that are capable of creating
curved (positive or negative) surfaces. These surfaces can be characterized by
concepts such as Gaussian curvature, which connects geometry and topology by
the Gauss-Bonnet-Euler formula

$\frac{3}{\pi}\int_{S}KdS=2N_{4}+N_{5}-N_{7}-2N_{8}.$ (1)

Here, $K$ is the local Gaussian curvature, which, integrated over the surface
$S$, gives the total Gaussian curvature. $N_{k}$ is the number of $k$-membered
rings along $S$. Note that $N_{6}$ does not appear on Eq. 1 since hexagonal
rings do not contribute to curvature. Even curved structures with only
hexagons, which is the case of nanotubes 2, are characterized as having zero
Gaussian curvature (see Figure 1).

Figure 1: The Gauss-Bonnet-Euler theorem (Eq. 1) establishes the relationship
between the sign of Gaussian curvature and the ring distribution in curved
structures. Six-membered rings do not contribute to the presence of curvature,
as in graphene and nanotubes. The resultant surface is flat when squares and
octagons are present in equal numbers, such as in biphenylene networks. A
closed buckyball surface is obtained with the insertion of pentagonal rings.
Negatively curved carbon structures can be built by introducing either seven-
or eight-membered rings or a combination of other types of rings. Here, two
examples of structures with negative Gaussian curvature are shown: a saddle-
shaped molecule [7.7]Circulane Cs, and P8bal Schwarzite. Four-, five-, seven-,
and eight-membered carbon rings are colored black, red, green, and purple,
respectively.

It is well known that the addition of five-membered rings on a graphene sheet
can form stable structures. In fact, pentagonal rings alongside hexagons form
fullerenes, such as $C_{60}$ 1. According to Eq. 1, this yields positive
Gaussian curvature, as illustrated in Figure 1. Conversely, the presence of
non-hexagonal rings may not result in curved structures if the right-hand side
of Eq. 1 vanishes. This is the case of biphenylene carbon network 44, with an
equal number of four- and eight-membered rings along its plane, also shown in
Figure 1.

Negatively curved structures can be built by combining different carbon rings
in such a way that the right-hand side of Eq. 1 becomes negative. The simplest
way of doing this is by introducing only heptagons or octagons. An example of
the former includes saddle-shaped molecules such as a class called Circulanes
45. The first proposed schwarzites by Mackay and Terrones in 1991 5 contained
only octagonal rings as the geometrical curvature generators (shown in Figure
2). Whereas other Schwarzites containing five- and seven-membered rings
(proposed by Lenosky and coworkers 46) also possess negative curvature since
they are placed in such a way that yields a negative total Gaussian curvature
(see Eq. 1).

The solutions to the well-known mathematical problem of finding surfaces with
minimal area and periodic along all three Cartesian directions are called
Triply-Periodic Minimal Surfaces (TPMS). Pioneering in this field, Herman
Schwarz 47, found different families of TPMSs such as Primitive (P) and
Diamond (D) (see Figure 2). Almost a century later, Alan Schoen 48 found other
families of surfaces that obey the properties required to be TPMSs, such as
Gyroid (G) and I-graph Wrapped Package-graph (IWP). The first work that came
up with the idea of decorating TPMS with graphitic (three-fold coordinated)
atoms was that of Mackay and Terrones in 1991 5. In Figure 2, we can observe
cubic unit cells of P, D, G, and IPWg families. Depending on the case, the use
of supercells is necessary to obtain size-converged properties, as is the case
for mechanical ones 8, as shown in Figure 2.

Figure 2: (Left) Unit cells of P, D, G, and IPWg families of TPMS/Schwarzites.
(Right) Supercells for P and G families show the presence of eight-membered
rings placed among several hexagons in blue. Reproduced with permission from
reference 8. Copyright 2020 Elsevier.

Within a given schwarzite family, there are several different structures that
differ mainly through their octagonal/hexagonal ring ratio, as can be seen in
Figure 2. Structures with no hexagons separating two laterally neighbor
octagons receive the suffix -688, as in P688 and G688, where the prefixes P
and G refer to their families, such as primitive and gyroid, respectively.
Structures with one hexagon between two consecutive octagons receive the
suffix 8bal since eight-membered rings introduce curvature, and it balances
both disconnected regions inside and outside pores in equal volumes. Note that
other nomenclatures exist, such as Miller et al. 6, used 8-0 and 8-1, which
substitutes 688 and 8bal, respectively.

##  3 Mechanics of Schwarzites

The larger the hexagonal/octagonal ring ratio, the larger the pores in a given
schwarzite structure. There is a monotonic decrease in mechanical properties,
such as in the Young and Bulk modulus values, as the pore size increases 6, 8.
This is consistent with the fact that a larger pore size yields lower-density
structures. On the other hand, these larger pore structures withstand larger
deformation levels before ultimately breaking 8. Under compressive loading,
they possess a more well-defined stress plateau (Figure 3 (a)) where the
appearance of kinks in this region is attributed to abrupt pore collapses
(shown on the inset of Figure 3 (a)). Some structures can be compressed up to
80% of strain before ultimately breaking.

Under stretching, fracture characteristics trends can be differentiated by
their respective families instead of pore size. In the stress-strain curve of
Figure 3 (a), G schwarzites tend to break before the P, and they are more
susceptible to the formation of Linear Atomic Chains (LACs) (or Carbynes), as
can be seen on the inset. The influence of LACs on the stress-strain response
can be seen for strain levels after structural breaking as non-zero stress
fluctuations. Carbyne is a very attractive material due to its high tensile
strength 49 and tunable electronic properties 50, 51. The appearance of such
LACs as a result of material fracture has been experimentally verified 52, 53,
54, 55. Although these chains are more energetically stable than other carbon
allotropes, their high chemical reactivity (due to the presence of many
unsaturated triple bonds) limits their fabrication on larger scales and, thus,
the appearance of such LACs after schwarzite fractures would be hindered by
this effect. As already shown in previous works 7, the family type plays a
significant role in the fracture pattern and, subsequently, in the stress-
strain curve shape. Fracture patterns of P, G, and D families are shown in
Figure 3 (b). For P Schwarzites, the fracture mechanism is dictated by the
appearance of a single crack that propagates throughout the structure, leading
to total rupture (as schematically shown in Figure 3 (b)), which occurs due to
the sparse distribution of atomic stress. This behavior results in stress-
strain curve shapes with a single peak followed by a drop to zero stress. The
presence of regions with high atomic stress separated by small stress-value
regions, shown for D and G types, favors the nucleation of many cracks prior
to total rupture. When the first crack is created, the stress is partially
released and then restarts to increase until the fracture propagates to the
nearest crack, where the total stress decreases again, yielding a sawlike-
shape stress-strain curve (shown in Figure 3 (b)).

