Instability of toroidal nematics
aa r X i v : . [ c ond - m a t . s o f t ] A p r September 26, 2018 Liquid Crystals Toroidal˙Nematics
To appear in
Liquid Crystals
Vol. 00, No. 00, Month 20XX, 1–17
RESEARCH ARTICLEInstability of toroidal nematics
Andrea Pedrini and Epifanio G. Virga ∗ Dipartimento di Matematica, Universit`a di Pavia, via Ferrata 5, 27100 Pavia, Italy ( Received 00 Month 20XX; final version received 00 Month 20XX ) Toroidal nematics are nematic liquid crystals confined within a circular torus and subject to planardegenerate anchoring on the boundary of the torus. They may be droplets floating in an isotropic en-vironment or cavities carved out of a solid substrate. A universal solution of Frank’s elastic free energyis an equilibrium configuration for the nematic director field, irrespective of the values of the elasticconstants, whose vector lines are the coaxial parallels of the torus. We explore the local stability of thisconfiguration and identify a range of parameters where the main drive towards instability does not comefrom the surface-like elastic constant K being large, but from the the ratio K /K of the twist to bendelastic constants being small, which also makes our study relevant to chromonic liquid crystals. Keywords:
Frank’s elastic energy; universal solutions in liquid crystals; chromonic liquid crystals;linear stability.
1. Introduction
Nematic liquid crystals are characterized by a natural, undistorted state, where the director n isuniform in space (in an arbitrarily chosen direction). The most elementary measure of distortion istherefore the spatial gradient ∇ n . The simplest formula for the elastic free energy density (per unitvolume) was put forward by Frank [1]: it is the most general quadratic expression in ∇ n invariantunder rotations and complying with the nematic symmetry, embodied by the director reversion, n
7→ − n . Frank’s free-energy functional F is given by F [ n ] = Z B n (cid:2) K (div n ) + K ( n · curl n ) + K | n × curl n | (cid:3) + K [tr( ∇ n ) − (div n ) ] o d V , (1)where B is the region in space occupied by the material and d V denotes the volume element.Classically, to render (1) more symmetric, a constant K is often introduced so that K = ( K + K ). K , K , and K are often referred to as the splay , twist , and bend elastic constants, respectively,by the names given to the three fundamental distortion modes characterized by the excitation ofonly the corresponding energy (see, for example, § § K is notoriously different from the others in at least two respects, (i) itweights an energy that can only be approximately isolated by the saddle-splay distortion [3, p. 121]and (ii) it can be converted into a surface integral over the boundary ∂ B of the domain occupied ∗ Corresponding author. Email: [email protected] eptember 26, 2018 Liquid Crystals Toroidal˙Nematics by the material. The latter property says that the saddle-splay energy is a null Lagrangian , whichdoes not contribute to the Euler-Lagrange equation obeyed by the equilibrium nematic textures[4]. Moreover, it can be shown that the K -energy, once converted into a surface integral, dependsonly on n and its surface gradient ∇ s n on ∂ B . Thus, we finally give F in (1) the following form, F [ n ] = 12 Z B (cid:2) [ K (div n ) + K ( n · curl n ) + K | ( ∇ n ) n | (cid:3) d V + Z ∂ B K [( ∇ s n ) n − (div s n ) n ] · ν d A , (2)where d A is the area element and ν is the outer unit normal to ∂ B .Ericksen [5] first remarked that the free-energy density associated with F in (1) is positive semi-definite, duly representing the cost incurred in distorting the natural uniform state, only if theelastic constants obey the inequalities K > K > , K > K , K > , (3)which are also referred to as Ericksen’s inequalities.Two different types of boundary conditions for n on ∂ B give the K -energy a special form:these are the strong and the planar degenerate anchorings. In the former case, n is prescribed onthe whole of ∂ B , and so the K -integral is the same for all competing equilibrium textures, andit can be altogether ignored. In the latter case, n · ν ≡ ∂ B (4)and, as remarked in [6], the K -integral can be rewritten as − K Z ∂ B (cid:0) κ n + κ n (cid:1) d A , (5)where κ and κ are the principal curvatures of ∂ B , and n i are the components of n along thecorresponding principal directions of curvature. It is clear from (5) that for K >
0, which is thestrong form of (3), whenever (4) applies the saddle-splay energy would locally tend to orient n on ∂ B along the direction of maximum (signed) curvature. We shall see below the form that (5) takesin the stability problem studied here.To close our hasty introduction to the fundamentals of the mathematical theory of nematic liquidcrystals and to place our study in a broader perspective, we recall the meaning of universal solutions in the hydrostatics of liquid crystals. They were first considered by Ericksen [7] for a general elasticfree-energy density delivered by an isotropic function W = W ( n , ∇ n ), not necessarily quadraticin ∇ n . Ericksen proved that a locally smooth director field n solving the Euler-Lagrange equationassociated with all such functions may only have rectilinear vector