BEADING OF SMECTIC DISCLINATION LINES

HAL Id: jpa-00216510

https://hal.archives-ouvertes.fr/jpa-00216510

Submitted on 1 Jan 1976

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.

BEADING OF SMECTIC DISCLINATION LINES

P. Cladis

To cite this version:

JOURNAL DE PHYSIQUE Colloque C3, suppldment au no 6, Tome 37, Juin 1976, page C3-137

BEADING

OF SMECTIC DISCLINATION LINES

P. E. CLADIS

Bell Laboratories, Murray Hill, New Jersey 07974, U. S. A.

R6sum6. - Une explication est proposke pour la formation des perles qui se developpent au cczur des cylindres de cristaux liquides smectiques B grands rayons (2 R

-

1 cm), orientes homeotropi- quement. On a trouve que le rayon des perles est donne par ro = n(2 Rt)llz, ob t = (Kl/B)112, K I est la constante Blastique associee B un changement de distance entre les couches. Si lors de I'etude de I'knergie libre en fonction du deplacement des couches, on garde les premiers termes non linkaires, on trouve qu'une combinaison d'hyperboles et d'elipses dBcrit la configuration des couches smectiques au voisinage du cceur. Cette configuration implique un montant d'tnergie trop impor- tant, donc elle Bvoluera soit vers des perles spheriques, soit vers une ligne suivant la valeur du rayon,

R, du cylindre.

Abstract. - The beading which has been observed to occur on the core of homeotropically oriented large cylinders (2 R

-

1 cm) of smectic liquid crystals is accounted for. The main results are that the bead radius ro = 742 R1)1/2 where 3, = ( K I / B ) ~ I ~ , K1 is the elastic constant of splay and B the elastic constant associated with a change in the layer spacing. In the limit of large layer deformation, hyperbolae and ellipses of revolution are found to describe the smectic layer configura- tion in the vicinity of the core. It is finally concluded, that these, being too energetic, relax even- tually to spherical beads or a line defect depending upon the tube diameter.

1 . Introduction.

-

Recently we have observed [I, 21 that when a smectic liquid crystal is oriented in a cylinder so that at the boundary, r = R, the director n is homeotropic (n = (1, 0, 0) in cylindrical coordi-

A A A

nates, (r, 8 , ~ ) ) the smectic layers appear to be concen- tric cylinders (Fig. 1) ; the sample is optically very clear and a disclination line is observed at the center of the cylinder. The disclination line looks like a very fine thread and is easily observed on our large samples (2 R

-

1 cm), however what appears as a smooth

FIG. 1. - A cylindrically oriented smectic A.

thread to the naked eye, on closer inspection, is in reality a string of beads, all of uniform diameter. Shining an unfocussed, unpolarized, He-Ne laser onto the disclination core results in : 1,) a splitting of the beam into three polarized exit beams [2] and 2) a diffraction pattern associated with the beaded core which enabled us to measure the bead diameter, 2 r,. The diffraction pattern is observed on all three exit beams and has the same polarization as the exit beam with which it is associated. Our measurements of the bead sizes seem t o indicate that there is a relationship between the bead diameter 2 r , and the sample size, 2 R. For example, beads of about 30 p are observed when the sample diameter is about 0.8 cm, whereas when the sample diameter is 70 p, the beads are only about 2 p. Another related observation is that on the larger samples (2

R

> 2 mm) we have only and always observed a beaded core. As the sample diameter became smaller, the occurrence of beads became more and more rare so that the 2 p beads mentioned above have been observed in only

--

1

%

of our small samples. If we assume that the total energy per unit length of the smectic cylinder is given by F= Fe,

+

F,,,,,,

where F,, = K,(div n)' is the Frank 133 elastic energy for a smectic and F,,,, = 2 nor,, o being a surface energy and r, being the core size which is found by minimizing F with respect tor,, then we would conclude that in fact

r,

is independent of R and depends only on K,, the splay elastic constant, and o. Since this does not correspond with our most recent observa- tions [2], we propose that in order to account for the beads, one must take into account the possibility 141

C3-138 P. E. CLADIS

that the smectic layer spacing, which is do at equili- brium, may change as the smectic layers are wrapped around smaller and smaller radii. Changing the layer spacing costs energy but if it ultimately enables one to replace a djsclination line with a discrete set of point defects, it may result in a configuration of total lower energy provided the sample radius R is larger than about 0.2 mm.

