## Abstract

An elastic membrane model of smectic A liquid crystal deformation is derived *ab initio* via a variational approach. The well-determined nature of the resulting nonlinear model equations reveals that the deformed states of the liquid crystal lamellae can only adopt privileged geometries. These are shown to generalize classical and novel ‘integrable’ geometries associated with Willmore, linear Weingarten and ‘membrane’ O surfaces. The main result establishes that, remarkably, the membrane model admits layered parallel Dupin cyclide structures of the kind originally observed by Friedel and Grandjean in their pioneering experiments of 1910 and subsequently elaborated upon by Friedel in 1922 and later by Bragg.

## 1. Introduction

*Focal conic domains* constitute the characteristic textures of a large class of mesomorphic phases, the so-called smectic phases, whose molecular structure is layer-like. In a now classical paper, Friedel & Grandjean (1910) recognized empirically the essential geometric properties of these focal conic domains, namely (cf. figure 1):

the appearance of a pair of singular lines (clearly visible in light microscopy) which are nothing but conjugate conics;

the law of corresponding cones, which sets down the geometric conditions satisfied when two focal conic domains are in contact.

The phases that display such intriguing textures were termed *liquides à coniques focales* by Friedel and Grandjean. This emphasized that the focal conic domains are indeed a characteristic feature of these liquids and identified them with Lehmann's (1904) *fliessende Kristalle*. Friedel and Grandjean's interpretation of their optical microscopy observations were all the more remarkable in that, at the time, they were unaware of the lamellar (shell) structure of these liquids. This signal advance was made following the observation (Grandjean 1916) of *gouttes en gradins* (droplets made of terraces), which were readily interpreted in the sense of Perrin's soap molecular layers. The term *smectic*, derived from the Greek word *σμηγμα* for soap, was then coined and popularized by Friedel (1922) in a celebrated paper, which takes stock of all these advances, not only in the field of smectics but also in that of other mesomorphic phases; namely, nematics, cholesterics and, to a lesser extent, blue phases. It is this paper, which, for the first time, alluded to Dupin cyclides associated with focal conic domains as the privileged surfaces along which the molecular layers deploy. The first published analytical treatment of Dupin cyclides in this context is due to Bragg (1934). Following a long oblivion, the smectic focal conic domain geometry was re-examined by Bouligand (1972) while a number of new important observations were provided by Williams (1976).

The Oseen–Frank elastic free-energy density of liquids endowed with a director unit vector field * n* valid for all types of mesomorphic phases has the form (Oseen 1933; Frank 1958)(1.1)where the constants

*K*

_{i}and

*K*

_{ij}are elastic moduli. The elastic free energy of a deformed smectic phase consisting of approximately

*parallel*lamellae may be derived from the above Oseen–Frank energy (cf. Kléman & Lavrentovich 2003). It reads(1.2)where

*σ*

_{1}and

*σ*

_{2}denote the principal curvatures of the lamellae and

*d*is the lamella thickness which is assumed to be

*d*

_{0}in the ground state. We make contact with equation (1.1) in four steps. Let

*be the field of unit normals to the lamellae. Then: (i) we have the two identities*

