Nematic versus ferromagnetic shells: new insights in curvature-induced effects
Gaetano Napoli, Oleksandr V. Pylypovskyi, Denis D. Sheka, Luigi Vergori
LLifted textures in nematic shells
Gaetano Napoli, a) Oleksandr V. Pylypovskyi,
2, 3, b) Denis D. Sheka, c) and Luigi Vergori d) Dipartimento di Matematica e Fisica ”E. De Giorgi”, Universit`a del Salento,Lecce (Italy). Helmholtz-Zentrum Dresden-Rossendorf e.V., Institute of Ion Beam Physics and Materials Research,01328 Dresden, Germany Kyiv Academic University, 03142 Kyiv, Ukraine Taras Shevchenko National University of Kyiv, 01601 Kyiv, Ukraine Dipartimento di Ingegneria, Universit`a di Perugia, Perugia (Italy) (Dated: 1 March 2021)
Magnetic materials and liquid crystals are examples of materials with orientational order which give rise totextures whose complexity is as beautiful as challenging to study. Their confinement in curved layers causesthe emergence of geometry-induced effects that are not usually observed in flat layers. In this paper we drawa parallel between ferromagnetic and nematic shells, which are both characterized by local interaction andanchoring potentials. Although similar curvature-induced effects (such as anisotropy and chirality) occur, thedifferent nature of the order parameter, a vector in ferromagnets and a tensor in nematics, yields differenttextures on surfaces with the same topology as the sphere. In particular, on a spherical shell the texturesof ferromagnets are characterised by vortices with integer topological charge, while the textures of nematicsmay admit also half-integer charge vortices.Recent advances in the theory of curvilinear mag-netism have highlighted a range of fascinating geometry-induced effects in the magnetic properties of materials .When confined in thin curved domains, effective physi-cal features arise from the interplay between the curvedgeometry and magnetic texture.According to continuum micromagnetic descriptionof the ferromagnetic media, the magnetic textures canbe well described by the vector order parameter m = M / | M | , which is the normalized magnetization vector.The energy of the ferromagnet typically includes: (i)a short-range exchange energy that penalizes the non-uniformity of the magnetization; (ii) an anisotropic termthat models the existence of directions preferred by themagnetization; (iii) an non-local term describing thelong-range magnetostatic interactions.When ferromagnets are confined in thin curvilinearlayers, the magnetic energy can be decomposed toreveal the emergence of geometry-induced anisotropyand geometry-induced chiral interaction with emergentDzyaloshinskii–Moriya interaction (DMI) as characteris-tic example. With this decomposition, a number of neweffects in ferromagnetic spherical shells have been stud-ied including topological patterning and magnetochiraleffects, for the review see Refs. Recent advances in experimental techniques have alsomade possible the manipulation of these effects for thedesign of new functional materials and applications forspintronics, shapeable magnetoelectronics, magnonics,biomedicine, and soft robotics . a) Electronic mail: [email protected] b) Electronic mail: [email protected] c) Electronic mail: [email protected] d) Electronic mail: [email protected]
Soft matter also provides an area in which the inter-play between geometry of the substrate and the orderparameter plays a crucial role. One example is providedby liquid crystal (LC) shells . These are microscopic col-loidal particles coated with a thin layer of nematic LC,and have potential applications as the topological de-fects (which may occur on them) can be engineered toemulate the linear, trigonal and tetrahedral geometriesof carbon atoms . This feature opens up the possibil-ity to design meso-atoms with special optical propertieswhose valence and directional-binding can be controlled.Photonic lattices made of LC shells are the new frontierfor the manufacture of a new-generation optical crypto-graphic devices .Nematic LCs are aggregates of rodlike molecules.Within the classic theory of nematics , the average mi-croscopic molecular orientation is described by a sole vec-tor order parameter n called the director . In the 1960s deGennes introduced the order-tensor theory which baseson the orientational probability distribution and providesmeasures of the degree of orientation and biaxiality. In itssimplest form, this theory uses as state variable a second-order symmetric traceless tensor Q = s ( n ⊗ n − I ), withthe scalar parameter s being the degree of orientation that vanishes at points where there is no privileged direc-tion. Contrarily to the director theory, de Gennes theoryallows the study of nematic-isotropic phase transitions.Most theories of nematic shells are based on energyfunctionals defined on surfaces expressed in terms ofvector or tensor order parameters. Theories inwhich the director field n is purely tangential have theflaw that topological defects, i.e. points where the direc-tor is not uniquely defined, inevitably arise on surfaceswith the same topology as sphere. Unavoidably, the en-ergy blows up in a neighbourhood of a defect. This flawcan be overcome by introducing a theory in which thetensor order parameter Q is tangential, i.e. the normal a r X i v : . [ c ond - m a t . s o f t ] F e b to the surface is an eigenvector of Q corresponding to thezero eigenvalue. In this framework, melting points (pointsat which the degree of orientation vanishes) occur, andthe energy is finite over the entire domain. A differentway to avoid singularities within the director theory isto release the constraint that the director field is tangen-tial (as