Generating carbon schwarzites via zeolite-templating

Efrem Brauna, Yongjin Leeb,c, Seyed Mohamad Moosavib, Senja Barthelb, Rocio Mercadod, Igor A. Baburine, Davide M. Proserpiof,g, and Berend Smita,b,1

aDepartment of Chemical and Biomolecular Engineering, University of California, Berkeley, CA 94720; bInstitut des Sciences et Ingenierie´ Chimiques (ISIC), Valais, Ecole´ Polytechnique Fed´ erale´ de Lausanne (EPFL), CH-1951 Sion, Switzerland; cSchool of Physical Science and Technology, ShanghaiTech University, Shanghai 201210, China; dDepartment of Chemistry, University of California, Berkeley, CA 94720; eTheoretische Chemie, Technische Universitat¨ Dresden, 01062 Dresden, Germany; fDipartimento di Chimica, Universita` degli Studi di Milano, 20133 Milan, Italy; and gSamara Center for Theoretical Materials Science (SCTMS), Samara State Technical University, Samara 443100, Russia

Edited by Monica Olvera de la Cruz, Northwestern University, Evanston, IL, and approved July 9, 2018 (received for review March 25, 2018)

Zeolite-templated carbons (ZTCs) comprise a relatively recent ZTC can be seen as a schwarzite (24–28). Indeed, the experi- material class synthesized via the chemical vapor deposition of mental properties of ZTCs are exactly those which have been a carbon-containing precursor on a zeolite template, followed predicted for schwarzites, and hence ZTCs are considered as by the removal of the template. We have developed a theoret- promising materials for the same applications (21, 22). As we ical framework to generate a ZTC model from any given zeolite will show, the similarity between ZTCs and schwarzites is strik- structure, which we show can successfully predict the structure ing, and we explore this similarity to establish that the theory of known ZTCs. We use our method to generate a library of of schwarzites/TPMSs is a useful concept to understand ZTCs. ZTCs from all known zeolites, to establish criteria for which In particular, we will focus on the role of the template in deter- zeolites can produce experimentally accessible ZTCs, and to iden- mining whether a stable ZTC will form. At present, the selection tify over 10 ZTCs that have never before been synthesized. We of zeolite templates for this process has been based on trial show that ZTCs partition space into two disjoint labyrinths that and error, and in this work, we provide a rationale for why can be described by a pair of interpenetrating nets. Since such the three commonly used zeolite templates—FAU, EMT, and a pair of nets also describes a triply periodic minimal surface beta [when possible, we use the International Zeolite Associ- (TPMS), our results establish the relationship between ZTCs and ation (IZA) three-letter code to represent a particular zeolite schwarzites—carbon materials with negative Gaussian curvature structure (29)]—have been successful (22, 30, 31). In addition, that resemble TPMSs—linking the research topics and demon- our approach yields suggestions for over 10 additional zeolite strating that schwarzites should no longer be thought of as purely structures that can be used for ZTCs that have not yet been hypothetical materials. synthesized. In Silico Generation of ZTC Structures schwarzite | zeolite-templated carbon | triply periodic minimal surface | template carbonization | microporous carbon Unlike zeolites, ZTCs are not crystalline; instead, they are non- periodic orderings of atoms on periodic surfaces. Consequently, we cannot simply take the crystal structure as a starting point he search for new forms of carbon has not stopped with the Tdiscovery of fullerenes and graphene. Indeed, there are over Significance 500 unique triply periodic hypothetical carbon structures present in the Samara Carbon Allotrope Database, with the vast majority having been proposed in the last decade (1). Many of these hypo- Nanocarbons can be characterized by their curvature—that thetical carbon allotropes take the form of schwarzites, which are is, positively curved fullerenes, zero-curved graphene, and carbon structures that mimic a triply periodic minimal surface negatively curved schwarzites. Schwartzites are fascinating (TPMS) (1–6). The interest in synthesizing these novel materials materials but have not been synthesized yet, although dis- is motivated not only by the scientific beauty of these surfaces ordered materials with local properties similar to schwarzites but also by the predictions that they have unique electronic, (“random schwarzites”) have been isolated. A promising syn- thetic method allows for the interior surfaces of zeolites to magnetic, and optical properties that may make them use- 2 ful for applications such as supercapacitors, battery