We attack generalized Thomson problems with a continuum formalism which
exploits a universal long range interaction between defects depending on the
Young modulus of the underlying lattice. Our predictions for the ground state
energy agree with simulations of long range power law interactions of the form
1/r^{gamma} (0 < gamma < 2) to four significant digits. The regime of grain
boundaries is studied in the context of tilted crystalline order and the
generality of our approach is illustrated with new results for square tilings
on the sphere.Comment: 4 pages, 5 eps figures Fig. 2 revised, improved Fig. 3, reference
  typo fixe

A. Cacciuto

A. Travesset

D. R. Nelson

D. L. D. Caspar

E. B. Saff

E. J. Davis

H. G. Flegg

H. W. Kroto

J. J. Thomson

J. P. Hirth

L. Bonsall

M. Bowick

P. Leiderer

P. M. Chaikin

R. Cotterill

T. Erber

T. C. Lubensky

English

arXiv

We attack generalized Thomson problems with a continuum formalism which exploits a universal long range interaction between defects depending on the Young modulus of the underlying lattice. Our predictions for the ground state energy agree with simulations of long range power law interactions of the form 1/r^{gamma} (0 \u3c gamma \u3c 2) to four significant digits. The regime of grain boundaries is studied in the context of tilted crystalline order and the generality of our approach is illustrated with new results for square tilings on the sphere

Bowick, Mark

Cacciuto, Angelo

Nelson, David R.

Travesset, A.

