Developable domains in hexagonal liquid crystals

HAL Id: jpa-00209299

https://hal.archives-ouvertes.fr/jpa-00209299

Submitted on 1 Jan 1980

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Developable domains in hexagonal liquid crystals

M. Kleman

To cite this version:

M. Kleman. Developable domains in hexagonal liquid crystals. Journal de Physique, 1980, 41 (7),

pp.737-745. �10.1051/jphys:01980004107073700�. �jpa-00209299�

Developable domains in hexagonal liquid crystals (*)

M. Kleman

Laboratoire de Physique des Solides (**), ^Bât. 510, Université Paris-Sud, ⁹¹⁴⁰⁵ Orsay and S.E.S.I., C.E.N., ⁹²²⁶⁰ Fontenay

Roses, France

(Reçu ^{le 17 décembre} 1979, révisé le 29 février, accepté ^{le 3}

1980)

Résumé.

Les configurations ^d’une phase mésomorphe ^du type ^cristal liquide hexagonal

possédant pas

d’énergie élastique mais seulement de l’énergie de courbure sont caractérisées géométriquement par l’existence de repères ^locaux

distordus par rapport ^à celui propre

cristal parfait, ^et par l’absence de déformation de

divergence (div ^t

0). Ces contraintes conduisent à

grande ^{classe de} solutions, ^sans déformation de torsion

(t. rot ^t

0),

déformation de flexion (t

^{rot t} ~ 0). ^On démontre que

solutions

torsion

térisent par l’existence de plans physiques perpendiculaires

alignements moléculaires

colonnes (pour ^les disquotiques)

tuyaux (pour ^les lyotropes) ^{de la} structure. Ces plans enveloppent

^surface développable, type de solution d’abord mis

évidence par Bouligand. Nous étudions

détail

propriétés géométriques ^et

montrons comment obtenir tous les domaines développables ^à partir ^d’une représentation plane ^{de la} configura-

tion spatiale. Nous donnons aussi des méthodes de calcul de l’énergie. Cette théorie est illustrée par deux exemples simples : l’un où le domaine développable équivaut ^à

disclinaison, l’autre où il équivaut ^à

point singulier.

Abstract.

Configurations without strain (as opposed ^to curvature) ⁱⁿ

mesomorphic phase ^{made of} long ^thin

rods (like hexagonal liquid crystals)

characterized by local frames of reference undistorted with respect ^to the ground ^{state and} by absence of splay deformation (div ^t

0). These constraints lead to

important ^class

of solutions, without twist deformation (t. rot t

0), with bend deformation (t

^{rot t ~} 0). ^{It is shown that}

this class of solutions is characterized by the existence of physical planes perpendicular ^{to the} rods, enveloping

a

developable surface. This is the solution first devised by Bouligand. ^We study ^its geometrical properties ⁱⁿ

details and solve the problem ^of obtaining all the so-called developable ^domains by using ^a plane representation

of the space configuration. ^{We also} give ^{methods to} calculate its energy. Two simple examples

given, ^{one in}

which the developable ^{domain is} equivalent ^to

disclination, ^another

in which it is equivalent ^to

singular point.

Classification

Physics ^Abstracts

61.30 - 61.70

61.30

61.70

The recent discovery of mesomorphic phases ^whose building ^{molecules are disk-like} [1] (the ^so-called

discotic phases) ^{and are stacked in} cylindrical ^rods

which are themselves hexagonally packed [2], ^raises

a number of interesting questions concerning ^the arrangements ^{of these} cylindral rods and the _energy attached to various arrangements. ^Such questions

could as well have arisen a long ^time ago with the well-known hexagonal phases ^of soaps and lipid-

water systems, first discovered by Luzzati et al. [3].

Although ^some observations of textures either by optical microscopy [4] ^or by ^electron microscopy ^of

freeze-etched specimens [5], have been made in such systems, ^{and have} provided ^us with standard pictures

of these textures, ^there is however no systematic description ^{of the} specific features of the organization

of hexagonally packed long ^thin rods, ^{when the} long

range order is perturbed. ^{As it is} well known now

by ^the example of nematics and other classical meso-

morphic phases, ^{such a} description ^has necessarily ^to

use altogether geometrical concepts (for the distor- tions with short wave vectors) ^and topological

concepts (for ^the understanding ^{of the} singularities

and defects of the order parameter ^at long ^wave

vectors [6]).

