Geometric frustration and compatibility conditions for two dimensional director fields
GGeometric frustration and compatibility conditions for two dimensional director fields
Idan Niv and Efi Efrati ∗ Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Isreal (Dated: June 27, 2017)The uniform director field obtained for the nematic ground state of the hard-rod model of liquidcrystals in two dimensions reflects the high symmetry of the constituents of the liquid [1]; It is amanifestation of the constituents’ local tendency to avoid splaying and bending with respect to oneanother. In contrast, bent-core (or banana shaped) liquid-crystal-forming-molecules locally favor astate of zero splay and constant bend. However, such a structure cannot be realized in the plane [2]and the resulting liquid-crystalline phase is frustrated and must exhibit some compromise of thesetwo mutually contradicting local intrinsic tendencies. The generation of geometric frustration fromthe intrinsic geometry of the constituents of a material is not only natural and ubiquitous but alsoleads to a striking variety of morphologies of ground states [3, 4] and exotic response properties [5].In this work we establish the necessary and sufficient conditions for two scalar functions, s and b todescribe the splay and bend of a director field in the plane. We generalize these compatibility condi-tions for geometries with non-vanishing constant Gaussian curvature, and provide a reconstructionformula for the director field depending only on the splay and bend fields and their derivatives. Last,we discuss optimal compromises for simple incompatible cases where the locally preferred values ofthe splay and bend cannot be globally achieved. The curved geometry of bent-core liquid crystal form-ing molecules is naturally associated curving of the direc-tor field along the long axis of the molecule, while mini-mizing the volume per particle locally favors packing themolecule parallel to one another across the long direc-tion rendering them uniformly spaced. Mathematicallythese tendencies are associated with a preferred state of aconstant non-zero bend and vanishing splay, respectively[6]. However, as is well known [2] (and derived below),there exists no director field in the plane that displaysa vanishing splay and constant non-zero bend. The in-accessibility of this uniformly bent ground state is, atleast partially, responsible for the plethora of locally sta-ble morphologies observed for bent core liquid crystalsincluding columnar, smectic and polar chiral phases [7–9]. To adequately describe the order parameter of suchliquid crystals one has to employ a third rank tensor inaddition to the vector and the second-rank tensor orderparameters [7]. However, much of the rich structure ofthese phases, and in particular the notion of geometricfrustration already arises when considering the directorfield alone.The case of bent core liquid crystals constitutes a par-ticular example of what has been recently termed frus-trated assemblies [3] and has seen numerous experimentalrealizations using colloidal particles [10, 11]. While onecould pursue the individual equilibria shapes through di-rect molecular simulation, unravelling the general princi-ples governing the assembly of such frustrated structuresrequires a coarse grained continuous geometric descrip-tion. Such a geometry based description could identifyuniversal modes of frustration associating seemingly dis-parate systems with the same type of geometric frustra-tion, and thus with similar morphologies and solutions. ∗ efi[email protected] In general, in such a coarse grained description the geo-metric frustration is associated with a mismatched geo-metric charge [11, 12], and one of the central challengesin formulating such a description is to properly identifythe geometric charge associated with the mutually in-compatible local preferences of the building blocks of theassembly.
FIG. 1. (a) The intrinsic geometry of a straight rod favorsa state with vanishing splay and bend. This can be real-ized by a uniform director field. (b) The intrinsic geome-try of a banana shaped rod favors a state of vanishing splayand constant bend. This state, however, cannot be achieved.Two possible compromises are presented; a state with van-ishing splay (equidistant layers) necessitates bending varia-tions as observed in the concentric circular director of thebottom panel. A constant bend state results inevitably innon-vanishing splay as seen in the non-uniform spacing in thetop panel.
