Cumulative geometric frustration in physical assemblies
CCumulative geometric frustration in physical assemblies
Snir Meiri and Efi Efrati ∗ Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel (Dated: January 26, 2021)Geometric frustration arises whenever the constituents of a physical assembly locally favor anarrangement that cannot be realized globally. Recently, such frustrated assemblies were shownto exhibit filamentation [1], size limitation [2], large morphological variations [3, 4] and exoticresponse properties [5, 6]. These unique characteristics can be shown to be a direct outcome ofthe geometric frustration. There are, however, geometrically frustrated systems that do not exhibitany of the above characteristics. The epitome of frustration in condensed matter physics, the Isinganti-ferromagnet on a triangular lattice, is one such system. In this work we provide a frameworkfor directly addressing the frustration in physical assemblies and their expected energy scalingexponents. We also present a new spin model that exhibits cumulative geometric frustration, anduse the newly devised framework to analyze it.
I. INTRODUCTION
The ground state of an assembly of identical parti-cles endowed with short range interactions is expected toreflect the symmetries of space and of its constituents.Hard discs with short range attraction in the plane willpack tightly in a six-fold symmetric order, such that thecenters of the discs will form the vertices of a unilateraltriangular lattice. This uniform order, however, cannotpersist if the discs are packed in curved space, e.g. thesurface of a large sphere. In such a curved space the sumof the internal angles in each of the formed triangles devi-ates from the preferred value of π by an angle deficit thatscales as ∆ ∼ d ρ , where d is the inter-particle distanceand ρ is the radius of the large sphere. While this angledeficit is identical in all the triangles, it leads to spatialgradients in the packing fraction of the bulk ground state[7]. These inevitable strain gradients are associated witha super extensive elastic energy contribution, in whichthe elastic energy per particle grows as the area of thedomain increases [8], and favors the formation of narrowfilamentous domains over isotropic bulks [1, 8].The above phenomenology of frustrated assemblies wasverified numerically and analytically for defect free crys-tals growing in a uniformly curved geometry [9], as wellas observed experimentally in a system of colloids con-fined to a spherical interface and endowed with very shortrange attraction [1]. The tendency to form filaments, thesuper extensive elastic energy and the line tension in-crease with domain size can all be attributed to the mis-match between the attempted (vanishing) Gaussian cur-vature of the material manifold, and the non-zero Gaus-sian curvature of the ambient space in which it is em-bedded. This provides a natural geometric charge thataccumulates and serves as a source term for the spatialstrain gradients in the material Similar behavior appearin many other systems including filament bundles [10],liquid crystals [11], chiral stiff rod-like colloids [12] and ∗ efi[email protected] twisted molecular crystals [13]. For some of these sys-tems, the geometric charge associated with the frustra-tion has not been identified. Recent works aim to providea unified framework to describe geometric frustration inthese diverse systems [7].There are, however, geometrically frustrated systemsthat do not exhibit any of the above traits. One suchexample is the Ising anti-ferromagnet on a triangular lat-tice, the epitome of frustration in condensed matter. Inthis system spins of values ± H = J (cid:88) (cid:104) i,j (cid:105) s i · s j . (1)While the minimal energy per edge in this system is − J ,the minimal energy per triangular facet which includesthree such edges is also − J as at least one of the edgesmust connect co-aligned spins. There are, however, anabundance of states that achieve this lower bound for theenergy and thus constitute equivalent ground states ofthe system. The total energy of the ground state is thusextensive, and the system does not favor fillamentationnor complex structures.Both systems described above display geometric frus-tration: Geometric constraints prevent the local groundstate from being realized uniformly throughout the sys-tem. However, the resulting phenomena differ substan-tially between the two cases suggesting that the two be-long to different classes of geometrically frustrated sys-tems. While the first features super extensive behaviourwhich drives the assembly towards fillamentation, thelatter features an extensive ground state energy, and ahighly degenerate ground state. What then distinguishesbetween the different types of frustration?In this work we distinguish between two classes of ge-ometric frustration, cumulative and non-cumulative, andprovide a framework that allows to distinguish betweenthe two without explicitly solving the ground state of thesystem. We also use this classification scheme to studythe frustration in several specific systems including a newspin model displaying cumulative frustration. a r X i v : . [ c ond - m a t . s o f t ] J a n II. DISTINGUISHING CUMULATIVE FROMNON-CUMULATIVE FRUSTRATION