This _paper will deal essentially ^with geometrical concepts ; ^recent observations in the polarizing ^micro-

scope of inverted hexagonal phases ^of complex detergents [7] ^and of discotics [8, 9] ^{have led to models}

for the repartition of molecular cylinders. ^In parti- cular, ^{it has} been assumed [8] and checked on some

examples that the distortions of low _energy ^are such that all the long thin rods are perpendicular ^{to a}

one-parameter familÿ ^of planes (in ^{which case the} long thin rods suffer only bend). ^{We shall} justify ^this assumption ^on well defined energetical grounds. ^This

condition is akin to the condition of parallelism ^of

smectic layers ^{in smectic} phases, which lead to the (*) ^Work partially supported by ^{DRET contract} 79/352.

(**) ^Associé

^C.N.R.S.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01980004107073700

738

existence of cofocal domains [10] ; ^as in this last case, it allows for a vanishing ^strain energy (strain ^as opposed ^to curvature).

We shall therefore discuss in details the case of pure bend, ^{for which we} shall extend Y. Bouligand’s

results [8]. ^The discussion of other cases, ^as well as

of topological concepts, will be left for a forthcoming

paper. We also leave for another publication ^a

detailed discussion of experimental observations of

defects, ^{in the} light ^{of the} present study.

We shall assume that the reader is familiar with the structural properties ^of hexagonal phases [1, 2, 3].

1. Free energy density ^and general considerations.

Let us call t a unit vector along ^a long ^{thin rod} (1) ;

t is a function of position ^{r. We} symbolize ^{the rod} by

a curve T along ^its central axis. The free _energy

density pf ^{contains a term of curvature} pic ^{and a}

term of strain pis. By analogy with other liquid crystals [11] ^{we have}

The strain _energy density ^reads [12] :

Let us first assume that the rods are infinite (the

concentration of chain ends vanishes) ^{and that mass} density of the medium is constant p

po. Therefore

we have [13] :

and the term of compressibility vanishes also in (2).

We shall investigate ^{the more drastic case in which} pfS ^vanishes completely ; i.e. the rods are at constant distances and the hexagonal packing ^of rods is pre- served (angularly ^and metrically) locally. ^These geo- metrical properties imply ^the following :

For the metrical properties : ^Consider any surface 1

generated by ^{a set} of rods which are neighbors ² by ² (we shall call r lines the lines representing ^the rods).

This surface can be defined unambiguously ^because

the distances between neighboring ^{r lines are small,} compared ^{to their} length, ^{and we can} safely ^{use the}

continuous limit when this distance _goes to zero. In such a 1 all the rods are parallel (in ^{the sense of} parallelism ^{on a} surface). Hence the orthogonal trajectories ^{G of T lines are} geodesic ^{lines on E. It}

(1) ^We

t, rather than

(the classical notation for the direc-

tor), ^because

will have here the meaning ^{of the} principal ^normal

at

point

along

^rod.

can be shown that these metrical properties imply,

as stated already ^{above :}

Angular properties : ^{Two surfaces} Il ^and L 2 ^which

intersect along ^{a common r line make a constant} angle ; hence, according ^to Joachimsthal’s theo-

rem [14], ^{either r is a line of curvature on both} LI

and L 2’ ^{or is on none. Hence we have to} distinguish

two cases :

Let us study ^{the case when r is a line of curvature}

on Il ^and 12. ^It is therefore a line of curvature on

any E passing through ^{T and we shall assume that}

there is a domain in which _{any r} is a line of curvature.

The G lines, ^{which are} orthogonal trajectories ^{to the} lines, ^are also lines of curvature of the L’S. Being ^at

the same time lines of curvatures and geodesic lines, they ^are plane (no torsion). Therefore there exist H

planes ^{which are} perpendicular ^{to the whole set of}

r lines of the specimen, ^and any straight line in such Il planes intersects orthogonally ^{the T} lines. More- over, since there are surfaces normal to the T lines,

we also have

and the _energy density ^{reduces to a} pure ^term of

bending. We have here obtained, by demonstrating

the existence of planes II, the fundamental possible geometrical feature of an hexagonal phase ^{in which}

the strain _energy density vanishes. This result was

proposed without demonstration by ^Y. Bouligand [8].

J. Mérucci [7] ^insisted only ^on the existence of straight

lines perpendicular ^{to the rods.}

Let us now assume that we are in the less interesting

case (at ^least energetically speaking) when all the r lines of a given domain of the sample ^{are not lines}

of curvature of the E’s. We shall show that this case

is impossible.