In two dimensional Riemannian geometry only onesuch geometric charge exists and corresponds to the in-tegrated Gaussian curvature. Frustration in this partic- a r X i v : . [ c ond - m a t . s o f t ] J un ular case occurs when the assembling constituents areattributed one value of Gaussian curvature, while thetwo dimensional space in which they assemble is asso-ciated with a different valued Gaussian curvature. For aconstant curvature difference, as the assembly grows itaccumulates the geometric mismatching charge propor-tionally to its area. However, the energy associated withthis increasing charge as well as the deformations it givesrise to typically grow much faster than the area result-ing in size limitation of the assembled structure favoringnarrow assemblies of large aspect ratio [11–13].Focusing on director fields in two dimensions we aim toformalize the notion of frustration for such systems andaddress the possible ways to resolve this frustration. Inthis work we provide an explicit formula for the Gaussiancurvature associated with given splay and bend fields,thus mapping the geometrically frustrated problem ofbent-core liquid crystals (or any other local splay/bendtendency) in two dimensions to the much studied realmof optimal embedding of manifolds of mismatched Gaus-sian curvatures. A problem encountered in elasticity [13–15], crystal growth of curved surfaces [11, 12] and thebundling of twisted filaments [16, 17]. I. PRE-COMPATIBILITY CONDITIONS INTHE PLANE
We start by considering a director field ˆn in the plane,setting ˆn = (cos( θ ) , sin( θ )) and ˆn ⊥ = ( − sin( θ ) , cos( θ )).In this case the splay, s , and bend, b , of the director fieldmay be related to the directional derivatives of the angle θ that the director forms with the x -axis. s = ∇ · ˆn = θ y cos θ − θ x sin θ = ˆn ⊥ · ∇ θ,b = | ˆn × ∇ × ˆn | = | ˆn · ∇ ˆn | = | θ x cos θ + θ y sin θ | = | ˆn · ∇ θ | . (1)When the bend is bound away from zero we may replacethe non-negative definition of the bend used above withthe signed bend b = ˆn · ∇ θ . For simplicity we assumethroughout what follows that the bend is non-negative.The above relations may be inverted to give the deriva-tives of θ as a function of the bend and splay. ∇ θ = b ˆn + s ˆn ⊥ = (cid:18) cos θ − sin θ sin θ cos θ (cid:19) (cid:18) bs (cid:19) . (2)Equations (2) may be considered as two first order non-linear partial differential equations for θ given the splayand bend fields. These equations may be integrated togive the director only if the mixed second derivatives of θ commute, i.e. ∂ x ∂ y θ = ∂ y ∂ x θ . In other words we verifythat the field ∇ θ is indeed conservative. This conditionmay be compactly written as ∇ × ( b ˆn + s ˆn ⊥ ) = ∇ · ( s ˆn − b ˆn ⊥ ) = 0 . When the definition of the bend and splay as expressedin (2) are substituted into this equation we obtain the pre-compatibility equation: b + s + ˆn · ∇ s − ˆn ⊥ · ∇ b = 0 . (3)This equation provides a necessary condition for the ex-istence of a director field with bend b and splay s . It iscalled pre-compatibility as it contains explicitly (through ˆn and ˆn ⊥ ) the function θ whose existence is sought.Nonetheless it already captures the textbook version ofincompatible fields [2]. Considering only uniform fields,then all gradients in equation (3) vanish and we obtain b + s = 0, implying that in this case only the trivialsolution s = b = 0 is admissible. II. PRE-COMPATIBILITY INNON-EUCLIDEAN GEOMETRIES
We now come to consider the more general setting inwhich the director field is defined on a surface S (not nec-essarily Euclidean). There is a natural local orthogonalcoordinate system, ( u, v ), induced by the director field inwhich the u -parametric-curves (along which v = const )are everywhere tangent to ˆn , and the v -parametric-curves(along which u = const ) are everywhere tangent to ˆn ⊥ .See figure 2 and appendix A for more details on the ex-plicit construction.With respect to the parametrization of S given by r ( u, v ), we have ∂ r /∂u ≡ r u = α ˆn , and ∂ r /∂v ≡ r v = β ˆn ⊥ . (4)There is a gauge freedom for each of the arc length func-tions which we eliminate by setting α ( u, v = 0) = 1 and β ( u = 0 , v ) = 1. Up to the choice of the location ofthe origin, this results in a unique Riemannian metricreading dσ = α du + β dv . (5)With respect to this parametrization the directionalderivatives along the director and its normal are asso-ciated with the arc-length differentiation along the coor-dinates u and v respectively:( ˆn · ∇ ) f = 1 α ∂f∂u (cid:12)(cid:12)(cid:12)(cid:12) v , and ( ˆn ⊥ · ∇ ) f = 1 β ∂f∂v (cid:12)(cid:12)(cid:12)(cid:12) u . (6)We note that for a general director field on a curvedsurface one could define two distinct bend measures. Thefirst, purely intrinsic and measures the geodesic curvatureof the director’s integral curve on the surface, κ g . Thesecond measures the curvature of the director field inte-gral curve in three dimensions, i.e. | ∂ σ ∂ σ γ | = κ , where