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Elastic light scattering by smectic A focal conic defects
N.A. Clark, A.J. Hurd
To cite this version:
N.A. Clark, A.J. Hurd. Elastic light scattering by smectic A focal conic defects. Journal de Physique, 1982, 43 (7), pp.1159-1165. �10.1051/jphys:019820043070115900�. �jpa-00209490�
Elastic
light scattering
by smectic A focal conic defectsN. A. Clark and A. J. Hurd
Department of Physics, Duane Physical Laboratories, University of Colorado, Boulder, Colorado 80309, U.S.A.
(Reçu le 27 novembre 1981, révisé le 2 mars 1982, accepté le 4 mars 1982)
Résumé. 2014 On induit, par dilatation, dans une préparation smectique A homéotrope, des réseaux quasi périodiques
de défauts coniques focaux (PFC). On montre, que ces réseaux, dans l’intensité de la lumière diffusée, provoquent
une répartition multi-bande en fonction du vecteur d’onde de diffusion. La diffusion, dont la source provient des
discontinuités des lignes confocales, est traitée, quantitativement pour donner deux déterminations indépen-
dantes et homogènes de la distance focale PFC en fonction de la dilatation appliquée. L’analyse permet de démon- trer, dans le smectique A, l’existence de défauts coniques focaux avec une distance focale aussi petite qu’une simple
couche smectique.
Abstract. 2014 Quasi-periodic arrays of focal conic (PFC) defects, induced by dilation of a homeotropic smectic A sample, are shown to produce a characteristic multi-band scattered light intensity distribution as a function of
scattering vector. The scattering, which is caused by the confocal line discontinuities is quantitatively analysed to yield two independent, self consistent determinations of PFC focal length as a function of applied dilation. The existence of smectic A focal conic defects having focal lengths as small as a single smectic layer is demonstrated.
Classification
Physics Abstracts
61.30J
1. Introduction. - The equilibrium smectic A (SA)
structure is a stack of planar fluid layers, taken to be
normal to z, which can flow over one another and
are readily curved, but which are not readily altered
in thickness [1]. This structure exhibits a marked
elastic anisotropy which results in a variety of novel
elastic effects, including the dilation induced undula- tion instability [2] and a wavevector (K)-dependent susceptibility for periodic layer displacements, which
becomes large near Kz = 0 [3]. An additional conse-
quence of this SA structure is the spontaneous gene- ration of metastable focal conic defects. These defects
are characterized by SA layers which are continuously
curved and have a thickness very nearly equal to the equilibrium value everywhere except along pairs of
line discontinuities. The discontinuities are confocally
related conic section curves, where the layers have coni-
cal cusps or become multiply connected [4]. In this
paper we demonstrate that such discontinuities can
produce significant light scattering and that the gene- ration and structure of focal conic defects can be
effectively probed using light scattered by their line
discontinuities.
In our study, focal conic defects were generated by
dilative stress applied normal to the layers, as shown
in figure 1. The SA, in this case CBOOA (p-cyano- benzylidene-p-octyloxyaniline) at 80 OC, was prepared
between flat glass plates spaced by t = 330 gn-4 with the surfaces treated with HTAB (hexadecyl trimethyl
ammonium bromide) to enforce parallel layer orienta-
Fig. 1. 2013 Layer distortion of a homeotropic smectic A sample under increasing applied dilation, bt. (a) 6t = 0;
(b) uniform dilation for 0 bt 2 A; (c) smoothly undu-
lated texture for 2 7rA bt bt,,c; (d) PFC array for bt > btpFC 1.7(2 nA), cut through a row of parabolae.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019820043070115900
1160
tion (homeotropic). A small applied dilation bt induces
a uniform layer expansion and increased layer-dila-
tional elastic energy density (Fig. lb). This induced strain can in principle be relaxed by layer tilt, a possible
response in the SA because the layers are free to flow
over each other. A uniform tilt of y = (2 btlt)112 will completely relax a small layer strain. However, because
of the boundary condition, a uniform layer tilt is not
favoured and the system responds with a periodic layer
undulation having alternate regions of layer tilt and
dilation (Fig. Ic). The threshold dilation, bt,,,, and
undulation wavelength in the layer plane, h§, are given by [2] :
where A is the de Gennes length, (KjB)1/2, and where B
is the layer-compressional elastic constant and K is
the layer-curvature (splay) elastic constant [3].