**n***σ*

_{1}+

*σ*

_{2}=±div

*and*

**n***σ*

_{1}

*σ*

_{2}=div(

*div*

**n***+*

**n***×curl*

**n***)/2; (ii) The relation*

**n***.curl*

**n***=0 holds since*

**n***is perpendicular to the set of lamellae; (iii) The term*

**n***K*

_{3}(

*×curl*

**n***)*

**n**^{2}/2, which describes a deviation from the parallelism of neighbouring lamellae, is small compared with the

*B*term in equation (1.2) and is neglected; and (iv) The

*K*

_{13}term is the only term to exhibit second-order derivatives of

*and is usually not taken into account.*

**n**The particular geometry of focal conic domains gives rise to a type of defects which results from a competition between two types of elastic contributions—*strain* elasticity (here the compressibility of the layers) and *curvature* elasticity (here the curvature of the layers)—whose ranges of action are different (Kléman 1982). Thus, let *R* be the typical size of a domain submitted to some compression or tension. The strain energy is of the order *BR*^{3} while the curvature energy is of the order *KR*. The quantity *λ* defined by *λ*^{2}=*K*/*B* constitutes a microscopic length comparable to the layer thickness *d*_{0}. The ratio of the two energies *f*_{strain}/*f*_{curvature}≈(*R*/*λ*)^{2} is larger than 1 as soon as *R*>*λ*. One therefore expects that the only distortion that would affect a layered medium on a macroscopic length *R* is a curvature distortion. Thus, the layers curve while keeping a constant interlayer distance not significantly different from *d*_{0} thereby essentially conserving parallelism. Such a constraint imposes drastic conditions on the shape of the singularities and, in focal conic domains, these are conjugate ellipses and hyperbolae. In view of the above, in the present paper, we proceed with the free-energy density representation(1.3)The term involving the Gaussian curvature (like those corresponding to *K*_{13} and *K*_{24} in equation (1.1)) is a divergence term and, accordingly, does not contribute to the Euler–Lagrange equations associated with the free-energy density (1.3). However, it does play a leading role when the topology of the lamellae has to be taken into account. For instance, it contributes differently to the elastic energy of a toric layer to that of a spherical layer by virtue of the Gauss–Bonnet theorem (do Carmo 1976). Furthermore, if equation (1.3) is brought into the form(1.4)where and can have any sign (but *K*>0), then the following conclusions may be drawn for a single layer.

If ,

*Λ*>0, then the ground state of (1.4) satisfies the relations*σ*_{1}=*σ*_{2}=0. The*planar*layer is stable.If ,

*Λ*<0, then the energy decreases without limit for |*σ*_{1}+*σ*_{2}| large and*σ*_{1}=*σ*_{2}. One expects some stabilization for a*spherical*layer (micelles). The Gaussian curvature is positive.If ,

*Λ*>0, then the energy decreases without limit for vanishing mean curvature and negative Gaussian curvature . One expects some stabilization for a*minimal*surface shape of the layer.

The above considerations apply not only to a single lamella, but also to a bulk phase of layers (if the *B* term is neglected). It is to be expected that, even in a smectic phase, the nature of the textures depends crucially on the signs of and *Λ*. In all the *thermotropic* smectics investigated yet, the only parts of the Dupin cyclides physically present are those of negative Gaussian curvature (cf. figure 2), which may indicate that (Boltenhagen *et al*. 1991). These are the special focal conic domains that we shall investigate in this paper. On the other hand, a number of *lyotropic* lamellar phases show complementary behaviour, namely only the parts are present (Boltenhagen *et al*. 1992). The geometries here involve nested spheres (*spherulites*).

The present paper is concerned with how classical elastic shell theory as developed by Love some 80 years ago bears upon modern liquid crystal theory. In this connection, an elastic membrane model was adopted by Kléman (1976) to derive a nonlinear system of equations governing the deformation of a single layer in a smectic phase. Rogers & Schief (2003), on the other hand, have recently established remarkable links between the equilibrium of shell membranes and soliton theory. Here, these two developments are brought together not only to solve a classical problem related to the foliation of smectic layers but also to indicate hidden integrable structure in the Kléman system. The elastic model of a smectic phase membrane relies on the identification of the variation of the potential energy of a thin shell, subjected to an appropriate infinitesimal deformation, with the variation of the total free energy of the smectic membrane. This correspondence leads to ‘constitutive’ relations for stress resultants and stress couples, which, on insertion into Love's equilibrium equations, produce the nonlinear system as originally derived in another manner by Kléman (1976). A particular reduction of the Kléman system determines the celebrated integrable Willmore surfaces, which, accordingly, may be interpreted as liquid crystal lamellae generated by an appropriate tension distribution. This class is subsumed in a ‘generalized Willmore equation’, which also reduces to the governing equation for linear Weingarten surfaces in a formal limit. The Kléman system ‘a fortiori’ includes the integrable class of O surfaces (Schief & Konopelchenko 2003) obtained from classical shell membrane theory in Rogers & Schief (2003).

In general terms, the well-determined nature of the Kléman system shows that liquid crystal lamellae deformations described thereby can only admit privileged geometries. As stated earlier, the pioneering experimental work of Friedel & Grandjean (1910), Grandjean (1916) and Friedel (1922), in particular, revealed smectic A liquid crystal geometries consisting of parallel layers of Dupin cyclides. Kléman (1976) introduced an admissible ansatz for the tension distribution which leads to a single Dupin cyclide solution of the elastic membrane model which turns out to be conformally equivalent to the Clifford–Willmore torus (Willmore 1965).