suggested by recent experimental observations ),and to assign an energy cost to the out-of-plane compo-nents of n . This alternative will be explored in this paperand will allow us to compare the theories of ferromagneticand nematic shells.In theories based on vector order parameters in whicha privileged direction is defined everywhere on the shell,there is a fundamental difference between the response offerromagnets and nematics that is due to the orientability of the state variables. For nematics, the head-tail sym-metry of the molecules is translated at the macroscopiclevel by the assumption that n must be indistinguish-able from − n . Roughly speaking, it is more appropriateto regard the director as a line instead of an arrow .We may then conclude that no vector order parametercan describe properly the mechanical response of nemat-ics. A tensor order parameter has to be introduced. Forthis reason, in our attempt to study the equilibrium tex-ture of a nematics on a curved shell S , we here use thesecond-order alignment tensor N = n ⊗ n instead ofthe director field n . Introduced the local orthonormalbasis B = { e , e , ν } , where e and e are the princi-pal directions on the surface S and ν = e × e is theunit normal to S , and denoted n = n e + n e + n ν ,the components of N form the symmetric 3 × N ij = n i n j ) i,j =1 , , . Observe that n and N are topolog-ically different from each other. In fact, the set of unitvectors is in one-to-one correspondence with the pointsof the two-dimensional unit sphere S , while the set ofsecond-order tensors in the form a ⊗ a , with | a | = 1, isin bijective correspondence with the real projective RP plane which, as known, is diffeomorphic to the quotientspace of S under the equivalence relation induced bythe antipodal map. We also note that using N as a statevariable for nematics is equivalent to fixing s = 1 in the Q − theory.We shall see that, given an energy density, taking thealignment tensor N as the state variable allows a richervariety of equilibrium textures than on using the direc-tor n as a macroscopic descriptor of the orientation ofnematic molecules. To justify this assertion, consider aspherical shell. Due to the topology of the sphere, atequilibrium, ferromagnets (whose orientation is describedby a unit vector) form vortices with integer topologicalcharges, while nematics (whose orientation is more ap-propriately described by a second-order tensor) may formalso vortices with half-integer topological charge. Thedifference in the possible topological charges of a vortexbetween the two theories stems from the different ori-entability of the state variables: arrows for ferromagnets,lines for nematics.We start by considering an ordinary 3D nematics con- fined in a thin layer. In the bulk the energy due to thedistortion of the molecular field is given by the celebratedFrank formula , while at the boundary a surface en-ergy term promotes a tangential alignment of the directorfield. As a result of a perturbation analysis conducted insection I in the supplementary information, such an en-ergy density approximates to the effective surface energydensity w s ( N , ∇ s N ) = k |∇ s N | | {z } ≡ E ex + % N · ( ν ⊗ ν ) , (1)where k > % repre-sents the anchoring strength. The sign of % determinesthe direction of the easy axis. The easy axis is normal tothe shell if % is negative, tangential if % is positive. De-noting q i ( i = 1 , ..,
6) the components of N with respectto the local basis B in the Voigt notation (see sectionII in the supplementary information for details), we candecompose for the exchange energy density of a nematicsas E ex = k ð γ q i ) | {z } ≡ E + k H ij q i q j | {z } ≡ E A + k D ( γ ) ij κ γ L ( γ ) ij | {z } ≡ E DM , (2)where γ = 1 , i, j = 1 , ...,
6, and the symbol ð denotesthe modified tangential derivative .Except for the different dimensions of the vector spaceswhich q = ( q , ..., q ) and the unit magnetization vectorbelong to, the exchange energy density (2) is formallysimilar to the one for magnetic systems .This decomposition allows to separate the intrinsic ef-fects from the spurious effects due to the embedding ofa 2D surface into a 3D Euclidean space. In fact, theterm w in the exchange energy density (2) involves onlycovariant derivatives and hence represents the intrinsicpart of the exchange interaction. This term is zero if andonly if the alignment tensor is uniform in the tangentplane. The other two terms, instead, account also for theextrinsic curvature of the shell.Since H is a symmetric 6 × E A in (2) couples the extrinsic curvature with thealignment and induces anisotropic effects. At each point, E A reaches its absolute minimum when the texture ispurely tangential and the director field is aligned alongthe principal direction with the smallest modulus of cur-vature. To explain this point, observe that E A can berewritten as E A = k [ κ (1 − q ) + κ (1 − q )] /