lines, which are either(i) parallel straight lines,(ii) straight lines precessing in a pure twist with director cosines (cos µz, sin µz, µ is aconstant,(iii) lines orthogonal to a family of concentric spheres or(iv) lines orthogonal to a family of coaxial cylinders.Marris [8, 9] proved that the extra universal solutions afforded by Frank’s expression for the elasticfree-energy density, which then solve the Euler-Lagrange equation for F in (1) for all values of We write the curvature tensor as ∇ s ν = κ e ⊗ e + κ e ⊗ e , where e and e are unit vectors along the principal directionsof curvature of ∂ B . eptember 26, 2018 Liquid Crystals Toroidal˙Nematics Frank’s constants, are fields obtained by translating uniformly in space planar fields whose vectorlines are either(v) concentric circles or(vi) coaxial circles intersected orthogonally by all members of another family of coaxial circles.Family (v) comprises a configuration that shall particularly interest us here. Family (vi) exertsquite a classic fascination, as both pencils of circles featuring as vector lines of n are Apolloniancircles (see, for example, § toroidal nematics . They are theoretical constructs, which in realitymay be equally approximated by either toroidal droplets produced in an isotropic environment ortoroidal cavities carved out of a rigid substrate, both inducing a degenerate planar anchoring (withno preferred surface orientation).We build on a previous pioneering study [6], from which we mainly drew our inspiration, andwhich we think have contributed to improve. The merit of [6] is having shown the role of K as adriving force behind the destabilization of a universal solution. However, the method through whichsuch a role is explored is too special to be credit with a universal meaning; the second variationof F is simply shown to become negative for a cleverly chosen, but particular test function. Wewere similarly unable to characterize the sign of the second variation of F , which would haveafforded a complete local stability analysis, but we have succeeded in extending significantly thepool of destabilizing modes, thus refining the physical interpretation of the destabilizing causes.Our instability criterion systematically improves on that found in [6].The paper has the following structure. Section 2, which is preparatory in nature, is concernedwith the representation of nematic director fields in the unusual toroidal geometry. In Sect. 3,we compute in general the second variation of F on the universal solution (v) and, following afortunate intuition of [6], we specialize it to divergence-free test functions. In Sect. 4, we benefitfrom this representation of the second variation and compute it for both the test function used in[6] and a new class of test functions, which eventually afford an improvement on their instabilitycriterion. Finally, in Sect. 5, we collect the main conclusions of our study and try and extractsuggestions from the new perspective gained here on how to attack the true challenge that stillfaces us: the non-linear stability of toroidal nematics and the minimizers they fall in, once theiruniversal solution has been destabilized. A technical appendix closes the paper, where all tedious,but necessary mathematical details are relegated for the perusal of the interested reader who wishesto follow all our steps.
2. Toroidal nematic fields
This section is mainly descriptive in nature; it is concerned with the construction of a family ofnematic director fields n inspired by the toroidal symmetry of the domain B where the free energy F will be studied. Toroidal coordinates and frame
Figure 1 depicts a circular torus B , which is to be thought of as either a droplet or a cavity whoseboundary enforces the degenerate planar boundary condition on n , requiring only that n · ν ≡ n otherwise unspecified. In Fig. 1, O is the center of B with minor and majorradii R > R > R , respectively. B is axially symmetric about the z -axis of a standard(orthonormal and positively oriented) Cartesian frame ( e x , e y , e z ) in three-dimensional space. The3 eptember 26, 2018 Liquid Crystals Toroidal˙Nematics Figure 1.: The geometric representation of the surface of a circular torus B , to be identified witheither a droplet or a cavity, obtained by revolving a circle of radius R around the z -axis. Theposition of a point P in B can be parameterized by r , ϕ and ψ , as in (6). On the left: geometricrepresentation of the entire surface and the center line with radius R (in red), with azimuthalangle ϕ . On the right: geometric representation of a cross-section with radius R of the toroidalsurface (in blue), with radial coordinate r and polar angle ψ .position vector p of a generic point P within B issued from the origin O can be expressed as p := P − O = ( R + r cos ψ ) cos ϕ e x + ( R + r cos ψ ) sin ϕ e y + r sin ψ e z , (6)where r ∈ [0 , R ] is the radial coordinate , ϕ ∈ [0 , π ) is the azimuthal angle and ψ ∈ [0 , π ) is the polar angle .At any point P in B , we introduce the (orthonormal and positively oriented) toroidal frame ( e r , e ϕ , e ψ ) conjugated with the toroidal coordinates ( r, ϕ, ψ ). As shown in Fig. 2, the toroidalframe can be expressed in the Cartesian