Crystalline assemblages of identical sub-units packed together and
elastically bent in the form of a torus have been found in the past ten years
in a variety of systems of surprisingly different nature, such as viral
capsids, self-assembled monolayers and carbon nanomaterials. In this Letter we
analyze the structural properties of toroidal crystals and we provide a unified
description based on the elastic theory of defects in curved geometries. We
find ground states characterized by the presence of 5-fold disclinations on the
exterior of the torus and 7-fold disclinations in the interior. The number of
excess disclinations is controlled primarily by the aspect ratio of the torus,
suggesting a novel mechanism for creating toroidal templates with precisely
controlled valency via functionalization of the defect sites.Comment: 4 pages, 4 figure

D. Mumford

D. R. Nelson

E. J. Snijder

J. Polchinski

Luca Giomi

Mark J. Bowick

R. Saito

English

arXiv

Crystalline assemblages of identical sub-units packed together and elastically bent in the form of a torus have been found in the past ten years in a variety of systems of surprisingly different nature, such as viral capsids, self-assembled monolayers and carbon nanomaterials. In this Letter we analyze the structural properties of toroidal crystals and we provide a unified description based on the elastic theory of defects in curved geometries. We find ground states characterized by the presence of 5-fold disclinations on the exterior of the torus and 7-fold disclinations in the interior. The number of excess disclinations is controlled primarily by the aspect ratio of the torus, suggesting a novel mechanism for creating toroidal templates with precisely controlled valency via functionalization of the defect sites