2, with q , q ∈ T = { ( u, v ) ∈ R : u, v ∈ [0 , , u + v ≤ } ,and check that, depending on the magnitude of the mod-uli of the principal curvatures, it reaches its minimum in T for q = 1 or q = 1. As a particular case, since anyunit vector tangent to a sphere is principal, at each pointon a spherical shell E A attains its absolute minimum atany tangential alignment. In any case, the curvature-induced anisotropy term is activated whenever the sur-face is curved. It combines with anchoring potential,strengthening its effects if % > % < , while the occurrence of the energy term E DM is completely novel in the theory of nematic shells.This term is responsible for a curvature-induced chi-ral interaction. The Lifshitz invariants of N , L ( γ ) ij = q i ð γ q j − q j ð γ q i , combines with the principal curvatures κ γ , with D ( γ ) ij being skew-symmetric matrices with con-stant entries, giving rise to an anisotropic DMI withgeometry-dependent coefficients. A term similar to E DM occurs in the Landau-de Gennes theory of cholestericLCs (which are intrinsically chiral materials) and iscaused with molecular chirality being responsible for two-dimensional modulated states and may form vortex orskyrmion lattices . In the supplementary informationwe prove that in terms of the components of the align-ment tensor E DM reads E DM = k ε ijα ε αβ L βγ N ih ð γ N hj , (3)where L βγ are the components of the extrinsic curvaturetensor. Comparing (3) with the chiral term in the energydensity (1) in we realize that the role played by the(constant) helicity q in cholesterics is here substituted bythe second-order tensor τ = ε αβ L βγ e α ⊗ e γ . This tensorcan be regarded as a sort of helicity tensor which accountsfor the anisotropy induced by the extrinsic curvatures.The components of τ with respect to the local basis B form the 3 × τ ] = κ − κ . (4)From (4) it is easy to deduce that τ is skew-symmetricon spheres, symmetric on minimal surfaces.A direct inspection shows that the essential ingredientfor the DMI, apart from curvature, is the existence of anout-of-plane component of the texture. The DMI termfavours rotation of the alignment around the normal di-rection, where the sense of rotation, in both nematic andmagnetic shells, is determined by the sign of the princi-pal curvatures. This term is instead suppressed in purelytangential or purely normal textures. The influence ofsurface geometry and easy axes on local interaction termsis sketched in Figure 2 in the paper by Sheka et al. .In spherical ferromagnetic shells, the interplay betweenthe DMI term and the normal anisotropy ( % <
0) is re-sponsible of the emergence of skyrmions states as pertur-bations of hedgehogs-like ground states . Very recently,LCs skyrmions have been realized as micron sized soli-tons in a chiral nematic material confined between twoparallel substrates . Note that, purely tangential field are admissible onsurface of genus one, while are not allowed on surfaceof genus zero. Thus, on a spherical shell, except for thehedgehog configuration, all terms of the local interactionare unavoidably activated.We now address the equilibrium textures on nematicspherical shells by the numerical minimization of the en-ergy functional (1) under appropriate constraints on thevariables q ’s (see the Supplementary Information for de-tails). We have used a finite differences scheme withconstant steps α = π/
40 for colatitude and longitude,and ∆ r = 2 λ , with λ = p k/ (2 % ) being a characteristiclength which measures the scale over which the align-ment tensor varies on the shell at equilibrium, for theradius r ∈ [2 λ, λ ]. The texture with the lowest energyresults from the interplay between the exchange and an-choring energies. For small enough values of the ratio r/λ the exchange energy dominates, which leads to ax-isymmetric textures that are almost uniform far from twosmall regions around two antipodal points where the ra-dial component ( q ) is pronounced (Figures 1(a,b,c)).This configuration corresponds to the 3D onion texturein ferromagnetic spherical shells .As the ratio r/λ increases the role played by the an-choring energy becomes more and more important, thenematics is forced to align tangentially to the shell,and more complex textures occur. We find that for r/λ ≥ ξ ? ≈ / / / r/λ and decreases. Wetested also textures with a single lifted site with charge+2, and with two lifted sites with charges +3 / / r/λ ∈ [2 , using only the Frank 2D theory,Bates et al. using a microscopic lattice model, Kralj etal. and Nitschke et al. using Landau-de Gennes 2Dtheory.Numerical results show that, although the energy den-sity can be regarded as being the same in the two theo-ries with n and N , the class of solutions admitted withinthe tensor theory is wider than that of vector theory.Thus, while ferromagnetics can admit only vortices ofinteger charge, nematics can also admit those of half-integer charge. However, there are other examples in theliterature of LCs, where this aspect has been addressed.Among them, Ball and Zarnescu discuss the case ofa nematic in a planar configuration, confined in a finiteplane region with two holes. Reporting the words of the r/λ E n e r g y E / k Onion Tetrahedral Trigonal c1 c2c3 c4
Onion configuration(b) q (longitudinal) (c) q (radial)0 1Tetrahedral configuration(d) q (longitudinal) (e) q (colatitudinal) c2c4c3 c2c4c3 Trigonal configuration(f) q (longitudinal) (g) q (radial) +1 / (h) Charge +1 / FIG. 1.
Equilibrium textures on nematic spherical shells. (a) Energy of onion, trigonal and tetrahedral texturesas function of the ratio r/λ . Insets show the configurations of the nematic director field. Red spots mark the positions ofdisclinations of the tetrahedral texture. Longitudinal ( q ) and radial ( q ) components of the onion configuration are representedby Kavrayskiy VII map projection in (b,c), where areas with different colours are separated by isolines. Longitudinal ( q ) andcolatitudinal ( q ) components of the tetrahedral configuration are displayed in (d,e). Trigonal configuration with two +1 / / Authors: ’...it turns out that there are continuous, evensmooth, line fields for which it is impossible to ‘assignarrows to the lines’; that is, one cannot make a choiceof a vector out of each line in such a way that no dis-continuity is created’. Clearly, in order to avoid infiniteenergy, discontinuous configurations are not allowed.When trying