frame ( e x , e y , e z ) through the equations e r := cos ϕ cos ψ e x + sin ϕ cos ψ e y + sin ψ e z , e ϕ := − sin ϕ e x + cos ϕ e y , e ψ := − cos ϕ sin ψ e x − sin ϕ sin ψ e y + cos ψ e z , (7)see Appendix A.1 for the analytic details that Fig. 2 cannot convey. In toroidal coordinates, thevolume and area elements ared V = r ( R + r cos ψ ) d r d ϕ d ψ and d A = r ( R + r cos ψ ) d ϕ d ψ . (8)In the toroidal frame, the director field can be expressed as n = n r e r + n ϕ e ϕ + n ψ e ψ . (9)In particular, taking α and ϑ to be the angles defined as in Fig. 2, we can write n r = cos ϑ, n ϕ = cos α sin ϑ, and n ψ = sin α sin ϑ . (10)As complicated as the problem of minimizing F in the torus B may be, we know at least that thedirector field n in (9) with n r = n ψ ≡ F , which may or may4 eptember 26, 2018 Liquid Crystals Toroidal˙Nematics Figure 2.: On the left, the toroidal frame ( e r , e ϕ , e ψ ) at a point P within the torus B : e r has thesame direction of the vector P − Q , e ϕ is orthogonal to the plane containing both the z axis and P , and e ψ is tangent to the disk of radius r and centre Q on that very plane. On the right, theangles α and ϑ used to express the director field n in the toroidal frame.not be a minimizer. As recalled in the Introduction, n ≡ e ϕ is a universal solution of F , whichhere we shall call the axial configuration , for brevity. Toroidal fields
Here the stability of the axial configuration will be probed within a large, but restricted classof director fields which share one essential feature with the perturbed field: they are everywheretangent to either the torus B or any of its inwards. The axial configuration is generated by setting α ≡ ϑ ≡ π : its vector lines are the parallels of the torus, an achiral pattern, invariant undercentral inversion about O . An intrinsic way to measure the chirality of the vector lines of n iscomputing their helicity Ω, Ω := n · curl n . (11)For the axial configuration, clearly Ω = 0.As in [6], we shall also take the director field to be independent of ϕ , subject to the condition n = e ϕ for r = 0, and with no radial component (i.e., ϑ ≡ π ). Therefore n = cos α e ϕ + sin α e ψ , with α = α ( r, ψ ) and α (0 , ψ ) ≡ . (12)Specializing (A4) of Appendix A.2 to the field in (12), we easily see that the vector lines of thelatter possesses the helicity Ω = − α ,r − R r ( R + r cos ψ ) sin α cos α . (13)The latter formula makes it clear that reversing the sign of α also reverses the sign of Ω thusproducing a director field n with opposite chirality. Correspondingly, the vector lines of two con-jugated director fields as in (12) that only differ by the sign of α are mapped into one another bya central inversion about O . We learn from Truesdell [11, p. 332] that this definition of Ω was introduced by Zhukowsky and named abnormality byLevi-Civita. We shall use the modern name of helicity . Here and in the following, f ,x denotes the partial derivative of a function f with respect to one of its variables, x . eptember 26, 2018 Liquid Crystals Toroidal˙Nematics Let η := R R ∈ [0 ,
1] (14)be the ratio between the radii of the torus. By performing the change of coordinate σ := rR = rR η , (15)the angle α becomes a function of σ , and ψ and, as shown with more details in Appendix A.3,Frank’s elastic free-energy functional takes the following reduced form, F [ α ] πR = Z Z π ( K [ ησ sin ψ sin α − (1 + ησ cos ψ ) cos α α ,ψ ] σ (1 + ησ cos ψ )+ K [ σ (1 + ησ cos ψ ) α ,σ + sin α cos α ] σ (1 + ησ cos ψ )+ K [ ησ sin ψ cos α + (1 + ησ cos ψ ) sin α α ,ψ ] σ (1 + ησ cos ψ )+ K (cid:2) ησ cos ψ + sin α (cid:3) σ (1 + ησ cos ψ ) ) d σdψ , − K Z π sin α (1 , ψ ) d ψ , (16)where the last integral reflects the general surface energy (5) and its ability to destabilize the axialconfiguration for K > F . By requiring n to be divergence-free, not onlywe keep the perturbing director field in the same family as the axial configuration which we intendto perturb, but we also spare an energetic contribution, the one coming from the splay constant K , which may be dominant, as is, for example, the case for the newly discovered chromonic liquidcrystals, for which typically K ≈ K ≈ K [12]. By requiring that div n = 0, also with the aidof (A3), we obtain the following differential equation α ,ψ = ησ sin ψ tan α ησ cos ψ , (17)which can be easily integrated, delivering α = arcsin a ( σ )1 + ησ cos ψ , (18)where a ( σ ) is a real function of σ only. For (18) to obey the inequality | sin α | a must be suchthat | a ( σ ) | ησ cos ψ for all ψ ∈ [0 , π ], whence it follows that | a ( σ ) | − ησ . (19) We were not able to prove that a divergence-free field n represents the optimal way to probe the stability of the axialconfiguration. At this stage, the example of chromonics is only meant to be suggestive. They indicate that nematic liquidcrystals with high elastic anisotropies are not just a theoretical curiosity, but real-life materials. eptember 26, 2018 Liquid Crystals Toroidal˙Nematics In the following section, we shall make use of (18) in (16) and we shall write F in the quadraticapproximation for α , so as to convert it into the second variation of the elastic free-energy evaluatedat the axial configuration.
