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Helicoidal instability in cholesteric capillary tubes
F. Lequeux, M. Kléman
To cite this version:
F. Lequeux, M. Kléman. Helicoidal instability in cholesteric capillary tubes. Journal de Physique, 1988, 49 (5), pp.845-855. �10.1051/jphys:01988004905084500�. �jpa-00210761�
Helicoidal instability in cholesteric capillary tubes
F. Lequeux and M. Kléman
Laboratoire de Physique des Solides associé au CNRS, Université de Paris Sud, bât. 510, 91405 Orsay, France (Reçu le 18 mai 1987, révisé le 15 janvier 1988, accepté le 19 janvier 1988)
Résumé. 2014 Un échantillon où les couches cholestériques sont des cylindres concentriques et dont le c0153ur est
non singulier présente, soumis à une dilatation radiale, une instabilité hélicoïdale de même hélicité que celle du
cholestérique : elle consiste en une ondulation hélicoïdale du c0153ur, le c0153ur étant une région à double-torsion
parfaite. Cette instabilité se développe en textures que nous décrivons. Nous avons fait une analyse élastique approchée du problème en termes d’effets de frustration et de dilatation et nous retrouvons le signe de
l’hélicité quelle que soit l’approximation faite. Nous calculons dans une approximation du c0153ur de type puits
carré le seuil de l’instabilité et discutons le cas de surfaces libres où K24 a une contribution importante ; en
introduisant la contribution des couches périphériques, nous obtenons un accord qualitatif avec les expériences. Une instabilité de même nature mais étendue existe dans S3, où la double torsion n’est pas frustrée. L’instabilité hélicoïdale que nous avons étudiée serait donc la version localisée, dans R3 de
l’instabilité d’une phase à double torsion uniforme dans S3.
Abstract. 2014 Cholesteric samples, with cylindrical concentric layers, and with a non-singular core, show a
helical instability with the same helicity as that of the cholesteric, when subjected to a radial dilation. It consists in a helical undulation of the core, the core being a perfect double-twisted zone. This instability develops into
textures which we describe. We make an elastic analysis of the problem, which involves frustration and dilation. The correct sign of the helicity of the instability is obtained in any approximation. We calculate the
instability threshold for the core region within a square-well approximation and discuss the free surface case
where K24 plays an important role. Adding the layers contribution, we obtain a good qualitative agreement with the experiments. An instability of the same nature exists unlocalized in S3, where double-twist is unfrustrated : the helical instability studied here, would therefore be a localized version in R3 of a uniform
instability of a double-twisted cholesteric in S3.
Classification
Physics Abstracts
61.30
1. Introduction.
The study of cylindrical samples has proved useful to
understand the defect structure of liquid crystals :
we recall the experimental discovery of singular points in nematics [1]. In cholesterics, the situation is
even richer and various experiments have shown the appearance of numerous textures of defects whose
relationship to the geometry has not yet been fully investigated [2, 3].
In this paper, we study one of the simplest effects
relevant to this geometry, namely the existence of a new type of instability previously observed by one of
us [4]. In the unperturbed geometry which shows double twist as in building blocks of blue phases
models [5], the director is perpendicular to the radial
vectors, and its orientation 41 (r) in the orthoradial
plane depends only on r, with qi (0) = 0. ql (r) is entirely fixed by its value at some boundary. The instability appears at zero amplitude for a given
value of the axial inverse pitch t =
( ddt/1) r
dr r r=0= . . Itconsists in an undulation of the curved layers and of
the double twist zone which has an helical pattern of
a well-defined chirality. In spite of the layers undu-
lation which could be at first sight interpreted as a
dilation instability of the layers in the cylinder, it is specific to a chiral medium. Also, it is a precursor of the formation of a helical texture with defects ; the ingredient of the analysis is a quantity which
measures the frustration of the pitch on the axis (it is
in general different from q, the natural inverse
pitch). The geometry is naturally frustrated and we can modify this frustration by varying the pitch of
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01988004905084500
the confined system. An essential feature of the
instability is its localization, a character which makes
our analysis difficult and approximate.