to force a field of vectors or lines on asurface S with the topology of the sphere, discontinuitiesare unavoidable. Topological defects arise and, accordingto the Hopf-Poincar´e theorem, the sum of the topolog-ical charges of all defects on S must be equal to 2, theEuler-Poincar´e characteristics of a sphere. The topolog-ical charge of each defect is an integer multiple of .However, it must be said that defects might not be theonly points at which the solution lacks continuity. In fact,while it is always possible to have a continuous field oflines on the whole sphere (of course, excluding the pointswhere the defects are located), the same is not true forfields of vectors. In particular, in the presence of a de-fect with fractional charge there are unavoidably curvesthrough which the vector field is discontinuous. An ex-ample can be found in the work of Vitelli and Nelson .So, assuming the points where the defects are excluded,there would be more continuous configurations allowedwith lines (nematics) than with arrows (ferromagnets).When the field of lines or vectors can escape from thetangent plane, topological defects no longer exist. How-ever, since out-of-plane escape has a certain energy cost,the texture tends to lie on the surface when possible. Fortopological reasons this is not possible at all points on asphere and, therefore, uplifts reverts to localized vorticesthat can have different structures depending on the the-ory. In other words, the texture around a lifting pointmimics that of topological defects. In conclusion, we have introduced a novel theory fornematic shells in which the optical axis is assumed not tobe purely tangential. Accordingly, we have observed theoccurrence of novel curvature-induced effects that havebeen observed within theories for ferromagnetic shells.In particular, the extrinsic curvature can be seen as asource of chirality responsible for the onset of skyrmion-or meron-type nonlinear waves. On the other hand, itmust be said that the difference in the orientability of theorder parameters yields a class of equilibrium solutionsfor nematics wider than that for ferromagnets.We believe that the impact of this work goes beyondthe liquid crystals. In particular, we expect that thetheory can be extended for curvilinear antiferromagnets.They are known to be described by several vector orderparameters. In the simplest case, the σ -model allows an-tiferromagnet to be described by the N´eel vector . Be-ing director by definition, in many cases it behaves simi-larly to liquid crystals supporting textures with fractionaltopological charges like disclinations . Recently, theconcept of a curvilinear antiferromagnetism was intro-duced in . One can expect that curvilinear antiferro-magnets admit novel textures similar to predicted in ne-matics. ACKNOWLEDGMENTS
D. S. acknowledges the financial support from theMinistry of Education and Science of Ukraine (project19BF052-01). The work of G. N. and L. V. has beenfunded by the MIUR (Italian Ministry of Education, Uni-versity and Research) project PRIN 2017, ”Mathematicsof active materials: From mechanobiology to smart de-vices”, project n. 2017KL4EF3. R. Streubel, P. Fischer, F. Kronast, V. P. Kravchuk, D. D. Sheka,Y. Gaididei, O. G. Schmidt, and D. Makarov, “Magnetism incurved geometries (topical review),” Journal of Physics D: Ap-plied Physics , 363001 (2016). E. Y. Vedmedenko, R. K. Kawakami, D. Sheka, P. Gam-bardella, A. Kirilyuk, A. Hirohata, C. Binek, O. A. Chubykalo-Fesenko, S. Sanvito, B. Kirby, J. Grollier, K. Everschor-Sitte,T. Kampfrath, C.-Y. You, and A. Berger, “The 2020 magnetismroadmap,” Journal of Physics D: Applied Physics , 453001(2020). D. Sheka, P. Pylypovskyi, O.V.An Landeros, and et al., “Non-local chiral symmetry breaking in curvilinear magnetic shells,”Commun Phys , 128 (2020). P. Fischer, D. Sanz-Hern´andez, R. Streubel, and A. Fern´andez-Pacheco, “Launching a new dimension with 3D magnetic nanos-tructures,” APL Materials , 010701 (2020). D. Makarov, M. Melzer, D. Karnaushenko, and O. G. Schmidt,“Shapeable magnetoelectronics,” Applied Physics Reviews ,011101 (2016). T. Lopez-Leon, V. Koning, K. B. S. Devaiah, V. Vitelli, andA. Fernandez-Nieves, “Frustrated nematic order in spherical ge-ometries,” Nature Physics , 391–394 (2011). D. R. Nelson, “Toward a tetravalent chemistry of colloids,” NanoLetters , 1125–1129 (2002). Y. Geng, J. Noh, I. Drevensek-Olenik, , R. Rupp, G. Lenzini,and J. P. F. Lagerwall, “High-fidelity spherical cholesteric liquidcrystal bragg reflectors generating unclonable patterns for secureauthentication,” Scientific Reports , 26840 (2016). E. G. Virga,
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2, 3, b) Denis D. Sheka, c) and Luigi Vergori d) Dipartimento di Matematica e Fisica ”E. De Giorgi”, Universit`a del Salento,Lecce (Italy). Helmholtz-Zentrum Dresden-Rossendorf e.V., Institute of Ion Beam Physics and Materials Research,01328 Dresden, Germany Kyiv Academic University, 03142 Kyiv, Ukraine Taras Shevchenko National University of Kyiv, 01601 Kyiv, Ukraine Dipartimento di Ingegneria, Universit`a di Perugia, Perugia (Italy) (Dated: 1 March 2021)
In this supplementary material we derive the surface Frank-Rapini-Papoular energy density as a result of aperturbation analysis of the usual 3D energy functional under the assumption that the region occupied bythe nematics is very thin. We then decompose the energy density as the sum of the four contributions thatare discussed in details in the Letter. Finally, by means of the numerical minimization of the energy densitywe find equilibrium textures on a spherical nematic shell with two, three and four disclinations.