electrodes, be templated with sp carbon, but theoretical study of these catalysis, gas storage, and separations (7–11). zeolite-templated carbons (ZTCs) has been limited because of The synthesis of these materials has been challenging (10, their noncrystalline structures. In this work, we develop an 12). Amorphous carbons consisting of local regions with neg- improved molecular description of ZTCs, show that they are ative Gaussian curvature have been observed [e.g., Bourgeois equivalent to schwartzites, and thus make the experimental and Bursill (13) and Barborini et al. (14)]. These materials have discovery of schwarzites ex post facto. Our topological char- been noted to be similar to “random schwarzite” models that acterization of ZTCs lends insights into how template choice have local properties similar to schwarzites (15), but they lack allows for the tunability of ordered microporous carbons. the long-range ordering required for characterization of their Author contributions: E.B. and B.S. designed research; E.B., Y.L., S.M.M., S.B., R.M., I.A.B., topologies by the periodic nets that describe TPMSs (16–18). and D.M.P. performed research; and E.B., S.M.M., S.B., D.M.P., and B.S. wrote the paper. A true breakthrough has been the development of the tem- Conflict of interest statement: The authors, the University of California, and EPFL have plate carbonization process, in which a carbon-containing pre- filed a provisional patent application on some of the results contained herein. cursor is introduced into a porous structured template and This article is a PNAS Direct Submission. carbonized, followed by removal of the template (19–23). Vary- This open access article is distributed under Creative Commons Attribution- ing the template structure can bring about a wide variety NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND). of carbon materials, including one-dimensional nanotubes and Data deposition: All files containing atomic coordinates are available for download from nanorods, 2D graphene stacks, and 3D ordered mesoporous or the Open Science Framework at doi.org/10.17605/OSF.IO/ZHDB2 and from the Materials microporous carbons (21, 23). Cloud at doi.org/10.24435/materialscloud:2018.0013/v1. Of interest for this work is when a zeolite is used as a tem- 1 To whom correspondence should be addressed. Email: berend.smit@epfl.ch.y plate, with the resulting liberated carbon material referred to This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. as a zeolite-templated carbon (ZTC) (21, 22). A motivation of 1073/pnas.1805062115/-/DCSupplemental. this work is the discussion in the literature about whether a Published online August 14, 2018.

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FAU tion the by templated that ZTCs far. experi- of so subset synthesized the been (small) with have the comparison on by reported method information our mental of justification a the on in on found the details be enable More can to ZTCs. algorithm templates novel zeolite of other generation to high-throughput it extended so readily inexpensive, computationally be experi- relatively require can is not it does are and it that data carbons is mental approach all this of an when advantage The stops using process rejected MC or sp The accepted criterion. are optimal energy their moves find to Proposed carbons added positions. the addi- allow performs to it moves and MC surface, tional zeolite the on carbons unsaturated 2 oaodsm ftelmttoso h xsigmtos we methods, existing the of limitations the of some avoid To w ute lutaiemtrasaeZCLAadZTC- and ZTC-LTA are materials illustrative further Two ZTCs the studied (27) al. et Parmentier and (31) al. et Kim strtd n omr ufc idn ie r available. are sites binding surface more no and -saturated, sp 2 hbiie abnaos(2 7 1.Accord- 31). 27, (22, atoms carbon -hybridized N etw focus we Next Methods. and Materials mtyproioe hswstknas taken was this -methylpyrrolidone; sp 2 hbiie abnaosnx to next atoms carbon -hybridized IAppendix SI n au oesr opeees eas eeae T for ZTC a generated also We completeness. ensure we to value and ing zeolites, zeolites D IZA IZA current a the all nearly of with for ZTCs set generate test to on a went then from ZTCs made having D ZTCs. the defect-free on for templates 5 limit as Hence, lower experimentally pores. a some dif- as to to seen barriers access kinetic restrict and can larger fusion be practice, can In deposi- precursor exist. possible limitations the diffusion however, 5 smallest no the that of and is cutoff precursor tion carbon The atomic rows). that two assumption defects bottom these of 2, parent none (Fig. a contained from that model formed ZTC ZTCs three-periodic the