If bt is increased to btPFC = (1.7 + 0.2) bt,,, a second instability is observed, characterized by the appearance of parabolic focal conic (PFC) defects as indicated schematically in figure Id [4]. PFC’s are a degenerate
focal conic case where the line discontinuities form
interlocking parabolae which lie in perpendicular planes and pass through each others’ foci (Fig. 2) [4].
The appearance of PFC’s indicates the onset of a strain condensation process; that is, the dilative strain, and
to some extent the curvature strain, are reduced in the bulk of the sample by the formation of line disconti- nuities around which both layer dilation and curvature strain are presumed to be large [5]. The PFC structure
which appears upon dilation has been studied by microscopy and is discussed in detail in reference [4].
The idealized structure, indicated in figure 2, is a
square PFC lattice, with the defect cores (where the parabolae and foci intersect) in the sample midplane,
and parabolae of adjacent defects meeting in groups of four at the sample surfaces. Typical plane sections passing through the parabolae are shown in figures 1
and 2. In practice, PFC arrays are not perfectly perio- dic, with 3 to 5 parabolae meeting at points on the
surface (cf. Fig. 10, reference [4]).
We have employed light scattering to study the PFC
array structure. Light scattering due to layer displa-
cement along z, uz(r) in a SA arises because the struc- ture is optically uniaxial, with dielectric anisotropy
As = 8zz - 8xx. As the layers tilt, the molecules remain locally normal to them so that a region with layers tilted by an angle C from z in the x-z plane will develop an off diagonal dielectric tensor component
be.,z = As. C = As(OulOx). The result, for light incident
with wavevector ks, is depolarized scattering of wave-
ve,ctor ks with the amplitude
where u(K) is the Fourier component of the layer displacement field uz(r) having wavevector K = ki - ki,
and Kr is the component of K which is normal to z. This
Fig. 2. - Three dimensional structure of a PFC array in a
homeotropic smectic A sample. Adjacent parabolae intersect
on the sample surfaces. The sample thickness is t, the PFC lattice parameter is L, and the parabola focal length is f.
The satellite scattering discussed in this paper arises from the meeting of the adjacent parabolae as they approach
the surfaces. The meetings on the bottom surface occur over the centres of the primitive unit cells of the square lattice of top surface meetings. This arrangement leads
to interference between top and bottom scattered light
that produces a splitting of the scattered light about K,, = 0
at (Kr), = 2 ir/L.
scattering in purely depolarized in that ð8xz is the only
non zero component of 6eij induced by au/ax.
For small amplitude layer displacements, u(K) is governed by an elastic free energy which has both
layer-dilational and splay-curvature terms [3]
For values of K,, such that K, ÀKr2, the restoring
force against layer undulations involves only splay elasticity. However, for K,, > ÀKr2, the layer-com- pression term dominates and strongly suppresses
layer displacement. Therefore, intense light scattering
occurs for wavevectors such that K. AK ’ K,
as has been observed to arise from both static surface
impurities and thermal fluctuations [6, 7]. In the case
of the undulation instability, the boundary condition requires K,, = nlt, and energy minimization requires Kr = (K.,IA)’I’ leading to the condition for the critical
wavelength, Ac = 2 n/Kc, given in equation (2). The
undulation instability is therefore heralded in the scattered intensity I (K) by the appearance of scattered
light spots at (K_, = 0, Kr = (7r/At)1/2 ). The instability
threshold dilation, bt,,, has been used to determine A
via equation (1), yielding A = (16 ± 0.5) A for CBOOA
at T = 80 °C [8].
