S2 analysis of cholesteric elasticity

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S2 analysis of cholesteric elasticity

F. Lequeux

To cite this version:

F. Lequeux. S2 analysis of cholesteric elasticity. Journal de Physique, 1988, 49 (6), pp.967-974.

�10.1051/jphys:01988004906096700�. �jpa-00210784�

S2 analysis of cholesteric elasticity

F. Lequeux

Laboratoire de Physique ^des Solides, ^Bât. 510, ^Centre d’Orsay, ⁹¹⁴⁰⁵ Orsay Cedex, ^France (Reçu le 30 novembre 1987, accepté ^sous forme définitive ^{le 8} fgvrier 1988)

Résumé. 2014 On montre que dans le ^cas d’une élasticité isotrope (K1 = K2

K3), ^les structures cholestériques ^à

une dimension ont un analogue ^mécanisme simple : ^{elles sont} représentées par le ^mouvement d’une charge glissant ^sans frottement en présence ^d’un champ magnétique ^{uniforme sur la} sphère image S2. ^{Etendant cette} approche ^{à des} structures à plusieurs dimensions et écrivant une règle ^{de flux, nous} présentons sommairement trois instabilités élastiques analysées ^sur S2.

Abstract.

We show that, in the case of isotropic elasticity, ^{the one} dimensional cholesteric structures can be

simply integrated : ^{on the} image sphere S2, they ^{can be} pictured ^as the motion of a charge gliding ^without

friction in the presence of ^{a constant} magnetic field. We can write a flux law, ^{even for a} multidimensional cholesteric field. We present briefly ^{how this} representation gives ^an interesting approach of three elastic instabilities.

Classification

Physics ^Abstracts

61.30

1. Introduction.

Cholesteric liquid ^{cristals are uniaxial} liquid crystals

and can be described, ^{from an elastical} point ^of view, ^{as a field} n (r ) from the real space R3 to the space S2 which is the unit sphere ^of R3 (if ^{one takes} n I =1). ^The equivalence between n and - n can be taken into account by using P2 instead of S2. Since

we do not consider defects but only ^continuous media, ^we prefer ^{to use} S2.

The free energy of ^a cholesteric has been given by

Frank [1] :

(where q is the natural inverse pitch).

This energy is constituted by ^{three terms which}

measure splay (K1 ), ^twist (K2) ^{and bend} (K3 ) [2]. ^In

the special ^case where the three ^constants Kl, K2, K3 ^are equal, the energy is :

or

In this _case, the energy becomes highly symmetric

and numerous studies have been done on this

isotropic elasticity, the latest developments being

studies of those systems ^{in the} curved space

S3 [3].

The classical representation of this energy is in terms of splay, ^{twist and bend.}

In the special case q

0 (i. e. nematic) ^Thurston [4] ^has proposed ^a representation of the energy. We extend his approach ^to cholesterics. We limit our-

selves to isotropic elasticity. Although ^{it could} easily

be extended to anisotropic elasticity, ^{it would be}

cumbersome and of no peculiar ^interest.

Let us explain ^{the basis of our} representation.

One can represent ^{the field on the} image space

S2 ^as follows : one can draw, ^on S2, ^{the curve} (n (r )) image ^{of the} R3-curve (r). ^In fact this appears to be interesting ^{when we} take, ^{for the} R3-curves (r), ^lines parallel ^{to one} of the three axes of a

classical reference frame Ox, y, ^z and we limit ourselves to this case.

We limit ourselves to the problem ^{of a static}

cholesteric in a rigid ^{box with a} strong anchoring, (i.e. where the field n is given ^on the faces of the

box), and without external fields. Consequently, ^we

will never take surface energy ^terms into account.

After explaining ^our method, ^we will show on some examples how this method gives (we hope) ^a

clear analysis ^{of some elastical} problems.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01988004906096700

968

Our method gives only ^a visualization of the energy, which seems to us to be very useful. It allows

us to do some approximations ^{or to} guess ^some textures without a complete ^classical calculation.

This is important ^{in this} problem ^where there is in fact only ^one dimensionless parameter qL (where

L is the size of the box).