Here, it is established that, remarkably, the theoretical liquid crystal model of Kléman based on classical elastic shell theory in fact admits a class of solutions that exhibit the layered Dupin cyclide structure of the form observed by Friedel in his empirical work of 1922 and subsequently discussed by Bragg (1934). Explicit expressions are presented for the corresponding tension distributions, including a reduction to the original Kléman ansatz, which produces the Dupin cyclide minimizing the Willmore functional. These expressions may be regarded as ‘constitutive laws’ relating the tensions and the principal curvatures.

## 2. The liquid crystal model

### (a) Preliminaries

The free-energy density for a *single* layer of a smectic phase, which may be referred to as a membrane, bears a resemblance to that devised long ago for an elastic shell (Love 1927; Timoshenko 1961; Novozhilov 1964). Thus, the expression (1.3) is also the free-energy density of a slightly bent *thin plate*, with the correspondence(2.1)where *h*⇔*d*_{0} is the thickness of the plate, *E* is Young's modulus and *ν* is Poisson's ratio. The parameter *κ*, usually designated by *D*, is called the flexural rigidity. ‘Thin’, in this context, means that the curvatures are small compared with the inverse thickness *h*^{−1}. Therefore, the principal curvatures *σ*_{1} and *σ*_{2} can be approximated in terms of the normal displacement *ζ* of the midsurface of the plate so that *σ*_{1}+*σ*_{2}= *ζ*_{xx} + *ζ*_{yy} and , where subscripts denote derivatives with respect to the Cartesian coordinates *x* and *y* of the unbent plate. Such a condition is, by definition, not obeyed in the case of thin *shells*, where account should be taken not only of the flexural deformation but also of the strains in the thickness of the shell. The energy carried by the strains (∼(*ζ*/*R*)^{2}), if any, is so large compared with the flexural part (∼(*ζ*/*R*)^{4}) that preference is given in practice to pure flexural deformations which have to be *inextensional* if the strains are vanishing. We are then again left with equation (1.3) but the principal curvatures *σ*_{i} are replaced by their changes Δ*σ*_{i} with respect to the undistorted shell. Inextensibility also means that the Gaussian curvature is invariant and, accordingly, the free-energy density of an inextensional shell adopts a form somewhat different from equation (1.3).

The above applies to solid materials but the smectic media to be investigated here are different. Thus, a smectic membrane is made of molecules elongated perpendicular to the midsurface, not of atoms. There is therefore a dominant perpendicular anisotropy. Let us compare the elastic properties of such a system at high temperature, in the ‘membranous’ state, and at low temperatures, in the solid-state. We allow deformed states that carry elastic deformations and curvatures and assume that one can change temperature but only in such a way that the principal curvatures are conserved. As a result, the midsurface of the shell in the solid-state is applicable on the membranous state (assuming here that there is no thermal dilatation). However, in order to preserve the high temperature shape, forces and torques have to be applied resulting in the presence of supplementary strains and stresses. Starting now from the solid-state and heating up the system, the stretching components of the deformation decrease steadily, up to a point where they totally disappear in a finite relaxation time due to viscosity. In the final state, the only deformation component that remains originates in the *splay* of the chains (something that is absent in a true atomic solid), thus preserving some part of the previous elastic deformation in the full solid-state. This would not be so in the isotropic liquid state.

The free-energy density (1.3) constitutes a vestige of the history of the solid-state of a molecular film with chains and not with point-like atoms. This has a number of consequences:

The correspondence (2.1) between the material constants in the solid-state and in the flexural state is still valid—the membrane is not in a full three-dimensional isotropic liquid state—and it now even applies in the case of large curvatures.

The effective Poisson ratio of the medium under consideration does not take its liquid state value of one-half for a three-dimensional medium or one- for a two-dimensional medium. Rather, it is a material constant of the membrane.