I. PARAMETRIZATION OF THE REGION OCCUPIEDBY THE NEMATIC SHELL. DIFFERENTIALOPERATORS
Let us assume that the nematic shell occupies a thinregion V of thickness h around a regular compact surface S . Let ν s be the normal unit vector field to S . Weparametrize points in the bulk through a coordinate set( u, v, ξ ) such that p ( u, v, ξ ) = p S ( u, v ) + ξ ν s ( u, v ) , (S1)where p S is the normal projection of p onto S , and | ξ | ,with ξ ∈ [ − h/ , h/ p from the samesurface. Such a coordinate set is well defined in a finiteneighbourhood of S . Next, we introduce the principalcurvatures κ s ( p S ) and κ s ( p S ) of S at p S ∈ S , and as-sume that h (cid:28) min p S ∈ S (max {| κ s ( p S ) | , | κ s ( p S ) |} ) − = ‘. (S2)For every fixed ξ ∈ [ − h/ , h/ S ξ = { p S + ξ ν s ( p S ) : p S ∈ S } locatedat distance | ξ | from S with the unit normal vector field ν : p ∈ S ξ ν s ( p S ). In this way, the unit vector field ν is defined on the entire region V . The second-ordertensor ∇ ν is symmetric. Its eigenvectors are ν (with anull eigenvalue) and the unit vector fields e i : p = p S + ξ ν s ∈ V e is ( p S ) ( i = 1 , , with e s and e s denoting the principal directions fieldson S . The spatial gradients of the eigenvectors ν , e and e are : ∇ ν = − κ s − ξκ s e ⊗ e − κ s − ξκ s e ⊗ e , (S3a) a) Electronic mail: [email protected] b) Electronic mail: [email protected] c) Electronic mail: [email protected] d) Electronic mail: [email protected] ∇ e = − e ⊗ Ω + κ s − ξκ s ν ⊗ e , (S3b) ∇ e = e ⊗ Ω + κ s − ξκ s ν ⊗ e , (S3c)where Ω = − (Ω e + Ω e ),Ω = − (cid:20) Ω s − ξκ s + ξ ∇ s κ s · e s (1 − ξκ s )(1 − ξκ s ) (cid:21) , Ω = − (cid:20) Ω s − ξκ s − ξ ∇ s κ s · e s (1 − ξκ s )(1 − ξκ s ) (cid:21) , (S3d)and Ω s and Ω s are the geodesic curvatures of the linesof curvature on S .The differential operator ∇ s in (S3d) is the surface gra-dient. To introduce this differential operator in the mostgeneral setting, let Φ be a smooth scalar, vector or tensorfield defined on S . Then the surface gradient of Φ is thetensor field resulting from the composition of the usualgradient of Φ and the projection onto the tangent planeof S , P = I − ν s ⊗ ν s , namely ∇ s Φ = ( ∇ Φ) P . (S4)In analogy with the usual gradient operator, for a givenvector field u we define the surface divergence and thesurface curl of u as the trace of ∇ s u and twice the axialvector corresponding to the skew-symmetric part of ∇ s u ,respectively. In formulas we havediv s u = tr ∇ s u = ∇ s u · P , curl s u = − ε ∇ s u , where ε denotes the Ricci alternator. II. DERIVATION OF THE SURFACEFRANK-RAPINI-PAPOULAR ENERGY
Let n be a unit vector, called the director, which rep-resents the average alignment of the molecules of the ne-matics. The celebrated Frank formula for the elastic a r X i v : . [ c ond - m a t . s o f t ] F e b energy density (per unit of volume) associated with thedirector distortion consists of four terms2 w OZF = K (div n ) + K ( n · curl n ) + K | n × curl n | + ( K + K )[( ∇ n ) · ( ∇ n ) T − (div n ) ] (S5)where the constants K , K , K , and K are called thesplay, twist, bend, and saddle-splay moduli, respectively.To ensure a stable undistorted configuration of a nematicliquid crystal in the absence of external fields or confine-ments, the three moduli K i ( i = 1 , ,
3) must be non-negative, whereas the elastic saddle-splay constant mustobey Ericksen’s inequalities : | K | ≤ K , K + K ≤ K . In the attempt to capture also the effects of the bound-aries ∂S ± h/ on the orientation of the nematic director,we consider the energy functional resulting from the sumof the Frank energy and the Rapini-Papoular surface an-choring energy W = Z V w OZF ( n , ∇ n )d V + % Z S h/ ∪ S − h/ ( n · υ ) d a, (S6)where the constant % represents the anchoring strengthand υ is the unit outward normal to S ± h/ . The anchor-ing energy in (S6) penalizes the deviation of the moleculesat the boundaries S ± h/ from tangential directions if % ispositive, from the normal direction if % is negative.Under the assumption (S2) on the smallness of thethickness of the region V occupied by the nematics, theenergy functional (S6) approximates to a surface inte-gral over S , with the integrand representing the surfaceFrank-Rapini-Papular energy density. To prove this, wefirst assume that the director field n does not vary alongdirections normal to S , that is it