Our method gives ^{neither a new method for} minimizing ^{the Frank} energy ^{nor a new} mathematical formulation but only ^make visual the energy of some

textures.

We will see that this visualization of the energy

can give ^an interesting approach and discussion of

some simple problems. ^In part 2, 3, 4, 5, ^we explain

the basis of our method. In 5, ^{we use it in some} very

simple ^cases.

2. Method.

As said previously, ^{we take} Kl = K2 ⁼ K3.

Let us first take a one dimensional-field, ^{i.e. a} field depending only ^{on z. We take} the orthonormal frame (0, x, y, z) with the coordinates (x, y, z). ^On S2, ^{a curve} (n (z ) ) is defined with z as a parameter.

We call velocity ^{on the curve} an , making ^an

az

analogy ^{between z and time.}

The curve on S2 is described with a velocity equal

to the gradient aznj.

Let us take S2 ^as the unit sphere ^{of R3 with the}

usual orthonormal _{frame ex,} ey, ^{ez. Of course, we} take our frame (x, ^y, z) of the real space equal ^{to our}

frame (ex, ey, ez) ^{of the space} image, ^{but we take}

different notations for real and image space in order to avoid some confusions.

On S2, ^{we can} express splay, twist and bend of a one dimensional field n(i ).

Multidimensional field.

If the field depends ^{on d} coordinates, ^the image

on S2 is a mapping ^{of a} part ^of S2 constituted by

d sets of _curves, each one being ^a one-dimensional

curve with the d-1 other coordinates as par- ameters. For example, ^for two-dimensional field

(x ^{and z for} instance) ^{we have two sets of curves :}

and

In this ^case the coupling between the two sets of

curves is complex ^because implicit (one ^{set of curves}

can be deduced of the other). Nevertheless, ^{we shall}

see that two dimensional structures can be interpre-

ted on S2.

3. Energy ^{of a curve} n (z).

The density of the energy of a structure can be written (after (1)) :

and can be understood as the sum of the energies

associated with the curves (in ^x, y ^or z). ^This

associated energy is :

(respectively x ^and y).

Let us consider the one-dimensional problem. ^The

energy ^{is the sum of a} kinetic _energy, (the square ^of the velocity), ^{and a kinetic moment of the} moving point ^{n about the axis} ez. The trajectory ^can easily ^be integrated by minimizing the free energy F.

4. One dimensional problem, mechanical analogy.

A material point ^{P of mass m and} charge e, moving

without friction on S2, in the presence of a magnetic

field B, ^{has a} Lagrangian ^L equal ^to (6) :

We can make the analogy

And the Lagrangian ^{of P is} equivalent ^{to the local}

energy of the field n (z ). Equilibrium configuration ^is

a minimum of the integral ^of F, ^and consequently, ^a

minimum of the action of the point ^{P ; the curve} n(z) ^{is the} trajectory ^{of a} point (m, e ) moving ^on S2, ^without friction, ^{in a} magnetic ^field B = -2e,-A

with the relation qm

eA.

The velocity ^{of the} point ^{P is constant in the}

modulus (because ^the magnetic ^{force does not} work), ^{so the value} an _{is constant in the space.}

az

We take n in spherical coordinates :

The kinetic energy is ^{constant :}

The Lagrangian ^{is :}

where

designates ^{the z} (or time) derivation.

.

Then the equations ^{of motion are :}

One derives :

where a is a constant.

The trajectory ^is restricted in a zone where 0 is such that

The integrals ^are

and

In the case where the trajectory goes through ^the point ez (or - ez), ^{we have a}

^{0 and} simply :

which is an elliptic integral of the first kind.

5. Flux law.

It is well know, ^{that the} trajectory ^{of a} particle moving ^{without friction on a surface is the} path ^{of a} perfect ^{elastic stretched over the same} surface. In order to extend the analogy ^{to our} system, ^we just

have to add the magnetic ^term, which appears ^to be

a flux term.

So, ^{we can} build this other mechanical analogy :

The energy is equal ^to the energy of ^an elastic loop (with ^a vanishing length ^at rest) plus the flux of ez through ^the loop, ^{with the} adequate coefficients.