Since the angle ^{between two E’s is constant all} along r, ^the geodesic ^torsion 1/r ^{of r is the same on}

both I’s, ^{and in fact on all I’s} passing through ^r.

1/t ^is simply ^{related to} the twist of the rod. We ^can indeed show that

The non-vanishing of the twist indicates that there

are no surfaces perpendicular ^{to the} rods. But the r lines have orthogonal trajectories ^{G, as in the} preceding ^{case, which are still} geodesic lines of the L’S, for the same reason as above, ^i.e. because the T line

are parallel ^on any 1. This implies ^{that the} osculating planes of the G lines contain the director t. There- fore the center of curvature of a G line is the point ^of

intersection of two straight ^lines bearing ^two neigh- boring ^{t directors t and t’ on two different r lines,}

when t and t’ approach ^{each other} infinitesimally

along ^{the G} line. If such a point of intersection is at

finite distance, ^this implies ^some elastic distortion of

the two-dimensional hexagonal packing. ^{But we}

have excluded this case. Therefore the radius of

curvature of a G line is infinite and the G lines are

straight. ^{r lines are} therefore still lines of curvature and there is only ^one class of solutions to our problem.

We can express this result ⁱⁿ other words, by stating

that there is a class of distortions of an hexagonal liquid crystal ^{in which} angular properties ^{are con-} served, but metrical properties ^{are not.}

We shall here restrict to the case where angular

and metrical properties ^are conserved. For reasons

which will appear shortly, ^{we call} developable ^domains

the regions ^{in which} eqs. (3) ^and (4) ^{are satisfied.}

2. Description ^{of the F} lines ; reciprocity ^{with a}

fixed line of singularity.

^Two neighboring ^{r lines} (r 1 ^and T 2) ^have parallel tangents ^at their intersec- tion with a given ^II plane (Fig. 1).

Fig. ^1.

^Two neighboring ^r lines at their intersection with

il

plane.

It is _easy, by differentiating

t

(where ^{v is an unit vector} which, in the Frenet trihe- dron ni, bi, tl, ^{reads v}

ni ^{cos m +} bi sinw), ^to

obtain

with

1 / T1 ^and 1 /R are the standard torsion and curvature

of T1 ⁱⁿ Pl. si and _s2 are curvilinear abscissae on T1,

and T2.

Let us express that t1

t2. ^One gets

Since ti

t2, ^the principal ^normals n, and _n2, and the binormals bi ^and b2 ^are parallel, ^{and we} get

Condition (7a) ^is equivalent to t. rot t

0. Eq. (8a)

means that the centers of curvature-C of the different ^r

curves intersecting ^{II are on a} straight ^{line D which is} perpendicular ^{to n} (Fig. 2).

Fig. ^2.

^{Centers of curvature C and} principal ^{normals of r}

in plane ^77.

It is clear that D is an axis of instantaneous rotation of the plane ^II (and of the lines T) ^{and as} such is the limit of the intersection of H with a neighboring plane

H’. In other words the set of planes ^II envelop ^a developable ^surface S; the lines D are the straight

lines of S. Call M the characteristic point ^{of D : it is}

the point ^where two neighboring ^{lines D and D’}

intersect when D’ tends towards D. The locus of M is a twisted curve L which is a singular ^{line on S} (the

so-called cuspidal edge of S). ^{A clear} description ^{of the} geometrical properties ^of developable surfaces is

given by Hilbert and Cohn-Vossen [15] (Fig. 3).

The physical ^{lines T exist in one of the} regions ^of

space delimited by ^{the two sheets of} S ; ^this region ^is

defined by ^the possibility ^of drawing ^{at least one real} plane ^P tangent ^{to S. For} example, ^{if S is a cone} (in

Fig. ^3.

^A developable ^{surface S} generated ^by straight ^{lines D}

and its cuspidal edge ^L which divides L into two sheets.

740

which case the cuspidal edge is reduced to the apex),

the physical region ^{is the external one.} It is clear in this

example ^{that we have to consider} physically only ^an

half of the plane II, ^{i.e. a} region ^bounded by ^{the line D.}

The system ^{contains a} surface of singularity ^S (to

which r lines are orthogonal) ^{and a line of} singularity

L. The name of developable ^domain appears ^now

justified.

The existence of L, S, ^{and D has} been demonstrated

by ^Y. Bouligand who however did not try ^to get formulae (7) ^and (8). ^{We shall now see that} 1/R ^and 1/Tare ⁱⁿ duality with similar quantities ^{relative to the} cuspidal edge ^L.