satisfies the property n ( p ) = n ( p S ) (S7)at each point p = p S + ξ ν s ∈ V . We then parametrizethe director field as n = sin θ cos ϕ e + sin θ sin ϕ e + cos θ ν , (S8)where, in view of ansatz (S7), the angles θ and ϕ satisfythe properties θ ( p ) = θ ( p S ) and ϕ ( p ) = ϕ ( p S ) (S9)for all p = p S + ξ ν s ∈ V .From (S3a)-(S3c) and (S8) the gradient of n is foundto be ∇ n = ( ν × n ) ⊗ ( ∇ ϕ − Ω ) − ν ⊗ ( ∇ ν ) n +( ν · n ) ∇ ν − ν − ( ν · n ) n | ν × n | ⊗ ∇ θ, (S10) where, due to (S9) and following the similar argumentsas in Appendix A in , ∇ θ = (cid:18) ∇ s θ s · e s − ξκ s (cid:19) e + (cid:18) ∇ s θ s · e s − ξκ s (cid:19) e , (S11)and ∇ ϕ = (cid:18) ∇ s ϕ s · e s − ξκ s (cid:19) e + (cid:18) ∇ s ϕ s · e s − ξκ s (cid:19) e . (S12)The scalar fields θ s and ϕ s on the right-hand sides ofequations (S11) and (S12) are the restrictions of θ and ϕ to the surface S , namely θ s : p S ∈ S θ ( p S ) and ϕ s : p S ∈ S ϕ ( p S ). From (S3a), (S3d) and (S10)–(S12)it is evident that although the director field is assumedto be constant along directions normal to S , its spatialgradient does depend on the spatial variable ξ in thenormal direction ν .At ξ = 0 the parametrization of n (S8) and (S9) givethe restriction n s of the director field to the surface S , n s = sin θ s cos ϕ s e s + sin θ s sin ϕ s e s + cos θ s ν s , (S13)while evaluating (S3a)-(S3d), (S11) and (S12) at ξ = 0gives the surface gradient of n s , ∇ s n s = ( ν s × n s ) ⊗ ( ∇ s ϕ s − Ω s ) + ν s ⊗ Ln s − ( ν s · n s ) L − ν s − ( ν s · n s ) n s | ν s × n s | ⊗ ∇ s θ s , (S14)where Ω s = − Ω s e s − Ω s e s , (S15a) L = κ s e s ⊗ e s + κ s e s ⊗ e s , (S15b)are the vector parametrizing the spin connection on S ,and the extrinsic curvature tensor of S , respectively. In-cidentally, we recall that the mean and Gaussian curva-tures of S are defined as H = 12 tr L and K = 12 (cid:2) (tr L ) − tr( L ) (cid:3) , (S16)respectively.Recalling the definition of the parameter ε (c.f. (S2)),from (S10)–(S14) we deduce that (see Napoli and Ver-gori for details) ∇ n = ∇ s n s + o ( (cid:15) ) as ε → , (S17)whencediv n = div s n s + o ( (cid:15) ) , curl n = curl s n s + o ( (cid:15) ) (S18)as ε → V reads d V = (1 − Hξ + Kξ )d A d ξ (S19)where d A is the area element on S . On the other hand,on the boundaries S ± h/ the unit outward normal andthe area element are υ = ± ν s andd A ± h/ = (cid:18) ∓ hH + h K (cid:19) d A, (S20)respectively. In view of (S2) the volume and area ele- ments(S19) and (S20) approximate tod V = d A d ξ + o ( ε ) and d A ± h/ = d A + o ( ε ) (S21)as ε → W = 12 Z V (cid:8) K (div n ) + K ( n · curl n ) + K | n × curl n | + ( K + K ) (cid:2) ( ∇ n ) · ( ∇ n ) T − (div n ) (cid:3)(cid:9) d V + Z S h/ % n · ν s ) d A h/ + Z S − h/ % n · ( − ν s )] d A − h/ = 12 Z h/ − h/ ( Z S (cid:2) K (div n ) + K ( n · curl n ) + K | n × curl n | (cid:3) (1 − Hξ + Kξ )d A ) d ξ + 12 Z h/ − h/ ( Z S ( K + K ) (cid:2) ( ∇ n ) · ( ∇ n ) T − (div n ) (cid:3) (1 − Hξ + Kξ )d A ) d ξ + Z S % n · ν s ) (cid:18) − hH + h K (cid:19) d A + Z S % n · ν s ) (cid:18) hH + h K (cid:19) d A = 12 Z h/ − h/ ( Z S (cid:2) K (div s n s ) + K ( n s · curl s n s ) + K | n s × curl s n s | (cid:3) d A ) d ξ + 12 Z h/ − h/ ( Z S ( K + K ) (cid:2) ( ∇ s n s ) · ( ∇ s n s ) T − (div s n s ) (cid:3) d A ) d ξ + Z S % n s · ν s ) d A + o ( ε )= h Z S (cid:2) K (div s n s ) + K ( n s · curl s n s ) + K | n s × curl s n s | (cid:3) d A + 12 Z S (cid:8) h ( K + K )[( ∇ s n s ) · ( ∇ s n s ) T − (div s n s ) ] + % ( n s · ν s ) (cid:9) d A + o ( ε ) as ε → . (S22)We have then proven that W ≈ W s = Z S w s d A , with2 w s = k (div s n s ) + k ( n s · curl s n s ) + k | n s × curl s n s | + ( k + k )[( ∇ s n s ) · ( ∇ s n s ) T − (div s n s ) ]+ % ( n s · ν s ) , (S23)where k i = hK i ( i = 1 , , ,
24) are, respectively, the re-scaled splay, twist, bend and saddle-splay constants withthe same physical dimensions as energy per unit area.Within the one-constant approximation ( K = K = K = K and K = 0 in (S5)) the surface energy density(S23) reduces to w s = k |∇ s n s | + % n s · ν s ) , (S24)with k = hK . III. SURFACE ENERGY DENSITY IN TERMS OF THEALIGNMENT TENSOR