If f is the length ^{of the} loop

L, ^the reciprocal length ⁱⁿ R3 and CP the flux of _e, through ^the loop

the total free energy is :

This formula is valid for any closed loop ^{n, and for}

an open loop, ^{the flux contains an} arbitrary ^additive

constant.

Note that for _any dimensional structure one has :

where S is the surface vector of the surface leaning

on n (i ).

Even if the velocity ^{is not constant, the value of}

the 2 qn . curl n ^{term for a curve} n (i ) is -4qOi, where 0 i is the flux of the vector ei through ^the

curve n (i ).

In general ^the velocity ^modulus is ^{not constant}

ðn ) ²

and one must integrate ( an ^{ai 2 over all the}

d set of curves.

The total energy is then

where the 0 i are the fluxes of ei through n(i ).

The coupling ^{between the x,} y and ^z directions is

implicit and in fact for 3 dimensional problems, ^{it is} quite ^{difficult to minimize} the energy of the field.

We will now discuss some particular ^{cases. The} boundary conditions are expressed by imposed points ^{or curves at} the extremities of the sets of the n curves.

Remark : The lowest energy situation, ^{for an}

infinite continuous medium, ^{is the classical one-} dimensional planar geometry :

described on S2 by ^the great ^circle n (z ) ^{normal to the}

direction Oz (the equator) swept many times, ^the velocity being equal to q, ^{as one can} easily verify :

on S2, it is the geometry which maximizes the

(negative) flux and the energy (given by Eq. (3)) imposes ^the velocity q.

6. Study ^{of some} simple configurations.

We now present ^some examples ^{where our} analysis

raises an interesting discussion.

6.1 TRANSITION CONSTANT FIELD/ONE DIMENSION- AL FIELD IN A HOMEOTROPIC GEOMETRY (OR TRANSLATIONAL INVARIANT CHOLESTERIC OR

TIC).

^{Let a} cholesteric sample be located between two parallel planes (in ^z

^{0 and z}

d) ^{where the} boundary condition is n

ez, (homeotropic ^anchor- ing). ^{One can observe} [7], by decreasing the natural

pitch ² 7r/q (by heating ^for example) ^{that the initial}

situation n

ez, ^{at a} certain threshold begins ^to

modulate along the Oz axis. On S2, ^the trajectory

970

n(z), initially ^{reduced to the} point ez then is described by ^{the relation} (Eq. (2)). n(z) ^{is then a}

lobe closed on the point ez of S2 (see Fig. 1).

Fig. ^1.

One dimensional structure between two parallel planes ^{with n normal to the} planes ^{on the} boundary. ^The

structure is a point ^{on S2 for} large pitch (n ^is constant) ^and develops ^{into a lobe} (while q increases) increasing ^{its flux}

about e.. On the bottom the classical representation ^{of the}

initial and the TIC structures.

We will first describe what happens ^{at the}

threshold between n constant field and n (z) rep- resented by ^{a lobe on} S2.

In order to get the smaller order (in size of the

lobe) ^terms in energy, ^we will approximate ^the sphere ^near ez by ^its tangent plane (z ^{= 1 in the} image space). ^{In this case, it is evident that the} n(z) ^{are circles} passing through ez (on ^the plane,

circles have a maximum ratio flux/perimeter). ^The

energy is then, (using Eq. (3)) :

where r is the radius of the circle n (z ).

For - ; , F ^is minimum for r ⁼ 0 and for

q >- ; , F is minimum for r - oo.

The threshold is then q - ’T _d

We can verify ^{that this} gives ^the good r2 term ^for

F, and that when Eo - ^0, equation 2 becomes :

The angle between ez and the circle gives ^an r4 contribution to the flux.

It is evident that the curvature of S2 (which

decreases the flux about the plane ^z = 1) ^stabilises

the lobes n(z ). ^{The r4 terms in F are} negative

because flux and the ratio flux/perimeter ^{are both} energetically unfavourable on S2 (about ^{on z} = 1).