2. Experimental details. - A Fraunhofer geometry
was employed to observe scattered light intensity distributions, I(K), as a function of scattering vector, K,
Fig. 3. - Experimental Fraunhofer scattering geometry showing SA sample, focussing lens, and film plane coor-
dinates (s, 0). The scattered light is indicated schematically (E - 0 at small s and O - E at large s), showing split
main rings ( ) and satellite bands (... ).
as shown in figure 3. The homeotropic SA sample was
illuminated by a broad, polarized, collimated 514 nm
argon laser beam (diameter = 3 mm) and I (K) was
recorded photographically in the focal plane of a lens (focal length = 16.7 cm). Because of the uniaxial
anisotropy, the incident and scattered polarizations
resolve into two normal modes, 0 (polarized normal
to 2), and E (polarized normal to 0 and ki). Since the scattering is purely depolarized, the only allowed
first-order (single) scattering events are E - 0 and
o -+ E.
The map of (K_,, K,) onto the scattering plane coordi-
nates (s, 0) is complicated by the refractive index
anisotropy. The geometry is detailed in reference [6].
For O - E or E - 0 scattering, the locus of points
for which K, = 0 is a circle of radius so, or s,o,
respectively centred on point 0, the intersection with the screen of a line which is normal to the sample plane
and passes through the centre of the lens (Fig. 3). For
0 -+ E scattering the point Kr = 0 is at the inter- section of the O - E circle with the plane of incidence
(0 = 0) and Kr increases with increasing 0, while K.,
is measured as a deviation from the O - E circle
along a radius. The components of the scattering
vector for 0 -+ E scattering are related to the film- plane coordinates {s, 0) indicated in figure 3 by the following equations :
where no and nE are the refractive indexes of the ordi- nary and extraordinary rays, respectively; qlj and os
are the angles from the sample normal of the incident
and scattered rays. These quantities are related to the
refractive indexes for light polarized propagating perpendicular (n,) and parallel (n,l) to the smectic A
director, the film-plane radial distance, s, and the lens focal length, h, according to the relations,
Here 132 is the quantity [(I/ni) - (I/nTI)] and SOE
is the radial coordinate of the Kz = 0 rings for 0 -+ E scattering. The E - 0 equations can be obtained by simply exchanging no and nE in equations (4a) and (4b) and sEO for SOE in equation (4f ).
The refractive indices used were n, = 1.51 and nll = 1.77, sEO/h = 0.217, and sOE/h = 0.304.
Typical photographs of the scattered intensity
distributions are shown in figure 4. The incident
laser beam was reduced in intensity by a small piece
of neutral density filter, as seen in photographs. The
Fig. 4. - Photographically recorded scattered light inten- sity distributions obtained with 5 145 A incident light pola-
rized at 45° to the plane of incidence for various dilations bt.
Both E -+ 0 (small s) and 0 --+ E (large s) scattering features
are evident. The direct laser beam was attenuated with
a small piece of neutral filter, revealing the focussed laser spot ((b), point P) : (a) bt=O, (b) bt=26o A, (c) bt=81o A,
(d) bt = 3 600 A; the coordinates are defined in figure 3.
An analyser is inserted in the scattered light in showing conoscopic fringes near the transmitted laser spot. The analyser easy axis is vertical, passing the 0 -+ E scattering
for 0 0 and E --+ 0 scattering for 0 > 0, as expected
for the respective single scattering processes. The band
passing through the laser beam is multiple scattered. The radial lines emanating from the focussed laser spot are stray light. The solid white arrows in (d) indicate the satel- lite bands. The open arrows in (b) inuicate the main ring splitting.
1162
laser was focussed in the plane of the screen so that
the dimension of the resulting diffraction-limited laser spot determined the K space resolution
(AK z 50 cm-1 ), which was much smaller than any of the features to be discussed.
The sequence of scattering events as the dilation is increased from bt = 0 to bt = 3 600 A is depicted
in figures 4a-d. At zero dilation (Fig. 4a) the scattering
is weak and confined to the two Kz = 0 rings. This scattering results from imperfections such as dust and
surface irregularities in the CBOOA preparation [6].