Figure 3: Mechanical properties of schwazites. (a) Stress-strain curves of
schwarzites under compression and stretching. Pore collapses responsible for
the presence of stress plateaus are shown on the inset, and the formation of
LACs after rupture. Reproduced with permission of reference 8. Copyright 2020
Elsevier. (b) Fracture patterns of different schwarzite families (types), with
their characteristic stress-strain curve shapes and local atomic stress
distributions. Reproduced with permission from reference 7. Copyright 2018
American Chemical Society. (c) Auxetic behavior under stretching of IWPg
schwarzite, where displacement vectors are represented as red arrows.
Reproduced with permission from reference 9. Copyright 2020 Royal Society of
Chemistry.

Negative Poisson’s ratio (also known as auxetic behavior) has also been
reported for some schwarzite structures 8, 9. In Figure 3 (c), it is shown
that the IWPg structure expands laterally while being uniaxially stretched.
This occurs due to re-entrant topology 56 present in the crystal structure of
IWP schwarzites, which holds for both compression and tension 9.

##  4 Thermal Conduction and Hybridization of Phonon Modes

Zero curvature sp2 carbon nanomaterials such as graphene and carbon nanotubes
possess extremely high lattice thermal conductivity of $\sim 1015$ and $\sim
1700$ Wm-1K-1, respectively 18. However, the presence of surface curvature
dramatically suppresses thermal transport, as reported for several schwarzites
18, 19 (called Negatively Curved Carbon Crystals (NCCCs)). Figure 4 (a)
illustrates this decrease in thermal conductivity as a function of the nominal
value of Gaussian curvature for several P schwarzites. It can be seen that the
type number increases with the relative number of hexagons in the unit cell
(similar to the idea illustrated in Figure 2), which translates into
structures with lower absolute values of curvature. A monotonic decrease in
conductivity is observed up to the case where the curvature is zero, which is
the case of graphene and nanotubes. More dense schwarzite models, such as Type
I in Figure 4 (a) (P688), provide better ways of sound propagation that are
reflected in higher group velocities of acoustic modes in the phonon spectrum
19. This is a possible mechanism that might explain the thermal conductivity
decrease upon increasing Gaussian curvature values (lowering density, shown in
the inset of Figure 4 (a)).

Figure 4: Thermal transport in schwarzites. (a) Decrease of thermal
conductivity caused by the presence of curvature for P schwarzites compared to
other carbon allotropes. Reproduced with permission from reference 19.
Copyright 2017 Royal Society of Chemistry. (b) Hybridization of acoustic
phonon branches in P688 schwarzite as a guest-host system. Reproduced with
permission from reference 20. Copyright 2018 Elsevier. (c) Non-equilibrium
molecular dynamics (NEMD) calculations of the thermal conductivity of G
schwarzites as a function of the unit-cell size. Reproduced with permission
from reference 57. Copyright 2017 Royal Society of Chemistry.

The porous structure, allied with the low thermal conductivity of schwarzites,
makes them suitable candidates for building guest-host systems for
thermoelectric applications. In such systems, chemically inert atoms (guests)
are placed inside the available pores of a crystalline framework (host) to
induce modifications in the vibrational modes of the lattice. An illustration
of such configuration can be seen on the inset of Figure 4 (b). There, a
generic inert atom interacts with the cage of a P688 schwarzite through
Lennard-Jones potential. The resultant effect is the anti-crossing bands at
the blue-hatched region of the phonon dispersion, which is attributed to the
coupling between longitudinal and transversal acoustic modes (mainly
responsible for phonon transport). This further reduces thermal conductivity
by decreasing the slopes of acoustic branches 20.

The results on thermal transport presented so far were obtained by the Green-
Kubo formalism, where transport coefficients are obtained from fluctuations of
a suitable quantity. For instance, thermal conductivity is the autocorrelation
of the heat flux. These results are collected from molecular dynamics
simulations of the system at dynamic equilibrium, which is denoted by
Equilibrium Molecular Dynamics (EMD). As we know, thermal transport is an out-
of-equilibrium phenomenon that requires the presence of a temperature gradient
(absent in EMD). Thus, it is also common to calculate thermal conductivity
from simulations where parts of the system are kept at constant temperatures
(heat source and cold sink), also known as Non-Equilibrium Molecular Dynamics
(NEMD). Figure 4 (c) shows an example of such a configuration, where a G
schwarzite supercell is subjected to a temperature gradient. Using different
unit-cell sizes (shown in Figure 4 (c)), size-dependence of thermal
conductivity is obtained as the ratio between the induced heat flux and the
corresponding chosen thermal gradient, according to heat Fourier’s law.
Similar to the case of EMD results for P schwarzites 19, thermal conductivity
is reduced by increasing the unit cell size (decreasing the density), as shown
in Figure 4 (c).