Our main result is that assuming that the fluctuations in the layer spacing is thermally driven, then in the limit of small deformations

-

2 r o = 2nJ2RiZ (1)

where ,I = ( K , / B ) ~ / ~ and B is the constant associated with a change in layer spacing. Given eq. (I), it can be shown that the beaded core is less energetic than the line core provided R

2

0.1 mm. An analysis of the smectic free energy in the limit of relatively large defor- mations of the layers indicates that the beads may pass through a stage of being ellipses, however, since this is shown to require too much energy compared to a line core, we conclude that the ellipses must eventually relax to a configuration where the equilibrium spacing is respected

-

either spherical beads or a line core.

2. Analysis of cylindrically oriented smectic.

-

We use the free energy of de Gennes [4]

where

S

= (d

-

do)/do is the relative change in the layer spacing. If q(r, z) = integer xdo defines the smectic layers, then the director n is given by [5]

and

d q

cp is taken to be a function of r and z only-and - r 0.

ae

Put

y7 = r - u(r, z)

,

(4) then eq. (2) becomes

[r(g)2-($)

( z ) 2 r + i ( $ ) 4 ] ] d r d z ,

Minimizing eq. (5) with respect to derivatives of u results in the Euler Lagrange equation

Eq. (7) shows that u(z) only is not in general a solution to the Euler Lagrange equation (see Appendix A). 2.1 LINEAR LIMIT. - In this limit, Cq. (7) becomes

We look for solutions of the form :

u = u, Z(x) cos qz (8)

where x = r/R, u, = cst. Then (7a) becomes

dx" x dx where

a2 = q4 A2 R2

.

The solution to eq. (76) is

where the functions I, and KO are the modified, Bessel functions of order zero and the constant of integration has been chosen to satisfy the boundary condition x = 1,Z(a) = 0. Since Zo is a linear combination of Bessel functions, defining

the 2, will have the same recursion formula as the K,.

We recall that as x -+ 0, 1, 4 1 and KO + co and

as x -+ co, KO -, 0 and I, + co. Thus,

u, = u, Z0(ax) cos qz (8b) and the total energy F in the range r,

<

r

<

R,

2 n .

O < z < - l S

BEADING OF SMECTIC DISCLINATION LINES C3-139

where we have not added the (positive but small

-

q2 A2) contribution from the surface terms. The last integral we estimate by noting that

i. e. KO is smaller than its asymptotic expansion thus

when we make the approximation rc E A. Conse- quently the solution (8b) is more energetic than the solution of smectic layers wrapped in concentric cylinders

-

where the energy is just A2 In Rlr,. In fact,

it seems that the system of concentric cylinders is eminently stable. This result is in contrast to that offered by KlCman and Parodi [S]. We will discuss this discrepancy later in Section 3 and now consider how to fix the constant, u,.

When we grew our samples [2], we applied no exter- nal forces to the smectic other than the boundary condition which we take to be strong anchoring. Thus, the only driving force to change u from a non-zero value must come solely from thermal fluctuations. Although the thermal fluctuations in u have a loga- rithmic singularity, de Gennes has shown [4] that they are in fact very small for finite samples

-

2-3

a

4 26 A, a molecular length.

Evaluating

<

u2

>

per unit volume and for a given q using eq. (8b) we find

which we see when a = 0,

J<

u2

>

= 0. Putting

u, =

J<

u2

>

leads to the condition that

or a s 0.5

-

0.75. For a to be much bigger than unity would require more energy than is provided by k, T. With this value for a (i. e. eq. (I)), we are able to account reasonably well for our observed values of the bead diameter. For example, taking A = 26

a,

a = 0.5 predicts that when 2

R

= 0.8 cm, the bead size

2 r ,

-

29 p. In fact we observed [2] beads ranging in size from 26 to 33 y. Also, when 2 R = 0.3 cm, a = 0.5 predicts 2 r, = 17 p. We observed [2] bead sizes ranging between 14-18 p

-

but also sometimes 7-9 p

which suggests the possibility that higher Fourier components may occur for u even though it is certainly an even more energetic configuration. Finally for the 7 0 y diameter tube with a = 0.5, we expect 2 r,

-

2.6 p. We found [2] 2 r,

-

2 y in the rare cases where beading of the core was observed.

Thus we have seen that thermal fluctuations could lead to a periodic deformation of the smectic cylinder but since this is a more energetic configuration than the

non-periodic one, it shouId relax eventually to the less energetic configuration where the layers lie on concen- tric cylinders. This is because the splay energy lost by tilting the director into the z direction is too small when compared to the energy required to dilate the layers. One is, therefore, Ied to conclude that the driv- ing force for bead formation is the g&n in energy eventually acquired when the line defect is changed to a row of point defects. In this case, the core energy for the line is replaced by the much smaller core energies for the points but in order for even this to pay off, we seem to require a large enough sample

-

which we will now try to illustrate.