Bowick, Mark

Giomi, Luca

Syracuse University Research Facility and Collaborative Environment

Syracuse University SURFACE Physics College of Arts and Sciences 1-23-2008 Toroidal Crystals Mark Bowick Department of Physics, Syracuse University, Syracuse, NY Luca Giomi Syracuse University Follow this and additional works at: https://surface.syr.edu/phy  Part of the Physics Commons Recommended Citation Bowick, Mark and Giomi, Luca, "Toroidal Crystals" (2008). Physics. 142. https://surface.syr.edu/phy/142 This Article is brought to you for free and open access by the College of Arts and Sciences at SURFACE. It has been accepted for inclusion in Physics by an authorized administrator of SURFACE. For more information, please contact surface@syr.edu. arXiv:0801.3484v1  [cond-mat.soft]  23 Jan 2008Toroidal CrystalsLuca Giomi and Mark J. BowickDepartment of Physics, Syracuse University, Syracuse New York, 13244-1130Crystalline assemblages of identical sub-units packed together and elastically bent in the form ofa torus have been found in the past ten years in a variety of systems of surprisingly different nature,such as viral capsids, self-assembled monolayers and carbon nanomaterials. In this Letter we analyzethe structural properties of toroidal crystals and we provide a unified description based on the elastictheory of defects in curved geometries. We find ground states characterized by the presence of 5-fold disclinations on the exterior of the torus and 7-fold disclinations in the interior. The numberof excess disclinations is controlled primarily by the aspect ratio of the torus, suggesting a novelmechanism for creating toroidal templates with precisely controlled valency via functionalization ofthe defect sites.Toroidal micelles can be formed in the self-assembly ofdumbbell-shaped amphiphilic molecules [1]. Moleculardumbbells dissolved in a selective solvent self-assemblein aggregate structures determined by their amphiphiliccharacter. This process yields coexisting spherical andopen-ended cylindrical micelles which evolve slowly overthe course of a week to more stable toroidal micelles.Toroidal geometries also occur in microbiology in the vi-ral capsid of the coronavirus torovirus [2]. The torovirusis an RNA viral package of maximal diameter between120 and 140 nm and is surrounded, as other coronaviri-dae, by a double wreath/ring of cladding proteins.Carbon nanotori form another fascinating and techno-logically promising class of toroidal crystals [3] with re-markable magnetic and electronic properties. The inter-play between the ballistic motion of the π electrons andthe geometry of the embedding torus leads to a colossalparamagnetic moment in metallic carbon tori [4]. Thehigh magnetic susceptibility, together with large ring ra-dius (up to 0.03 µm for single tubes of 1.4 nm in diameter[5]), implies that the mobility of the electrons in the high-est occupied molecular orbital is several times larger thanthat of other ring-shaped molecules such as benzene [6].A unified theoretical framework to describe the struc-ture of toroidal crystals is provided by the elastic theoryof defects in a curved background [7, 8, 9, 10]. Thisformalism has the advantage of far fewer degrees of free-dom than a direct treatment of the microscopic interac-tions and allows one to explore the origin of the emergentsymmetry observed and expected in toroidal crystals asthe result of the interplay between defects and geometry.Since defective regions are natural places for biologicalactivity and chemical linking, a thorough understandingof the surface topology of crystalline assemblages couldrepresent a significant step toward a first-principle de-sign of entire libraries of nano and mesoscale componentswith precisely determined valency that could serve as thebuilding blocks for mesomolecules or bulk materials viaself-assembly or controlled fabrication.The embedding of an equal number of pentagonal andheptagonal disclination defects in a hexagonal carbonnetwork was first proposed by Dunlap in 1992 as a pos-FIG. 1: (Color online) Isolated defects and scar phases inthe (r, V ) plane. When the number of vertices V increasesthe range of the screening curvature becomes smaller thanone lattice spacing and disclinations appear delocalized in theform of a 5− 7− 5 grain boundary mini-scar.sible way to incorporate positive and negative Gaus-sian curvature into the cylindrical geometry of nan-otubes [11]. In the Dunlap construction the curvature isachieved by the insertion of “knees” in conjunction witheach pentagon-heptagon pair arising from the junction