D connec- Of a sheet rows). with flat two zeolite or top some bridges 2, contain single-atom (Fig. linkages as the tors such or defect, structures, of carbon 2D sort or 1D smaller travel in (35)] can ing directions that 5 three probe about in spherical than channels largest zeolite’s the (D the of diameter through size sphere the as free define periodic triply largest tem- vapor zeolite. to chemical the throughout difficult the diffuse for be to necessary precursor will (CVD) are deposition pores channels large small diameter. since with large plate zeolites insufficiently of single-atom addition, weak pores In of in zeolite presence linkages in the of formed to absence bridges due triply the or remain to dimensions that due three stress ZTCs all to yield subject know will once templates we periodic zeolite ZTC-LTL, all and not ZTC ZTC-LTA of that diversity From wide the possible. IZA demonstrating the topologies 2, from Fig. taken in templates (29) zeolite database using generated we ZTCs ZTCs. the of Accessibility of ZTCs. Experimental possible descriptions other explore we structural section, this reasonable in ZTCs; very known ZTC-generating our that gives shown have method we section, previous the In Discussion and Results (21). nanotubes carbon oxide producing aluminum for that used way been pure same has the templating producing much (34)—in for purifica- methods postsynthetic method tion require synthetic typically more templating alternative fullerenes—which zeolite an carbon that of be suggest arrays might observations of consists These solvation. structure nanotubes. strong or ZTC-LTL compression be our by to caused Similarly, unlikely forces are withstand bridges to enough single-atom as experimen- observed, characteristics tally physical the explain may bridges atom positions). atom carbon occupied partially connecting Right lines Kim from (cyan model zeolite (31) XRD-derived FAU al. the parent and et the atoms), silicon atoms), yellow carbon and (black oxygen (red model atomistic our is Shown 1. 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CHEMISTRY Fig. 2. Library of ZTCs synthesized in silico. Shown is a subset of the ZTC structures (black carbon atoms) and their parent zeolites (red oxygen and yellow silicon atoms). The top two rows of ZTCs were generated from parent zeolites with Df,3p less than 5 A,˚ and they either lack linkages in some dimensions or they contain linkages with defects; we consider these ZTCs either unlikely to be experimentally synthesizable or unlikely to be stable subject to stress. The bottom two rows of ZTCs were generated from parent zeolites with Df,3p greater than 5 A,˚ which we propose as experimentally synthesizable and stable structures. The TPMS each structure resembles is given in parentheses. ZTC-RWY does not resemble a TPMS but rather consists of a body-centered cubic packing of fullerenes.

E8118 | www.pnas.org/cgi/doi/10.1073/pnas.1805062115 Braun et al. Downloaded by guest on September 27, 2021 Downloaded by guest on September 27, 2021 aif hs hroyai n iei tblt rtracom- criteria that stability zeolites kinetic IZA and of thermodynamic list these exhaustive satisfy triply The as accessible frameworks. experimentally periodic be stated to as MD finite-temperature that, framework surface. interrupted smooth formed less an ZTC-IFU, a with was provides earlier, exception zeolite only a the from K; 1,000 as temperatures as at maintained elevated even defects nanoseconds, of parame- several free for cell ZTCs structures the unit their of the all to almost However, changes ters. significant broke exhibited collapsed, defects or containing summarize bonds, ZTCs in the We of simulations many (37). that these found bonds of breaking results carbon–carbon dynamic the of allow formation to field and Empirical force Reactive (AIREBO) Intermolecular Adaptive Bond-Order the for used simulations, (MD) we dynamics which the molecular performing than by less ZTCs or to molecule. close buckminsterfullerene energies isolated an have analysis. defects of structural ZTCs of energy our free these ZTCs supporting of the energies, that Several lower found have We to 3). tend (Fig. cell unit the in where hoy(F) n ecmae hi e-tmeege to energies per-atom their functional formula compared the density using we using diamond minima and energetic (DFT), local theory their to smooth less tures a give around. and wrap zeolite to atoms the carbon of that for space surface atoms void inter- oxygen the these into undercoordinated framework; extend have interrupted -ITV, frameworks and an -IFT, denoting rupted -CLO, dash were the ZTCs D produce with a not with could B we zeolites and which IZA A only polymorphs The