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ON LIGHT DIFFRACTION BY CHOLESTERIC
LIQUID CRYSTALS WITH A PITCH GRADIENT
S. Mazkedian, S. Melone, F. Rustichelli
To cite this version:
JOURNAL DE PHYSIQUE Colloqzre €1, supple'ment au no 3, Tome 36, Mars 1975, page C1-283
Classification
Physics Abstracts
7.130
ON LIGHT DIJWRACTION BY CHOLESTERIC LIQUID
CRYSTALS WITH A PITCH GRADIENT
S. MAZKEDIAN
Facolth di Ingegneria, Universith di Ancona, Italy S. MELONE
Facolta di Medicina, Universith di Ancona, Italy F. RUSTICHELLI
C. C. R. Euratom, Ispra, Italy
and Facolta di Ingeneria, Universith di Ancona, Italy
RBsumB. - On ktudie thkoriquement, a partir d'un modkle physique simple, la diffraction de la lumikre par un cristal liquide cholesterique avec un gradient de pas. La validit6 du modkle employk est justifik par analogie avec les proprietks de diffraction de neutrons par les cristaux oh il y a un
gradient du parametre du rkseau. I1 est montre en particulier que la bande de longueur d'onde pour laquelle la lumikre polariske circulairement droite est totalement rkflkhie augmente si le gradient augmente, jusqu'h ce qu'une valeur critique soit atteinte. 11 apparait que cette valeur critique du gra- dient dkpend seulement de la birefringence du cholestQique consider& Quand le gradient dkpasse cette valeur critique, le pouvoir rkflecteur intkgrk croit progressivement mais la rkflection totale n'est plus obtenue.
Abstract.
-
The light diffraction properties of nonabsorbing cholesteric liquid crystals with a pitch gradient were theoretically investigated using a simple physical model. The validity of the model employed has been justified by analogy with the neutron diffraction properties of crystals with a gradient in the lattice parameter. It was shown, in particular, that the wavelength width of total reflection for right circularly polarized light increases if the gradient increases, until a critical value is reached. We found that this critical gradient depends only on the birefringence characteristics of the cholesteric investigated. When the gradient becomes higher than the critical value, the integrated- reflecting power of the light increases progressively but total reflection is no more obtained.1. Introduction.
-
The optical properties of cho- lesteric liquid crystals have been investigated both theoretically and experimentally by several authors. The most recent contributions were given by Nitya- nanda [I], and by Shashidhara Prasad and Mad- hava [2], who treated theoretically and experimentallyrespectively, the problems of rotatory dispersion, circular dichroism and light diffraction by cholesteric liquid crystals. In their papers an exhaustive biblio- graphy can be found on the preceding work on this subject.
It is well known that cholesteric liquid crystals exhibit a very high optical rotatory power and selec- tive light reflections according to the Bragg law. At normal incidence, one circular component is totally reflected within a narrow band of wavelenghts Al. of the order of 200
A,
whilst the other passes through unchanged. In the neighbourhood of the region of reflection, the rotatory dispersion is anomalous. The first quantitative explanation of these phenomena was given by de Vries [3], who assumed for thecholesteric liquid crystal a model consisting of a helical arrangement of thin birefringent layers, with the principal axes of the successive layers turned through a small angle.
A simpler and more elegant explanation of the reflection properties and of the rotatory power of cholesteric liquid crystals was given by Chandrasekhar and Srinivasa Rao [4], by using a treatment very similar to that used in the dynamical theory, which represents the correct description of the diffraction properties of X-rays and neutrons by perfect crystals. In particular they obtained for the reflection curve of right circularly polarized light the characteristic shape of the Darwin curve, which is well known from X-ray and neutron diffraction by perfect crystals.
On the other hand a lot of work has been done both theoretically and experimentally in the fields of X-ray and neutron diffraction in order to investigate the diffraction pattern modifications induced by some regular imperfections of the crystal like an induced curvature or a lattice gradient. Taupin [5, 61, in parti-
Cl-284 S. MAZKEDJAN, S. MELONE, F. RUSTICHELLI
cular, has developed a theory concerning the dynamical X-ray diffraction by curved crystals and by disloca- tions. He checked his theory by carrying out some reflectivity X-ray experiments on curved crystals, at different curvature radii. Similar experiments were also performed by other physicists. An extension of the Taupin theory to dynamical neutron diffraction by ideally curved crystals was performed by Klar and Rustichelli [7]. A comparison between the results of these calculations and experimental data on neutron diffraction by curved Si crystals was performed by Beuf and Rustichelli [8].