Super-extensive energy is not-uncommon for systemswith long range interactions, where every particle inter-acts with all other particles. For example, a uniformelectric charge density ρ in a spherical domain of radius r is associated with the energy E ∝ ρ r ∼ M / , where M ∝ r is the total mass of the charges [14]. For shortrange interactions, one expects that every constituentwill affect only those in its immediate vicinity. However,the cooperative nature of the ground state in systemsdisplaying cumulative geometric frustration causes thesesystems to display long range behavior.We adopt the super-extensive energy as the definingcharacteristic of cumulative frustration. Systems withshort range interactions in which the energy per parti-cle at the ground state of the system grows with systemsize will be said to display cumulative frustration. Incontrast, if the local resolution of the frustration can bepropagated globally to the entire system, its ground stateenergy will be extensive and it will be associated withnon-cumulative frustration.To distinguish between cumulative and non-cumulativefrustration in continuous systems we examine theirHamiltonian and its ability to support ground state so-lutions with spatially uniform energetic cost. In practicethis requires finding new variables that (i) fully char-acterize the state of the system, and (ii) allow to ex-press the Hamiltonian in a local form, i.e. containingno spatial derivatives. These new variables are, how-ever, not independent of each other as they are derivedfrom common native variables and their gradients. Thisinterdependence manifests in functional constraints thenew variables must satisfy which we term compatibilityconditions. The structure and nature of the compati-bility conditions determine the class of frustration. Lo-cal (non-differential) compatibility conditions give rise tonon-cumulative frustration whereas if the compatibilityconditions contain differential relations they yield cumu-lative frustration.For any frustrated system the compatibility condi-tions preclude the simultaneous minimization of all theterms in the Hamiltonian. In systems that exhibit non-cumulative frustration the optimal compromise at smallscales can be repeated throughout the system to resultin a global solution whose energy scales linearly with sys-tem size. In contrast, in system that exhibit cumulativegeometric frustration realizing some optimal compromiseat some small region of the system precludes repeatingthe same compromise again in its vicinity necessitatinga more energetically expensive resolution of the frustra-tion. A. Estimating the super-extensive energy exponent
Consider a Hamiltonian that depends on n fields andtheir derivatives: H ( φ , φ , ..., φ n , ∇ φ , ..., ∇ φ n , ∇∇ φ , ... )We associate new fields ψ i with each of the fields andfields’ gradients that enter the Hamiltonian. The vari-ables ψ i bring the Hamiltonian to a local form (contain-ing no derivatives), yet must comply with k distinct dif-ferential constraints that relate them and their deriva-tives to each other: G i ( ψ , ψ , ..., ψ n , ∇ ψ , ..., ∇ ψ n , ∇∇ ψ , ... ) = 0We group the fields to a vector Ψ and identify the lo-cal ground state of the Hamiltonian with the value ¯Ψ.These preferred values could vary in space or assume con-stant values depending on the context. We expand theHamiltonian to second order in the generalized strain, ε = Ψ − ¯Ψ, which measures the deviations of these fieldsfrom their locally preferred values. We then expressthe compatibility conditions in terms of this generalizedstrain by substituting Ψ = ε + ¯Ψ. For systems displayingcumulative frustration the compatibility conditions willtake the form of partial differential relations. In general,for all frustrated systems the compatibility conditionspreclude the state ε = 0 from being globally achieved.Thus, expanding the strain in orders of the spatial coordi-nates, e.g. for two dimensions ε ≈ ε , + ε , x + ε , y + ... ,the compatibility conditions preclude setting all the coef-ficients ε i,j to zero. For small domains the lowest ordersterms dominate the rate of growth of the strain, and tominimize the energy we seek to eliminate as many loworder coefficients as the compatibility conditions allow.If already ε , cannot be set to zero, then the systemwill exhibit an optimal compromise with extensive en-ergy scaling. If this coefficient can be set to zero, thenthe first coefficient that cannot be set to zero ε i,j where i + j = γ will lead to a strain that scales as ε ∝ r γ , for small enough isotropic domains. The associated en-ergy in this case will scale as E ∝ (cid:90) d d r(cid:15) ∝ r d r γ ∼ M γd For example if the compatibility conditions amount toa single differential equation of order p , and the locallypreferred values of the fields ¯Ψ are smooth in the domainconsidered, then p ≤ γ . To determine the exact valueof γ we expand the compatibility condition in terms ofthe spatial coefficients of the strain. If the zero-orderequation is non-homogeneous in (cid:15) i,j then γ = p . If itis homogeneous, then the first non-homogeneous orderof the compatibility condition, q , will yield γ = p + q .In general when there are multiple compatibility condi-tions the lowest term in the vector (cid:15) i,j that cannot be setto vanish will determine the exponent γ . The reader isreferred to SM for more details.In general, the structure and differential order of thecompatibility conditions determine the rate of growth ofthe associated energy of the optimal compromise of afrustrated system. Note, that systems that display non-cumulative frustration are associated with local compat-ibility conditions that are characterized by γ = 0, whichindeed by the formula above yield E ∝ M . B. Frustration saturation