σ denotes the arc-length parameter along the integral curve γ . This measure depends on the specific embedding ofthe surface and relates to the geodesic curvature through κ = κ g + κ n , where κ n is the normal curvature of thesurface along the direction of ˆn . For the case of a directorfield in the plane, these two measures coincide. Seeking u v ↵ ( u, v ) ( u, v ) FIG. 2. Director coordinate system. We construct the coordi-nate system by following the integral line of the director fieldthis is the u axis. As we go on u we can draw lines going onthe normal direction which point in the v direction. Movinga v line along the u direction we see that its 2 sides couldmove at different paces. The velocity of the point ( u, v ) inthis motion is α ( u, v ). We rescale the v coordinate such that α ( u = 0 , v ) = 1. We repeat the process for the other directionwith β . to obtain a strictly intrinsic relation between the splayand bend, in what follows we identify the bend with thefirst, purely intrinsic definition of the bend. In agreementwith the flat case the splay is defined through s = ∇ · ˆn .Applied to the metric (5) these definitions yield s = β u αβ , and b = − α v αβ . (7)One can calculate the Gaussian curvature, K , of thesurface r directly from its metric [18] through K = − (cid:18) − α v αβ (cid:19) − (cid:18) β u αβ (cid:19) − α ∂ u β u αβ + 1 β ∂ v α v αβ == − b − s − α s u + 1 β b v == − b − s − ˆn · ∇ s + ˆn ⊥ · ∇ b. (8)Similarly to equation (3) this equation explicitly containsthe director ˆn whose existence is sought and thus is sim-ilarly termed the pre-compatibility condition. It consti-tutes a necessary condition for the existence of an embed-ding of director field ˆn with splay s and intrinsic bend b on a surface of Gaussian curvature K . We can now easilyidentify (3) with the particular case where K = 0.Considering again the special case of uniform splay andbend fields, where all gradient vanish, the explicit depen-dence of the precompatibility condition on ˆn is eliminatedand we obtain the compact form b + s = − K. (9)The case for K = 0, solved in the previous section ad-dmitted only the trivial solutions s = b = 0. For dome-like surfaces, where 0 < K , even the trivial solution is not admissible and we obtain that this geometry cannotsupport any director field with constant bend and splay.Last, for saddle-like surfaces, where K <
0, non-trivialsolutions with constant bend and splay exist, provided | K | is large enough such that the splay obtained through s = (cid:112) | K | − b is real. III. THE RECONSTRUCTION FORMULA ANDEULERIAN COMPATIBILITY CONDITIONS
The precompatibility condition (8) constitutes a nec-essary condition for the existence of a director field withbend b and splay s . Its satisfaction, however, does notnecessarily imply the existence of such a director, as inobtaining this form we have made explicit use of the def-inition of splay and bend fields: b = | ˆn · ∇ ˆn | , s = ∇ · ˆn . (10)We next rewrite the precompatibility condition as an ex-plicit reconstruction formula for ˆn as a function of thesplay and bend fields and their gradients. However, thisreconstruction will be meaningful only if it trivially sat-isfies the above definition for the bend and splay.We first seek to eliminate the unit vector ˆn ⊥ from (8).To do so we define the operator J to be a π/ ˆn = ˆn ⊥ , and identify that ˆn ⊥ · ∇ b = J ˆn · ∇ b = − ˆn · J ∇ b . We thus may reinterpret (8) as theprojection of the director on the vector V = ∇ s + J ∇ b ,.i.e. ˆn · V = ˆn · ( ∇ s + J ∇ b ) = − b − s − K . We nowdefine the two unit vectorsˆ V = ∇ s + J ∇ b | ∇ s + J ∇ b | , ˆ V ⊥ = J ˆ V = − ∇ b + J ∇ s | ∇ s + J ∇ b | . (11)Except for the case where V = , these two unit vec-tors span the local tangent space and thus may be usedto express ˆn [19]. Equation (8) provides the coefficientsexplicitly: ˆn = − b + s + K | ∇ s + J ∇ b | ˆ V ± (cid:115) − (cid:18) b + s + K | ∇ s + J ∇ b | (cid:19) ˆ V ⊥ . (12)Note that the sign ambiguity above is artificial as onlyone of the two branches yields the correct reconstructedvalues for b and s . However, in different cases differentvalues of the sign need to be assigned, and the sign isgenerally chosen to assure continuity for the calculatedfields.The compatibility conditions are now obtained by ex-plicitly substituting (12) into the definition of the splayand bend (10). This results in two non-linear, secondorder partial differential equations for s and b . The sat-isfaction of these equations assures the existence of a di-rector field ˆn for which the splay and bend are givenrespectively by s and b . In this case the director is givenexplicitly by the reconstruction formula (12) which de-pends only on the splay and bend fields and their gradi-ents. This is in contrast with the integral formulae forreconstructing a vector field from knowledge of it curland divergence fields as given by the Helmholtz theorem(see [20]). While the constraint for the unit length of