Consider in the same II plane the Frenet trihedron in P (a point ⁱⁿ H) ^{on the T line} passing through ^P

and in M (the characteristic point ^of D) ^{on the cus-} pidal edge ^{L. The} tangent ^{to L is} p

b, ^{the normal}

to L is _q

_n, the binormal is ^r

t, ^{where b and n}

are respectively ^the principal normal and the binormal in P along ^{r. Let us} write Frenet formulae for both

curves rand L. We call u the curvilinear abscissa on L, 1/p ^and 1 /s ^{the curvature} and torsion of L in M. We get which leads to

Table 1.

Frenet formulae ^{on a rod r and on the} cuspidal edge ^L of ^the developable ^domain.

There is complete reciprocity between T and L.

On each plane II(u), TIR

p/s ^{is a constant.}

One can analyze the instantaneous motion of each

plane ^{II as a torsion} about its normal t and a rotation about D. If the coordinate ^Q is considered to grow with unit velocity, ^{lines D rotate about r with} angular velocity d03C8/da

1/p. ^{Since t}

r, each plane ⁷⁷

suffers a torsion of opposite velocity - 1/p (we ^avoid

here the term of twist about t, just ^{since we are consi-} dering geometries in which what we usually ^call twist, i.e. t. rot t, ^is absent). ^{Normals to} planes ^{77 rotate}

about D at a rate dqJjda

1/t.

3. First results on the energy of a developable

domain. Angular components ^{of a domain.}

Equa-

tion (9) allows for a simple calculation of the bulk energy

We use curvilinear coordinates : in each plane II(6)

the y-axis ^is along D(Q). R ^{is the value} of the x-coordi- nate (see Fig. 2) ; ^{ds is the} elementary length along ^the

normal to ll(u). ^Hence

Call lx ^and ly ^typical sizes of the domain. We get

where we have

and b is some molecular length (a ^cut-off length ^near

the developable ^surface S). ^{A surface} energy ^term has to be added to (11), and it is clear that this term will make all the difference, ^since (11) ^{does not} depend

on the precise geometry ^{of S and L. But other remarks}

are interesting ^{at this} stage.

Let us notice that the bulk _energy depends linearly

on the strength ^{03A9 of the} line ; ^{this is unusual.} Also, the _energy does not depend ^{on the evident other} strength ^{of the} domain, ^{viz. :}

We call Qil ^the longitudinal angular component ^or longitudinal strength ^{of the} developable ^domain,

since the axis of the infinitesimal rotation do-/T ^is along ^the cuspidal edge L ; this axis is also perpendi-

cular to the rod and is of constant direction (along b)

in the moving ^frame t, n, b.

We call Ql. the transversal angular component (or

transversal strength) ^{of the} developable ^{domain, since}

the axis of the infinitesimal rotation da / p ^is along ^the

binormal to the cuspidal edge, ^{i.e. the} tangent ^{t to F.}

t is also a constant vector-in thé moving ^{frame and}

is a direction of rotational symmetry ^{of the} hexagonal phase, of order 3 or 6 according ^{to the case.} (b, ⁱⁿ general, ^{is not a} direction of rotational symmetry of the hexagonal phase.)

We shall discuss in the last paragraph of this paper the nature of a developable ^{domain in terms} of defects.

Let us first look to a very simple example.

4. A disclination built on a developable ^domain (in collaboration with P. Oswald).

^The simplest

type ^of developable ^{domain is one} in which the deve-

lopable ^{is a circular} cylinder ^of radius a ; ^the cuspidal edge ^{is a} point ^at infinity. ^The pattern of the rods is the

same in _any orthogonal section of the cylinder : ^the

rods are curved in evolutes of the circular section of the

cylinder (Fig. 4). Clearly, ^{such a} configuration ^is

that one of a disclination line of strength ^S

^{+ 1,}

of core radius rc > a (we ^{have to establish a relation-} ship between rc ^and a). ^{Let us limit the} sample ^{to a}

size Ro (measured along ^a tangent ^to the circle of radius a) ^and compute ^the energy per unit length.

It is simpler ^{here not to use} eq. (11), ^{but start from the}

formula of definition

which here leads, for the bulk _energy per unit length of cylinder, ^{to :}

(one will notice that the strength

does not appear in this expression ^{as foreseen in}

eq. (11) ; ^{this is because} any rod crosses many times a

given ^II plane, ⁱⁿ fact21n Q II ^{times so that the effective}

longitudinal strength ^is 2 n, which is also the strength

of the disclination).

Fig. ^4.