If we start from the membranous state—in which equation (1.3) is valid because it satisfies the necessary symmetry requirements and contains the full information about the actual deformation of the system—and decrease the temperature then, as stated above, new distortion terms appear but equation (1.3) subsists under suitable applied forces (subject to some necessary modification of the material constants) with the same values

*σ*_{1}and*σ*_{2}of the principal curvatures.Stress resultants and stress couples may be defined in a curved (liquid crystal) membrane in the same manner as they are defined in a shell. The equations of equilibrium are the same even if the curvatures are large. These equations constitute the starting point of the discussion below.

### (b) The governing equations

The preceding considerations apply to a smectic A phase that constitutes a stacking of planar parallel lamellae in the ground state. These lamellae remain parallel in the above-mentioned focal conic domains. In this section, we adopt a variational approach to retrieve a set of equations, which governs a *single* layer (membrane) of constant thickness *h* in a smectic phase as proposed by Kléman (1976). This is motivated by the fact that, according to the standard structural model, the molecules of smectic A liquid crystal phases may flow independently along the parallel layers. In the following, it is convenient to regard the membrane as being made of a one-parameter family of parallel sublamellae *Σ*^{(z)}, where *z* represents a foliation parameter. Accordingly, if * r*=

*(*

**r***α*,

*β*) denotes the position vector of the midsurface

*Σ*given in terms of (local) coordinates

*α*and

*β*then the position vector of the membrane is conveniently parameterized by(2.2)where

*is the unit normal to the midsurface*

**n***Σ*and therefore to all sublamellae

*Σ*

^{(z)}. The total free energy of the membrane is then given by(2.3)Here, for convenience, we have applied the scaling . The principal curvatures and the surface element d

*Σ*

^{(z)}of the sublamellae

*Σ*

^{(z)}are related to the corresponding quantities associated with the midsurface

*Σ*by(2.4)(e.g. Eisenhart 1960). The above relations are valid for any thickness

*h*of the lamella. Even though, as in the case of thin shell theory, we shall assume that

*hσ*

_{i}≪1, it is convenient to proceed as far as possible without this assumption and then subsequently apply the analysis to thin membranes.

The efforts exerted on the (midsurface of the) lamella are obtained by integrating the components *σ*_{ik} of the three-dimensional stress tensor over the thickness of the membrane. Here, we adopt the notation of Love (1927; cf. Novozhilov 1964; Kléman 1976) to designate the stress resultants *S*_{i}, *T*_{i}, *N*_{i} and the stress couples *G*_{i}, *H*_{i}. In the following, we assume that the lines of principal stress coincide with the lines of curvature on the sublamellae (cf. Helfrich 1981). Accordingly, if *α*, *β* denote curvature coordinates then *σ*_{12}=0 which implies, in turn, that *S*_{i}=0 and *H*_{i}=0. Thus, the only relevant stress resultants and couples are given by (Love 1927)(2.5)Because lines of curvature are orthogonal, the first fundamental form of the midsurface assumes the form(2.6)The equations of equilibrium for the resultant forces then read (Love 1927)(2.7)while those for the resultant moments become(2.8)The coefficients of the above system are parameterized in terms of the *geometric* quantities *A*, *B*, *σ*_{1}, *σ*_{2}, which are constrained by the Gauss–Mainardi–Codazzi equations(2.9)associated with the midsurface *Σ*. However, the system does not depend on the actual *constitutive equations* that relate the stress resultants and stress couples on the one hand to the curvatures (and their derivatives) on the other. We now establish these constitutive equations by considering the variation of the free energy (2.3) with respect to particular infinitesimal distortions.

#### (i) Infinitesimal deformations

We consider an infinitesimal *virtual* deformation of the membrane. If the deformation is arbitrary, then(2.10)where the unit tangent vectors * x* and

*are defined by(2.11)with first fundamental forms(2.12)of the sublamellae*

**y***Σ*

^{(z)}. By definition, the infinitesimal strains caused by an infinitesimal deformation are given by (Love 1927)(2.13)and(2.14)If we take into account Love's hypotheses that the tangential deformation of a thin shell is linear in

*z*, and that the normal deformation is independent of

*z*; that is(2.15)then the infinitesimal strains may be expressed solely in terms of midsurfaces quantities. In particular, because the quantities , and , are independent of

*z*, we obtain the relations(2.16)and . The latter is consistent with our analysis in the introduction according to which large-scale textures are made of parallel, uncompressed layers (no

*B*term in (1.3)). It is observed that Love's hypothesis embraces the stronger assumption that ‘normals remain normals’ when the shell is being deformed (see, e.g. Novozhilov 1964). In this case, the displacements

*δs*and

*δt*may be expressed explicitly in terms of

*δu*,

*δv*,

*δw*and the infinitesimal strain components and vanish by definition.