thedirector field, | ˆn | = 1 resulted in the non-trivial compati-bility conditions, it also allowed the reconstruction of thedirector without resorting to the use of integral formulas.Instead we can calculate the director from knowledge ofthe curl and divergence fields and their gradients alone.The actual form of the compatibility conditions fol-lowing this Eulerian formulation is very cumbersome andopaque, and only rarely admits an explicit solution. Itmay be useful for the specific cases where the bend andsplay fields are given explicitly and one only seeks to findif they indeed describe a valid director field. In orderto use the compatibility conditions to explicitly obtain asolution we formulate them using the director’s naturalcoordinate system. IV. LAGRANGEAN FORMULATION OFCOMPATIBILITY CONDITIONS
The vectorial formulation of the compatibility condi-tions appearing in the previous section can be appliedto any set of curvilinear coordinates parameterizing thesurface of Gaussian curvature K . However, if the coordi-nates are to follow the directions of the director field, asin figure 2, then knowing the way these coordinate behavein space is equivalent to solving for the director. Such aLagrangean description of the splay and bend fields isof great interest as it provides a natural parameteriza-tion for the director and may better capture the intrinsicnature of the attempted splay and bend fields. In thisparametrization, however, the compatibility conditionsassume a slightly different form.If a metric dσ = α du + β dv , satisfies (8), mean-ing its Gauss curvature as calculated from the metricmatches the curvature of its embedding space, K , thenone can find an isometric embedding of this metric that isunique up to rigid motions. The unknowns in the metricare related to the splay and bend through equation (7).We use the compatibility condition to relate α and β andthen eliminate one of them in (7). For example we mayuse β = b v K + b + s + α s u (13)to express α u and α v as a function of α, b and s [21].Here too the obtained equations may be interpreted asfirst order PDE defining α that permit a solution onlyif ∂ v α u = ∂ u α v . Whenever this solvability conditionis not trivially satisfied we can obtain from it a cubicpolynomial equation for α . The solution to this polyno-mial equation, together with β defined through (13) aresubstituted to equation (7) providing the compatibilityconditions for the s and b fields in the natural u and v coordinate system. V. USING THE COMPATIBILITYCONDITIONS TO OBTAIN EXPLICITSOLUTIONS
We now come to exploit the compatibility conditionsin order to obtain the equilibrium solution for the in-compatible case of bent core liquid crystals. For suchsystems, the nematogen favors a state of vanishing splayand constant bend. As was previously shown, for ex-ample in equation (9), such a state cannot exist in theplane. Thus every realized configuration will inevitablycontain some compromise between the mutually contra-dicting bend and splay tendencies. The specific equilib-rium will naturally depend on both the dimensions of theregion under consideration and on the relative magnitudeof the Frank energy constants. For a positively definedbend, b , the Frank free energy for a 2D bent-core liquidcrystal is given by [22] E = (cid:90) (cid:104) K s + K (cid:0) b − ¯ b (cid:1) (cid:105) dA. (14) A. Splay free director field
We begin by examining the case where K (cid:28) K .In this limiting case one expects that whenever possiblethe solution will display vanishing splay, and minimizethe remaining energy with respect to the bend amongall such splay free solutions. Setting s = 0, equation(7) reads ∂ u β = 0. We are allowed to set the initialvalue of β along the u = 0 curve, as explained in Fig-ure 2, β ( u = 0 , v ) = 1, which leads after integration to β ( u, v ) = 1. Thus, the natural coordinate system forms asemi-geodesic parametrization for the plane [23]. Whensubstituted into (8) we obtain for the bend the equation b v = b + K , which for the case K = 0 yields b = b − b v . (15)The metric coefficient is given by α = b /b = 1 − b v ,corresponding to a director oriented along concentric cir-cles as appearing in Figure 1(ii). The function b is inturn chosen such as to minimize the energy (14). Con-sidering a finite narrow region in space, 0 ≤ u ≤ L and − w/ ≤ v ≤ w/
2, where w/L (cid:28) w ¯ b (cid:28) b ( u ) ≈ ¯ b (cid:0) − w ¯ b / (cid:1) . The energyassociated with this state of non uniform bend to leadingorder in the width reads E ≈ K ¯ b w L (cid:18) − ¯ b w (cid:19) . Note that this energy grows super-extensively for do-mains of constant aspect ratio; if wL = A and w/L isheld constant then to leading order the energy per unitarea grows according to E/A ∝ A . However, if the widthis held constant and only the length is changed then E/A is also constant. Such behavior is very typical of geomet-rically frustrated assemblies and often leads to filamen-tation [3, 11–13, 16] .