We give a general description of the helical
instability and analyse it in the case of small radius
samples. A coarse grained energy is introduced to discuss the instability of the layered geometry zone and its contribution to the instability. We discuss a possible relationship between t and the saddle-splay
modulus K24, and we show finally that the instability
is unlocalized in a curved space (a 3d sphere
S3) in which double twist geometry is uniformly
satisfied. In conclusion, we stress the interest of such instabilities for various physical and biological sys- tems.
2. Experimental method.
2.1 SAMPLE PREPARATION. - A glass capillary tube, of inner diameter of 50 Rm is treated for
homeotropic anchoring and filled slowly, by capil- larity, with a mixture of a nematic (E8 produced by
BDH) and a little cholesterol benzoate (the mixture
is a left cholesteric). The tube is then sealed and observed immersed in oil with a polarizing micro-
scope ; it is settled on a special stage and can be rotated about its axis and translated. Except speci- fication, we operate at room temperature, but the sample can be heated on a Mettler stage.
We modify the pitch of the sample in two ways, either by heating the specimen (the natural pitch
decreases when the temperature increases) or by letting two mixtures of different concentration dif- fuse one into the other. This second method has been described in details in a previous paper [4] ; it is
however difficult to get quantitative results on the pitch. We have therefore turned to the first method,
which is quicker, more quantitative, and which has the advantage of showing up the desired effects in the whole sample (while in the first case we are
limited to zones of relevant concentration and frus-
tration). It gives qualitatively the same results than the first method.
We have filled the tube by capillarity with a
Fig. 1. - Cholesteric sample obtained by diffusion of cholesterol benzoate in the nematic phase of E8 ; the anchoring is homeotropic. (a) Homogeneous sample. The h-instability extends in all the sample. Polarizer and analyser parallel ; (b)
The cholesteric rich zone is on the left : evolution of the h-instability in a half-layer sample towards an H texture ; circular light.
cholesteric mixture with a pitch smaller than the
radius, and we have obtained a resulting geometry which is planar even if the sample is prepared for homeotropic anchoring (with silane of BDM). The
cholesteric and the homeotropic anchoring are in
fact incompatible, and the flow process ignores homeotropic forces. If we heat the sample to the isotropic phase and cool it back to the cholesteric
phase, the structure becomes very disorganized, and
the layers have quite random directions ; the anchor-
ing is lost. If, on the other hand the cholesteric phase
is obtained by diffusion of active molecules in an
already aligned nematic sample (with homeotropic
condition satisfied as in [2, 3]) we observe that a
weak homeotropic anchoring remains (see Fig.1b) ;
there is a half pitch thick transition layer between
the inner planar orientation and the homeotropic boundary.
At equilibrium, after filling the tube and letting it
anneal for half an hour at room temperature (the
free singular points which move along the axis carry
supplementary layers, previously described in [2]),
we observe an equilibrium structure, made of cylin-
drical layers centered on the axis of the tube as
already noted in [2], and the core is non singular (the
molecules are aligned with the axis on the core). The
whole geometry is that of a S = + 1 disclination line surrounded by cylindrical cholesteric layers, whose
number depends on sample history. The bright lines correspond to molecules perpendicular to the axis ;
it is very conspicuous, although unoticed up to now to the best of our knowledge, that the distance
between the bright lines is nearly equal to the
distance between the first line and the axis, so that
the pitch is twice larger on the center of the cylinder.
We will interpret this fact later on.
2.2 TEXTURES OBTAINED UNDER HEATING. - We heat the sample from room temperature. The de- creasing of the pitch dilates the core and decreases
the thickness of the layers. We reach a threshold in temperature, Th, where the core shows up an helical
instability (Fig. 2a). We call it the h-instability. The helicity of this instability is left-handed alike the
cholesteric, and the period along the axis (i.e. the pitch of the screw) is about twice larger than the
cholesteric pitch. We check the helicoidal symmetry and its sign by rotating the capillary about its axis and observing a translation of the optical image. The amplitude of the h-instability increases and propa- gates in the whole sample with an amplitude decreas- ing from the axis to the surface of the tube where the
Fig. 2. - Heated samples. (a) h-instability. On the left a folded texture. Natural light ; (b) Alternance of « folded » and H textures, natural and convergent light.
anchoring is planar and permanent ; on the other hand, in diffusion processed samples with a strong homeotropic anchoring, the instability extends throughout the tube (see Fig. la).