Consequently ^{we have on S2}

We have a second order i la Landau transition : i. e, the lobe develops ^while increasing q ^for

q >- -i . ^{The classical} representation ^{of n on the real}

space is drawn in figure ^1.

6.2 MODULATION OF THIS STRUCTURE ALONG Ox

(DOUBLE ^{TWISTED OR} DT).

^{The latter structure} (the TIC), ^{at a certain} amplitude, presents ^another modulation along the Ox axis. Without taking ^details

into account, ^{we can} just explain ^it.

The initial lobe no(z), previously ^described (Eq. (2)), ^reaches (by increasing q) a zone near the equator ^{and then} develops ^{into a two} dimensional

configuration, ^{the structure} increasing its flux about the Ox direction.

Let us explain ^{how it} develops.

Curves at z

cst. are initially points, ^and spread

out in some small loops ^{centered on their initial} position, ^{in order to increase their flux about Ox.}

The more the loops ^{are normal to} Ox, ^the bigger they become, and their size vanishes for z

0 and

z

d (i.e. ^{at the} point ez) ^{in order to} satisfy ^the boundary conditions.

The curves for x constant can be deduced and are

similar to the initial lobe (see Fig. 2).

This complex situation exhibits a double-twist : there is twist about Oz and about Ox. We limit ourselves to this simple approach, ^{but will} develop ^a

more precise analysis ^{in a future} publication. ^We only present ^a picture ^{of the two} instabilities in

figure ³ (courtesy ^{of P.} Oswald).

6.3 ZERO LAYER CHOLESTERIC DILATION INSTA- BILITY.

Let us now consider a cholesteric between two parallel planes ^z

^{0 and z}

d, ^{with n on each} plane equal ^to ex. (This planar anchoring, ^{like the} homeotropic anchoring, ^{can be} imposed by adequate treatments on glasses).

For small q (i.e. large pitches), ^{n is constant}

between the plane ^{and is} represented by ^the point

ex ^on S2.

Let us now try ^{to estimate what} happens ^while

increasing q.

Fig. ^2.

^The previous ^lobe n(z) develops ^{in a 2-}

dimensional structure, creating ^small loops with flux about Ox (on the bottom the R3 representation).

In order to minimize its energy, the system ^will increase its flux. Near ex, the only ^{flux a lobe can} get is a flux about ex (the ^{surface at} ex contains

ey ^and ez). ^{Hence the structure will} develop n (x )

lobes.

For z

0 and z

d, ^the n (x ) ^{lobes must be the} point ex itself.

We can take n (x ) ^as circles centered on ex and of a

radius r(z ) ^with r(z

0)

r(z

d )

0 in order to

respect ^the anchoring.

Near the threshold, ^{we can} approximate (as ⁱⁿ 6.1) S2 by ^the plane ^x = 1, ^{and we take}

r ⁼ ro sin 7 . ^{In this case it is easy to} prove that this is the lowest energy situation.

The field on S2 and on R3 is represented ⁱⁿ figure 4. If the field is written n

cos cpex ⁺ sin cp sin Oey ^{+ cos Oez, we have :}

and

The _energy is written (in ^mean density using Eq. (3))

where 2 7T / k ^{is the} length in the real space along ^Ox

to sweep ^one time the n (x ) ^circles. (k ^{is the wave}

vector on R3).

As shown in figure 4, ^the n (z ) curves are arcs of great ^circles swept from ex and back ^to ex.

The structure can develop for a 2 F a2F ^{ar2 0 i.e.}

which gives the threshold (for ^{a free value of} k)

Remark : k has a definit sign, i.e., ^the n (x ) ^{are swept}

in the right direction. This will be developed ^{in the}

Fig. ^3.

Picture of the TIC and DT instabilities : the black zone (on ^the left) ^{is the n}

constant zone ; the white zone

(on ^the middle) is the TIC _{zone ;} the modulated zone is the DT zone. Optical microscopy, ^crossed polarizers, courtesy ^of

P.. Oswald.

972

Fig. ^4.

^Dilation instability ^{of a zero} layer cholesteric.