^{The rods}

evolutes of the circle of radius

The whole

configuration ^is equivalent ^to

disclination of strength unity, ^with

core

radius _rc

We have to add to eq. (12) ^{a term} of surface _energy

(the ^{core is} empty ^{in our} model). ^We write the surface tension as the sum of an isotropic ^term yo and of

anisotropic ^terms

where (Jc ^{is the} angle between the rod and the plane

tangent ^to the circle _{rc. We} have

If we limit ourselves to the first term of anisotropy,

we obtain

and the total energy has ^to be minimized with respect

to the two variables a and _r,. One gets :

Such a solution demands 0 1 2 yo - y 1 Yo.

We shall discuss this example ^{in more details, as}

well as the existence of lines of half integral strength (S

-1 ^{built on} developable ^domains, in another paper, in the light ^of experimental results obtained on

discotic mesomorphic phases.

5. Hpw ^{to obtain aU} developable ^domains. -‘ An important property ^{of a} developable surface S is of

course that it is developable ^{on the} plane. ^{In such a} mapping ^the cuspidal edge ^{L becomes a} line £’in the

plane. ^The straight ^{lines D} of S become the tangentes D

to C. The straight lines of the two sheets Si ^and S2 of S map ^{on one or} the other half infinite segments D 1

and D2 of D bounded by ^the point ^{of contact u1L on C.}

Lengths ^and angles ^are preserved ^{in these} operations ;

this remark implies ^{that the} curvatures of Land L at

corresponding points ^{are the same. But} torsion, ^which is an instantaneous rotation about the unit vector

tangent ^to L, ^{is not} preserved. ^{In the} sequel, ^{we call P}

the plane ^of mapping.

Let us now consider the inverse problem : ^{start from}

a curve £ in the plane ^{P and build a} developable ^surface

from it. The essential thing is that the result is not

unique. ^We quote Hilbert and Cohn-Vossen [14].

« If we proceed ^{the other} way around, starting ^{with a}

convex arc ô in the plane, ^and removing ^the portion

of the plane lying ^{on the concave} side of the _arc, then

we can bend the remaining part ^{of the} plane ^{in such a}

way that the curvature at every point ^{of the} space

curve d into which ô is transformed is the same as the curvature at the corresponding point of 03B4. As can be

proved analytically ^we can, at the same time, give ^the

space ^curve d _any torsion we wish. This type ^{of defor-} mation of a space curve, in which arc length ^{and first}

curvature are preserved while torsion is changed,

will be referred to simply ^{as a twisting of the curve in} question. »

Since _any torsion function i(6) ^can be chosen in order to build L, it is clear that an other way’of map-

ping ^the developable ^{domain on P is} to untwist it, ^i.e.

decrease its torsion function without changing ^its

curvature, until the torsion vanishes. Any ^interme-

diary situation in the course of such a process is a

742

developable domain which possesses the ^same repre- sentation in the plane ^{P. In} particular ^{the final state}

in P is a developable ^{domain in which all} planes ^are

reduced to the same plane ^{P. But this means also that}

all the hexagonal patterns ^{drawn on each} 17 map ^on the same pattern ^on P, ^{while at the same time each} physical ^{rod r becomes a} straight ^line perpendicular

to P (see Fig. 5) (2).

Fig. ^5.

Representation ^{of a} developable ^{domain in the} plane P ; the

L maps

£, and the straight lines of the two sheets of the

developable map

the set of tangents ^{to C. Each} plane 77 maps

on

the entire plane P, ^{while the 2} dimensional hexagonal patterns of the n’s map

unique hexagonal ^lattice

^{P. For} simplicity

we

have assumed that £ is

closed

This property ^{of the} hexagonal pattern will be used further on to build the families of rods pertaining ^{to a} given ^{C and a} given i(Q) ^and compute ^their energy, rod by ^rod (rather ^{than lI} plane by ^lI plane, ^{as it was}

done in a former paragraph). ^{But let us first turn our}

attention to the developable itself in more details, for various types of lines £ in the P plane.

a) If the line C is closed and _convex, the developable

maps ^on the exterior of C ; ^{it has two} sheets. Each sheet in space corresponds ^{on the} plane t’ ^{to a set of} half-tangents having ^{the same} properties ^{of conti-} guity ^with respect ^{to the} oriented ^{line £ ; for example}

all the forward half-tangents ^{will lift to the same}

sheet (see Fig. 5).