##  5 Schwarzites as Frameworks for Gas Adsorption

The search for nanoporous materials to capture and/or store natural molecular
gases is still an ongoing research topic that attracts the scientific
community’s interest due to the goal of going green. In particular, H2 and CO2
storage are the most investigated gases due to their relevance in hydrogen
fuel technology and carbon dioxide emission decrease 24. A systematic
understanding of how Gaussian curvature influences adsorption properties might
be crucial to design strategies to optimize molecule capture/storage
performance. It has been reported that the presence of regions with local
curvature both positive and negative (see Figure 5 (a)) enhances the binding
energy of some specific chemisorption sites for molecular hydrogen 26.
Although schwarzites are known for their negative Gaussian curvature, this
value refers to the total value averaged throughout their surfaces. In
contrast, locally, the curvature values can be positive and negative for some
regions. Namely, positively curved sites are more favorable to the formation
of chemisorption because the anti-bonding state of their $p$-bands is less
partially filled to those corresponding to the negatively curved sites 26, as
shown in Figure 5 (a).

Another molecular gas whose storage is a target of intense research is
methane. Its use in vehicular applications as natural gas fuel has been
focused on as an alternative to fossil fuels, significantly contributing to
the greenhouse effect. Several porous materials 58, 59 have been investigated
regarding their methane storage capabilities. The porous structure of
schwarzites has also gained attention from the scientific community in this
sense 60, 25. The smallest schwarzites from 4 different families have been
studied (Figure 5 (b)), with large-pore structures having higher adsorption
capabilities, as shown in the histogram of Figure 5 (b). This occurs due to
the well-known trend of larger pores, providing a larger surface area for a
greater adsorption capacity.

Figure 5: Gas adsorption in schwarzites cages. (a) Hydrogen storage in
positively and negatively curved sites alongside their relative p-band center
positions. Reproduced with permission from reference 26. Copyright 2022
Elsevier. (b) Methane adsorption for different schwarzite structures.
Reproduced with permission from reference 25. Copyright 2019 American Chemical
Society.

##  6 The Stability of Non-carbon Schwarzites

As it is well-known for the cases of diamond and graphene crystalline
structures, a version of schwarzites made of other chemical elements has
already been proposed in several works where the carbon atoms are replaced by
boron nitrde 16, silicon 61, 12, and, most recently, germanium 17. The
stronger sp3 character of silicon (Si) and germanium (Ge) bonding produces
significant structural differences in all carbon nanostructured analogs. For
instance, silicene and germanene membranes differ from purely flat graphene by
an out-of-plane buckling of the atomic structure 62, 63, 64, 65. This occurs
due to the presence of the pseudo-Jahn-Teller effect 66, 67, 68.

Figure 6: Non-carbon schwarzites. (a) P8bal, P688, and D688 BN schwarzites
with their energy-volume curves. Reproduced with permission from reference 16.
Copyright 2017 Royal Society of Chemistry. (b) Energy as a function volume per
atom of an fcc Si-168 schwarzite. Reproduced with permission from reference
12. Copyright 2010 American Institute of Physics. (c) Buckled structures of
D688, G688, and P8bal Ge schwarzites with their unit cell highlighted on white
squares. Reproduced with permission from reference 17. Copyright 2020 Royal
Society of Chemistry.

Similar to flat hexagonal boron nitride (BN), each boron atom only has
nitrogen atoms as nearest neighbors in BN schwarzites, as seen in Figure 6
(a). The presence of a single minimum indicates the structural stability of
all BN structures reported therein, where although they are metastable than
the usual hexagonal BN (hBN) and cubic BN, they are more stable than
previously reported T-B3N3. P8bal BN schwarzite is the most stable among all
three structures. Relative energetic stability between a silicon schwarzite
and the more usual silicon in a diamond structure 12 is of the same order of
magnitude as those of BN (Figure 6 (b). Recent studies considering germanium
schwarzites showed pronounced local structural distortions (compared to
carbon) due to the pseudo-Jahn-Teller effect for the largest structures 17
while showing relative stability compared to other germanium structures.

##  7 Possible Roadmaps to Schwarzite Synthesis

Although a well-controlled synthesis of schwarzites is yet to be achieved,
numerous efforts have been reported in the literature 31, 32, 34, 33.
Especially, in silico approaches have gained attention in the last few years
33 regarding the production of zeolite-templated carbon (ZTC). Several ZTCs
have geometry similar to schwarzites, including the generation of different
families, such as G and D 33 (see Figure 7 (a)). The claimed experimental
feasibility of such a protocol is mainly based on the idea of removing the
zeolite framework with acidic compounds while maintaining the whole carbon
structure at the end of the process.

Figure 7: Roadmap to schwarzites synthesis. (a) Illustration of in silico
schwarzites generation by Zeolite-Templating Carbon alongside two examples of
produced structures. Reproduced with permission from reference 33. Copyright
2018 National Academy of Sciences of the United States of America. (b)
Negatively curved spongy carbon modeled by D schwarzites. Reproduced with
permission from reference 29. Copyright 2003 Elsevier. (c) Spongy carbon
synthesis illustration, scanning electron microscope image, and stress cycle.
Reproduced with permission from reference 30. Copyright 2015 Nature.

Although experimental synthesis of crystalline schwarzites remains elusive,
their porous character motivated many approaches 27, 29, 30, 69 to use them as
models for graphitic random nanofoams. Their predominant sp2 character
provides them with the characteristic yielding and compliance of cell walls of
schwarzites. Namely, they can also be compressed to a high degree of
deformations before ultimately breaking 29, 30. Figure 7 (b) shows an example
of a graphitic nanofoam image from a scanning tunneling microscope synthesized
by a pulsed microplasma cluster source in the presence of a metal-organic
catalyst, and it can be deposited as a film by supersonic cluster beam
deposition. This form of carbon is characterized by interconnected thin layers
forming a spongy structure with meso- and macroporosity. It consists of a
robust, multiply-connected graphene sheet fully covalent along the three
dimensions 29. Assembly of ultra-large-area graphene sheets in solution via
solvothermal reactions (Figure 7 (c)) is another route that has been applied
to fabricate such carbon nanofoams 30. These graphene sponges shown in Figure
7 (c), with densities similar to air, display near-zero Poisson’s ratios along
all directions and are largely strain-independent during reversible
compression to giant strains. At the same time, they function as enthalpic
rubbers, which can recover up to 98% compression in air and 90% in liquids and
operate between -196 and 900oC. Furthermore, these sponges provide reversible
liquid absorption for hundreds of cycles and then discharge it within seconds
while still providing an effective near-zero Poisson’s ratio 30.