B. Constant bend director field
We now consider the opposite limit in which K (cid:28) K . Similarly we now expect that, whenever possible,the system will assume a state where b = ¯ b and minimizethe remaining energy with respect to the splay among allsuch constant bend solutions. Setting ¯ b = b = − α v /αβ we obtain from equations 7 and 8 ∂ u (cid:0) β ( s + ¯ b ) (cid:1) = 0 , ¯ bα = − ∂ u arctan( s/ ¯ b ) . Upon substiting s = ¯ b tan(Θ) we obtain β = c ( v ) cos(Θ),for some arbitrary function c ( v ), and ¯ b = − α ∂ u Θ. Thecompatibility conditions reduce to ∂ u (cid:0) Θ v + ¯ bc ( v ) sin(Θ) (cid:1) = 0 . Integrating the equation above produces an arbitraryfunction of u . Considering a narrow ribbon − w/ ≤ u ≤ w/ − L/ ≤ v ≤ L/ w/L (cid:28) w ¯ b (cid:28)
1, we may set the arbitrary function to zero andalso assume that s ( u = 0 , v ) = 0 [24]. We are thus led totan(Θ /
2) = A ( u ) B ( v ) , where setting the initial values of the metric coefficients α ( u, v = 0) = 1 and β ( u = 0 , v ) = 1 yields A ( u ) = − tan(¯ bu/
2) and B ( v ) = exp( − ¯ bv ). Similarly to the splayfree case, to leading order the Franck energy scales as E ≈ K b w L . C. Equal Frank coupling constants
Due to its symmetry, the particular case of K = K ,often termed the isotropic case, allows a particularly el-egant solution. In this case the Euler Lagrange equationtakes the form αs v + βb u = 0 . Differentiating (7) yields αs v = ∂ v ( αs ) − sα v = ∂ u ∂ v log( β ) + s b α β,βb u = ∂ u ( βb ) − bβ u = − ∂ v ∂ u log( α ) − s b α β, thus producing for the equilibrium condition ∂ u ∂ v log( β/α ) = 0, for which the most general so-lution reads α = A ( u ) η ( u, v ) , β = B ( v ) η ( u, v ) , for some three functions A ( u ) , B ( v ) and η ( u, v ). Werecall that any substitution U ( u ) and V ( v ) preservesthe nature of the parametric curves as pointing along and perpendicularly to the director field, respectively.Through such a transformation we loose the gauge free-dom we had in determining α and β along certain curves,but are allowed to set A ( u ) = B ( v ) = 1. This impliesthat there exists a conformal map such that the directorlines are the image of the cartesian x-parametric curves.The compatibility condition becomes − (cid:52) log( η ) η = K, while the splay and bend read s = η u η , b = − η v η . Considering the Euclidean case in a rectangular domainand setting λ = log( η ) we obtain that (cid:52) η = 0 in thebulk, and use the boundary terms from the Euler La-grange equations to close the boundary value problem.The central panel of Figure 3 displays such a solution. FIG. 3. Three limiting solutions for an attempted constantbend vanishing splay director in the plane. Zero splay (leftpanel), constant bend (right panel), and equal coupling con-stants (central panel). Top panels show the splay fields, andbottom panels show the bend field. Thin (black) lines denotethe director integral curves.
