Elastic energies of a directional medium

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Elastic energies of a directional medium

H.-R. Trebin

To cite this version:

H.-R. Trebin. Elastic energies of a directional medium. Journal de Physique, 1981, 42 (11), pp.1573-

1576. �10.1051/jphys:0198100420110157300�. �jpa-00209351�

Elastic energies ^{of a} directional medium

H.-R. Trebin

Institut für Theoretische Physik ^der Universität, ^D-8400 Regensburg, ^Federal Republic ^of Germany (Reçu ^{le 31}

1981, ^{révisé le} 22 juin, accepté ^le 7 juillet 1981)

Résumé.

Un champ tridirectionnel est employé

modèle pour

fluide général anisotrope. ^{La densité}

de l’énergie élastique ^est décrite par

polynôme ^{invariant dans les} composantes ^{du tenseur} de torsion. Les termes de l’énergie de surface sont dérivés à l’aide des conditions de compatibilité ^{pour le} champ tridirectionnel. Le modèle est appliqué

cholestériques de pas grand ^et petit, pour 3He-A, ^et pour les nématiques ^biaxes.

Abstract.

A tripod field is used

model for

general anisotropic fluid. The elastic energy density ^is expanded

invariant polynomial ^{in the} components of the contortion tensor. Surface energy ^terms

derived from compati- bility conditions for the tripod ^{field. The model is} applied ^to cholesteric liquid crystals ^of large ^{and small} pitch, ^to 3He-A, ^{and to} biaxial nematic liquid crystals.

Classification

Physics ^Abstracts

61. 30G

1. Introduction.

The elastic energy density ^{due to}

curvature strain was formulated phenomenologically by ^Frank [1] ^{for a} directional medium of local axial symmetry. ^{In the} present work, the elastic energy

density ^for mesomorphous phases of continuous translational symmetry, ^but arbitrarily ^{broken rota-}

tional symmetry ^is established. Following ^Kléman’s proposals [2], ¹ first consider a medium of orthonor- mal tripods. ^{As in the} early papers of Cosserat [3],

the contortion tensor is used as measure for curvature strain. The elastic _energy density ^is expressed ^as polynomial ^{in the} components ^{of this tensor of correct} local point symmetry. ^In the absence of defects, the contortion tensor must satisfy compatibility ^condi-

tions. It is these conditions which allow to express

part ^{of the} energy density ^as divergence ^{of a vector} field, resulting ^{in surface terms or} « nilpotent ^ener- gies ». ^{In case of constant} boundary conditions the

tripod field carries a topological quantum number,

the degree ^{of the} mapping, ^{which is} proportional

to the volume integral ^over the determinant of the contortion tensor. The theory ^is applied ^{to choles-}

teric liquid crystals, ^{where for} large pitch ^it yields

the Frank _energy [1], ^{for small} pitch ^Volovik’s large

scale continuous theory of cholesterics [4], ^{and to} 3He-A. For the elastic energy of the recently ^dis-

covered biaxial liquid crystals [5], ¹² bulk elastic constants are required.

2. Elastic _energy density.

^The directional medium is described by ^a field of orthonormal frames

(bl(X), b2(x), b3(x)). ^Each tripod is obtained from a

fixed triad of orthonormal vectors (êl, ê2, ê3) by ^a

rotation :

where R is an orthogonal ^{3 x} 3-matrix. Summation

over repeated indices is assumed implicitly. ^{As measure}

for curvature strain serves the contortion tensor K

of components

Gijk ^{are the} components ^{of the} completely antisymme-

tric third rank tensor. K contains all information about changes ^{of the} tripod field in linear approxima-

tion : when proceeding ^{from x to x + Ax, the vec-}

tors bi ^{are rotated} by the axial vector il of _compo- nents Q. 1

^Axi Klj, ^{so that} A-bi

^{Il x} bi.

When no defects are présent, the contortion tensor is restricted by the condition

for integration ^about any closed contour. Here the components ^{of fi are} differential forms

After application of Stokes’ theorem this leads ^to the

compatibility conditions (see for instance [2])

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198100420110157300

1574

In case of a weakly ^distorted anisotropic fluid, ^the

elastic energy density F(x) is formulated as polyno-

mial in the components ^{of K. For} simplicity, higher

order derivatives, ^{as in the work of} Nehring ^and Saupe [6], ^{will not} be included here. This polynomial

must be invariant under the operations ^{of the} point

group of the medium, applied ^{about x, the} symmetry

axes depending ^{on the} moving frame. Therefore it is convenient to write the polynomial ^{in terms of} compo- nents

of K in the moving frame, ^{and to} compose it of linearly independent homogeneous ^invariant polynomials pnq(Hrs) ^of degree q :

The coefficients c(q)n ^are the elastic constants. For the components Hrs ^{one obtains}

where abr

(br V) denotes the directional deriva- tive along ^axis b, ^{of the} moving tripod.