##  8 Macroscopic 3D-Printed Schwarzites

The challenge imposed by synthesizing schwarzites at an atomic scale prevents
exploring their exciting properties, as mentioned in this review. This
shortcoming has led to some pioneer works to consider the development of
schwarzite topologies at a macroscale with the use of additive manufacturing
techniques 22, 38, 39, 23, 41, 40. Sajadi et al. 22 explored a multi-scale
geometrical design of 3D-printed schwarzites (as seen in Figure 8 (a)), where
an interpolated surface generated from the atomic positions serves as the
basis for 3D printing of polymeric structures which are subsequently
mechanically compressed. They have observed that the mechanical
characteristics of schwarzites structures are similar regardless of the
material and length scale difference. This is a signature of the distinct
surface topology of each individual structure. The deformation study reveals
that Schwarzite structures have unique layered deformation and uniform
distribution of stress concentration. Such topology-dependent properties are
more pronounced when the scale factor is the same between both scales, as was
done by Gaál et al. 23. Figure 8 (b) shows 3D-printed schwarzites with the
same size scale relative to each individual structure. Also, some stress
collapses are observed in both length scales at roughly the same strain
levels. This work also considered different additive manufacturing polymers
such as Poly-Lactic Acid (PLA) and Acrylonitrile Butadiene Styrene (ABS). PLA
was found to be more well-behaved during mechanical compression and withstand
more significant strains without breaking 23. Another direction is to
hierarchically combine different schwarzite topologies to increase the
possibilities for their exceptional mechanical response. Some recent
strategies include the introduction of sp fragments into their cage-structures
70 (named as Schwarzynes) and the generation of polycrystalline domains with
tunable response 71. A recent structural application of schwarzites was
explored in damping enhancement and vibration isolation. Due to its excellent
mechanical strength, schwarzite exhibits an 80% higher damping ratio than the
solid base material. Also, it can withstand frequency shifts lower than
excitation, resulting in vibration isolation. In this way, vibration isolation
and increased energy dissipation result in lower force transmissibility 72.

Figure 8:  3D-printed schwarzites. (a) Atomic models and 3D-printed schwarzite
structures made of of PLA polymer. Reproduced with permission from reference
22. Copyright 2018 Wiley. (b) 3D-printed schwarzites with different sizes,
showing the same scale factor as atomic models. Reproduced with permission
from reference 23. Copyright 2022 Wiley. (c) The fabrication method of
cementitious schwarzite. Reproduced with permission from reference 42.
Copyright 2021 Elsevier. (d) Conformal coating strategy of epoxy-coated
ceramic schwarzite and schematic of mechanical performance with and without
coating. Reproduced with permission from reference 73. Copyright 2021 AAAS.
(e) Development of a schwarzite-based moving bed 3D printed water treatment
system for nanoplastic remediation. Reproduced with permission from reference
43. Copyright 2021 Royal Society of Chemistry. (f) before and after 3D printed
schwarzite and interaction of 3D printed schwarzite with Rhodamine B dye.
Reproduced with permission from reference 74. Copyright 2021 Elsevier.

Researchers have verified the feasibility of schwarzite in building and
construction areas, as the present building and construction materials possess
poor fracture strain. The PVA-coated schwarzite cement structure was developed
via 3D printing and investment casting. 3D printing technique was utilized for
the fabrication of a complex schwarzites mold structure using polyvinyl
alcohol, and then cement was cast in it. Afterward, PVA mold was dissolved in
a suitable solvent, resulting in PVA-coated schwarzite cement structure
(Figure 8 (c)). The specific energy absorption and fracture strain of
cementitious schwarzite structure is two times and 17 times, respectively,
greater than Cementitious solid structure 42. Similarly, epoxy-coated ceramic
Schwarzite structures were developed via 3D printing and conformal coating
(Figure 8 (d)) to enhance the structural stability of ceramic structures at
higher strains. Synergistic improvement in specific energy absorption (77
times) and fracture strain (13 times) was observed compared to the uncoated
ceramic schwarzite structure. They have observed that epoxy coating helps
delay crack propagation 73.

Due to their high surface area, schwarzites can be used for carrier
applications in different fields. 3D printed schwarzite carriers were used for
the remediation of nanoplastics (Figure 8 (e)). The selection of material is
based on the molar interaction constant; therefore, ABS-made schwarzite
carriers were used for the removal of polycarbonate due to electrostatic
interaction between them 43. Similarly, catalysis is another application where
surface area plays an important role. Dye degradation capability and water
cleaning efficiency of ZnO-decorated 3D-printed schwarzite structures were
studied. The accelerated removal efficiency ($\sim 80\%$) was experienced with
ZnO decorated schwarzite carrier against Rhodamine B (Figure 8 (f)) dye due to
the high surface area of schwarzite and catalytically active material (ZnO)
74. Bone regeneration is another field that could benefit from the structural
robustness of TPMS/Schwarzites topology 75, 76, 77. Their high surface area
also enhances the energy harvesting from rain droplets 78 and CO2 capture 79
along their surface, proving their multifunctional capabilities.