3. Second free-energy variation
For α ≡ F [0] delivers the energy associated with the axial solution, which has pure bend (seealso Appendix A.3), F [0] = 2 π R K (1 − p − η ) . (20)The extra elastic energy associated with a distortion other than α ≡ E [ α ] := F [ α ] − F [0] . (21)Taking the quadratic approximation for α in (16), we effectively compute the second variation of E at α ≡ E [ α ] πR = Z Z π ( K (cid:20) √ ησ sin ψ √ ησ cos ψ α − √ ησ cos ψ √ ησ α ,ψ (cid:21) + K (cid:20) √ ησ √ ησ cos ψ α + √ ησ √ ησ cos ψη α ,σ (cid:21) + K cos ψ − ησ ησ cos ψ α (cid:27) η d σdψ − K Z π α (1 , ψ ) d ψ (22)(see Appendix A.4 for more details). Enforcing in this context the divergence-free condition for n amounts to linearize (18) in a , which thus becomes α = a ( σ )1 + ησ cos ψ , (23)which is still subject to (19). For this latter to be valid up to σ = 1 and a (1) to remain free, thoughinfinitesimal, we shall hereafter take 0 η <
1. Making use of (23) in (22), we readily arrive at E [ a ] πR = Z Z π ( K (cid:20) − ησ cos ψ √ ησ ( √ ησ cos ψ ) a ( σ ) + √ ηση √ ησ cos ψ a ,σ ( σ ) (cid:21) + K cos ψ − ησ (1 + ησ cos ψ ) a ( σ ) (cid:27) η d σ d ψ − K Z π η cos ψ ) a (1) d ψ . (24)This form of E reveals that a must vanish as σ → a (0) = 0 , (25)7 eptember 26, 2018 Liquid Crystals Toroidal˙Nematics as a requirement of convergence for E . A number of manipulations, which also rely on (25), areneeded to convert (24) into a functional for the classical Bolza problem of the Calculus of Variationsin one dimension. All details are given in Appendix A.4, here we record only the final form of E : E [ a ]2 π R K = Z ( η σ ) + 4 η σ − η σ (2 + k ) (5 + η σ )2 σ (1 − η σ ) a ( σ )+ σ (1 − η σ ) a ,σ ( σ ) ) d σ + 1 + η − k (1 − η ) a (1) , (26)where we have scaled the elastic constants to K , assumed to be positive, k := K K and k := K K , (27)so that by (3) k > k . (28)The Euler-Lagrange equation for (26) is then12 σ [2(1 + η σ ) + 4 η σ − η σ (2 + k ) (5 + η σ )] a ( σ )= σ (1 − η σ ) a ,σσ ( σ ) + (1 − η σ ) a ,σ ( σ ) , (29)and, for 0 η <
1, the associated Robin’s condition for σ = 1 is given by(1 − η ) a ,σ (1) + (cid:0) η − k (cid:1) a (1) = 0 . (30)Unfortunately, equation (29) is too complicated to lend itself to an analytic solution, and so arigorous study of the sign of the second variation E in (24) could not be performed, and a completelinear-stability analysis of the axial configuration within a torus remains elusive. We shall becontended with exploiting (24) to explore the borders of linear instability. To this end, we usea simple consequence of (29) that must be valid for all solutions that do not vanish identically.Taking the limit as σ → + , we obtain the following asymptotic form of (29), a ( σ ) σ ≈ σa ,σσ + a ,σ , σ ≈ . (31)This is a homogeneous equation with solutions a = Aσ and a = Aσ − , with A an arbitraryconstant, only the former of which conforms with (25). Thus, the extremals of E that can make itnegative destabilizing the axial configuration are such that a ( σ ) ≈ Aσ, for σ ≈ , with A = 0 . (32)This condition, together with (30), will lead us in the following section to construct a test functionthat improves the linear instability analysis for toroidal nematics known from [6].8 eptember 26, 2018 Liquid Crystals Toroidal˙Nematics
4. Instability criteria
Before probing in a torus the stability of the universal solution in which all integral director linesare circle, we remark that since the functional in E in (26) is quadratic any perturbation a thatsucceeds in making E negative is accompanied by − a , which assigns the same value to E . Thoughmathematically this is nearly a trivial remark, physically it says that when the axial configurationbecomes unstable in a torus it gives way to two equally energetic (and so, equally likely) distortionsthat however differ in the sense their integral lines are wound around the torus. Such a chiralitydegeneracy , which was discussed at length in [6], is an intrinsic feature of our stability problem.In the following, we shall only concentrate on one variant of two possible destabilizing modes, theone for which A > E [ a ∗ ] <
0, it follows immediately from E being quadratic that it can be madeunbounded from below by just multiplying a ∗ by an increasing constant. A normalization conditionmust be added to our search for admissible test functions a ∗ : we shall take it to be a ∗ ,σ (0) = η . (33)In the light of this normalization, we may rewrite the linear test function chosen in [6] as a ∗ = a lin := ησ . By requiring E [ a lin ] <
0, we easily arrive at the inequality k > η − η + 6 − − η ) η k , (34)which, for any given η , represents the region above a straight line (with negative slope) in the strip0 k k , k ) (see Fig. 4). By use of (27), we readily reduce (34) to the form K − K K < k c ( η ) (35)with k c precisely given by (23) of [6] (after having set ξ := 1 /η ). The interesting physical inter-pretation of (35) is that the saddle-splay constant K screens the energy cost of twist, and doingso it facilitates the instability of the axial configuration, as it makes it happen for larger values of K compared to the case K = 0 (or strong anchoring is enforced on ∂ B ). As attractive as thisinterpretation may be, it geometrically relies on being the limit of stability in (34) a straight linein the plane ( k , k ) passing through the point (0 , a lin cannot possibly comply with Robin’s condition (30), which is anecessary condition for an extremum of E . The simplest way to make it valid for a test function a ∗ (together with (32) and (33)) is to choose a ∗ as the quadratic function a qua ( σ ) := ησ (1 − βσ ) , (36)where β is a constant that (30) determines as being β = 2 1 − k − η − k . (37)It is a simple matter to check that letting η vary in [0 ,
1) and k in [0 ,