b) If the line C is not closed and _convex, it is to be divided into segments £i ^delimited by termini, asymp- totes or inflection points. ^{Each such} segment gives

rise to two sheets of S. The plane W is divided by ^the

curve Ci ^{and its forward and reverse} end-tangents (the

forward end-tangent being only half-line of the

complete end-tangent which continues the curve)

into areas with different sheet-coverages (see Fig. 6).

c) ^The addition of sheet-coverages of various _seg-

ments C; ^{leads to the} following ^rules (which ^{can be}

checked easily by applying ^the construction of para-

graph b) ^{above to} successive segments) : the sheet-

(2) ^{We should not deduce, from the} reality ^of

analytical

process of untwisting, that it is possible ^{to decrease} continuously

to

the _energy of

developable ^domain by

physical process of the sort. Because such

process would ^not be conservative.

coverage increase by ^{2 at} every crossing ^{of the curve}

from inside to outside, ^and by ^{1 on} crossing ^{a forward} end-tangent from inside to outside (by ^{reference to}

the curve) ; it decreases by ^{1 on} crossing ^{a reverse} end-tangent ^{in the same direction.}

Fig. ^6.

^The segment ^C and its forward and

end-tangents

divide the exterior part (exterior ^with respect ^{to C and} end-tangents)

of the plane t into various sectors. The sector 1 is spanned by ^the

forward half-tangents ^{and lifts to}

^{sheet of} S ; the sector 2 is

spanned by ^the

half-tangents ^{and lifts to} another sheet.

Each of thèse sectoiss is hatched

different _way

this figure. ^There js

region ⁱⁿ plane (T which ^is

^{to both.}

6. Construction of a rod.

We start from a node t of the hexagonal ^{lattice in the P} plane. ^{We can} immediately ^{measure the} radius of curvature R(u)

of the rod r to which (T is lifted in the inverse mapping,

because lengths ^and angles ^are preserved. R(Q) ^{is the} length ^{of the} perpendicular dropped ^{from S to} 0(a),

the tangent ^{to £ at the} point m(u). ^{Also we can} say that we obtain all the values R(Q) ^{as the} distances from to the pedal-curve ^{of £ with} respect ^{to W} (Fig. 7). ^For example, ^{if L is a} circle, ^the pedal-curve of any point S

is a Pascal’s limaçon, ^whose equation ⁱⁿ polar ^coordi-

nates (with origin in S) ^reads

where a is the radius of £ and _ro the distance from S to the center ouf 12.

Fig. ^7.

a) If S is outside C, the pedal-curve ^{has two determina-}

tions corresponding ^{to the two sets of} tangents touching ^{C between}

the pair ^of tangents ^which pass through S : ^{the sets} touching respec-

tively

^the

side and the far side of £ with respect ^{to J.} b) ^{If S}

is inside C, ^{there is} only

determination.

We limit the discussion ^to the case that C is a closed

convex curve of one tum.

If there are two lines passing through T ^and tangent

to C, the rod is limited to the set of values of R( (J)

corresponding ^{to the} tangents ^{to £} which intersect

these two lines (for ^{which R}

0) ^{and whose} points

of contact with £ are on one only ^{of the two} segments of C delimited by ^the points ^{of contact of the} tangents

passing through (T. ^{There are therefore} many solutions, corresponding ^{to rods on} either side of D(a), ^{on a} given II (6) (see Fig. 2). ^{The set} of rods obtained by continuity ^{for all the} positions of S fills all the _space available, and is exclusive of the other set of rods cor-

responding ^{to the other} determination of the pedal-

curve. Which one is physical ^{is a} question ^of energy

(a similar situation occurs with cofocal domains,

for which there are two sets of possible material lines

sustaining ^the molecules).

One will notice that the rod lifted from T, ^{when W} is outside L, touches the two sheets of the developable (where R = 0) ^{and ends on them.} Now, ifT is inside £, and no line through J’ ^is tangent ^to £, ^the pedal-curve ^is unique ^{and the} corresponding ^{rod is} either closed

(if T(a) ^{is so} chosen that L is closed), ^{or infinite} (if s(J) ^{is so chosen} that L is infinite, ^for example t(03C3)

constant).

For example, ^{if £ is a} circle and T(6) _ TD (a ^cons- tant), ^{L is an helix} of pitch p

² 03C0a2 2013 . a2 The rods lifted

03C40

from W inside £ are distorted hélices ; the rods lifted from S outside £ are segments of distorted helices bounded by successive planes osculating ^{the helical} cuspidal edge. ^{A detailed} study ^{of this} object ^{will be} given ^elsewhere.