2.2 CORE ENERGIES OF BEADS VERSUS LINES.

-

We wish to compare the energy for the two cases shown

LINE CORE BEADED CORE

9

FIG. 2. - The spherically beaded core (right) versus the line core (left).

in figure 2. For the beaded case, we see that the total energy outside the beads can be approximated by

(2 T O ) n K , ln

El

since u , / ~ -g 1

.

We take the energy for the beads to be just 8 zKl(2 r,), where r O / z = l/q. This is just the elastic energy of a pure splay point defect [6]. We have then

neglecting the energy of point defects (or short line defects) which would appear between the beads. Thus for the beads to be stable, we want

2 r,

>

6-8 p (12b)

which from eq. (1) indicates R

>

0.2 mm for the

longest wavelength distortion, estimating r,

--

26-35

A,

I

-

26

A

since we have neglected so many contribu- tions to the energy of the bead configuration this is an overestimate. But, in order of magnitude it agrees very well with our observations. Many of our 2 mm dia- meter samples never showed the diffraction associated with beading and when one of them did, the beads were unexpectedly large [2]. The 5 mm diameter samples generally beaded and the 8 mm diameter sample always beaded.

Our conclusion is, therefore, that for large enough samples a beaded core is a lower energy configuration than a line core. Even though beads can be and have been observed on smaller samples a less energetic configuration for these samples is the line core. Unfor- tunately, such a complicated configuration as beads and the large layer deformations required, at least transiently, appears to be beyond the scope of the linearized eq. (7a), so we now look to see what the smallest order non linear terms have to say.

2.3 NON LINEAR EQUATION.

-

Here we study a solution to eq. (7), i. e.

Putting (13) into (7) we get

-

22

a = -

2 (also w,O)

(136)

p2

= 116 (also 1/2,0)

also we note div u

=

0. We see using eq. (13a) that

cp = r

-

u (i. e. a dilation of the layers) and cp = r

+

u

(i. e. a compression of the layers), correspond to a deformation of a right cylinder into hyperbolae (compression of layers) and ellipses (dilation of layers) as shown in figure 3. The fact that div u

=

0 is revealed by the appearance of a large conical hole in the case of the compressed layers due to the fact that the layers are being squashed as the hyperbolae approach the asymptotes. In the case of the ellipses where the layers are actually expanded and yet there is still a hole, this seems to be because layers would actually overlap in the hole. This is not shown in figure 3 where although z is taken to be multivalued, r is not. For each layer, Nd,, N is an integer, we have

D I L A T I O N

COMPRESSION

FIG. 3.

-

Deformation of cylindrical smectic sheet into ellipses and hyperbolae of revolution about the z-axis.

in units of I z do. Eq.'s (14) describes ellipses and hyperbolae the centers of which are located at half the distance between the tube center and the original distance N of the undistorted layer. The center of each ellipse (or hyperbola) is a distance _+ d0/2 from the centers of its nearest neighbors.

The amount of bend in this configuration is Bend Energy =

=

K,

1

(curl n)' dV

and we can control this by judiciously picking the maximum value for z, z,,,. A judicious choice of z,,,

would also ensure (at least) the convergance of the expansion of

1)

V q

)I

in terms of powers of the first derivative of u with respect to z and r upon which

eq. (7) is based.

BEADING O F SMECTIC DISCLINATION LINES required to change the layer spacing as much as these

deformations require, the bend energy is small relative to that.

This solution suggests the following configuration for the core - or the empty space in the cylinder axis. We pick the largest ellipse to fit into the spacei. e. if the maximum z is qnaX, then we fill the triangular shaped gap made by the hyperbolae with ellipses,

' I

' I , / ' I ' I / / , , /

/

(see Fig. 4). We can now fill the empty spaces that will exist by allowing the most compressed regions of the

FIG. 4.

-

Construction of beaded core before it has relaxed to the spherically beaded core shown in figure 2. Empty spaces cross hatched. These can be filled by allowing the compressed layers of the hyperboloid regions to relax. The plane of the drawing includes

the axis of revolution.

hyperbolae to relax. Certainly this configuration must be very energetic but it is interesting to note that hyperbolae and ellipses appear to be a natural conse- quence of assuming that the layer spacing is not an immutable truth -particularly so since the focal conics (hyperbolae and ellipses of Friedel [6]), are based on the assumption that the layer spacing is a fixed quan- tity (see for example Fig. 5). In fact, it was the observa- tion of focal conics that led Friedel to deduce that smectics were layered systems (i. e. translational sym- metry in one dimension) in the first place.