In what follows we shall consider the surface energydensity (S24), and, for simplicity of notation, we shallomit the subscript s to n , e , e and ν .We now show that the moduli of the surface gradientsof the director n and alignment tensor N = n ⊗ n arerelated each other.From the definition of surface gradient (S4), the com-ponents of the surface gradient of the alignment tensorare found to be( ∇ s N ) ijh = ∂N ij ∂x l P lh = ∂n i ∂x l P lh n j + n i ∂n j ∂x l P lh ≡ n i ; h n j + n i n j ; h . (S25)(In (S25), and in the subsequent formulas, the Einsteinsummation convention is used.) Since n i = 1 (the direc-tor is a unit vector), we deduce that n i n i ; h = ( n i ) ; h − n i n i ; h = − n i n i ; h = ⇒ n i n i ; h = 0 , which combined with (S25) yields |∇ s N | = N ijh = n i ; h n j + 2 n i n i ; h n j n j ; h + n i n j ; h = 2 n i ; h = 2 (cid:18) ∂n i ∂x l P lh (cid:19) = 2 |∇ s n | . (S26)In view of this identity, the surface energy density (S24)can be rewritten in terms of the alignment tensor as w s = k |∇ s N | + % N · ( ν s ⊗ ν s ) , (S27)On the other hand, with respect to the local basis { e , e , e = ν } on S , the surface gradient of the align-ment tensor reads ∇ s N = [ ∇ s N · ( e i ⊗ e j ⊗ e γ )] ( e i ⊗ e j ⊗ e γ ) , (S28)where, Latin indices range from 1 to 3, Greek ones from1 to 2 ( ∇ s N · ( e i ⊗ e j ⊗ ν ) = 0 for all i, j = 1 , , ∇ s N · ( e i ⊗ e j ⊗ e γ ) = ∇ γ N · ( e i ⊗ e j ) (S29)= ∇ γ N ij + N hj ( ∇ γ e h ) · e i + N ik ( ∇ γ e k ) · e j . In view of the symmetry of N we have that ∇ γ N · ( e i ⊗ e j ) = ∇ γ N · ( e j ⊗ e i ) , (S30)whence, with the aid of (S3) evaluated at ξ = 0, weobtain ∇ γ N · ( e α ⊗ e β ) = ð γ N αβ − ( N β δ αγ + N α δ βγ ) κ γ , (S31a) ∇ γ N · ( ν ⊗ e β ) = ð γ N β + κ γ ( N γβ − N δ βγ ) , (S31b) ∇ γ N · ( ν ⊗ ν ) = ð γ N + 2 N γ κ γ , (S31c)where ð γ N αβ = ∇ γ N αβ + ( ε αζ N ζβ + ε βζ N αζ )Ω γ , (S32a) ð γ N β = ∇ γ N β + ε βζ N ζ Ω γ , (S32b) ð γ N = ∇ γ N . (S32c)As known, the space of symmetric second-order ten-sors is isomorphic to the six-dimensional Euclidean vec-tor space. Specifically, on setting N = q , N = q , N = q , √ N = q , √ N = q , √ N = q , (S33)we can identify N with the vector q =( q , q , q , q , q , q ). Relations (S33) are known asthe Voigt notation. On using the Voigt notation itcan be proven that the space of second-order tensorsand the six-dimensional Euclidean vector space are alsoisometric, whence | N | = | q | , |∇ s N | = |∇ s q | and( ð γ N ij ) = ( ð γ q k ) .From (S31), (S32) and (S33) we obtain (after lengthyalgebra) |∇ s N | = [ ∇ γ N · ( e i ⊗ e j )] (S34)= ( ð γ q i ) + H ij q i q j + D ( γ ) ij κ γ L ( γ ) ij , where ð γ q = ∇ γ q + √ γ q , (S35a) ð γ q = ∇ γ q − √ γ q , (S35b) ð γ q = ∇ γ q , (S35c) ð γ q = ∇ γ q − Ω γ q , (S35d) ð γ q = ∇ γ q + Ω γ q , (S35e) ð γ q = ∇ γ q − √ γ ( q − q ) , (S35f) L ( γ ) ij = ( q i ð γ q j − q j ð γ q i ) , (S36) H = ( H ij ) i,j =1 ,..., is the symmetric sixth order matrix H = κ − κ κ − κ − κ − κ κ + κ ) 0 0 00 0 0 κ + 4 κ κ + κ
00 0 0 0 0 κ + κ ,, (S37)and D ( γ ) = ( D ( γ ) ij ) i,j =1 ,..., ( γ = 1 ,
2) are skew-symmetricmatrices the independent non-zero entries of which are D (1)15 = − D (1)35 = √ , D (1)46 = − , (S38a)and D (2)24 = − D (2)34 = √ , D (2)56 = − . (S38b)Finally, in view of (S26), on using the Voigt notationthe surface energy density (S24) becomes w s = k ð γ q i ) | {z } = E + k H ij q i q j | {z } = E A + k D ( γ ) ij κ γ L ( γ ) ij | {z } = E DM + % q . (S39)To find stable equilibrium configurations one has tominimize the energy functional W s = R S w s d A subjectto some constraints on the scalar fields q ’s. In fact, since q , ..., q are the components of the alignment tensor N = n ⊗ n they must satisfy the following inequalities andequations q i ≥ i = 1 , , , (S40a) q + q + q = 1 , (S40b) q q − q q q − q q q − q , (S40c)Inequalities (S40a) impose the non-negativeness of q , q and q as these components of N are the squares of thecomponents of the nematic director n . Since n is a unitvector, the trace of N must be equal to unity, whichjustifies (S40b). Finally, equations (S40c) together with(S40a) and (S40b) imply that the components of N forma matrix with rank equal to one as it has to be becausethe alignment tensor is of the form n ⊗ n .We conclude this section by observing that by using(S31) the energy density term E DM in (S39) can be writ-ten also as E DM = k ε ijα ε αβ L βγ N ik ð γ N kj . (S41)This expression allows a comparison between theDzyaloshinskii–Moriya interaction term E DM and the chi-ral term in the energy density (1) in (see the Letter). IV. SURFACE ENERGY DENSITY ON A SPHERICALSHELL