H transforms according ^{to a} polar ^{vector with}

respect ^{to the first} index, according ^{to an axial vector}

with respèct ^to the second.

For fixed (r, ^s, t) ^E cycl (1, 2, 3) ^the following ^rela-

tion results from the compatibility conditions (5) :

This term transforms like x. Hence the number of surface terms or « nilpotent energies » ⁱⁿ the elastic energy density ^is equal ^to the number of linearly independent polynomials ^{which can be} composed

of x2, y2 ^and z2, ^{and which are} invariant under the

point group of the medium.

3. Uniaxial systems.

^Linear combinations of H transforming according ^to the irreducible _repre- sentations of the symmetry group Doo ^are presented

in table I. For uniaxial systems the invariants can be

expressed ^{in terms} of the local symmetry ^{axis 1 =} b3

and the vector ^v of components vi

bl 1 - Dib2 (such ^that

v.br

Hr3)’ ^For superfluid 3 He-A, ^{v is} propor- Table I.

Linear combinations of ^H transforming according ^to irreducible representations of the symmetry group Doo.

tional to the superfluid velocity. The Mermin-Ho relation [2, 7]

is an immediate consequence of the compatibility

conditions (5).

Note that the integral

about a closed surface S yields the index N of all

point singularities ^{of the vector} field 1 enclosed by ^S.

The first degree ^invariant polynomials ^{are the linear}

combinations transforming according ^{to 1 ’. The}

second degree ^invariant polynomials ^{are linear com-}

binations of the direct products ^{of each} representation

with itself (Table II). ^{There are two} surface terms,

J4 and J5, transforming ^{as Z2 and X2 +} y2, respectively.

Table II. - Polynomials in the components of H

invariant under Doo.

In a small scale theory (large pitch) ^a cholesteric

liquid crystal is viewed as a nematic liquid crystal,

but lacking ^inversion symmetry, and thus of point

group Doo’ ^The symmetry ^axis points along ^{the direc-}

tor. Parametrizing the rotations R(x) by ^Eulerian angles (0(x), 9(x), 03C8(x)), ^{one finds}

0 and _ç denote polar and azimuthal angle ^of 1, 1/1 ^the

additional rotation about 1 to move the tripod ^into

its final position. ^For large pitch cholesterics 1/1,

and henceforth v, is a gauge variable. Though ^sub- jected ^to equation (10), ^{v is} arbitrary ^with respect ^to addition of the gradient ^{of a scalar} function, ^and

must not appear in the elastic _energy density. ^Hence

the elastic constants weighting J5 ^to J8 vanish,

leaving only the well-known splay, twist and bend curvature strains, apart from the surface term J4.

In a large ^scale theory ^as proposed by ^Volovik [4],

in lowest order of the pitch ^the director field is written

as bl(x) ^cos x(x) ⁺ b2(X) ^sin x(x) ^with Ox

qo 1, qo 1

denoting ^the pitch ^{wave vector.} Volovik inserted this form of the director field into the Frank (small scale)

elastic _energy and averaged ^{over the} rapidly varying phase x. By ^the averaging procedure ^the system ^obtains

a local cylinder symmetry Doo ^of symmetry ^{axis 1.}

So all terms of table II should contribute to the elastic energy density. However, since 1 is gradient ^{of a} phase,

curll vanishes, leaving for the second degree ^bulk energies those found by Volovik, namely Jl, J 7, J8.

The symmetry analysis ^{does not remove the linear}

term l. v. The elastic constant of this term vanishes

on the assumption ^{of a} stable local pitch, ^{so that in}

first order enlarging ^the pitch length (l.v > 0 for a

right-hand cholesteric helix) ^{costs the same} energy

as diminishing ^it (l. v 0). The surface term in the

large ^scale theory ^is 1/2 ^div (1 ^div 1). Note that the

large ^scale theory ^has only ^{a limited} range of appli- cability [4].

For superfluid ^3He-A the order parameter ^{is the} second rank tensor A

d _{Q (p,} where d is a real unit vector and _gP

b

⁺ ib2 ^a complex ^vector

related to the orbital motion of the Cooper pairs.

The orbital part ^{of the} gradient energy depends ^on

the spatial variation of _(p for which the contortion tensor is a natural measure. This _energy is invariant under rotations about 1 (gauge invariance), ^under

the inversion _(p

_(p (if simultaneously d ^{is trans-}

formed to - d), and under complex conjugation

(P

cp* ^which amounts to a reflexion in the plane spanned by b, ^and b3. In total the local symmetry

group is Dooh, for which the invariant polynomials

are those of second degree ^{in table II, in agreement}

with reference [7], although the surface term J 5 ^seems

not to be recognized ^as such in the literature.