The h instability evolves in the following way : the
+ 1 line of the core takes a left-handed compact helical shape (we call it the H-texture) whose pitch is
now equal to the natural pitch (Fig. 2b). We observe
a helical symmetry. This texture has been already investigated by Bouligand and Kleman [6] and is pictured figure 3b. Note that this texture occupies an
Fig. 3. - (a) Folding of a + 1 line from the h-instability
to the folded texture ; (b) Representation of the H-texture (after Kleman and Bouligand). Axial section. (Dashed
lines : n is parallel to the plane of the picture ; dotted
lines : n is normal to the picture.)
axial region of radial size of the order of the pitch.
This H-texture is (meta)stable when the temperature is now decreased, and disappears only at a tempera-
ture TD Th (hysteresis), by giving back the cylindri-
cal undisturbed texture.
Both H- and h-textures can relax towards a pure
cylindrical texture with singular points (as in [2]), by spontaneous nucleation of very mobile point de- fects ; in order to avoid this effect, we must heat the sample quickly enough (at least 3°/min) ; in other
words, both textures are metastable, and the
threshold temperatures cannot be determined pre-
cisely.
2.3 OTHER TEXTURES. - Apart H-texture and h-
instabilities, we observe two kinds of defects :
1) A folding of the + 1 line : whorls can appear on the h-instability, which therefore tilts and folds into a defect as drawn in figure 3a. Some folded textures
are shown figure 2b ;
2) Pairs of singular points described in [2] can
nucleate on the axis and add therefore a supplemen- tary layer. When moving, a singular point destroys
the H and the folded textures.
The fact that the h- and the H-periods are strongly
different involves interesting dynamic properties ;
one of the consequences is a frequent observation of
a quite regular alternation of the H-texture and the folded texture (see Fig. 2b).
3. Energetical and geometrical discussion.
3.1 EQUILIBRIUM CONFIGURATION. - In cylindrical coordinates, the director is :
Because of cylindrical and translational symmetry along Oz, ql and 0 depend only on r and vanish on
the axis (I/J (0), B(0) = 0). Near the axis, where 0 and qi are small, the free energy writes (after Frank [7]):
where
P being the natural pitch. This quantity is clearly
minimal on r = 0 for
a e ar
o = 0 and( ai ) _ q
0=0 is quite well verified exper-ar o 2 p
imentally (no splay). It is interesting to note that in
absence of surface forces, the axial twist is half of the natural twist.
A more detailed discussion needs the study of the
Euler-Lagrange solutions of the splayless free en-
ergy :
which minimizes for :
(the K24 term is a surface term and does not
contribute to equation (2). We will discuss this term in 3.3.2. Note that some authors use K2z + K24
instead of K24 for the saddle-splay surface).
We solved this equation numerically (in the case K2 = K3) using a second order Runge-Kutta
method. We must take two boundary conditions.
One is evident ; qi (0 ) = 0 which gives the continuity
of the field. The other is experimentally qi (R) = qfo (in the case of strong anchoring) but by calcu-
lation we must adjust the inverse pitch near the axis
t =
( 0 «/J)
ar r = . We have calculated the t/q value ofr = 0
Fig. 4. - Energetical diagram of the system with given boundary conditions wo = qi (R). In other words, the three curves qi (R) = wo + 7r, qi (R) = qio and t/J (R) = t/J 0 - 7T represent the energy of the tube with n - 1, n and
n + 1 half-layers. The point A separates two regions, one
in dilatation and one in compression, and both of equal energies. Any situation with n layers is represented by a point on the energy curve Fn with the natural pitch as the z-component. In a point like A, the singular point is in equilibrium, and the passage from B to C (3) is due to the
presence of a free moving singular point. If there is no
singular points, varying the inverse pitch (by heating or diffusion), the system evolves along a single curve and can
reach points like D. At D the cylindrical geometry is unstable and the helicoidal instability appears. The H- texture is very stable and develops (1) from the helicoidal
instability, and disappears (2) giving back the cylindrical geometry. But those helical patterns are metastable with respect to the motion along the axis of a singular point (3).