structures, (36). intergrown the beta two zeolite retains of and that composed 31), ZTC 30, is a (22, regularity zeolites generate structural few to zeolite’s the shown parent of been one already is has beta that zeolite because B polymorph beta features. these containing connectors, sheet ZTCs S1 flat Table represent and bridges squares single-atom red as while such defects, of free periodic ZTCs triply largest zeolite’s (C parent molecule the (D of diameter function sphere a free as diamond to tive 3. Fig. ru tal. et Braun eepc h Tsfe fdfcsadta r tbeunder stable are that and defects of free ZTCs the expect We the of stability finite-temperature the assessed we Finally, et erlxdtezoiefe he-eidcZCstruc- ZTC three-periodic zeolite-free the relaxed we Next, E e-tmeege fDTrlxdtreproi T oesrela- models ZTC three-periodic DFT-relaxed of energies Per-atom otisteifrainsoni hsfigure. this in shown information the contains steui eleeg and energy cell unit the is SZR 60 STW OSO OFF ERI GME ssona eeec nthe on reference a as shown is ) PUN ITR ITH AEI CHA AFX SAV BSV RHO AFY SOF LTA -EWT f,3p MFI -ITN -SVR .Teeeg fa sltdbuckminsterfullerene isolated an of energy The ). CON BOG IWR IWW UWY MSE ITG JSR MEL TUN UOV E ITT DFO ZTC,relative SBT POS SBS BEB BEA N IRR BEC We . S1 Table Appendix, SI atoms -IFU RWY f,3p ISV IWS = y EMT stenme fatoms of number the is rae hn4.8 than greater xs lecrlsrepresent circles Blue axis. N SAO atoms,ZTC E ZTC FAU − N IAppendix, SI tm,diamond atoms, E diamond for A ˚ -IRY , E C2/c BEB elt ru e e eu ZTC genus net net group Zeolite zeolites parent experimentally their the and of ZTCs symmetry accessible and Topology 1. Table of relation that the matches TPMS, ratio corresponding the their by until defined labyrinths labyrinths the two the varying the gradually of By surface constraints. volumes ZTC trans- volume the geometrical constant of to a area subject performing the by minimizing is numerically formation, TPMS associated 4). its (Fig. and labyrinth labyrinth, void zeolite the as the to as referred contained labyrinth removal other that their the labyrinth with before ZTC atoms the to zeolite refer by the defined We labyrinths two (39). surface the surface of zeolite one that the in atoms on zeolite atoms the leave carbon will the Depositing details). for lattice that (see manner matching a assignment a in unambiguous nets an in 14 labyrinth provides nets ZTC that remaining two labyrinth TPMS the obtained same the the We the with system. on by ZTCs traced 14 focus these be we of could cubic each 2); body-centered associate We a (Fig. ZTCs. of fullerenes consisting of labyrinths, instead two packing bounds structure through that Only the surface (16–18). running a 4) with (Fig. nets form disjoint TPMSs not of the does study two the by ZTC-RWY in into common described is space be as them, partition can that that labyrinths surfaces form the majority find ZTC we a surface. here this of surface—and resembles atoms closely ZTC carbon most (38)—the that which The TPMS surface posteriori. definition a a by on TPMS associate lie known a instead with was must ZTC We it (2–6). a resembled so schwarzite and a TPMS atoms, TPMSs carbon decorating by with created were schwarzites TPMSs. work, with theoretical ZTCs Accessible Experimentally of Comparison of templates as attention structures. closer synthesizable warrant potentially remain- therefore The 31). zeolites 30, ing (22, ZTC-beta namely, and been regularity: ZTC-EMT, structural already zeolite’s ZTC-FAU, have parent to that the encouraging ZTCs retain those is to shown predict It correctly 1). we (Table that SBT see and IWS, SBS, ITT, ISV, SAO, -IRY, RWY, IRR, POS, FAU, EMT, BEC, BEB, BEA, prises E P4 I P4 BEA SAO P4 ISV P4 BEC POS Fd FAU al aetesm pc ru steprn zeolite. parent the as group space same the have table P4 zeolite’s POS the of connected. if instead formed were have cages ‡ the would the and that occurred disrupting surface not † had the off, off describe pinching pack- we the cubic pinched Here body-centered fullerenes. a are of of consisting ing tunnels structure a connecting forming and labyrinth ZTC’s model *The 4,6T585 I4/mmm R P6/mmm IWS P6/mmm SBT P6 ITT IRR P6 SBS P6 -IRY Im EMT RWY h L PShsteP4 the has TPMS net. 