VI. DISCUSSION
In this work we identify the geometric charge thatcorresponds to the local intrinsic tendencies of the con-stituents of a frustrated two dimensional liquid crystal byproviding an explicit formula for the Gaussian curvatureassociated with given splay and bend fields. Through thismapping we obtain access to many optimal embedding re-sults that are formulated for different systems displayingtwo dimensional geometric frustration.Identifying the set of compatible intrinsic tendencieswill also allow to harness the incompatibility in liquidcrystals to produce simpler building blocks and moreelaborate structures. In Figure 4 a desired state charac-terized by spatial gradients of the splay and bend fieldsis achieved by uniform building blocks (that in particularfavor no spatial gradients). The uniform phase, charac-terized by the non-zero constant bend and splay, favoredby the building blocks is not compatible with the flatgeometry, and the closest compatible state, the desiredone, is realized instead. The Franck-free energy serves asa “metric” for the configuration space determining whichof the compatible states is closest to the attempted in-compatible one.
FIG. 4. A three dimensional projection of the infinite dimen-sional space describing the local geometry of a 2D liquid crys-talline phase. The splay, bend , and a combination of directedderivatives estimated at the center of a small square patch.The dimensions of the patch are smaller than the length scalesassociated with the bend and splay and their gradients, andthe Frank energy coefficients are assumed equal. Uniformbend and splay fields are mapped to the horizontal (blue)plane, and the compatible phases are mapped to the (orange)paraboloid. A desired state (necessarily compatible) possess-ing gradients is marked by a black sphere. The constant bendand splay configuration marked by the light sphere is cho-sen such that the desired state will be the closest compatiblestate.
We presented the compatibility conditions throughboth an Eulerian approach and a Lagrangean (intrin-sic) approach. In the former the splay and bend fieldsare given in terms of the lab frame cartesian coordinates,i.e. s ( x, y ) and b ( x, y ), and the derivation of the com-patibility conditions is more transparent. However, thistype of question is artificial as the splay and bend fieldsare provided as functions of the embedding coordinatesof the liquid crystalline phase, whose existence is soughtthrough the compatibility conditions. It is thus more nat-ural to consider the problem formulated in Lagrangeanform where the attempted splay and bend fields are areprovided as functions of a set of curvilinear coordinatesthat are oriented along and perpendicularly to the di-rector. While such a description could be more directlyrelated to the properties of the constituents of the liq-uid crystal, it renders the calculation itself more diffi-cult. The intrinsic formulation may prove useful also tocompatible systems as, for example, the Euclidean pre-compatibilty equation 3 must hold for all director fields in the plane. Recent examples where the present formu-lation provides further insight into the geometry of thedirector include the theory of thin nematic elastomers[25, 26], and in the implementation of the tensorial con-servation laws derived to describe nematic polymers [27].In many liquid crystalline systems, either boundaryconditions or the topology of the system lead to non-trivial order. In the frustrated case, even free boundaryproblems result in non-trivial solutions that in particu-lar depend on the relative value of the Frank couplingconstants. The limit K /K → b − ¯ b ) with re-spect to all splay free solutions. A similar scenario occursin the opposite limit where K /K →
0, and the bend-ing tendency is obeyed. The numerical results for bentcore liquid crystals support the results in these limits.It is, however, a-priori unclear when could one assumethe existence of such limits. Establishing the existenceof these limits requires and a more detailed and formalmathematical treatment, and it outside the scope of thiswork.Adequately describing the frustration of bent core liq-uid crystals in three dimensions and explaining the richstructures and strong optical activity such system dis-play still evades us. Na¨ıvely one could argue that the re-sults presented here support a scenario in which in orderto simultaneously obey a zero splay