oftubular segments of different chirality. The latter is con-ventionally specified by two integers (N,M) which iden-tify the direction along which a planar triangular latticeis rolled up in the form of a tube [12]. A junction betweenan (N, 0) and an (N,N) tube is obtained, for instance, byplacing a 7−fold disclination along the internal equatorof the torus and a 5−fold disclination along the externalequator [13, 14]. By repeating the 5 − 7 connection pe-riodically it is possible to construct an infinite numberof toroidal lattices with an even number of disclinationspairs and the dihedral symmetry group Dnh (where 2nis the total number of 5− 7 pairs).2Another class of crystalline toroids with dihedral an-tiprismatic symmetry Dnd was initially proposed by Itohat al [15] shortly after Dunlap. Aimed at producing astructure similar to C60 fullerene, Itoh’s original con-struction implied ten disclination pairs and the symmetrygroup D5d. In contrast to Dunlap toroids, the disclina-tions in the antiprismatic torus are not perfectly alignedalong the equator but rather staggered at some angulardistance δψ from the equatorial plane. A general clas-sification scheme for Dnd symmetric tori can be foundin Ref. 16. In this paper we will refer to the latticesthemselves, with 2n disclination pairs, by the symbolsTPn and TAn, with respective symmetry groups Dnhand Dnd.Within the elastic theory of defects on curved surfaces[7, 8, 9, 10] the original interacting particle problem ismapped to a system of interacting disclination defects ina continuum elastic curved background. Disclinations arecharacterized by their topological or disclination charge,qi, representing the departure of a vertex from a prefecttriangular lattice. Thus qi = 6− ci, where ci is the coor-dination number of the ith vertex. A classic theorem ofEuler requires the total disclination charge of any trian-gulation of a two-dimensional manifold M to be equal to6χM , where χM is the Euler characteristic of M . In thecase of the torus χM = 0, and thus disclinations must ap-pear in pairs of opposite disclination charge (i.e. 5−foldand 7−fold vertices with qi = 1 and −1 respectively) inorder to ensure disclination charge neutrality. The totalfree energy of a toroidal crystal with N disclinations canbe expressed asFel =12Y∫Md2xΓ2(x) + ǫcN∑i=1q2i + F0 , (1)where Y is the 2D-Young modulus and Γ(x) is the so-lution of the following Poisson problem with periodicboundary conditions:∆Γ(x) = Y ρ(x) , (2)where ∆ is the Laplace-Beltrami operator on the torusand ρ(x) is the total topological charge densityρ(x) =π3N∑k=1qkδ(x,xk)−K(x) (3)of N disclinations located at the sites xk plus a screen-ing contribution due to the Gaussian curvature K(x) ofthe embedding manifold. The first term in Eq. (1) rep-resents the long-range elastic distortion due to defectsand curvature. Its form resembles the potential energyof a system of electrical charges. In this analogy theGaussian curvature K(x) plays the role of a non-uniformbackground charge distribution while defects appear aspositively and negatively charged point-like particles. AsFIG. 2: (Color online) Elastic energy of a 5 − 7 disclinationdipole constrained to lie on the same meridian, as a functionof the angular separation. In the inset, illustration of a circu-lar torus of radii R1 > R2. Regions of positive and negativeGaussian curvature have been shaded in red and blue respec-tively. A standard parametrization of the torus is obtainedby considering the angles ψ and φ.a result, disclinations arrange themselves so to approxi-mately match the Gaussian curvature. The second termin Eq. (1) is the defect core-energy representing the en-ergy required to create a single disclination defect. Thisquantity is related to the short-distance cut-off of theelastic theory and is proportional to the square of thetopological charge times a constant ǫc. Finally F0 is anoffset corresponding to the free energy of a flat defect-freemonolayer.Using standard analysis the function Γ(x) can be writ-ten in the formΓ(x) =π3N∑k=1qkΓd(x,xk)− Γs(x) , (4)where Γs(x) represents the stress field due to the curva-ture of the torus and is given by:Γs(x)Y= log[r +√r2 − 12(r + cosψ)]+r −√r2 − 1r, (5)where r = R1/R2 is the aspect ratio of the torus. Thefunction Γd(x,xk) is the stress field at the point x arisingfrom the elastic distortion due to a defect at xk and isgiven byΓd(x,xk)Y=κ16π2(ψk − 2κξk)2− 14π2κ(φ− φk)2+14π2rlog(r + cosψk)− κ4π2Re{Li2(αeiψk)}+12πlog∣∣∣∣ϑ1(z − zkκ∣∣∣∣2iκ)∣∣∣∣ , (6)where ξ = κ tan−1(ω tanψ/2) is a conformal angle aris-ing from the mapping of the torus to