Above the representation ^{and on} the bottom the classical

representation. The initial situation is n

ex everywhere.

case of a many-layer ^dilation instability. ^{One can} verify using ^Euler’s equations, ^{that at the} threshold,

our model is good. Experiments ^{are in} agreement and will be discussed in another publication.

6.4 ANALYSIS OF THE UNDULATION DILATION OF A PLANAR CHOLESTERIC.

Let us now consider a

cholesteric laying ^{between two} parallel planes

z

0 and z = d with n, on each plane, ^{constant and}

contained in the plane (planar anchoring). ^The

initial situation is the natural helical situation of

cholesterics, with nc layers :

This is pictured ^on S2 by ^the equator swept ⁱⁿ n,, times.

When one dilates this system (or ^{decreases the} pitch ^{2 7T} /q), ^{one observes an} undulation of the

layers [8]. Usually ^the analysis ^{uses an} approximate layered energy, known ^as coarse-grained energy [9].

We now present ^another analysis, ^{based on} S2 configurations.

Let the system develop ^{into a two} dimensional structure n (x, z ). ^{One can} easily prove that in dilation the n (x = cst., z ) ^{curves cannot decrease} (with ^{a small} perturbation Sn(z)) ^their energy around their initial equatorial configuration no (z )

because their (negative) ^{fluxes are maxima} (for

instance one can verify ^that

for any 5 n, ^with Fz equal ^to the energy of the

n(z) ^{at x}

xo curves).

Thus the only ^{term which can} decrease is the flux term about ex.

We choose Ox to be parallel ^to n(z

d/2) (for

symmetry reasons), and look for the simplest ^struc-

ture decreasing the energy. We call no the initial one

dimensional structure.

We must create a flux about Ox i.e. little loops n (x, ^z

zo ) ^{near the} points ^of S2, ex and - ex. ^The

loops ^{have the} symmetry ^{of the} system. They ^must

be ellipses ^{of axes} ez and erp

^{no A ez. Let a and} f3 ^be the radii of the ellipses along erp ^and ez. The flux ^must have the same sign ^around

ex and - ex, i.e., ^the orientation of the n(x) ^{must be}

the same near ex and - ex. ^{But since} efP

ey ^at

ex, and efP = - ey ^{at - ex, we must} change ^the sign ^of

« or B. ^The symmetry ^{n, - n of the} cholesteric

implies ^{that we take « and -} f3 in - ex for

« and f3 in ex.

This imposes ^that /3 ^must be modulated by ^the

cosine _ex. no.

Moreover, a ^and B ^must vanish for z

0 and

z

d (in ^{order to} satisfy ^the anchoring), ^{and we can}

write that :

and

The only approximation ^{of this} construction is that in reality ^{the cos} 7rz/d and the no - ex modulations

are coupled ^{and resonate} (as ^{a more} precise analysis

will show in a forthcoming publication, ^{but for a} many-layer system, this effect is not important). ^The

modulation is given by ^the perturbation ^vector

Bn ;

and is drawn in figure ^5.

The energy ^can be calculated directly ^on S2 by (Eq. (4)), ^or by the classical Frank energy (Eq. (1))

without forgetting ^the condition n ⁺ Sn I ^{= 1 at the}

second order in a and 8.

One easily finds that the variation of energy due ^to Sn is :

2 ^7T

where t

n, - 2 7r ^{is the twist of the initial planar}

structure.

Minimizing ^this quadratic ^{form in a} and f3 (we

can for instance write that the determinant of the

quadratic ^{form 6F} (a, B ) ^{vanishes at the} threshold),

Fig. ^5.

Dilation undulation representation ^{on S2 :} initially, the situation is described by ^the equator. ^Little ellipses develop ^near ex and - ex with the ^same sign ^about

Ox. This picture ^must be modulated in amplitude ^from

z

0 to ^z = d in order to respect ^the boundary conditions.

one gets ^{for the} threshold :

This analysis ^{is more} precise ^{than the coarse} grain

calculation. It gives the second order for k2 in

7T/d:

and the first order term which is in general given

with the wrong coefficient in the classical analysis. ^In addition, ^we show that the ellipses, when q ^{= t, are} incompatible ^{with the} layer conservation hypothesis

« ⁼ t/k ^which is the basic concept ^{of the} layered energy (as ^d becomes very large, q - t ^{-+ 0}

and f- t/k ; ^{for many} layered systems ^the 18

hypothesis ^of layer conservation is valid).