The super extensive energy associated with cumulativegeometric frustration cannot persist indefinitely, as it willgive rise to an arbitrarily large energy per particle. Typ-ically, physical assemblies will be associated with a finitelocal frustration resolution energy. One could simultane-ously satisfy all the compatibility conditions by choosingsome constant state Ψ ∗ that is far from ¯Ψ. This willresult in an extensive, albeit very high, energetic cost.The compatible uniform state of lowest energy per parti-cle, e , defines the frustration escape energy e ∗ . In somesystems the energy per particle may tend to e ∗ asymp-totically [15]. In other systems, such as the Ising anti-ferromagnet on triangular lattice, e ∗ is achieved at a fi-nite scale, termed the frustration escape scale. For theIsing anti-ferromagnet, the frustration escape scale is asingle triangle facet as a certain oriented compromise isfitting for lattice tessellation. Considering the system atscales larger than the frustration escape scale yields non-cumulative behavior.Last, as mentioned before, frustration often leads tothe formation of filaments. While for sufficiently smallisotropic domains the energetic scaling is super-extensivefor such systems ,filamentous growth of finite width maydisplay extensive energy scaling (SM). Thus, filamenta-tion or the breaking up of the assembly to domains mightalso result in non-cumulative behavior. Which of theabove occurs first is determined by the details of the sys-tems and the ambient conditions. C. Continuous frustrated assemblies
Consider the embedding of an elastic disc with the ge-ometry of S in the plane. One may think of this prob-lem as flattening an infinitely thin spherical cap betweentwo flat glass plates and examining the in-plane stresses.Within the framework of metric elasticity, which is par-ticularly suited for such residually-stressed systems, theelastic energy reads E = (cid:90) (cid:90) A αβγδ ( a αβ − ¯ a αβ )( a γδ − ¯ a γδ ) d ¯ A, (2) where Greek indices assume the values { , } , ¯ a =1 , ¯ a = ¯ a = 0 and ¯ a = sin( r ) is the locally pre-ferred reference metric, and the elasticity tensor A andthe area element d ¯ A are material properties that dependonly on the reference metric. The two dimensional met-ric, a , fully describes the configuration of the system, yetonly metrics of vanishing Riemannian curvature can de-scribe the sought planar solution. The latter constraintleads to a compatibility condition in the forms of a sin-gle non-linear second order partial differential equationin the components of a , see SM. Consequently, γ = 2and the energy associated with embedding an isotropicdomain grows as E ∝ M . Similar results are obtainedfor embedding a flat surface in uniformly curved spacesuch as growing a defect free crystal on the surface of asphere [1, 9].Considering embedding a piece of the uniform andmaximally symmetric positively curved manifold S inEuclidean space E leads to an elastic energy of the formof (2), with a , ¯ a , A and d ¯ A replaced by their three dimen-sional variants. In three dimensions the elastic compati-bility conditions consist of three non-linear second orderdifferential equations in the components of the metric.This too yields γ = 2 which in turn results in E ∝ M / .Next, we consider the following general Hamiltonian oftwo fields in the plane: H = (cid:90) [ K (Ψ − Ψ ) + K (Ψ − Ψ ) ] dA, (3)where K and K are constant coefficients. Withoutknowledge of the form of the compatibility conditions forΨ, the ground-state energy-scaling cannot be addressed.For the case of a bi-stable ribbon known as a snap-bracelet [16], Ψ = { κ , κ } are the principal curvatures ofthe ribbon considered as a surface and ¯Ψ = { , − } . Thelowest order compatibility condition is given by Gauss’equation κ · κ = 0, see SM for more details. Thus γ = 0 and the system is dominated by a constant energyper unit mass.In contrast, for two dimensional liquid crystals charac-terized by a local unit director field ˆ n = (cos( θ ) , sin( θ )),we may identify (3) with the Frank free energy whereΨ = ˆ n ∧ ∇ θ = ∇ · ˆ n = s is the local splay, andΨ = ˆ n · ∇ θ = b is the bend of the director field. Thecompatibility condition in this case reads [11]: b + s + ˆ n · ∇ s − ˆ n ⊥ · ∇ b = 0 . (4)Considering the case of bent-core liquid crystals, a phaseknown to display geometric frustration [17], where ¯Ψ = { ¯ s, ¯ b } = { , b } the first order compatibility condition (4)is non-homogeneous in the strain (which in turn neces-sitate non-vanishing gradients) yielding γ = 1, and cor-respondingly the optimal compromise for small domainsscales as E ∝ M [11]. D. Bent core spins on a triangular lattice