the minimum of F, varying qio and assuming
( aF
= 0, i.e. free boundary condition (in the( a 41 R
case K24 = 0). It is interesting to comment that when the size increases, t/q globally decreases : the bend energy of the region where 7rn + 7r relaxes
2
partially while decreasing t/q. It is very difficult to
measure t because the optic is very complex but the experimental t/q ratio at equilibrium is well accord-
ing to the previous calculation (t/q = 0.35). It
appears clearly that in dilation, the core
( gi 2
dilates while the external layers contract, as observed experimentally. We give an idea of the value of the energy in figure 4, for various values of q, and for
1/1 (R) = qfo + nw (n integer). At equilibrium singu-
lar points are free, so qio can vary of nir. If we are in B, we are metastable, the equilibrium value being C.
The equilibrium situation is plotted along the thick
line.
In our experiments, we reach points like point C
and the h-instability appears in a metastable zone of this figure.
3.2 SMECTIC APPROACH OF THE INSTABILITY. -
Layered systems like smectic show, when submitted to a dilation, undulation instabilities in plane as well
as in cylindrical geometry [11]. Cholesterics are, at scales larger than the pitch, layered systems, and
Fig. 5. - h-instability. On the right, the initial configur-
ation : left-handed strength lines (thick or dotted) are
helices of axis Oz (right cholesteric). On the left, the perturbed situation : the + 1 line takes right-handed
helical shape of axis Oz.
show accordingly a planar instability in dilation. The
coarse grained smectic free energy for cholesterics
[10] uses the following expression for the penetration length :
The origin of such instabilities is that the system can relax by tilting the layers. In a planar geometry, the
wave vector of the undulation is given by :
where X is the dilation rate and A the penetration length. The undulation threshold is given by
X = 2 7rA L where L is the thickness of the system.
In cylindrical geometry, the corresponding theory
has been worked out in [12]. The undulations appear of course only in dilation, and have a wavelength k given by :
at a threshold X = A /R + 0 (A 2/R2) where R is the radius of the cylinder.
Of course, they do not display helical symmetry, since this coarse-grained theory of the cholesteric does not include chirality. Our analysis of the h- instability is, contrarywise, relative to the elasticity
at a scale of the order of, or smaller than the pitch.
This is why it will not reveal the smectic instability, although it would be possible to include it, but at an
enormous cost in calculations. Note however that the smectic approach is valid for the peripheral layers. Our helical instability is on the contrary due
to the core region.
3.3 GENERAL APPROACH. - We now use the Frank energy, i.e. the complete cholesteric energy. Because the cylindrical structure (no) is a local minimum of
F, for any perturbation 5n of the field, the first order in 6 n of the energy variation 5F = F(no + 8 n) - F(no) will always vanish ; we must consider the second order terms in s n. We take no . 6n = 0 and renormalize n to the second order in 5n:
Choice of 8 n.
We only consider 8 n with a helical symmetry. In fact, only one mode does not vanish on the axis, this
mode is :
Fig. 6. - Representation of 6n for different values of z in the modes (n =1 ) and (n = 2). The perturbation 5n (drawn on an axial circle) rotates of 2 1T when kz increases of 2 w.
The other modes are :
The small vector field 6n of the perturbation is
similar to the field of a disclination of strength
S = 1- n (see Appendix I).
The development of the free energy is complex (see Appendix II) and there is a coupling term
between n and - n modes (Eqs. 11.3 and 11.4 in
Appendix II).
Note (see Appendix I) that the modes are in
r11 - n1. The only mode which does not vanish on the axis (because it does not carry a singularity of the
disclination type) and which is of interest for us, is the mode n = 1 (S = 0 ).