FIG. 5.

-

A particularly symmetric example of the focal conics of Friedel found in a drop of EMBAC [ethoxybenzylidene (ethyl) 4-methoxybenzylidene-4'-aminocinnarnate] a t 106 OC. Reconstruction of layers for one of the figures of the drop shown on right in terms of the focal conics of Friedel. o. a. indicate the optical axis. The two views shown are : (top left) section perpen- dicular to the line in the plane AB (right) ; section including line L

in the plane of the line L. L is the locus of apices of cones of revolution for which the circle is a common section. This photo taken in collaboration with R. C. Hewitt and G . N. Taylor.

We note also that if we start from

we can again find solutions to the Euler-Lagrange equation for u of

which are hyperbolae and ellipses of revolution around the z-axis. In this case, div u f 0, u = (0, 0, u), for- mally, but again by restricting the domain of the independent variables (r, z), other topologically interesting solutions may present themselves.

C3-142 P. E. CLADIS

into account the energy of dilatation (or compression). Suppose we compare the energy associated with a line core for that of the associated ellipses and hyperbolae of revolution above. We then find

and in order that the hyperbolae represent a less ener- getic configuration z,,,, can only be about 150 or about 6 molecular lengths ! Consequently if such a configuration did occur we would expect it to even- tually relax to either a line core (if the sample is smaller than, say, 2 mm diameter ; or a string of point defects for which the layer spacing is do, the equilibrium value. Both of these situations have been observed [I, 21 to occur (see Fig. 6). This kind olf a configuration must

T i

R > 0.2 mm

FIG. 6.

-

Here we schematize the results of this investigation.

A small undulation of the concentric smectic layers (linear limit) may eventually lead to the formation of an ellipsoidal deforma- tion of the core (non-linear limit). The wave length of this defor- mation is determined by the tube diameter. For large tubes

(R > 1 mm) and hence long wavelengths, a core consisting of spherical beads (and point defects) may be less energetic than a line core. The total energy of the smectic tube as a function of core radius is sketched. Whether or not the second minimum at the larger core radius is a n absolute minimum will depend upon

the size of the tube.

then be considered a kind of metastable state - which can occur and has been observed to occur. However, in time, it will evolve into one or the other energetically less demanding configurations (Fig. 2).

3. Discussion. - The undulation of the smectic layers is not a new concept and was first introduced by Durand [7] to account for the appearance of a transient buckling instability observed [8, 91 when a homoeo- tropically oriented smectic in the parallel plate geome- try was recovering from having been compressed (called uniaxial dilative stress [9]). In these experiments, however, the appearance of the undulations was a

transitory phenomena. In contrast, the beading of the core of our homoeotropically oriented cylinders appears to be an equilibrium phenomena but only on large samples. In small samples, beaded cores are rarely observed. Another interesting difference is that Durand's (linear) solution supposes weak anchoring on one of his rigid boundaries, i. e. his solution is not constant on the moving boundary but varies periodi- cally in the transverse direction of the layers (parallel to the glass boundaries). Although weak anchoring may be a reasonable hypothesis in their case [8], since we are able to grow such large well oriented samples using the surfactant of Kahn [lo] we do not believe that this is a reasonable assumption in our case. Further, on the smaller tubes (2 R

-

100-200 p) we have oriented a tube treated with Kahn's homoeo- tropic surfactant in the nematic phase by applying a large magnetic field in the axial direction. We then cooled the tube into the smectic phase. If we cooled the tube relatively quickly (> 0.2 0C per minute), we observed the appearance of the bend stripes [Ill (these stripes are perpendicular to the undulation stripes) and the honeycomb texture. However, if we remained within 5 OC of the nematic-smectic A transi- tion temperature, these eventually relaxed to a.perfectly oriented homeotropic tube [I]

-

even with the magne- tic field (15

kG)

still on. Furthermore, these analyses [7, 91 predict a periodicity of the smectic undulations of a = q2

RA

-

instead of 112 which we deduce here - i. e. a periodicity somewhat less than 112 of what we have observed.