1] makes β cover the interval[0 , a qua exhibits the graphs shown in Fig. 3. In particular, for < β a qua has its maximum for 0 < σ < Often, the integral of | a ∗ | is prescribed as a normalization condition. Here, we chose to prescribe instead the derivative of a ∗ at a point; the special value in (33) was taken to represent the ratio of the only two lengths present in the problem. eptember 26, 2018 Liquid Crystals Toroidal˙Nematics Figure 3.: Graph of the function a qua in (36) shown for β = 0 , , , , β delivered by (37), the graph of a qua is encapsulatedbetween the two boundary graphs shown here.With some labour, we reduce the inequality E [ a qua ] < k > K ( η, k ) H ( η, k ) , (38a)where K ( η, k ) := 4(1 − k ) h η ( η + 1) k + 4 (cid:16) − p − η (cid:17) (1 − k ) + η (cid:0) η − η − (cid:1)i , (38b) H ( η, k ) := − − k ) h η + 55 η + 18 η p − η − (cid:16) − p − η (cid:17)i − − k ) p − η (cid:2) (cid:0) k + η − (cid:1) η arcsin η + 4 η (cid:0) − η (cid:1)(cid:3) − − k ) η (3 + η )(1 − η ) + 3 η ( η − η + 6) (cid:0) − η (cid:1) − η (cid:0) − η (cid:1) (38c)Inequality (38a) is admittedly far less elegant than (34), but it also improves upon that substantiallyfor all values of 0 < η <
1, as the region of the plane ( k , k ) delimited within the admissiblestrip 0 k k the functions on the right side of (38a) and (34), we see that forall 0 < η < a lin and a qua , the latter is finer than the former, uniformly in thegeometric parameter η .Since the line that marks the limit of stability for a qua in the plane ( k , k ) is not a straightline, the neat interpretation afforded by (35) is clearly in jeopardy. However, it remains true that,for given η and K , an increase in K widens the instability domain, that is, it reduces the criticalvalue of K that must be exceeded to make the cost of bend larger compared to the competing costof twist. In this region, we may say that, as was perhaps expected, the instability of pure bend is10 eptember 26, 2018 Liquid Crystals Toroidal˙Nematics (a) η = 0 . η = 0 . η = 0 . η = 0 . η = 0 . η = 0 . η = 0 . η = 0 . η = 0 . Figure 4.: Regions of instability identified by the modes a lin and a qua . The green region (lightergrey, in print) represents inequality (34); the red spike (darker grey, in print) must be added to thegreen region to represent the more convoluted, but less restrictive inequality (38a).enhanced by the surface energy in (22) (which, for K > , is negative). However, we cannot saythat surface energy is the only drive behind this instability, because for k <
12 (1 + η ) , (39)which according to (37) makes β > , the destabilizing mode a qua exhibits its maximum twist inthe interior of the torus, not on its boundary.
5. Conclusions
In a broad sense, this paper is a contribution to the linear stability of universal equilibrium so-lutions for Frank’s elastic energy of nematic liquid crystals. We probed the stability of the axialconfiguration within a circular torus with planar degenerate anchoring conditions on its boundary.We found that the domain of instability for this solution is indeed broader than shown in a previousstudy [6].This is not the only outcome of our study. Not only does the new destabilizing mode a qua in (36)broaden systematically the instability domain detected in [6], but it also shows a new, unexpectedqualitative character of the instability. In the parameter range identified by (39), the maximumtwist of the destabilizing mode is not achieved on the boundary of the torus, which would assignto the surface-like elastic constant K the role of main drive of the destabilization, but it occurswell inside the torus, which exalts the role of the bulk-like elastic constants K and K . This newscenario, which our linear analysis unveiled for the germ of instability, is likely to herald a property11 eptember 26, 2018 Liquid Crystals Toroidal˙Nematics of the minimizers of the full-blown variational problem, where the energy functional has the form in(16). The analysis of this highly non-linear problem is presently underway, guided by the outcomesof the linear stability analysis presented here.Having shown that the surface-like elastic constant K is not the main driving force behindthe instability of the bend-rich axial configuration in a torus poses a new question. Can elasticanisotropy, and more precisely a sufficiently small value of the ratio of the twist to bend constants, K /K , be alone responsible for the instability of the universal axial configuration in a torus withstrong anchoring conditions on its boundary? This question, which we propend to answer for thepositive, will be addressed in a subsequent paper, in a fully non-linear setting.The mathematical character of our study should not prevent the reader from appreciating itspotential applications to novel materials. In most thermotropic nematic liquid crystals, Frank’selastic constants K , K , and K are nearly equal in value [13, p. 105]. However, in a wide classof lyotropic liquid crystals, called chromonics , which have only recently been discovered and fullycharacterized [12], K may be as small as one tenth the (almost common) value of K and K .This explains, at least heuristically, a number of recent puzzling experiments, which have shownan unexpected excess of twist deformation arising in the equilibrium textures [14–18] (see also thewitty review [19]). Now, both excess of twist and smallness of K are the leitmotifs of our paper.This makes chromonics the natural experimental test-bed for our analytical results. Acknowledgements
The work of A.P. has been supported by the University of Pavia under the PRG initiative, meantto foster research among young postdoctoral fellows. E.G.V. acknowledges the kind hospitality ofthe Oxford Centre for Nonlinear PDE, where part of this work was done while he was visiting theMathematical Institute at the University of Oxford. We are both grateful to A. Zarnescu for severalenlightening discussions on the subject of this paper while our project was in its early stages.