##  9 Conclusions and future work

Schwarzites represent an interesting case in science, where the original
concept was based on pure mathematics, and then the topological concepts were
translated into carbon allotropes with negative curvature. Atomic-based models
were proposed and investigated for their structural, electronic, mechanical
and transport properties. The atomic models were also translated into
macroscale ones that were 3D printed. Remarkably, some mechanical properties
were found to be material and scale-independent, indicating that the topology
determines most of the mechanical behavior. The negative Gaussian curvature of
schwarzites is a unique topology amongst the different carbon allotropes. The
appealing mechanical properties of schwarzites are due to complex topology and
uniform stress distribution. Thermal conductivity can be tuned by varying unit
cells of schwarzites. Gas adsorption in schwarzite cages can also be tuned by
systematically designing the unit cell size. Real-world applications of 3D
printed schwarzites have been already demonstrated for water remediation and
biomedical applications. Recently, researchers have built a multi-layered
schwarzite architecture named Petal Schwarzite, which shows unique mechanical
properties 80, 81. It is possible to take advantage of the excellent
mechanical properties and high surface area of schwarzite in various fields by
developing multifunctional schwarzite structures for energy harvesting,
thermal management systems, sound absorbers, sensors, microfluidic systems,
water filters, etc. The recent synthetic advances in zeolite-templated
materials make schwarzities closer to being experimentally realized. We hope
the present work will stimulate further studies on these remarkable
structures.

The authors acknowledge the Center for Computing in Engineering and Sciences
at Unicamp for financial support through the FAPESP/CEPID Grants #2013/08293-7
and #2018/11352-7.