#### (ii) Variation of the potential energy

As is well known (Novozhilov 1964), the variation of the potential energy of a thin shell subjected to an infinitesimal deformation with *σ*_{12}=0 and is given by(2.17)independently of the constitutive equations. Insertion of the infinitesimal strains (2.16) into the variation (2.17) of the potential energy and subsequent integration then produce

(2.18)on use of the definitions (2.5). In view of the variation of the free energy (2.3), it is necessary to express the variation of the principal curvatures in terms of the quantities arising in equation (2.18). Thus, the changes of the principal curvatures *δσ*_{i} may be shown to be (Novozhilov 1964)(2.19)with and given by equations (2.13)_{1,2} evaluated on the midsurface (*z*=0). These may be brought into the form(2.20)on use of the relations (2.16)_{2,4,6,8}.

#### (iii) Variation of the free energy

Our elasticity model of a smectic phase membrane requires the identification of the variation of the potential energy as obtained in the preceding with the variation of the free energy (2.3). To this end, it is first observed that expansion of the principal curvatures in terms of the small quantities *zσ*_{i} and subsequent integration yield(2.21)owing to the vanishing of terms linear in *z*. For thin membranes, it is therefore sufficient to consider the free-energy variation(2.22)For an arbitrary deformation of the membrane, the midsurface area element changes according to(2.23)where, once again, and are given by equations (2.13)_{1,2} evaluated on the midsurface. Accordingly, the variation of the free energy reads(2.24)with . Insertion of the expressions (2.20) for *δσ*_{i} into equation (2.24) and integration by parts now lead to an avatar of the free-energy variation, which is of the same structure as the variation of the potential energy. Indeed, on use of the Mainardi–Codazzi equations (2.9)_{1,2}, we obtain(2.25)Here, it is noted that d*Σ*=*AB* d*α* d*β*.

#### (iv) The constitutive equations

Even though, strictly speaking, the six quantities , , *δκ*_{1}, *δκ*_{2} , cannot be independent since they are defined in terms of the five midsurface ‘displacements’ *δu*, *δv*, *δw* and *δs*, *δt*, it is shown below that it is consistent to set aside any differential relations which may exist between these quantitites. Indeed, comparison of the terms proportional to and in the variations of the free and potential energies delivers(2.26)while the coefficients of *δκ*_{1} and *δκ*_{2} give rise to(2.27)It is then readily verified that, remarkably, the equilibrium equations (2.8) for the resultant moments are identically satisfied. The remaining equilibrium equations (2.7) for the resultant forces adopt the form

(2.28)As expected, the constant *ν* does not appear in the above equilibrium equations, but enters the expressions for the stress couples *G*_{i}. Up to an additional term, which involves a purely normal external force to be discussed in the following section, the equilibrium equations (2.28) were proposed by Kléman (1976) as a model for single layers of smectic liquid crystal phases.

### (c) Admissible deformations

It is evident that the equilibrium equations (2.28) are subject to(2.29)The constraints imposed by this condition on the equilibrium equations depend crucially on the infinitesimal deformations admitted by layers of smectic A phases. In this connection, it is enlightening to consider the unphysical case of arbitrary deformations, which leads to the additional relations(2.30)Insertion of the latter into the equilibrium equations (2.28) shows that the first two equilibrium equations are identically satisfied by virtue of the Mainardi–Codazzi equations (2.9)_{1,2} and the remaining equilibrium equation reduces to(2.31)where(2.32)constitutes the midsurface Laplacian and and denote, as usual, the mean and Gaussian curvatures, respectively. Hence, the scalar equation (2.31) represents an admissible reduction of the equilibrium equations. Its relevance is discussed in the following section.

With respect to compressibility, a liquid crystal layer exhibits the properties of an incompressible sheet of liquid so that the preservation of the local area d*Σ* constitutes a necessary condition on the class of admissible infinitesimal deformations. By virtue of equation (2.23), we therefore obtain the constraint(2.33)and the residual condition (2.29) reduces to(2.34)The implications of the latter are discussed in §4*b*.