Syracuse University Research Facility and Collaborative Environment

Syracuse University SURFACE Physics College of Arts and Sciences 10-19-2002 Crystalline Order on a Sphere and the Generalized Thomson Problem Mark Bowick Department of Physics, Syracuse University, Syracuse, NY Angelo Cacciuto Syracuse University David R. Nelson Harvard University A. Travesset University of Illinois at Urbana Follow this and additional works at: https://surface.syr.edu/phy  Part of the Physics Commons Recommended Citation Bowick, Mark; Cacciuto, Angelo; Nelson, David R.; and Travesset, A., "Crystalline Order on a Sphere and the Generalized Thomson Problem" (2002). Physics. 156. https://surface.syr.edu/phy/156 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:cond-mat/0206144v3  [cond-mat.soft]  19 Oct 2002Crystalline Order on a Sphere and the Generalized Thomson ProblemM. Bowick,1 A. Cacciuto,1 D. R. Nelson,2 and A. Travesset31 Physics Department, Syracuse University, Syracuse NY 13244-1130, USA2 Lyman Laboratory of Physics, Harvard University, Cambridge MA 02138, USA and3 Loomis Laboratory, University of Illinois at Urbana, Urbana IL 61801, USAWe attack generalized Thomson problems with a continuum formalism which exploits a universallong range interaction between defects depending on the Young modulus of the underlying lattice.Our predictions for the ground state energy agree with simulations of long range power law inter-actions of the form 1/rγ (0 < γ < 2) to four significant digits. The regime of grain boundaries isstudied in the context of tilted crystalline order and the generality of our approach is illustratedwith new results for square tilings on the sphere.PACS numbers: PACS numbers: 68.35.Gy, 62.60.Dc, 61.72.Lk. 61.30.JfThe Thomson problem of constructing the groundstate of (classical) electrons interacting with a repul-sive Coulomb potential on a 2-sphere [1] is almost onehundred years old [2] and has many important physicalrealizations. These include multi-electron bubbles [3],which may be studied by capillary wave excitations, orthe surface of liquid metal drops confined in Paul traps[4]. Although the original Thomson problem refers tothe ground state of spherical shells of electrons, one canalso ask for crystalline ground states of particles interact-ing with other potentials. Such a generalized Thomsonproblem arises, for example, in determining the arrange-ments of the protein subunits which comprise the shellsof spherical viruses [5, 6]. Here, the “particles” are clus-ters of protein subunits arranged on a shell. Other re-alizations include regular arrangements of colloidal par-ticles in colloidosomes [7] proposed for encapsulation ofactive ingredients such as drugs, nutrients or living cells[8] and fullerene patterns of carbon atoms on spheres [9]and other geometries [10]. An example with long range(logarithmic) interactions is provided by the Abrikosovlattice of vortices which would form at low temperaturesin a superconducting metal shell with a large monopoleat the center [11]. In practice, the “monopole” could beapproximated by the tip of a long thin solenoid.Extensive numerical studies of the Thomson problemshow that the ground state for a small number of par-ticles, typically M ≤ 150, consists of twelve positivedisclinations (the minimum number compatible with Eu-ler’s theorem) located at the vertices of an icosahedron[12, 13]. Recent results have shown that for systems assmall as 500 particles, however, configurations with ad-ditional topological defects [14, 15] have lower energiesthan icosahedral ones.These remarkable results for the Thomson problemraise a number of important questions, such as the mech-anism behind the proliferation of defects, the nature ofthese unusual low-energy states, the universality with re-spect to the underlying particle potential and the gener-alization to more complex situations.A formalism suitable to address all these questions hasbeen proposed recently[16]. Disclinations are consideredthe fundamental degrees of freedom, interacting accord-ing to the energy [16]H = E0 +Y2∫ ∫dσ(x)dσ(y)[(s(x) −K(x))1∆2(s(y) −K(y))],(1)where the integration is over a fixed surface with areaelement dσ(x) and metric gij , K is the Gaussian curva-ture, Y is the Young modulus in flat space and s(x) =∑Ni=1π3qiδ(x, xi) is the disclination density(δ(x, xi) =δ(x − xi)/√det(gij)). Defects like dislocations or grainboundaries are built from these elementary disclinations.E0 is the energy corresponding to a perfect defect-freecrystal with no Gaussian curvature; E0 would be theground state energy for a 2D Wigner crystal of electronsin the plane. Eq.