Several other experimental investigations on neutron diffraction by curved crystals were performed (see ref. [7] and [8]). Also, theoretical and experimental investigations were performed on crystals with a gradient in the lattice parameter, to be used as special neutron monochromators [9]. In particular Cu-Ge single crystals with a concentration gradient were grown with a total relative variation of the lattice spacing Ad/d z 1
%.
Furthermore it was shown experimentally [lo] on cholesteryl nonanoate ( C w that a divergence of the cholesteric pitch exists near the smectic-A phase transition. A relative variation of the pitch
AP/P z 80
%
was observed. The consequent possibi- lity of producing a variation of the pitch suggested a theoretical investigation on optical properties of nonabsorbing cholesteric liquid crystals with a pitch gradient.An helicoidal structure with a pitch gradient, which is analogous to that investigated in this paper, was observed in crustacean and insect cuticles by Bouli- gand [ll]. The total relative variation of the helix pitch was roughly hundred percent. This paper presents the diffraction properties of cholesterics with a pitch gradient and is based on a work on the optical properties of perfect cholesteric liquid crystals [4] and on the dynamical theory of neutron diffraction applied to curved crystals [7].
2. Light diffraction in perfect cholesteric liquid crystals. - The first quantitative explanation of the light diffraction and rotatory power in cholesteric liquid crystals was given by de Vries [3], who treated the phenomenon according to the formalism of conventional optics. He assumed for the cholesteric a model consisting of a helical arrangement of thin birefringent nematic layers, with the principal axes of the successive layers turned through a small angle
8.
However we are here interested in the later treatment of the problem given by Chandrasekhar and Srinivasa Rao [4], who used the method of the dynamical theory of X-ray and neutron diffraction to describe the optical properties of cholesterics. They treated first the problem of the rotatory power using the same model as utilized by de Vries. Then, by following a method similar to that used originally by Darwin, they calculated the dynamical diffraction curve and theamplitude extinction factor at normal incidence as function of light wavelength. They found confirmation of the validity of the Bragg Law
for the average value 1, of the diffracted wavelength band, consisting of right circularly polarized light, where p, is the refraction index for this kind of light and P is the pitch of the cholesteric. On the other hand
the left circular light is not reflected, but passes through the material suffering anomalous rotatory dispersion around the reflecting region. The range H I of the total
reflection of right circularly polarized light extends practically from
Q , being the reflection coefficient per pitch of the cholesteric structure. It follows that
The diffraction curve obtained is very similar to the characteristic Darwin diffraction curve, which is well known for X-ray and neutron diffraction. Furthermore they found that there is an extinction of the radiation inside the range of total reflection, as in the case of X-ray or neutron diffraction. The intensity falls off according to the law
where
By analogy with the X-ray or neutron diffraction case, the extinction length t,,, of the light which is reflected by a perfect cholesteric liquid crystal will be defined by
I(text) = lo e-"
.
( 5 ) From eq. (3), (4) and (5) we obtainEq. ( 5 ) shows that at a distance text below the surface of a perfect cholesteric liquid crystal the intensity of the right circularly polarized light, in the total reflection region, is reduced to a negligible propor- tion.
ON THE LIGHT DIFFRACTION BY CHOLESTERICS C1-285
Extinction length, total rej7ection width and critical gradient for the chole.rteric liquid crystal theoretically investigated in ref. [12], according to which the following values were assumed : Po = 4 000 A, y z yd = 1.5,
Ap = 0.075.
Extinction Total reflection Critical 1
width p graderit P
length (text) gradient
(F) I@') (4 (gradcrit P) (p-1)
- - - -
8.0 191 1.6 x 1 0 - 3 4.0 x 10- 3
3. Total reflection by cholesteric liquid crystals with
a pitch gradient. - Let us consider a cholesteric liquid crystal with a pitch gradient (CPG). It is supposed that the gradient is constant and perpendicular to the molecular planes of the crystal, i. e. parallel to the axis
Z of the helix, and defined, therefore, by
d P grad P = -
.
dZ
The aim of this section is to foresee the diffraction pattern and in particular the width of the totally reflecting range of wavelength, when white light is incident perpendicularly to the crystal planes, as a function of grad P. It will b~ shown that the width of the totally reflecting range increases progressively as the gradient increases until a critical value is reached. An expression for the critical gradient will be given as a function of the intrinsic properties of the cholesteric investigated. A model for the CPG is used according to which the CPG of thickness to is imagined to be decomposed in n perfect cholesteric crystalline lamel- lae, each having a thickness equal to text, but different pitches, with n given by
The pitch of the i'th crystal is supposed to be constant inside each lamella, and to be given by
Pi = Po
+
(grad P ) i , t,,, (9) where Po is the pitch at the upper surface of the CPG. In other words, it is supposed that the pitch, in the presence of a given gradient, varies stepwise instead of continuously. It is evident that this model can be valid only if the grad P is enough small as specified below. The variation dlo in the diffracted wavelength, which is induced by the gradient along a CPG lamella of thickness text, is obtained from eq. (1)This should be less than or equal to the total reflection width W given by eq. (2)
AP(text) which appears in eq. (10) is the variation of
P through the considered CPG lamella.