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CHEMISTRY To obtain some insights into the types of new structures that can be generated, it is interesting to consider the databases of hypothetical zeolites that have been developed to guide the discovery of new zeolites. In this work, we used Treacy and Foster’s silver hypothetical zeolite database, which contains −1 1,270,921 structures with energies within 48.2 kJ molSi of α- quartz, and Deem’s SLC hypothetical zeolite database, which −1 contains 331,172 structures with energies within 30 kJ molSi −1 −1 of α-quartz, with 48.2 kJ molSi and 30 kJ molSi being each database creator’s respective criterion for a structure’s exper- imental accessibility (49, 50). In principle, we can solve the inverse problem by generating the ZTCs for all these hypothet- Fig. 4. Labyrinth nets of ZTC-FAU. (Left) One of our ZTC-FAU models (black ical zeolites and tabulating the corresponding TPMSs, similarly carbon atoms), and (Right) its parent FAU zeolite (red oxygen and yellow to our approach for the known zeolites, but the large number of silicon atoms), both shown with the ZTC’s dia zeolite labyrinth net (blue) structures makes this impractical. and the ZTC’s dia void labyrinth net (green) (40). To avoid this enormous computational effort, we can signifi- cantly filter down the hypothetical zeolite databases. Namely, we can restrict the search to zeolites with D larger than 5 A˚ and between the surfaces becomes apparent. We demonstrate this f,3p process in Fig. 5 for ZTC-FAU and in SI Appendix, Fig. S2 for ZTC-EMT. We show the TPMSs we associate with each of the experimen- tally accessible ZTCs in Table 1, where it can be seen that many of the ZTCs resemble known TPMSs. We identify ZTC-FAU as resembling the Schwarz D TPMS and ZTC-EMT as resembling the Schoen G-W TPMS (originally referred to as g–g0) (16), with the latter TPMS also resembling ZTC-IRY and ZTC-SBS. ZTC- EMT and ZTC-SBS are illustrated in Fig. 2, where the structural similarities between these two ZTCs are readily apparent, mak- ing evident the reason for their associations with the same TPMS. Nonetheless, it is clear that the two ZTCs are not identical: The two surfaces have different labyrinth volume ratios and unit cell parameter to carbon–carbon bond length ratios, and the car- bon networks are distinct; for example, they are composed of unequal fractions of hexagonal, heptagonal, and octagonal rings with varying degrees of strain. These disparities will lead to dis- tinctive electrical and mechanical properties, giving materials with different performances for select applications. Other TPMSs we have found to resemble ZTCs include Schwarz CLP (41), Schoen H0-T (16), Schoen tD (16, 42), rPD of Koch and Fischer (42), and mDCLP of Fogden and Hyde (43), the last three of which are lower symmetry variants of the Schwarz D TPMS. To the best of our knowledge, TPMSs with the labyrinth nets of ZTC-IWS and ZTC-ISV have not been reported in the literature, which may be because they are of the less well-studied nonbalanced surfaces (a TPMS that divides space into two congruent labyrinths is known as a bal- anced TPMS, whereas a nonbalanced TPMS has noncongruent labyrinths) (44, 45). Since we have made atomistic models of ZTCs from all known noninterrupted zeolites with sufficiently large Df,3p, to the best of our knowledge the TPMSs discussed in this work represent an exhaustive list of all schwarzites that can be made by templating the presently known zeolites with carbon.

Rational Design of Schwarzite Templates. So far, we have shown how to determine which schwarzite will be generated by templat- ing a given zeolite. Equally interesting is the inverse problem: Fig. 5. Numerical minimization of ZTC-FAU’s surface area. The process determining which template will produce a schwarzite resem- begins with the vertices of the surface defined by the carbon atoms and bling a given TPMS. If a TPMS of, say, type CLP is desired, triangulation of the faces (first image), followed by further refinement of Table 1 can be consulted to find that zeolites POS and BEC the triangulation by subdividing the original triangles and minimizing the can be used as templates. However, if the TPMS is not in Table surface area subject to a constant volume constraint (second image). The 1, a novel material would be needed. Fortunately, new zeolites ratio of the two labyrinths’ volumes in the constraint is slowly varied while are continuously being synthesized, and the ability to rationally continuing to minimize the surface area (third and fourth images) until it is in unity (fifth image), equal to that of the Schwarz D TPMS. The resulting design synthetic methods to achieve desired zeolites is begin- surface is visually very similar to the Schwarz D TPMS (final image). The side ning to become available (46–48). Furthermore, one need not of the surface touching the ZTC’s zeolite labyrinth is colored blue, and the restrict oneself to zeolites as multiple new material classes are side touching the ZTC’s void labyrinth is colored green. See also SI Appendix, being developed for use as hard-templates (20, 23). Movie S1.