and constant bendthe nematogens organize into curved surfaces of negativeGaussian curvature K = − ¯ b , that are then optimallypacked to fill a volume. The optical activity in this casecould be argued to be the outcome of the strong man-ifestation of handedness observed along asymptotic di-rections of hyperbolic surfaces, and predicted to locallyaverage to zero if all directions are considered [28]. How-ever, this na¨ıve approach not only ignores the non-trivialpacking of surfaces of constant Gaussian curvature, butalso underestimates the actual bend of the director inthree dimensions. As discussed in section II the truebend in three dimensions b D relates to the bend func-tion used here, b , through b D = (cid:112) b + k n , where k n isthe normal curvature of the surface along the directionof the director. Thus, mapping the three dimensionalproblem to a collection of two dimensional problems oncurved surfaces requires some adaptations. However, wehope that the present work will lay the foundations forthe geometric formulation of frustrated liquid crystals inthree dimensions. [1] L. 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Bre-mer, IEEE Transactions on Visualization and ComputerGraphics , 1386 (2013).[21] The above elimination assumes that either s u (cid:54) = 0 or b v (cid:54) = 0. The where case both vanish corresponds to the V = 0 discussed earlier and in appendix ?? .[22] This expression for the energy can be deduced by takingthe two dimensional and polarized limit of the polar ne-matic theory of bent core liquid crystals, as for exampleappears in [29], and identifying the polar vector with theperpendicular to the director, p (cid:107) ˆn ⊥ . For non-positivelydefined b , it is natural (albeit less convenient) to formu-late the bend term using squares: ( b − ¯ b ) .[ ? ].[23] D. Struik, Lectures on Classical Differential Geometry:Second Edition , Dover Books on Mathematics (DoverPublications, 2012).[24] . [25] H. Aharoni, E. Sharon, and R. Kupferman, PhysicalReview Letters , 257801 (2014).[26] T.-S. Nguyen and J. V. Selinger, arXiv:1612.06486 [cond-mat] (2016).[27] D. Svenek, G. M. Grason, and R. Podgornik, PhysicalReview E , 052603 (2013).[28] E. Efrati and W. T. Irvine, Physical Review X , 011003(2014).[29] Z. Parsouzi, S. M. Shamid, V. Borshch, P. K. Challa,A. R. Baldwin, M. G. Tamba, C. Welch, G. H. Mehl, J. T.Gleeson, A. Jakli, O. D. Lavrentovich, D. W. Allender,J. V. Selinger, and S. Sprunt, Physical Review X ,021041 (2016). Appendix A: Director aligned coordinate systemand precompatibility through Gauss Bonnet
Give a surface S and a director field ˆn on S we cometo construct a parametrization on S such that the para-metric curves are oriented parallel and perpendicular tothe director field. We first choose an origin, a point on S , that will be attributed the coordinates u = 0 and v = 0. From the origin we construct the integral curveof the director filed and the integral curve of the nor-mal to the director; these are termed the base curvesand are marked by (black) solid and dashed curves infigure 5. We parametrize the base curves by arclength; u measures arclength along the director integral curveand v measures arclength along the integral curve of thedirector’s normal. Provided s + b is bounded thenwe know that all ingetral curves in the domain do notcurve too much, and we can find a domain on S suchthat for every point in the domain the director, and di-rector’s normal integral curves that start at the point,each cross one of the base curves. The point inheritsits coordinates from the base curve’s arclength coordi-nate at the point of crossing. For example, in Figure 5we follow the director’s integral curve (blue dash-dottedline) from the point until we cross the dashed base-lineat the point (0 , v ). Following the director’s normal in-tegral curve (red dotted line) leads to the intersectionwith the second base curve at ( u , u , v ). Note that keepingthe v -coordinate constant and varying u thus follows adirector integral curve, and that correspondingly the v -coordinate parametric curve (along which u is constant)are director’s normal integral curves. Last we note thatby construction here α ( u,