the periodic plane,3FIG. 3: (Color online) Phase diagram for Tp configurationsin the plane (r, ǫc/AY ). For r ∈ [3.68, 10.12] and ǫc ∼ 0 thestructure is given by a T10 configuration with symmetry groupD5h.z = ξ + iφ, and α, κ and ω are dimensionless constantsdepending on the aspect ratio r:α =√r2 − 1− r , κ = 2√r2 − 1 , ω =√r − 1r + 1.Finally ϑ1 is a Jacobian theta function and reflects thedouble periodicity of the torus [17].To analyze the elastic energy Eq. (1) we first considerthe simplest case of two opposite sign disclinations lyingon the same meridian of a torus of area A = 4π2R1R2.The elastic free energy of this system is shown in Fig.2 as a function of the angular separation between thetwo disclinations. The minimum is obtained for the pos-itive disclination along the external equator of the torus(where K(x) is maximally positive) and the negativedisclination along the internal equator (where K(x) ismaximally negative). The picture emerging from thissimple case suggests a procedure to systematically con-struct optimal defect patterns for an arbitrary number ofdisclination pairs, by placing the same number of equallyspaced +1 and −1 disclinations along the internal andexternal equators respectively. We name this configura-tion with the symbol Tp, where p stands for the totalnumber of disclination dipoles.A comparison between the free energy of different Tpconfigurations as a function of aspect ratio and core en-ergy is summarized in the phase diagram of Fig. 3. Westress here that only Tp graphs with p even have an em-bedding on the torus corresponding to lattices of the TPp2class. Nevertheless a comparison with p−odd configura-tions can provide additional information regarding thestability of p−even lattices. The defect core energy en-tering Eq. (1) is equal to 2pǫc. Although dependenton the microscopic details of the system, the constantǫc/AY ∼ 10−5 for a crystal of roughly 103 atoms. Inthe range r ∈ [3.68, 10.12] and ǫc ∼ 0, the structure isdominated by the T10 phase corresponding to a doublering of +1 and −1 disclinations distributed along the ex-ternal and internal equators of the torus at the verticesof a regular decagon. The corresponding lattice has D5hsymmetry group.That this structure might represent a stable configura-tion for polygonal carbon toroids has been conjectured bythe authors of Ref. 13, based on the argument that the36◦ angle arising from the insertion of ten pentagonal-heptagonal pairs into the lattice would optimize the ge-ometry of a nanotorus consistently with the structureof the sp2 bonds of the carbon network (unlike the 30◦angle of the 6−fold symmetric configuration originallyproposed by Dunlap). In later molecular dynamics simu-lations, Han [18] found that a 5−fold symmetric lattice,such as the one obtained from a (9,0)/(5,5) junction, is infact stable for toroids with aspect ratio less then r ∼ 10.The stability, in this case, results from the strain energyper atom being smaller than the binding energy of carbonatoms. We have shown here, from first principles, that a5−fold symmetric lattice indeed constitutes a minimumof the elastic energy for a broad range of aspect ratiosand defect core energies.For small aspect ratios the 5−fold symmetric configu-ration becomes unstable and is replaced by the T9 phase.This configuration, however, does not correspond to apossible triangulation of the torus. In this regime, weexpect the minimal energy structure to consist of tennon-coplanar disclination pairs as in the antiprismaticFIG. 4: (Color online) Phase diagram of a 5−fold symmet-ric lattice in the plane (r, δψ). For small δψ and r in therange [3.3, 7.5] the prismatic TP5 configuration is energeti-cally favored. For r < 3.3, the system undergoes a structuraltransition to the antiprismatic phase TA5.4TA5 lattice. The latter can be analyzed by introducinga further degree of freedom, δψ, representing the angu-lar displacement of defects from the equatorial plane. Acomparison between the TP5 and TA5 configurations isshown in Fig. 4 for different values of δψ. For smallδψ and r ∈ [3.3, 7.5] the prismatic TP5 configuration isenergetically favored. For r < 3.3, however, the latticeundergoes a structural transition to the TA5 phase. Forr > 7.5 the prismatic symmetry of the TP5 configurationbreaks down again. In this regime, however, the elasticenergy of both configurations rapidly becomes higher be-cause of the lower curvature and defects disappear.In the regime of large particle numbers, the amountof curvature required to screen the stress field of an iso-lated disclination in units of lattice spacing