As the sample is dilated, spots due to the undulation mode appear when bt > btc at
on the Kz = 0 rings, growing in intensity as (bt - ðtc) [8]. These spots persist until 6t = 1.7 bt,
at which dilation there are two changes in the pattern.
First, the intensity of the undulation mode scattering
saturates, indicating that the undulation ceases to grow for 6t > bt,. Second, two new features develop
in the scattered intensity distribution : there appear pairs of satellite bands which separate from each of the main (K,, - 0) bands as Kr increases (Figs. 3, 4c, and 4d), and a splitting of the main band for a range
of Kr (Figs. 3, 4b, and 4c). As bt is increased the satellite bands deviate further from the K,, -- 0 rings for a given K, and the particular Kr at which the main
rings split monotonically decreases. The satellite bands satisfy their respective polarization conditions
for E - 0 and O - E processes - i.e. they arise from single scattering. Equations 4a and 4b and the photo- graphs were used to obtain measurements of Kz vs. Kr
for the satellite intensity maxima. A plot of these data, figure 5, shows : (a) that Kz is proportional to Kr along
the maxima (K,, = Kr/a), (b) that the constant of proportionality, a, increases with increasing dilation,
and (c) that K.,IK,, is small compared to unity. In addition, the scattered intensity is found to be inde- pendent of sample orientation about z.
Fig. 5. - Location of the satellite band maxima for various dilations bt : x : 3 600 A ; 0 : 3 600 A ; + : 810A; A :
390 A ; fl : 260 A. The slope of a given curve gives a for that
bt.
3. Analysis. - All of the observed features of
I(K), including some additional features to be pre- sented below, can be interpreted in terms of scattering by line discontinuities in PFC’s. Although the scatter- ing of light by PFC’s can in principle be calculated in
terms of electromagnetic propagation in a spatially nonhomogeneous anisotropic dielectric, such a proce- dure is significantly complicated by the large defor-
mations and complex structure intrinsic to focal
conic defects. To avoid these difficulties and yet qualitatively analyse our light scattering data, we
have adopted an approximate form, suggested by the data, for the scattered light amplitude E(K). Fourier
transformation of E(K) then yields an effective distri- bution of scattering centres,
which we identify with particular features of the PFC array.
The data indicate that scattering is confined to a
cone of angle a about Kz ; the symmetry about K,,
is a result of the independence of the distribution of the
scattering to sample rotations about z. That is, E(K) is of the approximate form
The function b(Kr) is introduced in E(K) to account
for the dependence of the satellite scattering intensity pattern along its intensity maximum (K, = aKz).
The photographs indicate that b(K,) decreases monotonically with increasing Kr once the satellites become split from the K. = 0 rings, i.e. for sufficiently large Kr. For small (K,,, KJ, E(K) arises from long wavelength layer undulations and falls to zero as
K, -+ 0, according to equation (2). E(K) in equa- tion (5) was Fourier transformed analytically for a variety of simple monotonically decreasing b(Kr)’s
such as (b(Kr) oc Kr-l, Kr 2, etc...). Although the
details of the resulting scattering density n(r) depended
on the b(Kr) chosen, they possessed the common
feature that n(r) was strongly peaked on the real
space cone coaxial with z given by r = az. For example, taking b(Kr) = Kr- 1 yields
Hence, the K space distribution of satellite-scattered
light indicates scattering centres which are conically
distributed coaxial with z in real space, with maximum
density along a cone of angle a = r/z with a 1.
Returning to the ideal PFC array geometry of figures 1
and 2, the features best characterized by this descrip-
tion are the meeting of adjacent parabolic line defects at the sample surfaces.