## 3. Generalized Willmore surfaces

In the preceding section, a variational approach was used to establish a smectic membrane model under the assumption that no external forces are present. In general, in smectics, there exists a force that prevails along the normal to the smectic layers. Thus, if *Z* constitutes a constant normal force per unit area (‘normal pressure’), then inclusion of the corresponding pressure term leads to the equilibrium equations

(3.1)supplemented by the Gauss–Mainardi–Codazzi equations(3.2)In the formal limit *K*=0, the above system reduces to the classical equilibrium equations of shell membrane theory (Novozhilov 1964) subject to the assumptions that the lines of principal stress coincide with the lines of curvature and that the loading be purely normal and constant. Remarkably, the associated membranes have been proven (Rogers & Schief 2003) to have geometries within the integrable class of so-called ‘O surfaces’. The latter class of surfaces has recently been introduced and shown to admit Bäcklund transformations and Lax pairs (Schief & Konopelchenko 2003). The connection with integrability is facilitated by the observation that the structure of the equilibrium equations (3.1)_{1,2} with *K*=0 is precisely that of the Mainardi–Codazzi equations (3.2)_{1,2}.

In order to analyse the geometric properties in the case *K*≠0, it is convenient to make the change of dependent variables(3.3)which is motivated by the special reduction (2.30). Indeed, the equilibrium equations then become(3.4)so that, once again, the equilibrium equations (3.4)_{1,2} coincide with the Mainardi–Codazzi equations (3.2)_{1,2} if the substitution is made.

The above formulation readily reveals that the equilibrium equations may be regarded as a linear superposition of two classical integrable systems. On the one hand, if *K*=0 then, as pointed out above, system (3.4) is descriptive of integrable membrane O surfaces. On the other hand, if , then the equilibrium equations reduce to(3.5)as pointed out in the previous section. It is well known (Schadow 1922; Thomsen 1923; Blaschke 1929) that equation (3.5) is nothing but the Euler equation associated with the area functional(3.6)in conformal differential geometry. The critical points *Σ* of the above functional have come to be known as ‘Willmore surfaces’ in recognition of the important Willmore conjecture (Willmore 1965). Thus, subject to the residual condition (2.34), any Willmore surface may be interpreted as a liquid crystal lamella with tensions *T*_{1} and *T*_{2}, given by equations (3.3) with .

Since the rediscovery of the fact that the conformal Gauss map of Willmore surfaces constitutes an integrable harmonic map, Willmore surfaces have been the subject of extensive studies (see Willmore 2000 and references therein). It is also remarked that the derivation of the Willmore equation in the context of the preceding section is no coincidence. Indeed, the ‘bending energy’ or ‘virtual work’(3.7)associated with the deformation of curved plates was investigated in the nineteenth century by such luminaries as Germain (1811–1815, 1821), Poisson (1812) and Kirchhoff (1850). It is evident that the solutions of the ‘Willmore equation’ (3.5) also extremize the energy functional (3.7).

A larger class of liquid crystal lamellae is governed by the generalized Willmore equation(3.8)which is obtained from equation (3.4)_{3} by considering the admissible reduction(3.9)and *Z*=2*c*_{3}. It is noted that the remaining equilibrium equations (3.4)_{1,2} are identically satisfied by virtue of the Mainardi–Codazzi equations (3.2)_{1,2}. In the context of the theory of biomembranes, equation (3.8) has come to be known as the *shape equation*, which determines the equilibrium shapes of lipid bilayer vesicles. It constitutes the Euler–Lagrange equation associated with the minimization of the energy functional(3.10)where is a ‘spontaneous curvature’, subject to the preservation of the surface area and volume of the vesicles (Canham 1970; Helfrich 1973; Zhong-can & Helfrich 1987*a*, *b*). In the formal limit *K*=0, equation (3.8) defines ‘linear Weingarten surfaces’ (Eisenhart 1960), because the mean and Gaussian curvatures are linearly related. Generic linear Weingarten surfaces inherit their integrability from the fact that they are parallel to surfaces of constant Gaussian curvature (e.g. Rogers & Schief 2002).