Appendix A. Toroidal geometry and energy
In this Appendix, we collect a number of technical details used in in Sects. 2 and 3, which areomitted there for ease of presentation.
A.1
Construction and properties of the toroidal frame
Let ( e r , e ϕ , e ψ ) be the toroidal frame corresponding to the coordinates ( r, ϕ, ψ ) defined in (6).Letting ( r, ϕ, ψ ) depend on a parameter t makes p describe a trajectory in three-dimensionalspace. Differentiating this with respect to t , we obtain˙ p = ˙ r (cos ϕ cos ψ e x + sin ϕ cos ψ e y + sin ψ e z )+ ( R + r cos ψ ) ˙ ϕ ( − sin ϕ e x + cos ϕ e y )+ r ˙ ψ ( − cos ϕ sin ψ e x − sin ϕ sin ψ e y + cos ψ e z ) , from which, setting ˙ p = q r e r + q ϕ e ϕ + q ψ e ψ , we extract both the toroidal frame ( e r , e ϕ , e ψ ) asdefined in (7), and q r = ˙ r , q ϕ = ( R + r cos ψ ) ˙ ϕ , q ψ = r ˙ ψ , which readily deliver (8). It is immediateto check that (7) provides an orthonormal and positively oriented basis of the Euclidean space R .12 eptember 26, 2018 Liquid Crystals Toroidal˙Nematics Moreover, ˙ e r = cos ψ ˙ ϕ e ϕ + ˙ ψ e ψ , ˙ e ϕ = − cos ψ ˙ ϕ e r + sin ψ ˙ ϕ e ψ , ˙ e ψ = − ˙ ψ e r − sin ψ ˙ ϕ e ϕ . (A1)We recall that for any scalar field χ ( p ) expressed in the toroidal frame ( e r , e ϕ , e ψ ):˙ χ = ∇ χ · ˙ p = χ ,r ˙ r + χ ,ϕ ˙ ϕ + χ ,ψ ˙ ψ . Then ˙ e r = ∇ e r ˙ p , ˙ e ϕ = ∇ e ϕ ˙ p , and ˙ e ψ = ∇ e ψ ˙ p and, since ˙ p = ˙ r e r + ( R + r cos ψ ) ˙ ϕ e ϕ + r ˙ ψ e ψ ,it is immediate to check that ∇ e r = cos ψR + r cos ψ e ϕ ⊗ e ϕ + 1 r e ψ ⊗ e ψ , ∇ e ϕ = − cos ψR + r cos ψ e r ⊗ e ϕ + sin ψR + r cos ψ e ψ ⊗ e ϕ , ∇ e ϕ = − sin ψR + r cos ψ e ϕ ⊗ e ϕ − r e r ⊗ e ψ . (A2) A.2
Gradient, divergence, and curl of the director field
We consider a director nematic field n expressed in the toroidal frame as in (9). Recalling that forany vector field χ ( p ) := χ r e r + χ ϕ e ϕ + χ ψ e ψ ∇ χ = χ r ∇ e r + e r ⊗ ∇ χ r + χ ϕ ∇ e ϕ + e ϕ ⊗ ∇ χ ϕ + χ ψ ∇ e ψ + e ψ ⊗ ∇ χ ψ , (A2) yields ∇ n = n r,r e r ⊗ e r + n r,ϕ − cos ψ n ϕ R + r cos ψ e r ⊗ e ϕ + n r,ψ − n ψ r e r ⊗ e ψ + n ϕ,r e ϕ ⊗ e r + cos ψ n r + n ϕ,ϕ − sin ψ n ψ R + r cos ψ e ϕ ⊗ e ϕ + n ϕ,ψ r e ϕ ⊗ e ψ + n ψ,r e ψ ⊗ e r + sin ψ n ϕ + n ψ,ϕ R + r cos ψ e ψ ⊗ e ϕ + n r + n ψ,ψ r e ψ ⊗ e ψ . Therefore div n = n r,r + R + 2 r cos ψr ( R + r cos ψ ) n r + 1 R + r cos ψ n ϕ,ϕ − sin ψR + r cos ψ n ψ + 1 r n ψ,ψ (A3)and curl n = (cid:18) sin ψ n ϕ + n ψ,ϕ R + r cos ψ − n ϕ,ψ r (cid:19) e r + (cid:18) n r,ψ − n ψ r − n ψ,r (cid:19) e ϕ + (cid:18) n ϕ,r − n r,ϕ − cos ψ n ϕ R + r cos ψ (cid:19) e ψ . eptember 26, 2018 Liquid Crystals Toroidal˙Nematics Thus n · curl n = sin ψ n r n ϕ + n r n ψ,ϕ − n ψ n r,ϕ R + r cos ψ − R r ( R + r cos ψ ) n ϕ n ψ + n ϕ n r,ψ − n r n ϕ,ψ r − n ϕ n ψ,r + n ψ n ϕ,r , (A4) − n × curl n = ( ∇ n ) n = n r n r,r + n ϕ n r,ϕ − cos ψ n ϕ R + r cos ψ + n ψ n r,ψ − n ψ r ! e r + (cid:18) n r n ϕ,r + cos ψ n r n ϕ − sin ψ n ϕ n ψ + n ϕ n ϕ,ϕ R + r cos ψ + n ψ n ϕ,ψ r (cid:19) e ϕ + n r n ψ,r + sin ψ n ϕ + n ϕ n ψ,ϕ R + r cos ψ + n r n ψ + n ψ n ψ,ψ r ! e ψ . A.3
Frank’s energy