(1) restricted to a 2-sphere givesH = E0 +πY36R2N∑i=1N∑j=1qiqjχ(θi, ψi; θj , ψj)+ NEcore , (2)where R is the radius of the sphere and χ is a function ofthe geodesic distance βij between defects with polar co-ordinates (θi, ψi ; θj , ψj)[16]; χ(β) = 1 +∫1−cosβ20dz lnz1−z .Here 5- and 7-fold defects correspond to qi = +1 and −1respectively. In this letter we show that the continuumformalism embodied in Eq.(2) implies: (a) flat space re-sults for elastic constants can be bootstrapped into veryaccurate quantitative calculations for generalized Thom-son problems, thus providing a stringent test of the va-lidity of this approach; (b) new results for finite lengthgrain boundaries, consisting of dislocations with variablespacing, in the context of the 2π disclinations appear-ing in tilted liquid crystal phases[17]; and (c) sufficientpower and generality to determine the ground state forthe 8 minimal disclinations arising in square tilings of asphere.2γ a1(γ) (n, n) (n, 0)1.5 1.51473 1.51454(2) 1.51445(2)1.25 1.22617 1.22599(7) 1.22589(7)1.0 1.10494 1.10482(3) 1.10464(3)0.75 1.04940 1.04921(6) 1.04910(6)0.5 1.02392 1.02390(4) 1.02372(4)TABLE I: Analytical predictions (first column) for a largenumber of particles and values extrapolated from numeri-cal simulations (second and third columns) of the coefficienta1(γ), as defined in Eq.(4) for (n, n) and (n, 0) icosadeltahe-dral lattices. Similar accuracy holds for other values of γ.The Young modulus appearing in Eq.(2) and the en-ergy E0 may be computed in flat space via the Ewaldmethod [18, 19]. The result for M particles with longrange pairwise interactions given by e2/rγ (0 < γ < 2) is[20]Y = 4η(γ)e2A1+γ/2C,E0Me2= θ(γ)(4πAC)γ/2+πACRγ−2ρ(γ) , (3)where η, θ and ρ are potential-dependent coefficientswhose numerical values will be reported elsewhere [20]and AC is the area per particle. For M particles crys-tallizing on the sphere, AC = 4πR2/M , and combiningEqs.(2) and (3) gives a large M expansion for the groundstate energy,EG =e22Rγ[a0(γ)M2 − a1(γ)M1+γ2 + a2(γ)Mγ2 + · · ·],(4)where a0(γ) = 21−γ/(2 − γ) and the subleading coef-ficients ai(γ) depend explicitly on the potential and onthe positions and number of disclinations. The first (non-extensive) term is proportional to M2 and is usually can-celed by a uniform background charge for Wigner crystalsof electrons. The coefficient a1 is a universal function ofthe positions of the defects, up to a potential-dependentconstant. Using the results in [16], theoretical predic-tions for large M for icosadeltahedral lattices [5] of type(n, 0) and (n, n) are given in Table I for five values of γ.These predictions may now be compared with directminimizations of particles on the sphere in icosadeltahe-dral configurations by fitting the results to Eq.(4). InFig.1 we plot ε(M) versus 1/M for (n, 0) and (n, n)icosadeltahedral configurations for γ = 1.5 and γ = 0.5,whereε(M) =[2RγEG/e2 − a0(γ)M2]M1+γ/2. (5)The coefficient a1(γ) is determined by the intercept inthe M → ∞ limit (M = 10n2 + 2 for (n, 0) lattices andM = 30n2 + 2 for (n, n) lattices [5]).−1.5148−1.5146−1.5144−1.5142−1.5140−1.5138−1.5136−1.5134(n,0)(n,n)0 0.001 0.002 0.003 0.0041/M−1.0239−1.0237−1.0235−1.0233−1.0231−1.0229(n,0)(n,n)ε(Μ)γ=1.5γ=0.5ε(Μ)FIG. 1: Numerical estimate of ε(M) as a function of 1/M for(n, 0) and (n, n) icosadeltahedral lattices with γ = (1.5, 0.5).FIG. 2: Strain energy distribution (red/high, blue/low) for a(10,0) and a (6,6) configuration.3FIG. 3: A 10-arm grain boundary array emerging from a +6defect (purple square) at the north pole. Charge +1 disclina-tions are red and charge −1 disclinations are yellow.The continuum elastic interaction between disclina-tions in Eq.(2) is essential to obtain the correct limitingbehavior since it reflects contributions both from the en-ergy per particle in flat space as well as the energies of12 isolated disclinations. The defect core energy termin Eq.