0) = 1 and β (0 , v ) = 1. How-ever, One could reparametrize the base curve to otherthan arclength variables V ( v ) and U ( u ) while keeping theproperty that parametric curves are tangent and perpen-dicular to the local director.We can now formulate the precompatibility conditionas an immediate corollary of the Gauss-Bonnet theorem:We consider a closed curve composed of four parametric (0,v ) (u ,0) (u ,v ) FIG. 5. The origin, marked in (Red) cross, and base curvesmarked by (Black) dashed and solid curves. The (Blue) up-right arrows denote the local orientation of the director field. curves for example the trajectory(0 , → (0 , v ) → ( u , v ) → ( u , → (0 , (cid:90) (cid:90) K dA = − (cid:73) k g dl + (cid:88) i θ i − π We note that the integral curves are everywhere perpen-dicular thus all angles are right angles, and that directlyfrom the metric 5 one can show the geodesic curvatureof the director and director’s normal integral curves read k g = − b and k g = − s respectively. Thus we obtain (cid:73) k g dl = (cid:73) ( b ˆn − s ˆn ⊥ ) · d(cid:126)l. Substituting into the Gauss Bonnet theorem and usingStokes theorem we obtain (cid:90) (cid:90) [ K + ∇ · ( s ˆn − b ˆn ⊥ )] dA = 0 . As this integral equation must hold for all (and in par-ticular arbitrarily small) parametric quadrangles its in-tegrand must vanish locally and we recover the precom-patibility condition 8: K = ∇ · ( − s ˆn + b ˆn ⊥ ) . (A1) Appendix B: Sign ambiguity in the reconstructionformula
We start by considering the compatible splay and bendfields b = 1 and s = − x √ − x . The splay and bend gra-dients produce constant unit vectors : ˆ V = ( − ,
0) andˆ V ⊥ = (0 , − b + s + K | ∇ s + J ∇ b | = (cid:112) − x . Substituting into 12 gives ˆn = ( (cid:112) − x , ∓| x | ) . We note that the correct values for the bend are obtainedonly if we use ˆn y = −| x | for x < ˆn y = | x | for x > ˆn = (cid:0) √ − x , x (cid:1) is smooth, andrecovers the splay and bend functions. Appendix C: Euler-Lagrange and stability condition
We assume that the given director at equilibrium cor-responds to the splay and bend fields s and b . Thesein turn minimize the Frank energy with respect to allpossible director fields. Small variations of the directorcorrespond to locally small rotations by a small angle θ : ˆn (cid:48) = cos( θ ) ˆn + sin( θ ) ˆn ⊥ , where θ (cid:28)
1. Explicit sub-stitution in equation 1 shows that such an infinitesimalrotation leads to variations in the splay and bend accord-ing to s (cid:48) = s − b θ + θ v βb (cid:48) = b + s θ + θ u α We now calculate the energy difference to first order in θ using the unperturbed coordinate system u , and v . Note,that in particular this implies that the u direction will nolonger point along the perturbed director, and that thearea element remains unchanged.∆ E = 2 (cid:90) (cid:104) K s (cid:18) θ v β − bθ (cid:19) + K (cid:0) b − ¯ b (cid:1) (cid:18) sθ + θ u α (cid:19) (cid:105) αβ du dv. (C1)We first consider only differential rotations that vanishon the boundary of the region considered. Integrating byparts, and eliminating the boundary contribution leadsto the following Euler-Lagrange equation: K s v β + K b u α = K ˆn ⊥ · ∇ s + K ˆn · ∇ b = 0 . (C2) Appendix D: Application to compatible liquidcrystal systems
Recently, a geometric theory for thin nematic elas-tomer was presented [25]. The theory studied a thinnematic elastomer that displays a length shrinkage by afactor λ / along the director and a factor λ − ν t / in theperpendicular direction when activated. The work pro-vided a direct formula relating the Gaussian curvatureof the geometry induced by this differential shrinkage asa function of the directors angle and its first and sec-ond derivatives. As this quantity is clearly coordinateindependent, it must allow an intrinsic formulation usingthe bend and splay of the unperturbed field. Exploitingequation 3 we obtain the compact result K shrinked =( λ − λ − ν t ) (cid:0) s + ˆn · ∇ s (cid:1) =( λ − λ − ν t ) (cid:0) − b + ˆn ⊥ · ∇ b (cid:1) , where the last equality holds because the initial directorwas cast in a flat geometry.Another application can be found in the implementa- tion of the tensorial conservation laws derived to describenematic polymers [27]. In the limit of uniform densityand nematic degree the authors obtain the equation ∇ · [ ˆn ( ∇ · ˆn )] + ∇ · [( ˆn · ∇ ) ˆn )] = 0 . Considering directors with no z components we obtainan effectively planar two dimensional system, for whichthe flat precompatibility equation assumes a similar formonly with the opposite sign between the two terms. Thisimplies that each of the terms above vanish indepen-dently; s + ˆn · ∇ s = b − ˆn ⊥ · ∇ bb