For brevity, we shall write F = 12 ( F + F + F ) + F , where F [ n ] := K Z B (div n ) d V , F [ n ] := K Z B ( n · curl n ) d V , F [ n ] := K Z B | ( ∇ n ) n | d V and F [ n ] := K Z ∂ B [( ∇ s n ) n − (div s n ) n ] · ν d A .
We now compute F for the director field n described in Sect. A.2 relative to the toroidal frame( e r , e ϕ , e ψ ), under the simplifying assumption that ϑ = π (i.e. when n r = 0, n ϕ = cos α and n ψ = sin α ) and α = α ( r, ψ ). We obtain F [ α ] = K Z R Z π Z π [ r sin ψ sin α − ( R + r cos ψ ) cos α α ,ψ ] r ( R + r cos ψ ) d r d ϕ d ψ , (A5a) F [ α ] = K Z R Z π Z π [ r ( R + r cos ψ ) α ,r + R sin α cos α ] r ( R + r cos ψ ) d r d ϕ d ψ , (A5b) F [ α ] = K Z R Z π Z π ( [ r sin ψ cos α + ( R + r cos ψ ) sin α α ,ψ ] r ( R + r cos ψ )+ (cid:2) r cos ψ + R sin α (cid:3) r ( R + r cos ψ ) ) d r d ϕ d ψ , (A5c) F [ α ] = − K Z π Z π (cid:0) R cos ψ + R sin α ( R , ψ ) (cid:1) d ϕ d ψ . (A5d)14 eptember 26, 2018 Liquid Crystals Toroidal˙Nematics By using (14) and (15), and taking α = α ( σ, ψ ), we get d r = R η d σ and α ,r = R η α ,σ . Then, bycomputing the integral in ϕ , the four components of Frank’s energy become F [ α ] = 2 πR K η Z Z π [ ησ sin ψ sin α − (1 + ησ cos ψ ) cos α α ,ψ ] ησ (1 + ησ cos ψ ) d σdψ , F [ α ] = 2 πR K η Z Z π [ σ (1 + ησ cos ψ ) α ,σ + sin α cos α ] ησ (1 + ησ cos ψ ) d σdψ , F [ α ] = 2 πR K η Z Z π ( [ ησ sin ψ cos α + (1 + ησ cos ψ ) sin α α ,ψ ] ησ (1 + ησ cos ψ )+ (cid:2) ησ cos ψ + sin α (cid:3) ησ (1 + ησ cos ψ ) ) d σdψ , F [ α ] = − πR K Z π sin α (1 , ψ ) d ψ , and (16) follows at once. Letting α ≡ F [0] = πR K Z Z π η σ ησ cos ψ d σ d ψ = 2 π R K (1 − p − η ) , (A6)valid for all 0 η A.4
Quadratic approximation
Letting f , f , f , and f denote the integrands in the functionals F , F , F , and F , respectively,see (A5), we approximate them to the second order in α as f ≈ ησ sin ψ ησ cos ψ α + 1 + ησ cos ψησ α ,ψ − ψ αα ,ψ ,f ≈ ησ (1 + ησ cos ψ ) α + σ (1 + ησ cos ψ ) η α ,σ + 2 η αα ,σ ,f ≈ ησ ησ cos ψ + 2 cos ψ − ησ sin ψ ησ cos ψ α + 2 sin ψ αα ,ψ ,f ≈ α (1 , ψ ) , where both (14) and (15) have been used and a factor R has been pulled out of all integrals. Since2 sin ψ αα ,ψ = sin ψ ( α ) ,ψ , an integration by parts shows that Z π ψ αα ,ψ d ψ = − Z π cos ψ α d ψ and so the bending energy can be given the simple form in (22).15 eptember 26, 2018 Liquid Crystals Toroidal˙Nematics To derive (26) from (24), a number of integrals in ψ must be computed; they are recorded herefor completeness, so as to allow the interested reader to retrace all our steps: Z π (1 − ησ cos ψ ) (1 + ησ cos ψ ) d ψ = 2 π (1 + η σ ) + 2 η σ (1 − η σ ) , Z π − ησ cos ψ (1 + ησ cos ψ ) d ψ = 2 π η σ (1 − η σ ) , Z π