E8120 | www.pnas.org/cgi/doi/10.1073/pnas.1805062115 Braun et al. Downloaded by guest on September 27, 2021 Downloaded by guest on September 27, 2021 ru tal. et Braun had ZTCs and these 6 S1 (Fig. of TPMS G-W five the resembled that instead found we and hms structures, converged final algorithm ZTC-generating on our which for materials six n n PS ymthn hi epciesaegroups, space respective their matching by TPMSs G and P ing structure. ZTC atomistic full methods the the geometric generating generating inexpensive before by (35) using nets surface labyrinth likely zeolite the accessible determine first to ent P group of space supergroup a has with observed TPMS was G-W as (the TPMS to ZTC-POS zeolite from symmetry in increase D P a the had with 23 476 which found we structures, zeolite D large sufficiently P have (e.g., surface desired the would that material a find it. syn- to template TPMS how to demonstrate how G we suggest here, (51) Schoen not it; could al. thesize the they in et resembling but properties, Qin schwarzites schwarzite interesting example, has as a motivating that examined a shown been As mate- have (2–6). have of depth science they most the and in the 18), role relevant (17, a templates, rials play zeolite min- surfaces using the G these generated of be and since all can whether P, to surfaces H, as minimal the question imal interesting for TPMSs, an present CLP is It and not TPMSs. D are the zeolites generate corresponding can there but that that see we zeolites 1, exist Table examining already By and 41). mDCLP, 18, (deformations tD, (16, TPMS the TPMSs) like rPD G variants Schoen symmetry D, lower the admit Schwarz and which of TPMSs, five the P Only are and (18). these H, TPMS known; CLP, a are have for surfaces they genus targets minimal because possible of minimal called smallest choice the so natural 3, surfaces,” A genus minimal TPMSs. “minimal target the of is set we a for genus, problem zeolite low pro- the sufficiently within Hence, problem 1). of inverse (Table the is solve class. 5 material to TPMS or able known be 4, target all to 3, expect the from of low-genus that TPMSs genus form low the vided to a reg- that likely Thus, have very observe more cells. zeolites be we be unit to Indeed, to small zeolites TPMSs. tend and expect zeolites symmetry would high yet we without with cell, channels materials unit of ular number larger large con- a angle a and requiring prevent length the zeolites bond The of cell, (45). template. straints unit zeolite respectively the the 25, of the in cell in and channels unit more tunnels 21, correspondingly require more 19, which of have 15, formation 9, surfaces 3, genus genus Higher of with TPMSs of TPMSs number Pa known the Pm example, is group which For space genus, contains. its surface is the the mea- topology consider useful handles surface’s A must symmetry. a we their of to formed, sure addition which be in form TPMSs understand to of could To likely topology that group. more one space are only same the structures the be with not zeolites would from TPMS target the so group space the sides.] two is, surface’s surface—that the oriented distinguishes group–subgroup describes that that the a subgroup of the by to symmetry refer described the we (44); be groups can space of note TPMS pair [We 1. balanced Table in expecta- a given an findings that our zeolite, with parent consistent the is a of that or tion group the with space identical of the be group of either lat- space should supergroup The the ZTC TPMS. a that with target expectation associated an the TPMS from as comes group filter space ter same the have which iial,w a n orsodn eltsfrteremain- the for zeolites corresponding find can we Similarly, matching group space a with zeolites seek to is step first The inverse the solve to attempt us let points, these illustrate To and TPMS, a define uniquely not does group space a However, .TeZCrsmln h - PSehbt h same the exhibits TPMS G-W the resembling ZTC The ). ayit esadtu eebe h PS hl one while TPMS, H the resembled thus and nets labyrinth nld h ,CP,CPb (),P,and Pb, C(P)a, C(P)b, C(P), P, the include 3m 2;ti eutdmntae hti sexpedi- is it that demonstrates result this 6m2); f,3p rae hn5.0 than greater f,3p 2frteHTM)ta lomust also that TPMS) H the for 6m2 u fte16mlinhypothetical million 1.6 the of Out . .W vlae h first the evaluated We A. ˚ 2saegop of group, space 6m2 P6 Table Appendix, SI 3 mc hc is which /mmc, iebt,wihi opsdo w negonsrcue plmrhA [polymorph structures intergrown two of composed (53) is Database XRD- and to which Structure beta, the model, (31) Crystal lite Inorganic experimental in al. Mart the their atoms et of from with entry zeolite Kim taken comparison was from the which visual zeolite as BEB, better carbon-loaded taken for the was exceptions known allow of two which All (29); structure FAU, code. database derived were MC IZA the in-house this from an to taken using were structures performed zeolite was generation ZTC guiding in Methods success and Materials more to lead may con- efforts. synthetic which In experimental direct mind, a found. with in made be been pathway eventually have structures will ZTC our symmetry high trast, energies, high low and with can they structures stability, which that by true car- assuming method hypothesized