(2) contributes to the leading correction a2(γ).We find agreement to four significant figures for (n, 0)icosadeltahedral lattices and to five significant figures for(n, n) lattices. The small residual difference in energyfor (n, 0) and (n, n) configurations may be understoodfrom the differing strain energies shown in Fig.2. Thissmall discrepancy may be attributable to a line tensionassociated with ridges of minimum or maximum strainconnecting disclinations [20].We now turn to the study of grain boundaries on asphere using the model described by Eq.(2). We illus-trate the method [16] by considering just two 2π defects(appropriate to crystals of tilted molecules) with suitableboundary conditions [17]. To approximate the 2π discli-nation of tilted molecules in a hemispherical crystal [17]we replace the icosahedral configuration of twelve discli-nations with two clusters of six 2π/6 defects at the northand south poles. For simplicity, we use isotropic elastictheory and neglect nonuniversal details near the core ofthe +2π disclination. Upon adding just one dislocationof Burgers vector b (i.e. a +/ − 2π/6 disclination pairseparated by distance b), the minimum energy in Eq.(2)is achieved by a polar angle θ0(b) with ~b perpendicular tothe geodesic joining the north and south poles. For smallnumbers of dislocations, the minimum energy configura-tion consists of two polar rings of dislocations locatedat angles θ0(b) and π − θ0(b) relative to the north pole.The dislocations eventually organize into grain bound-aries centered on θ0(b), as shown in Fig.3. Remarkably,no other minima were found.Because the global minimization just described be-comes computationally demanding for more than thirtydefects, further minimizations focused on a reduced pa-rameter space specified by the orientations and distances00.511.5θm=5122 defectsm=5Grain Boundary typem=1062 defects122 defectsFIG. 4: Defect positions obtained from minimization (up tri-angle) and from Eq.(7) (diamonds).of the grain boundaries from the two +6 defects. Fol-lowing [16], the system dynamically chooses the averagelattice spacing a ≡ b that best accommodates the arraystructure by extremizing the energy of Eq.(2).For reasonable parameter values in our continuum de-scription, both two antipodal +6 disclinations and theicosadeltahedral configurations are indeed unstable to theformation of grain boundaries for sufficiently large sphereradius[14, 15, 16, 23]. In a flat monolayer, the spacingbetween dislocations in m-grain boundary arms radiatingfrom a disclination of charge s is given by [21]l =b2 sin(s2m) , (6)where b is the Burgers vector charge. The disclinationcharge is s = 2π6p (p = 1 or 6, corresponding to theThomson problem or tilted molecules, respectively). Togeneralize this result for symmetrical grain boundarieson a sphere, consider the Burgers circuit formed by anisosceles spherical triangle with apex angle φgb = s/mat a disclination at the north pole and centered on oneof the m-grain boundary arms [22]. If the altitude ofthis triangle is h, the net Burgers vector B defined bythis circuit is given by the geodesic distance spanningthe base of this triangle. A straightforward exercise inspherical trigonometry leads tocos(BR)=cot2(h/R) cos2(φgb/2) + cos(φgb)1 + cot2(h/R) cos2(φgb/2). (7)Writing B ≈ b∫ h0dh′/l(h′), where l(h) is a (variable)dislocation spacing [16], we can invert this formula andthus generalize Eq.(6) to the sphere.Results comparing our minimization with Eq.(7) areshown in Fig.4. Both approaches predict the same num-ber of dislocations within a grain and dislocation spac-ings which increase with θ [16]. The small discrepanciesin the positions of the dislocations are presumably dueto interactions between dislocations in different arms.The physics associated with Eq.(2) is remarkably gen-eral. We sketch here how the results above may be4adapted to a sphere tiled with a square lattice. A planarsquare lattice is described by three elastic constants (asopposed to the two Lamé coefficients in the triangularcase), leading to an energyH =λαβ,µν2∫d2xuαβuµν , (8)where the independent elastic constants are λ11,11, λ11,22and λ12,12. The same derivation as that leading to Eq.(1)now leads toH =12∫dσ(x)dσ(y)(K(x)−s(x))G(x, y)(K(y)−s(y)),(9)where G(x, y) = ( 1Y ∆2 + 2ǫ∇21∇22)−1, with ∇i the gradi-ent in direction i = 1 or 2, as defined by the local squarelattice. The fundamental defects are now ±π2disclina-tions. The elastic constants areY =λ211,11 − λ211,22λ11,11,ǫ = −λ11,22 + λ11,11λ211,11 − λ211,22+12λ12,12. (10)The interaction energy for disclinations in a square lat-tice becomes equivalent to Eq.