## References

  * Kroto et al. 1985 Kroto, H. W.; Heath, J. R.; O’Brien, S. C.; Curl, R. F.; Smalley, R. E. C60: Buckminsterfullerene. _Nature_ 1985, _318_ , 162–163
  * Iijima 1991 Iijima, S. Helical microtubules of graphitic carbon. _Nature_ 1991, _354_ , 56–58
  * Terrones and Terrones 2003 Terrones, H.; Terrones, M. Curved nanostructured materials. _New journal of physics_ 2003, _5_ , 126–126
  * Mackay 1985 Mackay, A. L. Periodic minimal surfaces. _Nature_ 1985, _314_ , 604–606
  * Mackay and Terrones 1991 Mackay, A. L.; Terrones, H. Diamond from graphite. _Nature_ 1991, _352_ , 762–762
  * Miller et al. 2016 Miller, D. C.; Terrones, M.; Terrones, H. Mechanical properties of hypothetical graphene foams: Giant Schwarzites. _Carbon_ 2016, _96_ , 1191–1199
  * Jung and Buehler 2018 Jung, G. S.; Buehler, M. J. Multiscale Mechanics of Triply Periodic Minimal Surfaces of Three-Dimensional Graphene Foams. _Nano Letters_ 2018, _18_ , 4845–4853
  * Felix et al. 2019 Felix, L. C.; Woellner, C. F.; Galvao, D. S. Mechanical and energy-absorption properties of schwarzites. _Carbon_ 2019,
  * Gong et al. 2020 Gong, H.; Liu, J.; Xu, K.; Wu, J.; Li, Y. Surface-topology-controlled mechanical characteristics of triply periodic carbon Schwarzite foams. _Soft matter_ 2020, _16_ , 4324–4338
  * Huang et al. 1993 Huang, M.-Z.; Ching, W. Y.; Lenosky, T. Electronic properties of negative-curvature periodic graphitic carbon surfaces. _Physical Review B_ 1993, _47_ , 1593–1606
  * Valencia et al. 2003 Valencia, F.; Romero, A. H.; Hernández, E.; Terrones, M.; Terrones, H. Theoretical characterization of several models of nanoporous carbon. _New journal of physics_ 2003, _5_ , 123–123
  * Zhang et al. 2010 Zhang, F.; Stojkovic, D. S.; Crespi, V. H. Theory of a three-dimensional nanoporous silicon lattice with unsaturated bonding. _Applied physics letters_ 2010, _97_ , 121906
  * Lherbier et al. 2014 Lherbier, A.; Terrones, H.; Charlier, J.-C. Three-dimensional massless Dirac fermions in carbon schwarzites. _Physical Review B_ 2014, _90_ , 125434
  * Weng et al. 2015 Weng, H.; Liang, Y.; Xu, Q.; Yu, R.; Fang, Z.; Dai, X.; Kawazoe, Y. Topological node-line semimetal in three-dimensional graphene networks. _Physical Review B_ 2015, _92_ , 045108
  * Owens et al. 2016 Owens, J.; Daniels, C.; Nicolaï, A.; Terrones, H.; Meunier, V. Structural, energetic, and electronic properties of gyroidal graphene nanostructures. _Carbon_ 2016, _96_ , 998–1007
  * Gao et al. 2017 Gao, P.; Chen, X.; Guo, L.; Wu, Z.; Zhang, E.; Gong, B.; Zhang, Y.; Zhang, S. BN-schwarzite: novel boron nitride spongy crystals. _Physical Chemistry Chemical Physics_ 2017, _19_ , 1167–1173
  * Tromer et al. 2020 Tromer, R. M.; Felix, L. C.; Woellner, C. F.; Galvao, D. S. On the structural stability and optical properties of germanium-based schwarzites: a density functional theory investigation. _Physical Chemistry Chemical Physics_ 2020, _22_ , 16286–16293
  * Pereira et al. 2013 Pereira, L. F. C.; Savić, I.; Donadio, D. Thermal conductivity of one-, two- and three-dimensional sp2 carbon. _New journal of physics_ 2013, _15_ , 105019
  * Zhang et al. 2017 Zhang, Z.; Chen, J.; Li, B. Negative Gaussian curvature induces significant suppression of thermal conduction in carbon crystals. _Nanoscale_ 2017, _9_ , 14208–14214
  * Zhang et al. 2018 Zhang, Z.; Hu, S.; Nakayama, T.; Chen, J.; Li, B. Reducing lattice thermal conductivity in schwarzites via engineering the hybridized phonon modes. _Carbon_ 2018, _139_ , 289–298
  * Feng et al. 2020 Feng, X.; Mu, H.; Xiang, Z.; Cai, Y. Theoretical investigation of negatively curvedD carbon based on density functional theory. _Computational Materials Science_ 2020, _171_ , 109211
  * Sajadi et al. 2018 Sajadi, S. M.; Owuor, P. S.; Schara, S.; Woellner, C. F.; Rodrigues, V.; Vajtai, R.; Lou, J.; Galvão, D. S.; Tiwary, C. S.; Ajayan, P. M. Multiscale geometric design principles applied to 3D printed schwarzites. _Advanced Materials_ 2018, _30_
  * Gaal et al. 2021 Gaal, V.; Felix, L. C.; Woellner, C. F.; Galvao, D. S.; Tiwary, C. S.; d’Ávila, M. A.; Rodrigues, V. Mechanical properties of 3D printed macroscopic models of schwarzites. _Nano Select_ 2021,
  * Damasceno Borges and Galvao 2018 Damasceno Borges, D.; Galvao, D. S. Schwarzites for Natural Gas Storage: A Grand-Canonical Monte Carlo Study. _MRS Advances_ 2018, _3_ , 115–120
  * Collins et al. 2019 Collins, S. P.; Perim, E.; Daff, T. D.; Skaf, M. S.; Galvão, D. S.; Woo, T. K. Idealized Carbon-Based Materials Exhibiting Record Deliverable Capacities for Vehicular Methane Storage. _The Journal of Physical Chemistry C_ 2019, _123_ , 1050–1058
  * Seok et al. 2022 Seok, J. H.; Jun, B.; Lee, C. H.; Lee, S. U. Theoretical investigations into the hydrogen evolution reaction of the carbon schwarzites: From electronics to structure-catalytic activity relationship. _Carbon_ 2022, _190_ , 136–141
  * Townsend et al. 1992 Townsend, S.; Lenosky, T.; Muller, D.; Nichols, C.; Elser, V. Negatively curved graphitic sheet model of amorphous carbon. _Physical Review Letters_ 1992, _69_ , 921–924
  * Barborini et al. 2002 Barborini, E.; Piseri, P.; Milani, P.; Benedek, G.; Ducati, C.; Robertson, J. Negatively curved spongy carbon. _Applied physics letters_ 2002, _81_ , 3359–3361
  * Benedek et al. 2003 Benedek, G.; Vahedi-Tafreshi, H.; Barborini, E.; Piseri, P.; Milani, P.; Ducati, C.; Robertson, J. The structure of negatively curved spongy carbon. _Diamond and related materials_ 2003, _12_ , 768–773
  * Wu et al. 2015 Wu, Y.; Yi, N.; Huang, L.; Zhang, T.; Fang, S.; Chang, H.; Li, N.; Oh, J.; Lee, J. A.; Kozlov, M. et al. Three-dimensionally bonded spongy graphene material with super compressive elasticity and near-zero Poisson’s ratio. _Nature Communications_ 2015, _6_ , 6141