synthesized, the the works be to from past regard that deviation without note schwarzites a we bon causes Finally, 5). also (3, this schwarzites relax- TPMS energetic However, the as quality, surface. of in TPMS than ation rather the quantity in on is exactly difference plac- from by atoms deviate generated carbon ZTCs were of ing schwarzites whereas atoms surface, carbon TPMS the the that are is ZTCs classes these that material a between suggests difference been conceivable work One have incarnate. our schwarzites schwarzites with concept, a Although carbon hypothetical select topology. ordered purely to pore give microporous how a particular TPMSs as obtain a problems about to such template theories for zeolite useful that ZTCs, found into insight have we schwarzites, complicated into self-assemble that structures. SDAs in of of that use zeolites synthesis the of require the synthesis turn the Thus, require (17). might concept TPMSs been high-genus also TPMS inverse– has the of surfactants using sort of described a self-assembly ZTCs The around making material. templates inverse (52), soft crystallize as zeolites act the which sometimes zeolite and SDAs direct ZTCs These frequently between synthesis. that be zeolite. here (SDAs) parent itself made agents its might be structure-directing synthesizing surface might of connection ZTC difficulty Some the genus tempting of large A indication existing symmetry. a an an the that replicating breaking is then simply speculation and by struc- cell genus zeolite unit 11. higher zeolite’s hypothetical be even generating would with envision surface tures can ZTC one corresponding fact, the In of genus the and Pm ie n a n tutrsta iesrae ihrelatively with zeo- hypothetical surfaces Foster’s give 225 and Treacy that lite example, structures For zeo- genus. find hypothetical high of can databases one the lites in while surfaces, low-genus materials template schwarzites. schwarzite. find desired and to a yield how ZTCs would demonstrated that of have topics with we associated research Finally, be the can linking ZTCs how TPMSs, shown models. have atomistic we realistic Furthermore, with to scientists and experimentalists computational known guide from provide to synthesizable serve be should library to this com- ought zeolites; a that present ZTCs to of us allows library and plete ZTCs generate known describes the to of correctly method structures method the silico Our structure. in zeolite an any developed from ZTCs have we work, this In Remarks Concluding problem. inverse the solving of capable indeed and having 6 TPMSs, (Fig. G and a did P structure had the final 2 resemble a and on converged 4 algorithm which ZTC-generating of our groups, space these D with 4 and 23 found f,3p yfcsn ntesrkn iiaiisbtenZC and ZTCs between similarities striking the on focusing By nitrsigosraini htalkonzoie form zeolites known all that is observation interesting An and 3m rae hn5.0 than greater 6 82hsoeo h ags ntcl oue (49), volumes cell unit largest the of one has 1852 I4 ´ ınez-I .Tu,w e htw are we that see we Thus, S1). Table Appendix, SI 1 2 u ftehpteia elt tutrs we structures, zeolite hypothetical the of Out 32. et ta.(4.“E”rfr oplmrhBo zeo- of B polymorph to refers “BEB” (54). al. et nesta ˜ .Fo hs,tefis aeil o which for materials first the these, From A. ˚ PNAS pcu | o.115 vol. and srs | ayit nets labyrinth o 35 no. | E8121

CHEMISTRY Fig. 6. ZTCs synthesized in silico from hypothetical zeolites with the same space groups as the H, P, and G minimal minimal surfaces. Shown are the ZTC structures (black carbon atoms) along with their parent zeolites (red oxygen and yellow silicon atoms). The TPMS each structure resembles is given in parentheses. The top two rows of ZTCs were generated from zeolites with space group P6m2, ZTC-221 2 6 was generated from a zeolite with space group Pm3m, and ZTC-8331018 was generated from a zeolite with space group I4132. Zeolite 221 2 6 was taken from Treacy and Foster’s silver hypothetical zeolite database, and the other zeolites were taken from Deem’s SLC hypothetical zeolite database (49, 50).

is referred to as BEA, but polymorph B lacks an IZA assignment (29)]. The carbon atom and, depending on the coordination of the selected carbon hypothetical zeolite structures were taken from Treacy and Foster’s silver atom, adding either one or two new carbon atoms. The new carbon atoms hypothetical zeolite database and Deem’s SLC hypothetical zeolite database were placed in an empty space in a shell around the selected carbon atom (49, 50). and were bonded to this atom. Additional bonds were added between The generation of a ZTC begins with a seed of four carbon atoms (a cen- the inserted atoms and other neighboring undercoordinated carbon atoms. tral atom bonded to three neighbors) placed in the zeolite’s pore space. Deletion moves were done by first selecting an existing undercoordinated Two types of MC moves were then attempted: moves that insert–delete carbon atom and removing the atom and its one or two bonds; if the carbon atoms and moves that change the carbon bonding network. Inser- deleted atom had two bonds, a new bond was created between the two tion moves were performed by first selecting an existing undercoordinated atoms to which it was formerly bonded.