Here we do not present ^the complete analysis ^of

the dilation instability, ^{because it is not our} present aim, ^{but we} will discuss it later in a forthcoming publication. ^This approach, ^{which is not a} complete

solution of the problem, is nevertheless interesting

and more precise ^{than the} existing analysis.

7. Conclusion.

We have shown that the cholesteric structure with an

isotropic elasticity, ^when considered as a mapping ^of S2, ^{can be} simply interpreted ^{from an} energetic point

of view.

Each image ^{of a} straight line of the flat space is ^a

curve on S2. We can associate to each curve an

energy which is the ^sum of a kinetic energy and ^a flux.

We hope ^{that our S2} method of energy analysis

can help ^to get ^{a better} understanding ^{of the}

cholesteric elasticity.

We are fully ^aware of the limits of our method,

and in particular of the fact that this method does not clearly bring ^new mathematical resolutions of the cholesteric elasticity, ^{but we are} convinced that it

can be simple and useful for a priori approaches, ^and approximations.

The anisotropic energy is cumbersome because the surface and the magnetic field of the mechanical

analogous ^are very complex, ^{but it is} quite simple ^to transpose isotropic results in an anisotropic analysis.

We hope ^{too that we} will be able to initiate new

interest in cholesteric elastical instability.

Acknowledgments.

We are greatly ^{indebted to} M. Kleman who has

suggested ^{and discussed this work and we thank P.}

Oswald for discussions about the T.I.C. and D.T.

experimental observations.

References

[1] FRANK, ^F. C., ^Discuss. Faraday ^{Soc. 25} (1958) 19 ; NERHING, J., SAUPE, A., ^{J. Chem.} Phys. ^54-1 (1971)

337.

[2] ^{See for} example ^DE GENNES, ^P. G., ^The physics ^of liquid crystals (Clarendon ^Press, Oxford) ^1974.

[3] ^From SETHNA, ^J. P., WRIGHT, C. D., MERMIN, ^N. D., Phys. Rev. Lett. 51 (1983) ^{467 to} PANSU, B., DANDOLOFF, R., DUBOIS-VIOLETTE, E., ^J.

Phys. ^{France 48} (1987) ^297.

[4] ^{THURSTON, R.} N., ALMGREN, ^F. J., ^J. Phys. ^France

42 (1981) 413 ;

THURSTON, ^R. N., ^J. Phys. ^{France 42} (1981) ^418.

[5] ^{In the case of} anisotropic energy, ^{one must} separate the three components (in K1, K2 ^and K3) ^{of the} gradient ^and multiply ^them by the convenient coefficient. This can be solved by taking ^a complex surface 03B6 instead of S2 in order to have

(~z)03B6 ^~ F - 2 qn · curl ⁿ (see [4]).

[6] ^LANDAU, L., LIFSHITZ, I., ^Field Theory (Mir ^Mos- cow).

[7] OSWALD, P., BECHHOEFER, J., LIBCHABER, A., LEQUEUX, F., Phys. Rev. A 36 12 (1987) 5832 ; LEQUEUX, F., OSWALD, P., ^{to be} published.

PRESS, ^M. J., ARROTT, ^A. S., ^J. Phys. ^{France 37}

(1976) ^387.

974

[8] CLARK, ^N. A., MEYER, ^R. B., Appl. Phys. ^{Lett. 22} (1973) ^{473 ;}

DELAYE, M., RIBOTTA, R., DURAND, G., Phys.

Lett. A 44 (1973) ^139.

[9] ^{DE GENNES, P. G., The} physics ^of liquid crystals (Clarendon ^Press, Oxford) chapitre 6, p. 245

(1974) ; (see ^also LUBENSKI, ^T. S., Phys. ^{Rev. A}

6 (1972) 452).