As it is very difficult to make a complete analysis,
we neglect the influence of the n = -1 mode near
the axis.
In the approximation qr ,1, one gets (after (11.3) Appendix II) :
where K 1 (o ) _ (k - q )2 - (q - t )2.
Because of sin qi terms, the approximation
r
k 1 (r ) ~ K 1 (0) is valid only for tr 7T /4, i.e. for the double twist zone. For tr > 7T /2, there is very little double twist and the coupling of the modes + 1 and
- 1 would give the classical layers undulation sol- ution as described in 3.2.
Let us discuss the core contribution.
We first consider strong anchoring, i.e.
* n (R ) # °.
3.3.1 Strong anchoring case. - In this case, it is easy
R
to show that
Jo
0 F, r dr will not depend on K24, aswell as the Euler equation. Equation (11.4) is a Schrodinger equation H-0 1 = Eo 1 where - h2 =1,
2 m
K1= V and E = 0 (H=-A+ K1). Let us call 1, m the eigenfunctions of H and Em their respective eigenvalues. As any 0 1 can be written 1=
A,Xam Ol,m assuming Ol,m I Ol,n) = Sm,n and
~1,m (R) = 0 (A is real, an are complexes) one gets easily :
11 J?
R
If all the Em are positive,
Jo
0 F, r dr becomesminimum when A = 0, the structure is stable with
regard to any 01. But if the smallest eigenvalue Eo is negative, the structure will be unstable for
ol = Aol’o, hence F = Fo + A 2 Eo + 0 (,k 4).
Approximation K 1 (r ) = K 1 (0 ) (square well ap-
proximation).
The threshold is given by Jo(, ,/- K, R ) = 0 and - K 1 R is the first zero of Jo. Because :
the k which minimizes K 1 is k = q. K 1 is negative for
k = q quand q -:F t ; this implies that the structure is
very stable for q = t and the mode will have a wave vector equal to q. The pitch of the instability and its sign will be those of the cholesteric. For a small
cylinder, we write 4, (R) = 4’R == tR and lq-tl R = 2.4 which gives :
For .p R 2.4 rad, we have only a threshold in dilation (t q ). For instance, if t/1R 7r , there is a
4
threshold only in dilation for t = 0.25 q. Because in
K 1 (r ), there are sin terms, our approximations
r
are good only for t/1 R 4 § the h-instability is
localized in the double twist zone.
This analysis explains the observed sign for the helicity of the h-instability.
3.3.2 Case 0 (R) :0 0 and (R) = 0. - In this
paragraph we will discuss some aspects of the
instability when the boundary conditions are free by
which we mean the only surface term is :
In this case, for a small cylinder the equilibrium
situation is :
the axial twist depends strongly on K24. (For energetical reasons [13], one has 0 K24 2 K,
hence to has always the sign of q.)
Let us write the energy of the helical perturbation
mode n =1 on this cylindrical structure.
This variation of energy is :
where
The Euler equation is as expected the same as (II.4), the 0 1 0 * term is K 1= g and is equal, for
r = 0 and cp 1 constant, to :
where f =
( 2 - Ar"24 ).
(to - t ) is proportional toK
the twist stress in the cylinder. to is given by (6).
So, in the hypothesis cPt = 0, we have as a general
conclusion that the favourable k is written :
If f is different from 0 and k = f + t, it is always possible to make F1 1 negative.
Therefore the cylindrical geometry for a very small cylinder is always instable in regard to mode
n = 1, except for t = to (6).
Boundary conditions can be considered as free when we have periodic boundary conditions, for example in blue phase which is well known [5], to be
made of a stacking of cylinders of the type we considered.
We will develop these considerations in the case of cholesteric polymer fibers and other related prob- lems, but we insist on the generality of relation (7).
3.4 DISCUSSION OF SOME LAYERS SAMPLE.
3.4.1 The dilation case. - For a large tube, in dilation, the core and the layers present an undu- lation instability. The sign of the helicity is given only by the core terms but the instability extends
over all the tube. In fact, the complete resolution of the problem would give a progressive coupling