For values of grad P so large that the condition (1 1)
is not verified one should expect that the model is no longer valid. This statement will be justified below by analogy with the neutron diffraction case. But first the implications of this model on the diffraction proper- ties of CPG will be discussed.
The condition (11) can be put in a different form :
by using equations (10) and (7) it is easy to see that this condition becomes
W
grad P d ---
Pd. text '
This relation constitutes a limitation for the grad P, due to the fact that W, y d , text are physical quantities which are characteristic of the considered cholesteric.
According to the above considered CPG model and the extinction properties in a perfect cholesteric, it is expected that, if inequality (12) is satisfied, the i'th crystal acts, concerning the reflection of light, as an infinitely thick perfect cholesteric having a pitch P i
given by eq. (9) : in other words it is assumed that it will give rise to a Darwin curve centered around the wavelength value obtained by eq. (I). Furthermore it is assumed that each CPG lamella will produce a diffraction pattern independently of the others. All these assumptions will be justified below by analogy with the neutron case.
As a consequence the total diffraction pattern of light by the CPG will be given by the superposition of n Darwin curves shifted with respect to each other, as shown in figure 1. The shift between the Darwin
FIG. 1.
-
Representation of the diffraction pattern of right circularly polarized light by a cholesteric liquid crystal with a pitch gradient (CPG). PH is the diffracted light power and POis the incident light power.
curves produced by two adjacent lamellae is given by eq. (10)
C1-286 S. MAZKEDIAN, S. MELONE, F. RUSTICHELLI as it can be seen from eq. (8) and (13) (see also Fig. 1).
It is obvious that eq. (14) is valid only if the thickness to of the CPG is larger than t,,,.
From condition (12) it follows that a critical value of grad P exists, representing the maximum value of the pitch gradient for which total reflection is still possible. The expression for this critical gradient is given by
W
gradcrit P = -
.
P d t e x t (15)
Figure 2 illustrates the diffraction pattern obtained when grad P assumes its critical value.
FIG. 2. - Representation of the critical gradient condition in cholesteric liquid crystal with a pitch gradient (CPG).
The condition for the critical gradient implies a variation AAo equal to W between the diffracted wavelengths at the two faces of a CPG lamella of thickness t,,,, so that the Darwin curves of two adjacent lamellae are adjacent. For values of pitch gradient higher than the critical gradient the width of the reflected wavelength band increases but the reflec- tion coefficient becomes lower than one, because the total extinction of a given wavelength is no longer possible in a given lamella. In fact the pitch variation over a length t,,, is in this case too large the two faces of the lamella to be able to diffract the same wave- length. A more rigorous justification of this idea was given in the neutron diffraction case and will be discussed in the next section.
By using the eq. (I), (2), (4) and (6), we obtain an equivalent expression for the critical gradient given by eq. (15) :
Q;
grad,,,, P = -
.
2 7L (16)
An approximate expression for Qo has already been given in reference [12] :
By using eq. (16) and (17) we find for the critical gradient
It appears that the value of the critical gradient depends practically only on the birefringence parame- ters of the cholesteric liquid crystal. The values of the critical gradient and of the corresponding relative pitch variation per unit length were calculated using the data reported in reference [12] and are presented in Table I.
4. Justification of the CPG model. - The validity of the model and of the various assumptions utilized in the previous derivation of the light diffraction properties of CPG will be justified in this section by analogy with a very similar model used in neutron diffraction studies on crystals with a gradient in the lattice parameter [13]. First of all, we recall the theore- tically proven equivalence [14] of the diffraction pro- perties of neutron and X-rays in their interaction with perfect crystals. Goldberger and Seitz [14] found that the system of equations for the amplitudes of neutron waves is identical with the corresponding system in the dynamical theory of X-ray diffraction, after an obvious readjustment of the physical constants. An experimental verification of this fact in a significant case is reported by Shull [15]. In parti- cular the diffraction patterns of X-rays and neutrons by perfect crystals are equal and given by the Darwin curve, which describes also the diffraction pattern of right circularly polarized light by a perfect cholesteric, as previously discussed. This equivalence between the diffraction properties by perfect crystals of X-rays, neutrons and light, constitutes the basis of the CPG model justification.