On a spherical shell of radius r the local basis formedby the principal directions and the unit normal is { e ϕ , e θ , − e r } , with e r , e θ and e ϕ being the radial, co-latitudinal and longitudinal directions, respectively, theprincipal curvatures are κ ϕ = κ θ = 1 /r and the vectorparametrizing the spin connection is Ω = (cot θ/r ) e ϕ . Therefore, for a spherical shell the energy density terms E , E A and E DM , and the whole energy density (S39)read, respectively, E = k r ( (cid:16) q ,ϕ csc θ + √ q cot θ (cid:17) + q ,θ + (cid:16) q ,ϕ csc θ − √ q cot θ (cid:17) + q ,θ + q ,ϕ csc θ + q ,θ + ( q ,ϕ csc θ − q cot θ ) + q ,θ + ( q ,ϕ csc θ + q cot θ ) + q ,θ + h q ,ϕ csc θ − √ q − q ) cot θ i + q ,θ ) , E A = k r (cid:16) q + 2 q + 4 q + 5 q + 5 q + 2 q − q q − q q (cid:17) , E DM = k r (h √ q q ,ϕ − q q ,ϕ − q q ,ϕ + q q ,ϕ ) − q q ,ϕ + q q ,ϕ i csc θ + h √ q (2 q − q − q ) − q q i cot θ + √ q q ,θ + q q ,θ − q q ,θ − q q ,θ ) − q q ,θ + q q ,θ ) , and w s = k r h ( q ,θ ) + (cid:16) q ,ϕ csc θ − √ q − + √ q cot θ ) (cid:17) + (cid:16) q ,ϕ csc θ − q − √ q cot θ + √ q cot θ (cid:17) + (cid:16) q ,ϕ csc θ + √ q − √ q + q cot θ (cid:17) + (cid:16) q ,θ − √ q (cid:17) + (cid:16) q ,ϕ csc θ − √ q cot θ (cid:17) + (cid:16) q ,θ + √ q − √ q (cid:17) + (cid:16) q ,θ + √ q (cid:17) + (cid:16) q ,ϕ csc θ + √ q (cid:17) + ( q ,ϕ csc θ − q cot θ + q ) + ( q ,θ + q ) + ( q ,θ − q ) i + % q . (S42) V. SIMULATIONS
To determine the equilibrium configuration of nematicmolecules, we numerically minimize the energy functional W s with energy density as in (S42). To do this, the con-tinual fields q l , l = 1 , φ = 2 π/M and ∆ θ = π/N , respectively, where N >
M > f ij represents the FDMcounterpart of the function f ( θ, φ ) on a sphere, then f ( i ∆ θ, j ∆ φ ) = f ij ,f ( θ = 0) = f , f ( θ = π ) = f N ,i = 1 , N − , j = 0 , M − . (S43) Integration of W s is performed using the double Riemannsum method. We minimize the function W tot = W s + W p using the conjugate gradients method with the penaltyfunction W p = Λ X ij ( (1 − q ij − q ij − q ij ) + (cid:20) q ij q ij − (cid:16) q ij (cid:17) (cid:21) + (cid:20) q ij q ij − (cid:16) q ij (cid:17) (cid:21) + (cid:20) q ij q ij − (cid:16) q ij (cid:17) (cid:21) ) + Λ (cid:26) (1 − q − q − q ) + h q q − (cid:0) q (cid:1) i + h q q − (cid:0) q (cid:1) i + h q q − (cid:0) q (cid:1) i +(1 − q − q − q ) + h q N q N − (cid:0) q N (cid:1) i + h q N q N − (cid:0) q N (cid:1) i + h q N q N − (cid:0) q N (cid:1) i (cid:27) , (S44)where Λ = 10 . To avoid saddle points of W tot , themultiplier q is replaced by 1 − q − q in the anchoringterm.To test stability of different configurations, the strictlyin-surface initial state is given q = cos σ, q = sin σ, q = √ σ cos σ,q = q = q = 0 (S45)for the sphere of each radius with the concrete configu-ration determined by the angle σ . For the onion config-uration it reads σ onion = π/ . (S46)The relaxed state is shown in Fig. S1. Note, that in gen-eral case, the axis of symmetry not necessary to coincidewith the direction along the sphere poles.The initial state with four charges 1 / σ / × = 12 arg (cid:18) ie iφ cos θ θ − √ e − iφ sin θ (cid:19) . (S47) The relaxed state is shown in Fig. S2.The initial state with one charge +1 and two charges1 / σ / × = (cid:20)
12 arg (cid:18) ie − iφ sin θ −
12 sin θ (cid:19)(cid:21) mod π. (S48) G. Napoli and L. Vergori, “Surface free energies for nematicshells,” Physical Review E , 061701 (2012). F. C. Frank, “On the theory of liquid crystals,” Discuss. FaradaySoc. , 19–25 (1958). J. L. Ericksen, “Inequalities in liquid crystal theory,” Phys. Fluids , 1205–1207 (1966). A. Rapini and M. Paapoular, “Distortion d’une lamelle n´ematiquesous champ magn´etique. conditions d’angrage aux paroix,” LeJournal de Physique Colloques , C4–54–C4–56 (1969). A. Duzgun, J. V. Selinger, and A. Saxena, “Comparing skyrmionsand merons in chiral liquid crystals and magnets,” Physical Re-view E , 062706– (2018). FIG. S1.
Onion configuration of the nematic director on the sphere of radius r = 4 λ ( λ = p k/ (2 % ) . (a) Flux linesof the director field on the sphere. (b–d) Components of alignmenttensor tensor N .FIG. S2. Tetrahedral configuration of the disclinations on the sphere of radius r = 10 λ . (a–d) +1 / N . Locations of disclinations in top panels arelabeled. FIG. S3.