## 4. Layered Dupin cyclides

### (a) The equilibrium and Gauss–Mainardi–Codazzi equations

Because the equilibrium equations (3.1) and the Gauss–Mainardi–Codazzi equations (3.2) constitute a well-determined system, it is evident that liquid crystal lamellae can only assume particular geometries. In his monumental work, Friedel (1922) deduced from experimental observations that smectic A liquid crystals in a confocal domain are composed of (pieces of) parallel layers of Dupin cyclides. With a particular ansatz for the tensions *T*_{1} and *T*_{2}, Kléman (1976) showed that for any given confocal domain, there exists only one Dupin cyclide that gives rise to a solution of the constrained equilibrium equations so that stacking of parallel Dupin cylides is not possible. It turns out that Kléman's ansatz for the tensions corresponds to and the Dupin cyclide is conformally equivalent to the particular torus of revolution (‘anchor ring’), which was used by Willmore (1965) to illustrate his conjecture. The latter states that the minimal value of the ‘Willmore functional’ (3.7) for embedded tori is attained by the above-mentioned anchor ring.

The main aim of the present paper is to show that for any given Dupin cyclide and prescribed normal loading *Z*, the equilibrium equations (3.1) are indeed compatible and, moreover, their solution may be found explicitly. In particular, this implies that the theoretical liquid crystal model proposed by Kléman is consistent with Friedel's empirical discovery of layered parallel Dupin cyclides. The restrictive result obtained with the Kléman ansatz of 1976 is retrieved as a special case of our present analysis.

Dupin cyclides are defined by the requirement that both families of lines of curvature consist of circles (Eisenhart 1960). Analytically, this condition is expressed by the constraints(4.1)These are compatible with the Mainardi–Codazzi equations (3.2)_{1,2} if and only if(4.2)Differentiation of the Gauss equation (3.2)_{3} and simplification by means of equations (4.2) then lead to expressions for the derivatives *A*_{βββ} and *B*_{ααα}, which we denote by(4.3)It is now assumed that a solution of the Gauss–Mainardi–Codazzi equations (3.2) subject to the constraints (4.1)–(4.3) is to hand. Equation (3.1)_{3} is then regarded as a constraint on the tensions *T*_{1} and *T*_{2} obeying the equilibrium equations (3.1)_{1,2}. Differentiation of (3.1)_{3}/(*AB*) yields(4.4)where the coefficients *f*_{i} are known functions of *A*, *B*, *σ*_{1}, *σ*_{2} and their derivatives. In the case of shell membranes (*K*=0), these relations have been shown to be compatible with the equilibrium equations (3.1)_{1,2} modulo (3.2)_{1,2}, (4.1) and (4.2). It turns out that this result carries over to the current situation (*K*≠0). However, in order to establish this assertion, it is required to make use of the Gauss equation (3.2)_{3} and its differential consequences (4.3). Indeed, it is readily verified that the compatibility conditions *T*_{iαβ}= *T*_{iβα} are satisfied for any solution of the geometric system (3.2), (4.1)–(4.3) corresponding to Dupin cyclides. Accordingly, the equilibrium equation (3.1)_{3} may be regarded as a first integral of the compatible Frobenius system (3.1)_{1,2}, (4.4) for the tensions *T*_{1} and *T*_{2} with the normal loading *Z* being the associated constant of integration.

Having established the compatibility of the equilibrium equations (3.1) for Dupin cyclides, which may be brought into the form(4.5)we now explicitly determine the tensions *T*_{1} and *T*_{2}. A convenient parameterization of Dupin cylides is given by (Dupin 1822)(4.6)where *a*^{2}=*b*^{2}+*c*^{2} and the pair of associated focal conics consists of the focal ellipse(4.7)and the focal hyperbola(4.8)If the focal conics are prescribed then any two associated confocal Dupin cyclides with parameters *μ*=μ_{1} and *μ*=μ_{2} are parallel and their distance is |*μ*_{2}−*μ*_{1}|. Hence, the parameter *μ* dictates the stacking of parallel Dupin cyclides in a confocal domain. The one-parameter family of parallel Dupin cyclides is orthogonal to the congruence of lines connecting any two points on the focal conics (e.g. Kléman 1977). A typical confocal domain is displayed in figure 2.