In fact a model for a crystal with a gradient in the lattice parameter, very similar to that utilized here for
FIG. 3.
-
Comparison between the neutron diffraction pattern by a curved crystal, as obtained in the dynamical theory, and the rectangular pattern foreseen by the lamella model. The patterns correspond to the critical curvature (C = 1). PH is the neutron diffracted power and Po is the incident neutron power. Y(0) represents a physical quantity related to the orientation of the crystal in respect to the primary neutron beam and A = 10 corresponds to a crystal thickness of five extinction lengths (seeON THE LIGHT DIFFRACTION BY CHOLESTERICS
the CPG, was used to describe the neutron diffraction by such a regularly deformed crystal [9, 131. The vali- dity of such a model was justified by reference to the work of Klar and Rustichelli 171, who carried out direct calculations for the neutron diffraction patterns by ideally curved crystals. Actually a model can be imagined for the curved crystal, which is analogous to the model for the crystal with a gradient in the lattice parameter and therefore to the CPG model :
the curved crystal is supposed to be decomposed in several lamellas of thickness equal to the extinction length, which differ each one in respect to the others for the orientation, instead than for the lattice para- meter [7].
An expression was derived for the critical curva- ture [7], (C = 1) by imposing the condition that the different Darwin curves associated to the different lamellas be adjacent, just as in the case of the critical gradient condition in CPG. It was shown [7] that the neutron diffraction patterns .obtained by the rigorous dynamical theory calculations, can be quite well explained by the simple lamella model under discus- sion. This fact was used to justify the validity of the analogous model for the crystal with a lattice para- meter gradient [9], and can be also used, therefore, to justify the CPG model for light diffraction. In order to give an idea of the degree of approximation associated to the lamella model, figure 3 shows for the case of critical curvature (C = 1) the neutron diffrac- tion pattern obtained by the dynamical theory, in comparison with the rectangular pattern obtained in the simplified model. Figure 4 shows the analogous comparison for a curvature (C = 0.1) ten times smaller than the critical curvature. For curvatures higher than the critical value, it appeared [7] that the width of the diffraction pattern increases, but total reflection is no Ionger obtained, in agreement with the model predictions.
[I] NITYANANDA, R., Mol. Cryst. Liqu. Cryst. 21 (1973) 315. [2] SHASHIDHARA PRASAD, J. and MADHAVA, M. S., MoI. Cryst.
Liqu. Cryst. 22 (1973) 165. [3] DE VRIES, H., Acta Cryst. 4 (1951) 219.
[4] CHANDRASEKHAR, S. and SRINIVASA RAO, K. N., Acta Cryst. A 24 (1968) 445.
[5] TAUPIN, D., Thkses Universite de Paris (1964).
161 TAUPIN, D., Bull. SOC. Frang. Miner. Cryst. 87 (1964) 469. [71 KLAR, B., RUSTICHELLI, F., NUOYO Cimento 13B (1973) 249. [8] BEUF, A., RUSTICHELLI, F., Acta Cryst., to be published.
FIG. 4.
-
Comparison between the neutron diffraction pattern by a curved crystal as obtained by the dynamical theory, and the rectangular pattern foreseen by the lamella model. The patterns correspond to the curvature C = 0.1, i. e. ten times smaller thanthe critical curvature.
5. Conclusions.
-
The diffraction properties of a cholesteric liquid crystal with a pitch gradient were investigated as a function of the gradient using a simplified model. It was shown in particular that the wavelength width of the total reflection for right circularly polarized light increases progressively as the gradient increases up until a critical gradient is reached. The expression for the critical gradient was derived as a function of the intrinsic properties of the cholesteric investigated. The validity of the model was justified by analogy to previous work on neutron diffraction by crystals with a gradient in the lattice spacing. Although the model used does not constitute a theory of light diffraction by cholesterics with a pitch gradient, we believe, however, that it can repre- sent in a satisfactory way the essential physical cha- racteristics of light diffraction by such regularly deformed cholesterics.rences
[9] RUSTICHELLI, F., Proc. of IAEA Symp. on Neutron Inelastic Scattering Grenoble (1972) Paper SM-155/F-2. [lo] PINDAK, R. S., HUANG, C. C., Ho, J. T., Phys. Rev. Lett.
32 (1974) 43.
[ l l ] BOULIGAND, Y., J. Physique Colloq. 30 (1969) C4-90. [12] CHANDRASEKHAR, S., SHASHIDHARA PRASAD, J., MoI. Cryst.
Liqu. Cryst. 14 (1971) 115.