Another point we would like to include here concerns the recent remarks of KlCman and Parodi 151 on the stability of smectic layers formed into concentric cylinders. The authors have found a solution to the linear equations u = cst z which would lead to a decrease in the total energy of any size cylindrical tube. We do not observe such a solution and account for its non-appearance by noting that, by adding the first tkrm in the non-linear limit, one would find that this is only true if R is of the order' of a molecular length. Their other solution, where the smectic layers are wrapped jelly-roll fashion, we find always to be very energetic for a smectic A

-

even in the linear limit. In contrast, our solution is never less energetic than the system of concentric cylinders, however, if we include core energies, we can make an argument for the total energy then being smaller for an undulated smectic tube with a string of point defects forming the core than a concen- tric cylindrical smectic with a line core. This we have seen, however, requires a large enough tube.

BEADING O F SMECTIC DISCLINATION LINES C3-143 fluctuations alone are responsible for an undulation in

the smectic layers, then the period of these undulations is linked to the sample size by the expression

Since our geometry is cylindrical such undulations would result in an alternating pressure along the discli- nation line transforming the line into a string of beads each with an associated elastic energy and a pair of point defects. Since the basic size of the beads is determined by the tube's linear response to the thermal fluctuations, the beaded configuration may be less energetic than the line configuration only if the bead diameter is larger than about 6 p - which will on average only occur in tubes larger than 0.2 mm

-

assuming 1

-

26

A.

In the limit of relatively large changes in layer spacing (non-linear limit), we found that solutions to the equilibrium equations were ellipses and hyperbolae of revolution. By arranging these ellipses and hyperbolae, we show that it is possible to arrive at a space filling configuration. Unlike the stable ellipses and hyperbolae of the Dupin Cyclides, which are anchored around line defects and for which the smectic layer spacing is fixed, our hyper-

bolae and ellipses define the smectic layers and the layer spacing changes within a layer. From energy considerations we argue that if these hyperbolae and ellipses do occur, they are very energetic and conse- quently would relax to a configuration where the equilibrium smectic spacing is respected (e. g. Fig. 2). Thus, at equilibrium we expect our. beads to be sphe- rical !

-

rather than ellipsoidal. They certainly look spherical. In all of our estimates of the energies, we have taken 1

-

26

A.

This is a typical value for CBOOA (N-p-cyanobenzylidene octyloxyaniline) about 2 OC below the nematic smectic A transition 1121. Since this is approximately the temperature at which we grew the large tubes. At the transition B + 0, there- fore 1 + co. Far from the transition 1

-

13

A.

For lipid bi-layers 1

-

150

A

[13].

Acknowledgments. Useful discussion with R. C. Dynes and W. F. Brinkman is gratefully acknow- ledged. Also, I have benefited from discussion on topics included in this paper (presented in part at the Europhysics Conference in Smectics, Les Arcs, France, December 1975) with M. KlCman, G. Durand,

0. Parodi, J. Prost, C. E. Williams, and Y. Bouligand. Appendix A. - If we put cp = r

-

u(r, 8, z) we get for the energy integral

Eq. (A. 1) shows us that to O(u2), ~ ( 0 ) only can lead to an increase in F. Similarly, the Euler Lagrange equation is

Thus a form of u(6, z )

-

qz

+

ere

(q, a cst), [S] is not a solution to the Euler-Lagrange Equation in this limit. References

[I] CLADIS, P. E., Phil. Mag. 29 (1974) 641.

[2] CLADIS, P. E. and WHITE, A. E. (to be published, J. Appl.

Phys., April 1976).

[3] FRANK, F. C., Discuss. Faraday Soc. 25 (1958) 19. [4] DE GENNES, P. G., J. Physique Colloq. 30 (1969) C 4-65. [5] KLEMAN, M. and PARODI, O., J. Physique 36 (1975) 671. [6] See for example : BOULIGAND, Y., CLADIS, P. E., LIEBERT, L.

and STRZELECKI, L., MoZ. Cryst. Liq. Cryst. 25 (1974) 233.

[7] DURAND, G., C. R. Hebd. Skan. Acad. Sci. B 275 (1972) 629.

[8] RIBOTTA, R., DURAND, G. and LITSTER, D., Solid State

Commun. 12 (1973) 27.

[9] CLARK, NOEL, A. and MEYER, R. B., Appl. Phys. Lett. 22

(1973) 493.

[lo] KAHN, F. J., Appl. Phys. Lett. 22 (1973) 386.

[ l l ] CLADIS, P. E. and TORZA, S., J. Appl. Phys. 46 (1975) 584. [12] CLARK, N.,, ((Pretransitional Mechanical Effects in a

Smectic A Liquid Crystal >> (preprint).