Last, as lattice models, and in particular spin latticemodels, are key tools for establishing theoretical under-standing of complex physical behavior we seek a latticespin model that displays cumulative geometric frustra-tion. The Ising anti-ferromagnet, as well as a varietyof frustrated binary spin models were recently shown tosupport only non-cumulative frustration [18]. In view ofthese results, and the central role assumed by a differ-ential compatibility condition in establishing the type offrustration we choose to study a continuous lattice spinmodel.The model consists of a non-linear adaptation of theXY-model, inspired by bent core liquid crystals. We con-sider classical continuous spins, quantified by the anglethey form with the x-axis, θ i , located at the vertices ofa triangular lattice. As opposed to the usual discussedmodels of ferromagnetism and antiferromagnetism, inwhich the spins tend to either co-align or anti-align, inthis case we design the spin interactions to favor onlyslightly mis-aligned orientations. We assume that the di-rections of the three spins are not too disparate, so thatthe average direction in each facet, ¯ θ = ( θ + θ + θ ) / = − θ − θ l sin(¯ θ ) ± θ − θ − θ √ l cos(¯ θ ) , Ψ = θ − θ l cos(¯ θ ) ± θ − θ − θ √ l sin(¯ θ ) , (5)where the +(-) signs above corresponds to upright (up-side down) triangles; see figure 1(a) left (right). TheHamiltonian with respect to these variables simply reads H = (cid:88) facets k Ψ + k (Ψ − b ) . We Numerically minimize the Hamiltonian to find thezero temperature ground state of the system. We ob-tain that the lowest energy states display non-uniformdeviations of the variables Ψ from their locally desiredvalues. Moreover, considering small enough domains( nlb / <
1, where n is the number of edges along thesides of the rectangular domain) yields super extensiveenergy that grows as the number of facets in the domainsquared. If, however, the system forms ribbons of finitewidth the energy per particle plateaus, see figure 1 (d).To understand the origin of the frustration in this sys-tem we need to consider the structure of the compati-bility conditions. On each individual facet, the values ofΨ and Ψ are not sufficient to fully determine all threespin degrees of freedom. However, for two adjacent trian-gles that share an edge the four Ψ values fully determinethe four spins at the vertices, see sm. Every facet be-longs to three distinct such triangle pairs, each uniquelydetermining the values of the spins at its vertices. Re-quiring these different prescribed values to mach yields the sought compatibility condition on the values of Ψin adjacent facets. Considering the continuum limit, inwhich l → = s and Ψ = b . The corresponding compatibility conditionon four adjacent facets yields, to leading order, equation(4). III. DISCUSSION
The framework provided in this work allows to distin-guish cumulative from non-cumulative geometric frustra-tion in physical assemblies. Within this framework thecompatibility conditions assume a central role in predict-ing the behavior of the system and in particular predictthe super-extensive energy growth rate. The compati-bility conditions allow to identify the set of admissiblestates, which in turn greatly reduces the dimensionalityof the problem. This dimensional reduction paves theway to better understanding frustrated assemblies, andmay guide the engineering of their response [19, 20].The scaling argument for the super-extensive energygrowth rate provided above holds only for small enoughisotropic domains. As was observed for crystals on curveddomains [1, 9, 21] and is shown in Figure 1, an assemblycan escape super-extensivity by forming non-isotropic,filamentous domains, at the cost of increased surfacearea. Some frustrated systems allow complete alleviationof the frustration in the bulk by a finite (albeit high) en-ergy [15]. Alternatively, a system may incorporate pack-ing defects into the assembly to attenuate the accumula-tion of energy due to frustration [22]. Which of the aboveoccurs first will determine the fate of the assembly anddepends on the details of the energy of the systems, andthe ambient conditions in which the assembly takes place[23].The variables with which we examine physical assem-blies are chosen such that they render the Hamiltonianlocal. Cumulative geometric frustration then arises fromnon-local compatibility conditions relating these vari-ables and their spatial derivatives. Using these prin-ciples we were able to engineer a lattice spin systemthat exhibits cumulative geometric frustration. The sys-tem shows non-uniform response that depends on the di-mensions of the domain examined and in particular wedemonstrate the system possesses a potential pathwayto filamentation. The great advantage of a controllablespin model is in its capability to unveil the elementaryphysical principles governing the behavior of frustratedassemblies, and in particular the path between cumula-tive to non-cumulative frustration.While there are distinct mechanisms for the generationof frustration, seemingly disparate systems whose com-patibility conditions share the same structure will displaya universal response to frustration. Thus, gaining betterunderstanding of the foot-prints and hallmarks of cumu-lative geometric frustration in liquid crystals and lattice-
FIG. 1. (a) Up-pointing and down-pointing triangles notations for vertices. (b) Sample of the lower left corner of the 34 × b − b (center)and color-map of s (right), all taken from a simulation result of a 34 ×
34 sites lattice. Colors are according to the color-barto the right. (d) Plot of energy per triangle vs. the length of the system (number of triangles in edge). Isotropic systems(width=length) of different sizes are marked in blue circles, and rectangular systems (length ≥ width) of 20, 30 and 40 triangleswide are marked as red diamonds, green asterisks and purple squares, respectively. Frank free energy constants are set K = K .Inset displays logarithmic plot of energy per triangle vs. the area of the system (total number of triangles). Linear scaling linewith area is added to guide the eye. spin-models may lead to a deeper understanding of therole of frustration in the shaping of complex naturallyoccurring assemblies of ill-fitting elements ranging fromamyloid structures to sickled hemoglobin fibers. ACKNOWLEDGMENTSAppendix A: Appendix A: Snap bracelet