The geometric quantitites *A*, *B* and *σ*_{1}, *σ*_{2} are readily shown to be(4.9)Now, on elimination of *T*_{2} in equations (4.5)_{1,3}, we obtain a linear first-order ordinary differential equation for *T*_{1}. Subsequent integration yields(4.10)where *g*(*β*) constitutes a function of integration. Similarly, *T*_{2} is necessarily of the form(4.11)where *f*(*α*) is a second unspecified function of integration. However, by construction, the functions *f* and *g* are determined by inserting *T*_{1} and *T*_{2} into the remaining equilibrium equation (4.5)_{3} and separating variables to obtain(4.12)where *C* represents a constant of separation. Accordingly, it is concluded that for any given one-parameter (*μ*) family of parallel Dupin cyclides and prescribed normal loading *Z* and flexural rigidity *K*, there exists a one-parameter (*C*) family of tensions *T*_{1} and *T*_{2}, which obey the equilibrium equations (3.1).

In general, the tensions *T*_{1} and *T*_{2} become singular whenever cos *β*=0 (cf. (4.16)); that is on the limiting cylinders displayed in figure 2*b*. However, there exists a unique value of the constant *C* for which these singularities are not present. Indeed, if we set(4.13)then *T*_{1} and *T*_{2} reduce to(4.14)and(4.15)Thus, if the tensions are assumed to be bounded, except on the focal conics where one of the principal curvatures *σ*_{1} or *σ*_{2} becomes singular, then *T*_{1} and *T*_{2} are uniquely determined. A Dupin cyclide together with the corresponding bounded tensions are depicted in figures 3 and 4.

It is noted that the expressions (4.11) and (4.12) for the tensions *T*_{1} and *T*_{2} may be brought into the form(4.16)which reduce to(4.17)in the case *Z*+2*C*=0. These may be regarded as ‘constitutive equations’ relating the tensions *T*_{i} and the principal curvatures *σ*_{i}. Comparison with equations (3.3) reveals that and vanish for all *α*, *β* if and only if *Z*=0 and *μ*^{2}=*b*^{2}/2+*c*^{2}. Accordingly, the corresponding Dupin cyclides constitute Willmore surfaces and are precisely those obtained by Kléman (1976). In the context of the shape equation (3.8) of biomembrane theory, these have been retrieved by Zhong-can (1990, 1993).

### (b) The residual condition

Up to now, we have set aside the residual condition (2.34), which potentially imposes constraints on the equilibrium equations (3.1). If we treat the liquid crystal layer merely as an incompressible sheet of liquid, then the infinitesimal strain component is non-vanishing and we obtain the additional requirement that . In this case, the equilibrium equations reduce to the shape equation (3.8) with *c*_{1}=0 and . However, the stress distributions (4.16) reveal that *Z*=*C*=0 and *μ*^{2}=*b*^{2}/2+*c*^{2}. Accordingly, parallel layerings of Dupin cyclide membranes do not exist in this setting. On the other hand, if the assumption of an ‘inextensible solid’ is made, then the infinitesimal deformations are isometric and hence . Accordingly, the residual condition (2.34) is identically satisfied. However, from a physical point of view, the assumption of inextensibility is not valid in the context of smectic liquid crystal phases. We therefore *postulate* that the class of admissible deformations is given by . This postulate is weaker than that of inextensibility but embraces incompressibility. Importantly, it renders the residual condition redundant and is therefore compatible with the existence of a parallel layering of smectic A membranes. The associated physical and geometric implications are currently being investigated.

## 5. Conclusions

In this paper, we have rederived Kléman's (1976) model of smectic liquid crystal phases by means of a variational principle and shown that, remarkably, it admits layered parallel Dupin cyclide solutions in accordance with Friedel's (1922) original empirical evidence. Because Friedel & Grandjean's (1910) *geometric* law of corresponding cones states how space may be packed with contiguous layered focal conic domains, we now require investigation of whether the *mechanical* relationships that exist between focal conic domains with common generators are compatible with the solutions presented in the present paper. The results of this investigation as well as other integrable reductions of the Kléman system (Burstall & Schief in preparation) will be published elsewhere.

## Footnotes

- Received November 9, 2004.
- Accepted April 20, 2005.

- © 2005 The Royal Society