  * Nishihara et al. 2009 Nishihara, H.; Yang, Q.-H.; Hou, P.-X.; Unno, M.; Yamauchi, S.; Saito, R.; Paredes, J. I.; Martínez-Alonso, A.; Tascón, J. M.; Sato, Y. et al. A possible buckybowl-like structure of zeolite templated carbon. _Carbon_ 2009, _47_ , 1220–1230
  * Nishihara and Kyotani 2018 Nishihara, H.; Kyotani, T. Zeolite-templated carbons - three-dimensional microporous graphene frameworks. _Chemical Communications_ 2018, _54_ , 5648–5673
  * Braun et al. 2018 Braun, E.; Lee, Y.; Moosavi, S. M.; Barthel, S.; Mercado, R.; Baburin, I. A.; Proserpio, D. M.; Smit, B. Generating carbon schwarzites via zeolite-templating. _Proceedings of the National Academy of Sciences of the United States of America_ 2018, _115_ , E8116–E8124
  * Boonyoung et al. 2019 Boonyoung, P.; Kasukabe, T.; Hoshikawa, Y.; Berenguer-Murcia, A.; Cazorla-Amorós, D.; Boekfa, B.; Nishihara, H.; Kyotani, T.; Nueangnoraj, K. A Simple ”Nano-Templating” Method Using Zeolite Y Toward the Formation of Carbon Schwarzites. _Frontiers in Materials_ 2019, _6_
  * Ikemoto et al. 2017 Ikemoto, K.; Kobayashi, R.; Sato, S.; Isobe, H. Synthesis and Bowl-in-Bowl Assembly of a Geodesic Phenylene Bowl. _Angewandte Chemie International Edition_ 2017, _56_ , 6511–6514
  * Ikemoto et al. 2018 Ikemoto, K.; Lin, J.; Kobayashi, R.; Sato, S.; Isobe, H. Fluctuating Carbonaceous Networks with a Persistent Molecular Shape: A Saddle-Shaped Geodesic Framework of 1,3,5-Trisubstituted Benzene (Phenine). _Angewandte Chemie International Edition_ 2018, _57_ , 8555–8559
  * Ikemoto et al. 2022 Ikemoto, K.; Akiyoshi, M.; Mio, T.; Nishioka, K.; Sato, S.; Isobe, H. Synthesis of a Negatively Curved Nanocarbon Molecule with an Octagonal Omphalos via Design-of-Experiments Optimizations Supplemented by Machine Learning. _Angewandte Chemie International Edition_ 2022, _61_ , e202204035
  * Qin et al. 2017 Qin, Z.; Jung, G. S.; Kang, M. J.; Buehler, M. J. The mechanics and design of a lightweight three-dimensional graphene assembly. _Science Advances_ 2017, _3_ , e1601536
  * Sajadi et al. 2019 Sajadi, S. M.; Woellner, C. F.; Ramesh, P.; Eichmann, S. L.; Sun, Q.; Boul, P. J.; Thaemlitz, C. J.; Rahman, M. M.; Baughman, R. H.; Galvão, D. S. et al. 3D printed tubulanes as lightweight hypervelocity impact resistant structures. _Small (Germany)_ 2019, _15_ , e1904747
  * Felix et al. 2022 Felix, L.; Ambekar, R. S.; Woellner, C.; Kushwaha, B.; Pal, V.; Tiwary, C. S.; Galvao, D. S. Mechanical Properties of 3D-Printed Pentadiamond. _Journal of physics D: Applied physics_ 2022,
  * Ambekar et al. 2020 Ambekar, R. S.; Oliveira, E. F.; Kushwaha, B.; Pal, V.; Machado, L. D.; Sajadi, S. M.; Baughman, R. H.; Ajayan, P. M.; Roy, A. K.; Galvao, D. S. et al. On the mechanical properties of atomic and 3D printed zeolite-templated carbon nanotube networks. _Additive Manufacturing_ 2020, 101628
  * Sajadi et al. 2021 Sajadi, S. M.; Tiwary, C. S.; Rahmati, A. H.; Eichmann, S. L.; Thaemlitz, C. J.; Salpekar, D.; Puthirath, A. B.; Boul, P. J.; Rahman, M. M.; Meiyazhagan, A. et al. Deformation resilient cement structures using 3D-printed molds. _iScience_ 2021, _24_ , 102174
  * Gupta et al. 2021 Gupta, B.; Ambekar, R. S.; Tromer, R. M.; Ghosal, P. S.; Sinha, R.; Majumder, A.; Kumbhakar, P.; Ajayan, P. M.; Galvao, D. S.; Gupta, A. K. et al. Development of a schwarzite-based moving bed 3D printed water treatment system for nanoplastic remediation. _RSC advances_ 2021, _11_ , 19788–19796
  * Fan et al. 2021 Fan, Q.; Yan, L.; Tripp, M. W.; Krejčí, O.; Dimosthenous, S.; Kachel, S. R.; Chen, M.; Foster, A. S.; Koert, U.; Liljeroth, P. et al. Biphenylene network: A nonbenzenoid carbon allotrope. _Science_ 2021, _372_ , 852–856
  * Rickhaus et al. 2017 Rickhaus, M.; Mayor, M.; Juríček, M. Chirality in curved polyaromatic systems. _Chemical Society Reviews_ 2017, _46_ , 1643–1660
  * Lenosky et al. 1992 Lenosky, T.; Gonze, X.; Teter, M.; Elser, V. Energetics of negatively curved graphitic carbon. _Nature_ 1992, _355_ , 333–335
  * Schwarz 1890 Schwarz, H. A. _Gesammelte Mathematische Abhandlungen_ ; Springer Berlin Heidelberg: Berlin, Heidelberg, 1890
  * Schoen 1970 Schoen, A. H. _Infinite periodic minimal surfaces without self-intersections_ ; Technical, 1970
  * Liu et al. 2013 Liu, M.; Artyukhov, V. I.; Lee, H.; Xu, F.; Yakobson, B. I. Carbyne from first principles: chain of C atoms, a nanorod or a nanorope. _ACS Nano_ 2013, _7_ , 10075–10082
  * Artyukhov et al. 2014 Artyukhov, V. I.; Liu, M.; Yakobson, B. I. Mechanically induced metal-insulator transition in carbyne. _Nano Letters_ 2014, _14_ , 4224–4229
  * La Torre et al. 2015 La Torre, A.; Botello-Mendez, A.; Baaziz, W.; Charlier, J.-C.; Banhart, F. Strain-induced metal-semiconductor transition observed in atomic carbon chains. _Nature Communications_ 2015, _6_ , 6636
  * Jin et al. 2009 Jin, C.; Lan, H.; Peng, L.; Suenaga, K.; Iijima, S. Deriving carbon atomic chains from graphene. _Physical Review Letters_ 2009, _102_ , 205501
  * Chuvilin et al. 2009 Chuvilin, A.; Meyer, J. C.; Algara-Siller, G.; Kaiser, U. From graphene constrictions to single carbon chains. _New journal of physics_ 2009, _11_ , 083019
  * Moras et al. 2011 Moras, G.; Pastewka, L.; Gumbsch, P.; Moseler, M. Formation and oxidation of linear carbon chains and their role in the wear of carbon materials. _Tribology letters_ 2011, _44_ , 355–365
  * Andrade et al. 2015 Andrade, N. F.; Aguiar, A. L.; Kim, Y. A.; Endo, M.; Freire, P. T. C.; Brunetto, G.; Galvão, D. S.; Dresselhaus, M. S.; Souza Filho, A. G. Linear Carbon Chains under High-Pressure Conditions. _The Journal of Physical Chemistry C_ 2015, _119_ , 10669–10676
  * Ashby 2006 Ashby, M. F. The properties of foams and lattices. _Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences_ 2006, _364_ , 15–30