becomes toolarge and disclinations are unstable to grain boundary“scars” consisting of a linear array of tightly bound 5−7pairs radiating from an unpaired disclination [8, 10]. Ina manifold with variable Gaussian curvature this processyields the coexistence of isolated disclinations (in regionsof large curvature) and scars. In the case of the torusthe Gaussian curvature in the interior is always largerin magnitude than that in the exterior for any R1 > R2and thus we may expect a regime in which the negativeinternal curvature is still large enough to support theexistence of isolated 7−fold disclinations, while on theexterior of the torus disclinations are delocalized in theform of positively charged grain boundary scars.To check this hypothesis we compare the energy of theTP5 lattice previously described with that of “scarred”configurations obtained by decorating the original toroidin such a way that each pentagonal disclination on theexternal equator is replaced by a 5 − 7 − 5 scar. Theresult of this comparison is summarized in the phase di-agram of Fig. 1 in terms of r and the number of verticesof the triangular lattice V (the corresponding hexagonallattice has twice the number of vertices, i.e. Vhex = 2V ).V can be derived from the angular separation of neigh-boring disclinations in the same scar by approximatingV ≈ A/AV , with AV =√32a2 the area of a hexagonalVoronoi cell and a the lattice spacing. When the as-pect ratio is increased from 1 to 6.8 the range of thecurvature screening becomes shorter and the number ofatoms required to destroy the stability of the TP5 latticedecreases. For r > 6.8, however, the geodesic distancebetween the two equators of the torus becomes too smalland the repulsion between like-sign defects takes over.Thus the trend is inverted.On the material science side, defects can be functional-ized to provide binding sites for ligands. The number ofexcess disclinations will then determine the valency of thesurface, which itself can serve as a building block for newmolecules and bulk materials [19]. The universality ofthe predictions presented here means that one has precisecontrol of the valency by tuning the aspect ratio of thetorus. This could lead to a very efficient scheme for cre-ating a variety of basic surfaces with well-defined valencywhich subsequently self-assemble into novel moleculesand bulk structures.We acknowledge David Nelson and Alex Travesset forstimulating discussions. This work was supported by theNSF through Grant No. DMR-0219292 (ITR) and byan allocation through the TeraGrid Advanced SupportProgram.[1] J. K. Kim, E. Lee, Z. Huang and M. Lee, J. Am. Chem.Soc. 128, 14022 (2006).[2] E. J. Snijder and M. C. Horzinek, in The Coronaviridae,edited by S. G. Sidell, 219 (Plenum Press, New York,1995)[3] J. Liu, H. Dai, J. H. Hafner, D. T. Colbert, R. E. Smalley,S. J. Tans and C. Dekker, Nature 385, 780 (1997).[4] L. Liu, G.Y. Guo, C. S. Jayanthi and S.Y. Wu, Phys.Rev. Lett. 88, 217206 (2002).[5] R. Martel, H. R. Shea and P. Avouris, Nature 398, 299(1999).[6] R. C. Haddon, Nature 388, 31 (1997).[7] A. Perez-Garrido, M. J. W. Dodgson, and M. A. Moore,Phys. Rev. B 56, 3640 (1997).[8] M. J. Bowick, D. R. Nelson and A. Travesset, Phys. Rev.B 62, 8738 (2000).[9] V. Vitelli, J. B. Lucks, and D. R. Nelson, Proc. Natl.Acad. Sci. USA 103, 12323 (2006).[10] L. Giomi and M. J. Bowick, Phys. Rev. B 76, 054106(2007).;[11] B. I. Dunlap, Phys. Rev. B 46, 1933 (1992); Phys. Rev.B 49, 5643 (1994); Phys. Rev. B 50, 8134 (1994).[12] R. Saito, G. Dresselhaus and M. S. Dresselhaus, PhysicalProperties of Carbon Nanotubes (Imperial College Press,London, 1998).[13] P. Lambin, A. Fonseca, J. P. Vigneron, J. B. Nagy andA. A. Lucas, Chem. Phys. Lett. 245, 85 (1995); A. Fon-seca, J. Hernadi, J. B. Nagy, P. Lambin and A. A. Lucas,Synth. Met. 77, 249 (1996); V. Meunier, P. Lambin andA. A. Lucas, Phys. Rev. B 57, 14886 (1998).[14] Z. Yao, H. W. Ch. Postma, L. Balents and C. Dekker,Nature 402, 273 (1999).[15] S. Itoh, S. Ihara and J. I. Kitakami, Phys. Rev. B 47,1703 (1993); S. Ihara, S. Itoh and J. I. Kitakami, Phys.Rev. B 47, 12908 (1993); S. Itoh and S. Ihara, Phys. Rev.B 48, 8323 (1993).[16] J. E. Avron and J. Berger, Phys. Rev. A 51, 1146 (1995),J. Chem. Soc. Faraday. Trans. 91, 4037 (1995).[17] D. Mumford, Tata Lectures on Theta I (Birkhau¨ser,Boston, 1983); J. Polchinski, String Theory, Vol. I (Cam-bridge University Press, Cambridge, 1998).[18] J. Han, Tech. Rep. NAS (1997); Chem. Phys. Lett. 282,187 (1998).[19] D. R. Nelson, Nano Lett. 2, 1125 (2002).

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