Groups of four adjacent parabolae meet at points
on the sample surface, as indicated in figure 2, to produce a layer dimpling which is approximately translationally invariant when moving along z, satisfy- ing the basic feature in the scattering that K_, Kr
as long as the parabola focal length, f, is small compar- ed to sample thickness, t, a requirement which is always satisfied, as indicated below. The points of
intersection of the ideal array form a two dimensional square array of lattice constant L, where
Parabolae meet approximately as straight lines at
the surface making an angle a with the z axis, where
Hence, the basic geometry of the intersections is established by the sample thickness, t, and the PFC focal length, f, and is related by the structure to the spacing, L.
Light scattered from an ideal array would exhibit the four fold symmetry of the lattice. However, in
our experiments a large (,;:t 10 mm2 ) sample area
was illuminated. Over this area, the PFC array is
polydomain, containing regions of differing lattice
orientations. In addition, a variety of lattice defects are
possible, involving the meeting of 3 or 5 parabolae (cf. reference [4], Figs. 8 and 10). These effects produce
rotational averaging of the scattering distribution about z, leading to the observed conical I(K). Thus,
we attribute the satellite scattering to the effective
conical distribution of scattering centres arising from
the intersections of the parabolae at the surfaces.
At first glance it may seem peculiar that the observed
scattering arises from those portions of the PFC line defects near the surfaces where the refractive index
discontinuity is the weakest. However, the approxi-
mate translational invariance along z of the structure
near the surfaces gives rise to constructive interference
along the satellite bands of scattering at different z.
The PFC core regions, although they have a much larger variation in n, are small in size compared to an optical wavelength and will scatter weakly and nearly isotropically.
We have further direct evidence that the scattering
arises from near the sample surfaces. If the scattering originates from near the two surfaces, then interference effects between the two sets of scatterers are possible.
In the ideal PFC array the parabola intersections
are on the sample surface and are arranged in an array of square cells of lattice parameter L with the inter- sections on the opposite surface lying in the centres
of the square cells. This arrangement leads to the
following interference effects. The scattered intensity
distribution function, E(K), arising from an isolated
intersection (approximated by equation (5 )) is modulat-
ed by an interference term [1 + exp(K,, t + K, L/2)].
This modulation of the main and satellite scattering
is evident in the structure near Kr = 0 where regions
for which Kz = n(2 x/t) (n = 0, +- 1, ± 2) are
enhanced in intensity. These bright regions of constant K, run across the satellite bands and along circles
centred at point 0 (Figs. 3 and 4) in the scattering plane, as do the conoscopic fringes, made evident in figure 4d by inserting an analyser. This interference
produces the second major feature of the scattering, namely the splitting for some Kr of the main rings.
Taking Kr = 2 nIL in the above interference term
produces a splitting : an intensity zero at K_, = 0
with a pair of intensity maxima at Kz = + 7r/t.
Thus, for a given dilation, a measurement of (Kr)S’
the Kr for which the splitting occurs, gives a measure-
ment of the PFC array lattice parameter L = 2 nl(K,),.
Furthermore, using equation (7a), we get a measure-
ment of the parabola intersection angle a, i.e.
a = 7r/[t(K,),,]. Since a can also be obtained from the satellite scattering intensity distribution, we have
two independent measures of a at a given dilation
and hence can check for self consistency. This test is
illustrated in figure 6. Here we have plotted the a
obtained from the satellite scattering data, i.e. from
the slope of the Kz(Kr) curves as in figure 5, plotted
versus the measured (K,.), for the various dilations.
The solid line is given by a = n/[t(Kr)sJ, using the
known sample thickness. There are no adjustable parameters. These two measures of a are in substantial, although not perfect, agreement : the a’s obtained from the satellite locations are slightly larger (- 10 %)
than those predicted on the basis of the PFC lattice
spacing. There are two likely ways to account for
this, both arising from the same effect. Either the
Fig. 6. - Plot of a = K,IK., from the satellite band maxima
vs.I(K,.),,, the location of the splitting of the main band.
From our scattering model and the PFC geometry we expect al(K,)s = nit. The solid line has slope nit as determined by the known sample thickness. The data indicate an
effective sample thickness slightly smaller than t, possibly
a result of adjacent parabolae collapsing into a single
line defect before intersecting the sample surface.