  * Jung et al. 2017 Jung, G. S.; Yeo, J.; Tian, Z.; Qin, Z.; Buehler, M. J. Unusually low and density-insensitive thermal conductivity of three-dimensional gyroid graphene. _Nanoscale_ 2017, _9_ , 13477–13484
  * Wu et al. 2016 Wu, K.; Chen, Z.; Li, X.; Dong, X. Methane storage in nanoporous material at supercritical temperature over a wide range of pressures. _Scientific Reports_ 2016, _6_ , 33461
  * Mercado et al. 2018 Mercado, R.; Fu, R.-S.; Yakutovich, A. V.; Talirz, L.; Haranczyk, M.; Smit, B. In silico design of 2D and 3D covalent organic frameworks for methane storage applications. _Chemistry of Materials_ 2018, _30_ , 5069–5086
  * Damasceno Borges et al. 2017 Damasceno Borges, D.; Maurin, G.; Galvão, D. S. Design of Porous Metal-Organic Frameworks for Adsorption Driven Thermal Batteries. _MRS Advances_ 2017, _2_ , 519–524
  * LaViolette and Benson 2000 LaViolette, R. A.; Benson, M. T. Density functional calculations of hypothetical neutral hollow octahedral molecules with a 48-atom framework: Hydrides and oxides of boron, carbon, nitrogen, aluminum, and silicon. _The Journal of Chemical Physics_ 2000, _112_ , 9269–9275
  * Ni et al. 2012 Ni, Z.; Liu, Q.; Tang, K.; Zheng, J.; Zhou, J.; Qin, R.; Gao, Z.; Yu, D.; Lu, J. Tunable bandgap in silicene and germanene. _Nano Letters_ 2012, _12_ , 113–118
  * O’Hare et al. 2012 O’Hare, A.; Kusmartsev, F. V.; Kugel, K. I. A stable ”flat” form of two-dimensional crystals: could graphene, silicene, germanene be minigap semiconductors? _Nano Letters_ 2012, _12_ , 1045–1052
  * Vogt et al. 2012 Vogt, P.; De Padova, P.; Quaresima, C.; Avila, J.; Frantzeskakis, E.; Asensio, M. C.; Resta, A.; Ealet, B.; Le Lay, G. Silicene: compelling experimental evidence for graphenelike two-dimensional silicon. _Physical Review Letters_ 2012, _108_ , 155501
  * Dávila et al. 2014 Dávila, M. E.; Xian, L.; Cahangirov, S.; Rubio, A.; Le Lay, G. Germanene: a novel two-dimensional germanium allotrope akin to graphene and silicene. _New journal of physics_ 2014, _16_ , 095002
  * Hobey 1965 Hobey, W. D. Vibronic interaction of nearly degenerate states in substituted benzene anions. _The Journal of Chemical Physics_ 1965, _43_ , 2187–2199
  * Bersuker 2013 Bersuker, I. B. Pseudo-Jahn-teller effect–a two-state paradigm in formation, deformation, and transformation of molecular systems and solids. _Chemical Reviews_ 2013, _113_ , 1351–1390
  * Perim et al. 2014 Perim, E.; Paupitz, R.; Botari, T.; Galvao, D. S. One-dimensional silicon and germanium nanostructures with no carbon analogues. _Physical Chemistry Chemical Physics_ 2014, _16_ , 24570–24574
  * Zhu et al. 2022 Zhu, C.; Shoyama, K.; Niyas, M. A.; Würthner, F. Supramolecular Substructure of C60-Embedded Schwarzite. _Journal of the American Chemical Society_ 2022,
  * Oliveira et al. 2022 Oliveira, E. F.; Ambekar, R. S.; Galvao, D. S.; Tiwary, C. S. Schwarzites and schwarzynes based load-bear resistant 3D printed hierarchical structures. _Additive Manufacturing_ 2022, _60_ , 103180
  * Ambekar et al. 2021 Ambekar, R. S.; Mohanty, I.; Kishore, S.; Das, R.; Pal, V.; Kushwaha, B.; Roy, A. K.; Kumar Kar, S.; Tiwary, C. S. Atomic Scale Structure Inspired 3D‐Printed Porous Structures with Tunable Mechanical Response. _Advanced engineering materials_ 2021, _23_ , 2001428
  * Herkal et al. 2023 Herkal, S.; Rahman, M. M.; Nagarajaiah, S.; Harikrishnan, V. V. J.; Ajayan, P. 3D printed metamaterials for damping enhancement and vibration isolation: Schwarzites. _Mechanical systems and signal processing_ 2023, _185_ , 109819
  * Sajadi et al. 2021 Sajadi, S. M.; Vásárhelyi, L.; Mousavi, R.; Rahmati, A. H.; Kónya, Z.; Kukovecz, A.; Arif, T.; Filleter, T.; Vajtai, R.; Boul, P. et al. Damage-tolerant 3D-printed ceramics via conformal coating. _Science Advances_ 2021, _7_
  * Kumbhakar et al. 2021 Kumbhakar, P.; Ambekar, R. S.; Mahapatra, P. L.; Sekhar Tiwary, C. Quantifying instant water cleaning efficiency using zinc oxide decorated complex 3D printed porous architectures. _Journal of hazardous materials_ 2021, _418_ , 126383
  * Hoel et al. 2023 Hoel, C. A.; Peterson, S. K.; Rose, R.; Telfeyan, E.; Boyd, L. A.; Her, J.-H.; Erno, D. J.; Bhushan, C.; Martinez, J. S.; Davis, B. M. et al. Processing vat polymerized triply periodic minimal surface scaffolds of hydroxyapatite. _Advanced engineering materials_ 2023, _25_ , 2200956
  * Hayashi et al. 2023 Hayashi, K.; Kishida, R.; Tsuchiya, A.; Ishikawa, K. Superiority of Triply Periodic Minimal Surface Gyroid Structure to Strut-Based Grid Structure in Both Strength and Bone Regeneration. _ACS Applied Materials & Interfaces_ 2023,
  * Pugliese and Graziosi 2023 Pugliese, R.; Graziosi, S. Biomimetic scaffolds using triply periodic minimal surface-based porous structures for biomedical applications. _SLAS technology_ 2023, _28_ , 165–182
  * Kumbhakar et al. 2022 Kumbhakar, P.; Parui, A.; Ambekar, R. S.; Mukherjee, M.; Siddique, S.; Pugno, N. M.; Singh, A. K.; Tiwary, C. S. Rain energy harvesting using atomically thin gadolinium telluride decorated 3D printed nanogenerator. _Advanced Sustainable Systems_ 2022, _6_ , 2200296
  * Ellebracht et al. 2023 Ellebracht, N. C.; Roy, P.; Moore, T.; Gongora, A. E.; Oyarzun, D. I.; Stolaroff, J. K.; Nguyen, D. T. 3D printed triply periodic minimal surfaces as advanced structured packings for solvent-based CO2 capture. _Energy & Environmental Science_ 2023, _16_ , 1752–1762
  * Bastos et al. 2023 Bastos, L. V.; Ambekar, R. S.; Tiwary, C. S.; Galvao, D. S.; Woellner, C. F. Mechanical Energy Absorption of Architecturally Interlocked Petal-Schwarzites. _arXiv_ 2023,
  * Ambekar et al. 2023 Ambekar, R. S.; Bastos, L. V.; Galvao, D. S.; Tiwary, C. S.; Woellner, C. F. Topological engineered 3D printing of Architecturally Interlocked Petal-Schwarzites. _arXiv_ 2023,