(1) only in the limit ǫ = 0.Although details such as the critical value of R/a for theonset of grain boundaries will differ, we expect that thephysics remains essentially the same. For small numbersof particles the ground state will consist of 8 q = +1disclinations. In the isotropic case (ǫ = 0) the groundstate is a distorted cube, with one face twisted by 45◦[20],similar to tetratic liquid crystal ground states on thesphere [24].We expect similar results for geometries other than thesphere. For isotropic crystals on a torus with the right as-pect ratio, for example, one might expect 12 5-fold discli-nations on the outer wall (where the Gaussian curvatureis positive) and twelve compensating 7-fold disclinationson the inner wall (where the Gaussian curvature is neg-ative) to play the role of the icosahedral configurationson the sphere. As more particles are placed on the torus,we expect grain boundaries to emerge from these discli-nations.There are some important issues left for a future pub-lication [20], such as a more detailed derivation of theasymptotic expansion Eq.(4) and a reliable determina-tion of the optimal number of arms within a grain bound-ary. We hope the calculations presented here will con-vince the reader of the usefulness of our approach.It is a pleasure to thank Alar Toomre for sharing withus his numerical work on the original Thomson prob-lem. The work of M.B. and A.C. has been supported bythe U.S. Department of Energy (DOE) under ContractNo. DE FG02 85ER40237. The research of D.R.N wassupported by the National Science Foundation throughGrant No. DMR97-14725 and through the Harvard Ma-terial Research Science and Engineering Laboratory viaGrant No. DMR98-09363. The work of A.T. has beensupported by the Materials Computation Center andgrants NSF-DMR 99-76550 and NSF-DMR 00-72783.[1] J. J. Thomson, Philos. Mag. 7, 237 (1904).[2] E.B. Saff and A.B.J. Kuijlaars, Math. Intelligencer 19, 5(1997).[3] P. Leiderer, Z. Phys. B98, 303 (1993).[4] E.J. Davis, Aerosol Sci. Techn. 26, 212 (1997).[5] D.L.D. Caspar and A. Klug, Cold Spring Harbor Symp.Quant. Biol. 27, 1 (1962).[6] C.J. Marzec and L.A. Day, Biophys. Jour. 65, 2559(1993).[7] M. Nikolaides, Thesis, Physics Department, TU Munich,2001.[8] A.D. Dinsmore, M.F. Hsu, M. Nikolaides, M. Marquez,A.R. Bausch and D.A. Weitz, Colloidosomes: Selectively-Permeable Capsules Composed of Colloidal Particles (tobe published).[9] H.W. Kroto et al, Nature 318, 162 (1985).[10] J.O.M. Sano, A. Kamino, and S. Shinkai, Science 293,1299 (2001).[11] M.J.W. Dodgson and M.A. Moore, Phys. Rev. B 55, 3816(1997) (arXiv:cond-mat/9512123).[12] E.L. Altschuler, T.J. Williams, E.R. Ratner, R. Tipton,R. Stong, F. Dowla and F. Wooten, Phys. Rev. Lett. 78,2681 (1997).[13] T. Erber and G.M. Hockney, Adv. Chem. Phys. 98, 495(1997).[14] A. Toomre, private communication.[15] A. Perez-Garrido and M. A. Moore, Phys. Rev. B 60,15628 (1999) (arXiv:cond-mat/9905217) and referencestherein.[16] M. Bowick, D. R. Nelson, and A. Travesset, Phys. Rev.B 62, 8738 (2000) (arXiv:cond-mat/9911379).[17] R. Pindak, S.B. Dierker, and R.B. Meyer, Phys. Rev.Lett. 56, 1819 (1986); by introducing a pressure differ-ence across the free standing liquid crystal film, one couldstudy the 2π disclinations and grain boundaries of thisreference in a hemispherical environment (R. Pindak andC.C. Huang, private communication).[18] L. Bonsall and A.A. Maradudin, Phys. Rev. B 15, 1959(1977).[19] D. Fisher, B. Halperin, and R. Morf, Phys. Rev. B 20,4692 (1979).[20] M. Bowick, A. Cacciuto, D.R. Nelson, and A. Travesset,in preparation.[21] J. P. Hirth and J. Lothe, Theory of Dislocations (Wiley,New York, 1982).[22] C. Carraro and D. R. Nelson, Phys. Rev. E 48, 3082(1993) (arXiv:cond-mat/9307008).[23] For the purposes of determining grain boundary stabilityit is useful to write the core energy term Ecore in Eq.(2)as NEcore = 12E5+N−122Ed and parametrize the physicsin terms of a dislocation core energy Ed of order 0.1Y a2.For antipodal +6 defects, NEcore = 2E+6 +N−22Ed.[24] T.C. Lubensky and J. Prost, J. Phys. II 2, 371 (1992).

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Crystalline Order on a Sphere and the Generalized Thomson Problem

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