Consider a thin, narrow and long ribbon of dimensions t (cid:28) w (cid:28) L (thickness, width and length respectively).Assume the ribbon possesses two equal and opposite ref-erence curvatures, yet a trivial reference metric [8, 16],i.e. ¯ a = (cid:18) (cid:19) , ¯ b = (cid:18) ¯ κ
00 ¯ κ (cid:19) . For small enough thickness the energy minimizing solu-tion is given by the bending minimizing isometry where a = ¯ a [24, 25]. For such isometries the bending energyfor a material of vanishing Poisson ratio reads: E ∝ (cid:90) (cid:90) (cid:0) (¯ κ − b ) + 2 b + (¯ κ − b ) (cid:1) d ¯ A, which is already local in the components of the secondfundamental form b αβ . The compatibility conditions pre-scribing the relations between the components of the sec-ond fundamental form read: b b − b = 0 , ∂ b = ∂ b , ∂ b = ∂ b . The first of these equations is the famous Gauss’ equa-tion, while the other two are the Peterson-Mainardi-Codazzi equations [26]. As Gauss’ equation is algebraic in the components of the second fundamental form, thelowest order of the compatibility conditions is zero.Indeed, the ground state of such a ribbon obeys theprinciple curvature directions of the reference curvatures,setting b = 0. Energy minimization respecting Gauss’constraint reads b = 0 and b = ¯ κ or b = 0 and b = ¯ κ , which in turn result in an extensive bendingenergy (to leading order). For more details the reader isreferred to [16], and its supplementary materials section. Appendix B: Appendix B: Compatibility conditionbent core spins on a triangular lattice
A single facet in the lattice has three spin degrees offreedom located at its vertices. The splay and bend as-sociated with the facet can be directly deduced from thevertices’ spin values Using equations (5). The converseis , however, under-determined; given prescribed valuesfor the bend and splay of a given facet there is a onedimensional space of solutions that corresponds to differ-ent choices of spins, as can be seen in figure 2 (e), whichdisplays two such curves that were computed numericallyfor certain choices of variables in an up-pointing and adown-pointing triangles. A pair of adjacent facets as infigure 2 (a), is associated with four distinct spin valuesand two pairs of intrinsic fields associated with the splayand bend of each of the facets. In this system the lat-ter fields fully determine the spin values. The under-determinacy present for a single facet is eliminated infacet pairs by the requirement that the resulting sharedspin values match. This can be considered as the cross-ing of the projections of the curves onto the plane of theshared spins. As can be seen in figure 2 (f), more thanone such crossing might exist.The spin values at the vertices of every two adjacentfacets are fully determined by the splay and bend val-ues of both facets. However, every single facet partici-pates in three distinct such pairs, as seen in figure 2 (a-c).The values ascribed to the spins of a facet obtained fromdifferent pairs should match, leading to a non-trivial re-lation between the splay and bend of the three facets.This over-determinacy settles with the observation thatsuch triplet have five spin degrees of freedom that aretransformed to six values of bend and splay. Three suchtriplets exists for every bulk element as seen in figure 2(d). Only two independent such relations exists per suchelement, due to transitive relations.We next consider a continuum treatment for this bulkelement. The locations of all the vertices in figure 2 whensetting the location of θ to the origin of the frame ofreference are given by: (cid:126)r = (0 , − l √ , (cid:126)r = ( l , l √ , (cid:126)r = ( − l l √ ,(cid:126)r = (0 , l √ , (cid:126)r = ( − l, − l √ , (cid:126)r = ( l, − l √ ,(cid:126)r = (0 , l √ , (cid:126)r = ( − l , − l √ , (cid:126)r = ( l , − l √ . Indices 1-6 above refer to the vertices and indices 7, 8and 9 refer to the locations of θ , θ and θ , respectively.The definitions of bend and splay are invertible andresult in: ∂ x θ = b cos θ − s sin θ and ∂ y θ = b sin θ + s cos θ .The line integral of the gradient of θ for every closed loopmust vanish. This condition results in:0 =( (cid:126)r − (cid:126)r ) · (cid:18) ∂ x θ ∂ y θ (cid:19) + ( (cid:126)r − (cid:126)r ) · (cid:18) ∂ x θ ∂ y θ (cid:19) ++( (cid:126)r − (cid:126)r ) · (cid:18) ∂ x θ ∂ y θ (cid:19) (B1)In the continuum limit where l is small and the changein the spin directions is also small, one can expand themean directions, splays and bends around their valuesat the location at θ , by assigning gradients in bend,splay and the mean direction. Plugging these expandedterms into equation B1 and expanding the expression inorders of l, the edge length, agrees to first order with thecompatibility condition of planar director fields, shownin equation 4. Appendix C: Appendix C: Embedding a geodesicdisc from S in E Consider a geodesic disc of radius R cut from S pa-rameterized through a polar semi-geosedic parameteriza-tion ¯ a = (cid:18) R sin( r/R ) (cid:19) , FIG. 2. (a)-(d) Notations for compatibility conditions. (a)-(c) Three types of pairs of facets in the lattice sharing anedge. (d) A quartet of connected facets in the lattice. (e)-(f)Graphic solutions for an upright pair for a choice of b = 0 . s = − . b = 0 . s = 0 .