The energy per rod is proportional ^to

dsldu ^is obtained from _eq. (9)

R(u) being given by ^the pedal-curve construction and

i(6) being arbitrary. ^{The curve r} itself is entirely

determined when we know its torsion, which is also obtained from _eq. (9) :

Finally ^the energy reads

Let us notice that, ^{if the} generating curve £ is

expressed ⁱⁿ polar coordinates with W as center of

coordinates (r

r(O)), we have

and finally

7. A singular point ^{as a} developable ^domain.

We indicate here some properties ^{of the geometry}

of the rods when the developable ^{S is a cone of half-} angle ^{a at its} apex. The representation ^{in the} plane ^P

is made of a point A and of all the straight ^lines passing through ^it (we ^{do not} keep only ^{a sector of} angle

2 n sin a, but all the plane, ^{because we shall not}

restrict the plane ^{II to} roll around the cone by only

one rotation of 2 n). ^{If S} represents ^a node of the

hexagonal lattice, ^the pedal-curve ^{is a} circle of dia- meter A (T

1 (Fig. 8). ^We also notice immediately

that the _space curve r is at a constant distance from A i.e. is on a sphere ^{of center A} and radius AS. The radius of curvature is R

1 sin / ; ^an analysis ^{of the}

instantaneous rotation of the plane, using eq. (9) ⁱⁿ conjunction ^{with d6}

p d03C8

^r d~ ^{leads to}

and ds

T d03C8. ^{The curve is therefore} entirely

known. The pedal-curve (the ^circle of diameter AS) represents ^{a curve} running normally (for colatitudes 0

a and rc - a) ^{to a} circular cross-section of the cone, and lying ^{on the} sphere of radius 1. All other rods are obtained by ^a rotational symmetry ^{about the} axis of the cone. Scaling ^{of l} generates ^the other rods.

Fig. ^8.

Representation ^{in the} plane ^{P and} spatial ^{sketch of}

developable domain reduced to

circular

the rod correspond- ing ^{to T’ is drawn.}

The paths of the rods are easily visualized (3) ^{on the}

sphere ^from axisymmetry ^and equispacing. Call f3 ^the longitude and 1À ^the angle ^{between r and a line of} longitude. Equispacing ^{of rods} require

(3) ^These paths

orthogonal trajectories ^{to the} family ^of great circles bitangent ^to the small circles 0

and 0

the

sphere (since great ^circles

^the geodesic paths ^{of the} sphere).

744

giving Il = 0 ^{at 0=(x} and P = + 1] - a) ^at

0 = 03C0/2. ^The sign depends ^{on the torsion which is} given ^{to the set of} rods, ^{which can be either} right-

handed or left-handed (Fig. 8) (a remarkable fact,

since the whole system ^is twist free !).

Now, ^{since the} elementary length ^{on a rod is} given by ds2 = l2(do2 ^{+ sin2 0} d03B22) ^the condition of equi- spacing ^{reads also}

This equation ^{can be} integrated. However the

expression of the solution is complex ^{and we shall}

not give ^{it here.}

8. Defect content of a developable ^domain.

Eq. (9) ^{and the} expressions given just ^below eq. (9)

define the instantaneous rotation of the plane ^{77. It is}

easy ^to attach densities of dislocations (in ^{the sense of} Nye [16]) ^to these rotations :

a) 11R ^{is a} density ^of edge dislocations parallel

to the directions b in the plane II, dç being ^{the dihedral} angle ^{between two} planes ^{and +} du). ^The Burgers’ ^{vector is} along ^t. This direction t is not a

direction of Burgers’ ^{vector of the} hexagonal lattice ; accordingly these densities of edge dislocations cannot

polygonize ^and form tilt wall boundaries.

Note also that these densities are prescribed ^{when £}

and i(6) ^{are known.}

b) - 1/T ^{is a} density ^{of screw} dislocations parallel

to the II planes, ^dm being ^the angle of rotation along ^t

between H(u) ^and 77(r ⁺ du). ^The Burgers’ ^vectors

are in the plane ^{n. There are} 3 directions of Burgers’

vectors in this plane ^{in the} hexagonal lattice ; ^hence

these densities can polygonize ^and relax in the form of twist wall boundaries of the hexagonal lattice.