4. Biaxial systems.

^Biaxial systems ^{are the} biaxial nematic liquid crystals ^{[5] of} symmetry D2n,

but also for cholesteric liquid crystals biaxial models

(of symmetry D2) ^exist [9, 10], ^{the three} inequivalent

axes being ^{the director, the helix axis, and the binormal.}

The irreducible representations ^of D2 according ^to

which the components ^{of H} transform, ^{are listed in}

table III. For symmetry D2 ^{there are} three first degree invariants, namely Hi i, H22, H33’ Those of second

degree ^are also invariant under D2h. They ^{are of the}

form Rii Hjj, HijHij ^and HijHji ^{(no summation}

Table III.

Components of H transforming according

to irreducible representations of D2

here over repeated indices). ^For comparison ^with

uniaxial symmetry different linear combinations of the invariants seem favourable. All the invariants for the small scale theory of cholesteric liquid crystals (D 00 plus gauge invariance) ^{are also} invariants for symmetry D2 (I1, Jl, J2, J3, J4 ^{of table} II). ^{Since the}

gauge invariance is broken, ^also invariants 12, Js

to J8 of table II occur. The additional invariants of table IV come from the fact that rotations about

axes b1

^and b2 ^are inequivalent. ^The corresponding

elastic constants for these latter should be small

compared ^to the former in case of weak biaxiality.

For biaxial nematic liquid crystals ^{12 bulk terms and}

three surface terms (J4, Js, J1 s) contribute to the cur- vature elastic _energy density.

Table IV.

Additional invariant polynomials ^{in case} of’ biaxial symmetry (D2)

5. Remarks on third degree ^invariant polynomials.

If the anisotropic ^{fluid is} constrained by ^constant boundary conditions (K - ^{0 for x 1 -+} oo), physical

space becomes topologically equivalent ^{to a three-} sphere S 3, because all points ^at infinity ^are identified.

Since the _space of rotations can also be viewed ^as part ^{of a} three-sphere ^{[11], the} tripod ^{field may be}

seen as mapping S 3 -> S 3. ^Then det K d3x ^{is the}

volume of the image sphere ^covered by ^the tripods

in the volume d3x. The integral

is an integer ^{and counts} how often the tripods sweep

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over the image sphere ^{as one moves} through ^{all of} physical space. This « degree ^{of the} mapping » (see

for instance [12]) ^{can also be} expressed ^as [11]

Relation (14) ^is applicable ^to systems ^{of uniaxial} symmetry, ^{in which case N} denotes the Hopf-index,

and also to cholesteric liquid crystals ^{even if the}

director field, is not constant at the boundary, ^but

forms ^a homogeneous ^helix. Simultaneously ^{det K}

is an invariant third degree polynomial ^{for systems}

whose point group contains only proper rotations

(like D2). ^{Since the} integer ^{N does not} change upon continuous variations of the rotation field R, ^this invariant polynomial ^{does not} contribute to the Euler-

Lagrange equations ^for the minimalization of the curvature energy, and thus constitutes a third order

« nilpotent » energy.

Acknowledgments.

¹ thank Prof. Kléman for valuable remarks regarding the number of compati- bility conditions.

References [1] FRANK, ^F. C., ^Discuss. Faraday ^{Soc. 25} (1958) ^19.

[2] KLÉMAN, M., ^Philos. Mag. ²⁷ (1973) ^1057.

[3] COSSERAT, ^{E. and} COSSERAT, F., Théorie des Corps déforma- bles, (Hermann ^{A. et} Fils, Paris) ^1909.

[4] VOLOVIK, ^G. E., Pis’ma Zh. Eksp. ^{Teor. Fiz. 29} (1979) 357, ⁵²³ [JETP ^{Lett. 29} (1979) 323, 475].

[5] Yu, ^{L. J. and} SAUPE, A., Phys. ^{Rev. Lett. 45} (1980) ^1000.

[6] NEHRING, ^{J. and} SAUPE, A., J. Chem. Phys. ⁵⁴ (1971) ^337.

[7] MERMIN, ^{N. D. and} Ho, ^T. L., Phys. Rev. Lett. 36 (1976) ^594.

[8] VOLOVIK, ^{G. E. and} MINEEV, ^V. P., ^Zh. Eksp. ^{Teor. Fiz. 72} (1977) ²²⁵⁶ [Sov. Phys. ^{JETP 45} (1977) 1186].

[9] WULF, A., J. Chem. Phys. ⁵⁹ (1973) ^6596.

[10] PRIEST, ^{R. G. and} LUBENSKY, ^T. C., Phys. ^{Rev. A 9} (1974) ^893.

[11] VOLOVIK, ^{G. E. and} MINEEV, ^V. P., ^Zh. Eksp. Teor. Fiz. 73

(1977) ⁷⁶⁷ [Sov. Phys. ^{JETP 46} (1977) 401].

[12] FLANDERS, H., Differential ^Forms (Academic Press) ^1963.