15. The blue curve marksallowed spin orientations considering the down-pointing tri-angle alone and the orange curve is obtained by consideringthe up-pointing triangle alone. The three dimensional curvesare shown in (e), where black diagonal dashed line marksequal orientations in all the spins. (f) The projections ontothe { θ , θ } plane; crossing points marked in red. where, for transparency we explicitly retain the curva-ture radius R associated with the constant Riemanniancurvature denoted by K = R − . Any planar embeddingmay be parametrized through the standard polar semi-geodesic coordinates which may be in turn recast usingthe reference coordinates through ds = dρ + ρ dθ = ρ (cid:48) ( r ) dr + ρ ( r ) dθ The covariant components of the strain thus read ε = 12 (cid:18) ρ (cid:48) − ρ − R sin( r/R ) (cid:19) . (C1)We next make use of the metric description of elasticity[8, 24], which is particularly suited for residually stressedsolids. For simplicity we consider an elastic media of van-ishing Poisson ratio, and renormalize the Young’s mod-ulus to unity to obtain E = (cid:90) r max (cid:90) π ε αβ ε βα R sin[ r/R ] dθdr, where the mixed index strain tensor ε βα = a αγ ε γβ reads¯ a − ε = 12 (cid:32) ρ (cid:48) − ρ R sin( r/R ) − (cid:33) . The elastic energy reduced to E = π (cid:90) r max (cid:32) ( ρ (cid:48) − + (cid:18) ρ R sin( rR ) − (cid:19) (cid:33) · R sin[ rR ] dr, Numerically minimizing the above functional yields E ∝ r max , as can be observed in figure 3.We next come to consider the compatibility conditionsthat must be satisfied for a metric a (and consequentlythe strain ε = ( a − ¯ a )) to describe a valid configurationin R . The necessary and sufficient conditions in thiscase are the vanishing of all components in the Riemanncurvature tensor. For two dimensions (2D) this yieldsjust one equation, proportional to the Gaussian curvatureof the metric a .The form of the strain that appears in equation C1 andthat is subsequently used in the numerical minimizationcomes from a configuration and thus is, by definition,compatible. To unveil the frustrated nature of this sys-tem we na¨ıvely write a = ε + ¯ a, making no assumptionsregarding the structure of the strain, ε . We expand thecompatibility condition expressed in terms of the strainin orders of the spatial coordinates. To zeroth order weobtain the non-homogeneous equation O ( (cid:15) ) + 1 R = 0 , where the O ( (cid:15) ) contains second derivatives of the strainand thus leads to γ = 2.We now repeat the above exercise with a different mea-sure of strain and a different elastic energy. As the sys-tem displays small strains, we expect all descriptions toagree and in particular expect to obtain a similar scalingexponent. The elastic energy we employ follows from adeformation gradient approach, which yields an explicitlysolvable Euler Lagrange equation. In this approach thebasic variable is the deformation gradient F = ∇ r , whichsatisfies F T F = g . The deformation gradient is com-pared with a reference value ¯ F that satisfies ¯ F T ¯ F = ¯ g ,and is defined up to a rigid rotation. This reference valueis termed the “virtual” deformation gradient [27, 28], asin frustrated systems it does not correspond to a gradi-ent of a configuration. Rigid motions, and in particularrigid rotations, lead to no elastic distortions yet vary F .This manifests in the elastic energy, which measures the distance of ¯ F − F from SO (2), eliminating the associatedfreedom of a rigid rotation from the elastic energy: E = (cid:90) dist (cid:0) ¯ F − F, SO (2) (cid:1) d ¯ A, where SO (2) denotes the family of orientation preservingrotations. This energy is better behaved mathematicallycompared with the metric description (as it penalizes lo-cal inversions), and thus is often favored as the startingpoint for formal Γ − limits calculations [29]. However, itis less accessible geometrically and often intractable ana-lytically rendering it difficult to apply in many