11 + ησ cos ψ d ψ = 2 π − η σ ) , Z π cos ψ − ησ (1 + ησ cos ψ ) d ψ = − π (5 + η σ ) ησ (1 − η σ ) , Z π ησ cos ψ ) d ψ = 2 π − η σ ) . Similarly, it is advisable to perform the following integration by parts: Z η σ (1 − η σ ) a ( σ ) a ,σ ( σ ) d σ = 1 + η (1 − η ) a (1) − Z η σ (5 + η σ )(1 − η σ ) a ( σ ) d σ , where use has also been made of (25). References [1] Frank FC. On the theory of liquid crystals. Discuss Faraday Soc. 1958;25:19–28.[2] Stewart IW. The static and dynamic continuum theory of liquid crystals. London: Taylor & Francis;2004.[3] Virga EG. Variational theories for liquid crystals. London: Chapman & Hall; 1994.[4] Ericksen JL. Nilpotent energies in liquid crystal theory. Arch Rational Mech Anal. 1962;10:189–196.[5] Ericksen JL. Inequalities in liquid crystal theory. Phys Fluids. 1966;9(6):1205–1207.[6] Koning V, van Zuiden BC, Kamien RD, et al. Saddle-splay screening and chiral symmetry breaking intoroidal nematics. Soft Matter. 2014;10:4192–4198.[7] Ericksen JL. General solutions in the hydrostatic theory of liquid crystals. Trans Soc Rheol. 1967;11(1):5–14.[8] Marris AW. Universal solutions in the hydrostatics of nematic liquid crystals. Arch Rational Mech Anal.1978;67(3):251–303.[9] Marris AW. Addition to “Universal solutions in the hydrostatics of nematic liquid crystals”. ArchRational Mech Anal. 1979;69(4):323–333.[10] Ogilvy CS. Excursions in geometry. Mineola, NY, USA: Dover; 1990. Unabridged and corrected repub-lication of the work originally published by Oxford University Press, New York.[11] Truesdell CA. A first course in rational continuum mechanics. 2nd ed. Vol. 1. San Diego: AcademicPress; 1991.[12] Zhou S, Nastishin YA, Omelchenko MM, et al. Elasticity of lyotropic chromonic liquid crystals probedby director reorientation in a magnetic field. Phys Rev Lett. 2012;109:037801.[13] de Gennes PG, Prost J. The physics of liquid crystals. 2nd ed. Oxford: Clarendon Press; 1993.[14] Davidson ZS, Kang L, Jeong J, et al. Chiral structures and defects of lyotropic chromonic liquid crystalsinduced by saddle-splay elasticity. Phys Rev E. 2015;91:050501.[15] Davidson ZS, Kang L, Jeong J, et al. Erratum: Chiral structures and defects of lyotropic chromonicliquid crystals induced by saddle-splay elasticity [Phys. Rev. E 91, 050501(R) (2015)]. Phys Rev E.2015;92:019905. eptember 26, 2018 Liquid Crystals Toroidal˙Nematics [16] Jeong J, Kang L, Davidson ZS, et al. Chiral structures from achiral liquid crystals in cylindrical capil-laries. Proc Natl Acad Sci USA. 2015;112(15):E1837–E1844.[17] Jeong J, Davidson ZS, Collings PJ, et al. Chiral symmetry breaking and surface faceting in chromonicliquid crystal droplets with giant elastic anisotropy. Proc Natl Acad Sci USA. 2014;111(5):1742–1747.[18] Lubensky TC. Confined chromonics and viral membranes. Mol Cryst Liq Cryst. 2017;646(1):235–241.[19] Masters A. Chromonic liquid crystals: more questions than answers. Liq Cryst Today. 2016;25(2):30–37.[16] Jeong J, Kang L, Davidson ZS, et al. Chiral structures from achiral liquid crystals in cylindrical capil-laries. Proc Natl Acad Sci USA. 2015;112(15):E1837–E1844.[17] Jeong J, Davidson ZS, Collings PJ, et al. Chiral symmetry breaking and surface faceting in chromonicliquid crystal droplets with giant elastic anisotropy. Proc Natl Acad Sci USA. 2014;111(5):1742–1747.[18] Lubensky TC. Confined chromonics and viral membranes. Mol Cryst Liq Cryst. 2017;646(1):235–241.[19] Masters A. Chromonic liquid crystals: more questions than answers. Liq Cryst Today. 2016;25(2):30–37.