This can occur when the curve C presents singular points ^{at which the} tangent ^{to C suffers an} abrupt change A03C8 : ^{the two} corresponding planes tangent ^to the developable ^{surface are} rotated about their normal

one with respect ^{to the other} by ^{the same} angle A03C8,

which can be managed ^{in the} specimen by the appea-

rance of a twist wall. The same phenomenon ^can

occur for curves C possessing asymptotic ^directions making ^an angle ô.t/1 : ^{if the} planes ^{II relative to two} asymptotic directions are vanishingly ^{close a twist}

wall is present ^{at their common location} (Fig. 9).

Note that the presence and magnitude ^{of twist}

walls can be directly ^{inferred from the} inspection ^of

the plane ^P.

Fig. ^9.

Situations of the plane ^{P which}

give ^{rise to}

^twist

wall in _space.

In conclusion we see that developable ^{domains are}

not directly ^{related to} quantized strengths. ^However Nye densities of edge ^{and screw} dislocations (or,

in another language, densities of tilt and twist walls)

add their strengths ^{in order to build} wedge ^{and twist}

disclinations along ^the singularities ^where 1 /R ^and IIT ^vanish [17]. ^These singularities build here the

developable ^S. It may therefore happen, according

to the configuration, ^{that some} quantization ^of Qjj ^II

or Q.L appears necessary. But this is not a prerequisite,

and the quantization might appear ^{on some} other

quantity (in ^our example ^of paragraph 4, Qjj ^{II is not} quantized).

9. Conclusion.

We have here limited our study

of configurations ^{in a} mesomorphic phase ^{made of}

ordered rods to the case when splay ^{and twist are} absent, and when elastic distortions vanish. Of course

this is a pretty ^{limited case,} but which seems to compare very well with the first observations we have made. It might therefore be that the coefficients of twist is large ^{in such} materials ; the elastic distortions cost certainly ^{more than curvature} deformations,

as in usual liquid crystals. However other defects than

developable domains will involve all terms of _{energy ;} this will be the subject ^of forthcoming papers. Note

finally that the results of the present paper apply

not only ^to hexagonal packings ^{of rods} (the ^{most usual} experimental case) ^{but also to other} regular packings

of rods.

Acknowledgments.

We are grateful ^{to Prof.}

Sir Charles Frank, F.R.S., O.B.E., ^{for a} critical read-

ing ^{of a} first version of this _paper, for corrections of some errors, and for suggestions ^which improved greatly ^the presentation ^of paragraphs ^{6 and 7. We}

also thank Dr. Y. Bouligand for communication of

results before publication.

References

[1] CHANDRASEKHAR, S., SADASHIVA, SURESH, Pramana 9 (1977)

471.

BILLARD, J., DUBOIS, ^J. C., NGUYEN HUN TINH and ZANN, A.,

Nouveau Journal de Chimie 2 (1978) ^535.

[2] LEVELUT, A. M., ^J. Physique Lett. 40 (1979) ^L-81.

[3] LUZZATI, V., MUSTACCHI, H., SKOULIOS, ^{A. and} HUSSON, F., Acta Crystallogr. ¹³ (1960) ^660.

[4] ROGERS, ^{J. and} WINSOR, ^P. A., J. Colloid Interf. ^{Sci. 30} (1969)

500.

[5] ^{BALMBRA, R.} R., CLUNIE, J. S. and GOODMAN, J. P., ^{Proc. R.}

Soc. (London) ^285A (1965) ^534.

[6] KLEMAN, M., Points, Lignes, ^Parois (Editions ^de Physique, Orsay) ^1977.

[7] MERUCCI, J., ^Private communication, ^1976.

[8] BOULIGAND, Y., Private communication, ¹⁹⁷⁸ [9] KLEMAN, M., unpublished.

[10] FRIEDEL, G., ^Ann. Phys. ¹⁸ (1922) ^273.

[11] ^DE GENNES, ^P. G., Physics of Liquid Crystals (Oxford ^Univer- sity Press) ^1974.

[12] KATS, F. I., Sov. Phys. J.E.T.P. 48 (1978) ^918.

[13] ^DE GENNES, ^P. G., ^Mol. Cryst. Liq. Cryst. ^{Lett. 2} (1977) ^177.

[14] DARBOUX, G., Théorie Générale des Surfaces (Chelsea ^Pub.

Cy, Bronx, N.Y.) ^1972.

[15] HILBERT, ^{D. and} COHN-VOSSEN, S., Geometry ^{and the} Imagi-

nation (Chelsea ^Pub. Cy, Bronx, New York) ^1957.

[16] NYE, ^J. F., ^{Acta Metall.1} (1953) ^153.

[17] KLEMAN, M., ^Philos. Mag. ²⁷ (1973) ^1057.