practicalsettings. In the present case, the high symmetry of theproblem and expected solution implies that the radialand azimuthal directions keep their orientations. As aresult we identify the identity as the member of SO (2)closest to ¯ F − F and obtain the simple energy E = π (cid:90) r max (cid:18) ( ρ (cid:48) − + ( ρR sin( rR ) − (cid:19) R sin[ rR ] dr. This leads to a linear Euler Lagrange equation( ρR sin( rR ) − − ddr (cid:0) ( ρ (cid:48) − R sin[ rR ] (cid:1) = 0 , supplemented by the boundary conditions ρ (0) = 0 and ρ (cid:48) ( r max ) = 1; the solution for which reads: ρ ( r ) = − R tan (cid:16) u R (cid:17) (cid:16) cot (cid:16) r max R (cid:17) log (cid:16) cos (cid:16) r max R (cid:17)(cid:17) + cot (cid:16) u R (cid:17) log (cid:16) cos (cid:16) u R (cid:17)(cid:17)(cid:17) . Figure 3 shows that the elastic energy C of this con-figuration leads to a similar energy growth exponent.The compatibility condition for F , however, lead to afirst order set of equations, as they arise from ∇ × F = 0.To obtain the exponent γ from this approach we write thedeformation gradient in terms of the elastic strain F = ε + ¯ F (note that this ε is not symmetric). We then writethe compatibility conditions without any assumptions asto the form of ε . For transparency we will use a Cartesiancurl operator, which requires that we express ¯ F in itsCartesian form¯ F cart = O T ¯ F J = κ − / x √ κ + Ry sin (cid:16) √ κR (cid:17) xy √ κ − Rxy sin (cid:16) √ κR (cid:17) xy √ κ − Rxy sin (cid:16) √ κR (cid:17) y √ κ + Rx sin (cid:16) √ κR (cid:17) , where O = (cid:18) cos( θ ) sin( θ ) − sin( θ ) cos( θ ) (cid:19) is the rotation from Carte-sian to polar directions, J = ∂ ( r,θ ) ∂ ( x,y ) is the Jacobian ma-trix associated with the coordinate transformation and FIG. 3. Numeric results and analytic solutions of embedding isotropic geodesic domains with positive Gaussian curvatures inEuclidean space. (a) Results of the embedding a geodesic disc from S in E . (b) Results of the embedding a geodesic spherefrom S in E . Numeric results for the minimization of the energy functional resulting from the metric description are markedin blue filled circles while the results attained in the case of the deformation gradient description are marked in red diamonds.Analytic solution is marked in dashed line. Notice the logarithmic scale in both mass and energy. Scaling lines of M and M / are added to guide the eye. κ = x + y . The equations ∇ × F = 0 read0 = (cid:18) ∂ y ε − ∂ x ε ∂ y ε − ∂ x ε (cid:19) + x (cid:18) ∂ x ∂ y ε − ∂ x ∂ x ε t + ∂ x ∂ y ε − ∂ x ∂ x ε (cid:19) + y (cid:18) − t + ∂ y ∂ y ε − ∂ x ∂ y ε ∂ y ∂ y ε − ∂ x ∂ y ε (cid:19) + O ( x + y ) , where all the derivatives of the strain components areestimated at the origin, x = y = 0. Note that in this casethe zero order of the compatibility condition is homo-geneous, making the first order (which includes secondderivatives of the strain) the leading non-homogeneousterm. We may thus claim that γ = 2 in this case as well. Appendix D: Appendix D: Embedding a geodesicdisc from S in E Similarly to the procedure carried out for the 2D casewe begin by considering a finite domain cut from S pa-rameterized by spherical coordinates. With respect tothese coordinates the reference metric reads¯ g = R sin( r/R ) sin( φ )
00 0 R sin( r/R ) . Again, we assume that the embedding preserves thespherical symmetry and thus can be given in terms ofa single radial function ρds = dρ + ρ sin( φ ) dθ + ρ dφ = ρ (cid:48) ( r ) dr + ρ ( r ) sin( φ ) dθ + ρ ( r ) dφ . The resulting elastic energy thus reads E = π (cid:90) r max (cid:18) ( ρ (cid:48) − + 2( ρ R sin( rR ) − (cid:19) · R sin[ rR ] dr. The minimal values of the above energy for various valuesof r max are presented in Figure 3, and follow E ∝ M / ,where M (reference volume) is the mass of the regionconsidered.Similarly to the calculation carried out for the 2D case,in order to obtain the form of the compatibility con-ditions one needs