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A ˚ = .TeZCui el eerepli- were cells unit ZTC The A. ˚ (25|b 01e.Ti oc edwas field force This eV. 0.0081 ∆E A, ˚ θ i | 2 steeeg difference energy the is 0 + hbiie abn and carbon, -hybridized = E 0.5)) 120 h a 0 where , ◦ −1 , k h buckmin- The . f,3p r × = × Γ-centered bv 5 above 5 eV, 20.559 10 π 2 |b × orbitals, −6 i | grid 2 sthe is and , A. ˚ h utpelbrnh ecie ythe by dia described labyrinths multiple the eerhadinvto rgam GatAreet668,MaGic). 666983, Agreement 2020 (Grant Horizon programme Union’s innovation European and the the research under from (ERC) support Council Flagship Research acknowledges Graphene European B.S. the NECT-ICT-604391). under (Contract support I.A.B. for program 1106400. Commission DGE European Grant under the Fellowship thanks Research support Graduate (“MARVEL”) acknowledges NSF R.M. Materials an (SNSF). from Novel Foundation Revolu- Science of Materials’ National Swiss Discovery (NCCR) the of and Research Design in Computational Competence by tion: funding of acknowledges Center S.B. Forschungsge- National 1570). Deutsche the SPP the Program Priority by the (DFG, funding meinschaft acknowledges by University ShanghaiTech S.M.M. of supported support. platform for Fund Facility computing DE-AC02- Startup high-performance Contract Research User the under University and ShanghaiTech Energy Science Com- of the Department thanks of Scientific Y.L. US 05CH11231. the Research Office of Energy Science DOE This of National Office DE-SC0001015. a the Award Center, of under Energy, puting of Sciences resources of Department Energy used helpful US part Basic an research the Technologies, for Science, as by Energy of funded supported Boyd Clean Office Center to was Peter Research Relevant Frontier research Separations and Energy Gas dis- This TPMSs, for helpful topology. Center and for the crystal Falvo, Slater implementation allotropes on Cyril Ben field carbon discussions helpful), Hoehnerbach, force particularly on Markus AIREBO were cussions Kohlmeyer, the Berger Axel Richard in and Plimpton, bugs J. eliminating (Steven for list ing ACKNOWLEDGMENTS. Surface and LAMMPS, CP2K, with used of we nets files the input Evolver. contains and 1, that Table database in ToposPro ZTCs all a ZTCs, DFT-relaxed and laxed Cloud Materials the ( and with D and Schwarz (doi.org/10.17605/OSF.IO/ZHDB2) Framework the (72). website of Brakke’s files Ken input from taken which were for TPMSs G-W (71), Schoen 2.70 version Evolver face Kα Cu the using 56 3.9 were version patterns Mercury XRD Powder with 35). calculated (29, database 1.35 IZA the to in set calculated diameters both radii atomic oxygen and ellei h ayit;freape T-A’ odlbrnhntcnbe can net labyrinth I void a ZTC-SAO’s equally by example, can for traced choices the labyrinth; other properly captures the purpose, that in our nets lie for of well well choice labyrinths valid the a of gives labyrinth geometry method given a our provides trace although to method so symmetry used be nets’ Our (18), can the nets (69). captures different it general, (4,6T585) and In available nets, embeddings. the were of assignment two unambiguous nei- first an when the nomenclature of (TTD) ther Database Types when Topological ToposPro’s nomenclature and (RCSR) Resource Structure (e.g., according available Chemistry names Reticular assigned the were the to nets by resulting replaced were strong The quadrilateral two centroid. a and of quadrilateral’s making centroid, boundary cube’s rings the the forms strong by sum replaced six whose were rings example, replaced cube for a was of above; rings edges described the of up as set vertex the a single labyrinth, bounds a each their by that within edges contained shared was wholly with net rings face a strong when Similarly, several the ring. contain to the to connected to found was replaced connected which formerly was centroid, that were ring’s ring (66)] that the the cycles vertices at labyrinth, smaller located vertex of its single sum within a a contained by not wholly is found face that was cycle to a net [a incident bounds a ring be When strong to contracted. a found contain was was to edge edge the an vertices, simpli- When coordinated were two manners. nets following labyrinth the the both some in and Subsequently, fied In disconnected, added. is surface. were that ZTC net edges labyrinth the missing a cross to leads that construction edges this inter- cases, two deleting into of by net net dual subnets dual the tak- decomposing penetrated the and determining by (66), (67), ToposPro obtained zeolite using were parent tiling labyrinths this the of two tiling ZTCs’ natural the the describe ing that nets two The of three total of chain a a with and thermostated was fs it thermostats. 100 and fs, of 1,000 of constant constant Nos damping damping and A Nos constants time thermostats. A lattice a chained change. three three with the to used with allowed was change, ensemble them mostat to NPT between the allowed angles in constants three ns the lattice 1 three final the a isothermal–isobaric and the only in with ns ensemble 1 additional (NPT) an ensemble, (NVT) canonical the :srcuefie falnonre- all of files structure doi.org/10.24435/materialscloud:2018.0013/v1): oooia nlsswscnutdwt ooPovrin5302(66). 5.3.0.2 version ToposPro with conducted was analysis Topological ehv aetefloigfie cesbewt h pnScience Open the with accessible files following the made have We D The 7) ueia ufc ramnmzto a odce ihteSur- the with conducted was minimization area surface Numerical (70). 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