to consider a na¨ıve approach wherethe form of a is not presumed to come from an embed-ding. The compatibility conditions again correspond tothe vanishing of all components of the Riemann curvaturetensor, yet for three dimensional (3D) manifolds thereare six independent components (from which we can con-struct three independent scalar equations). The resultingrelations, much like the case for 2D, are second order dif-ferential equations whose zeroth order (expanded in thespatial coordinates) yields a non-homogeneous relation.Thus, here as well we obtain γ = 2 and the exponent 7 / E = π (cid:90) r max (cid:18) ( ρ (cid:48) − + 2( ρR sin( rR ) − (cid:19) R sin[ rR ] dr. Figure 3 presents the minimal values of the above energyfor various r max values, following the same exponent asthe metric description.Much like the 2D case, the compatibility conditionshere form a linear set of equations as they too arise from ∇ × F = 0. However, for 3D these result in nine equa-tions. We again seek to implement the curl in Cartesiancoordinates. We thus write ¯ F cart = O T ¯ F J , where O = cos( θ ) sin( φ ) sin( θ ) sin( φ ) cos( φ ) − sin( θ ) cos( θ ) 0cos( θ ) cos( φ ) sin( θ ) cos( φ ) − sin( φ ) and J = ∂ ( r, θ, φ ) ∂ ( x, y, z ) , are the rotation matrix transforming between Cartesianand spherical directions, and the associated Jacobian ma-trix, respectively. The zeroth order of the nine equationsthat arise from ∇ × ( ¯ F cart + ε ) = 0 yields only homoge- neous first order differential equation for the strain, muchlike the case for 2D. The next order (linear in the coordi-nates) yields non-homogeneous equations and thus givesas well γ = 2. [1] G. Meng, J. Paulose, D. R. Nelson, and V. N. Manoha-ran, Science , 634 (2014).[2] G. M. Grason, arXiv:1909.05208 [cond-mat] (2019),arXiv:1909.05208 [cond-mat].[3] S. Armon, H. Aharoni, M. Moshe, and E. Sharon, SoftMatter , 2733 (2014).[4] M. Zhang, D. Grossman, D. Danino, and E. Sharon,Nature Communications , 3565 (2019).[5] I. Levin and E. Sharon, Physical Review Letters ,035502 (2016).[6] D. Grossman, E. Sharon, and E. Katzav, Physical Re-view E , 022502 (2018).[7] G. M. Grason, The Journal of Chemical Physics ,110901 (2016).[8] E. Efrati, E. Sharon, and R. Kupferman, Soft Matter ,8187 (2013).[9] S. Schneider and G. Gompper, EPL (Europhysics Let-ters) , 136 (2005).[10] I. R. Bruss and G. M. Grason, Proceedings of the Na-tional Academy of Sciences , 10781 (2012).[11] I. Niv and E. Efrati, Soft Matter , 424 (2018).[12] T. Gibaud, E. Barry, M. J. Zakhary, M. Henglin,A. Ward, Y. Yang, C. Berciu, R. Oldenbourg, M. F. Ha-gan, D. Nicastro, R. B. Meyer, and Z. Dogic, Nature , 348 (2012).[13] A. Haddad, H. Aharoni, E. Sharon, A. G. Shtukenberg,B. Kahr, and E. Efrati, Soft Matter , 116 (2019).[16] S. Armon, E. Efrati, R. Kupferman, and E. Sharon,Science , 1726 (2011).[17] R. B. Meyer, in Molecular Fluids, Les Houches Lectures,1973 , edited by Balian R. and Weill G. (Routledge, 1976).[18] P. Ronceray and B. Le Floch, Physical Review E ,052150 (2019).[19] E. Y. Urbach and E. Efrati, Science Advances ,eabb2948 (2020).[20] I. Griniasty, H. Aharoni, and E. Efrati, Physical ReviewLetters , 127801 (2019).[21] D. M. Hall, I. R. Bruss, J. R. Barone, and G. M. Grason,Nature Materials , 727 (2016).[22] G. M. Grason, Physical Review E , 031603 (2012).[23] M. F. Hagan and G. M. Grason, arXiv:2007.01927 [cond-mat, physics:physics, q-bio] (2020), arXiv:2007.01927[cond-mat, physics:physics, q-bio].[24] E. Efrati, E. Sharon, and R. Kupferman, Physical Re-view E , 016602 (2009).[25] M. Lewicka and M. R. Pakzad, ESAIM: Control, Opti-misation and Calculus of Variations , 1158 (2011).[26] Gray Alfred, Abbena Elsa, and Salamon Simon (CRCPress, 1997) second edition ed., p. 600.[27] A. Green and P. Naghdi, International Journal of Engi-neering Science , 1219 (1971).[28] A. Hoger, Journal of Elasticity , 125 (1997).[29] G. Friesecke, R. D. James, and S. Muller, Archive forRational Mechanics and Analysis180