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Smectic-A flexo-, piezo- and ordo-electricity
J. Fournier, Geoffroy Durand
To cite this version:
J. Fournier, Geoffroy Durand. Smectic-A flexo-, piezo- and ordo-electricity. Journal de Physique II, EDP Sciences, 1992, 2 (5), pp.1001-1006. �10.1051/jp2:1992103�. �jpa-00247687�
Classification Physics Abstracts
61.30G 61.308
Short Communication
Smectic-A flexo-, piezo- and ordo-electricity
J-B- Fournier and G. Durand
Laboratoire de Physique des Solides, Universit4 Paris-Sud, Bitiment 510, 91405 Orsay, France
(Received 17 December 1991, revised 20 January 1992, accepted 11 February 1992)
Rdsumd. L'existence de distributions de charges inhomogbnes le long des mo14cules de cristaux liqnides donne naissance I des contributions purement smectiques £ la flexo-, piezo- et
£ une nouvelle ordo-electricit4, proportionnelles au carr4 du paramktre d'ordre smectique.
Abstract. Inhomogenous charge distributions along mesogenic molecules of liquid crystals give contributions of a pure smectic origin to flexo-, piezo- and a new ordo-electric polarization
which
are proportional to the square of the smectic order-parameter.
Non ferroelectric liquid crystals such as nematics or smectics [ii present the inversion sym- metry. In the presence of a slow deformation, this symmetry can be broken and strain-induced
polarizations may appear. For nematics, the well-known flexo [2, 3]-polarization is created by gradients of the director field n. Gradients of the order-parameter S, e-g- surface rough-
ness induced melting, are responsible for the ordo [4]-polarization. In the case of smectics-A, since the smectic layers are perpendicular to the director, flexo-electricity corresponds to layer
curvature. One can also define a piezo-polarization [ii associated with the gradients of the
smectic layer spacing. In the past [5], for smectics-A, nematic-like contributions to the flexo- and ordo-polarizations have been considered. In this paper we discuss the elsects of a charge modulation along the mesomorphic molecules. Coupled to the smectic ordering, this modula- tion is shown to give pure smectic contributions to the smectic-A flexo-, piezo-electricity and
a new ordo-electricity. This new ordo-electric elsect is proportional to the squared modulus of the smectic order-parameter as expected from a pure symmetry basis.
In usual thermotropic smectics-A, rod-like molecules are segregated inside equidistant planes of molecular spacing a
~- 30 Ji. In addition to this unidimensional (quasi) ordering, the molecules are aligned almost parallel to each other, the common direction (nematic direc- tor n, n~
= 1) being normal to the smectic layers. The de Gennes-Landau order-parameter [I]
is defined as the (complex) amplitude of the spatial modulation associated with the smectic
ordering. Let us call ks = 21rla the layer wave vector, and z the piling direction (smectic
1002 JOURNAL DE PHYSIQUE II N°5
planes along z, y). Since we consider smectics-A, we will strictly restrict ourselves to n locally perpendicular to the layers (parallel to z in the ground state). The mass density modulation,
for instance, written as
bm = bmo
~fi cos k~(z + u) (1)
defines the complex order parameter ~fiexp(ik~u). The amplitude of the modulation (bmo( is defined by the normalization condition
~fi = for a saturated smectic. As a convention, we
assume &mo > 0 if the mass excess is localized near the center of the molecules. Hence from
(1), the amount of order is described by
~fi and the displacement of the layers by u, which may also exhibit slow spatial variations ~fi(r) and u(r). In this case the new smectic layers, defined by z + u(r) = Cte, are generally tilted and dilated. The local wave vector modulus (k( and the
local normal to the layers n become then, to first order in u and its derivatives,
~ul
(k( ~- k~(i + a~u)
,
n ~- o~u j2)
In smectics, all physical quantities are in principle modulated and proportional to ~fiexp(ik~u)
at lowest order. Since usual mesogenic molecules bear permanent electrical dipoles and
quadrupoles, it is reasonable to introduce a net electrical charge modulation:
Plr) = Poll(r) COSkS(z + u(r)) (3)
Let us estimate po for a standard molecule of length a
~- 30 Ji, width w
~- 5 Ji, polarizable
core length z
~- 10 Ji and quadrupolar density e(
~- 3 x 10~ cgs (nematic quadrupolar flexoelectricity [6]). For the elongation of a charge q over the core, we expect (-e)
~- qz~law~
and po '~ qlaw~. Then po is found to be of order po '~
(-e)/z~
~- -3 x 10~° cgs, which corresponds to the fraction II10~~ of one electronic charge over the molecular volume. po is an intrinsic molecular quantity, defined as the amplitude of the fundamental charge modulation along the molecules. Comparing ii) and (3), as the mass excess is usually at the center of the
molecules, po has the sign of the excess of charges near the core.
In the same way, the polarizability tensor is expected to be written as
a;j = ajj + baij
,
bajj jr) = ba)j (r) ~fi(r) cos k~(z + u(r)). (4)
Where j is the background nematic contribution and da( the amplitude of the smectic modulation measuring the dilserence between the molecule cores and tail polarizabilities. Note that ba)j jr) can be slowly space dependent as will be shown later on. ba)j should be large
since aromatic groups are much more polarizable than aliphatic chains. As such groups are frequently close to the core of the molecules, ba)~ is usually positive. From uniaxial symmetry,
ba)j is given by ill, I referring to the layer piling direction kn)
b~lj(~) " b~l(~) biJ + (b~)(~) b~l(~)) ~i(~)~Jl~) (5)
where bj; is the Kronecker symbol. In (5) both baj, do[ (the difference of which is propor- tional to the nematic order S(r)) and the normal to the layers n(r) have been assumed space
dependent.
Let us first calculate the electrical field modulation associated with p(r). We shall then build the strain-polarizations by coupling this wave with the polarizability wave. As the density p
arises from molecular charges, it is convenient to treat them as dielectric internal charges.
Then in the absence of an external field, there is no electric displacement, I-e- D = 0, and the internal electrical field E is given by the vacuum Maxwell equations: div E = 41rp, curl E
= 0.
Note that p and E, although varying rapidly across the layers, are in fact quasi-macroscopic quantities, since they are averaged over large distances inside the layers. In the reciprocal
space, the components of E are simply related to the components of p:
4gr k
~~ " @~~M ~~~
For a uniform smectic with
~§ and u constant, E reduces to a sine wave along z of amplitude 41rp/k, where p
= polfi and k
= ks. In general pk has non zero components in the vicinity of k = ~ks ~ q, q « k~ being the inverse length-scale of the slow gradients of ~fi(r) and
u(r) defined in (3). Following our quasi-macroscopic analysis, we can write the field-induced polarizations as
pi = ai; E; (7)
It is important to stress that we are looking for electrical effects giving rise to polarizations which
are macroscopic, I-e- which occur at the gradient wave vector q. As long as & is assumed uniform, such gradients induce components of the electrical field E at + k~ ~ q and the corre- lated polarizations also appear at ~ ks ~ q. These electrical elsects are not truly macroscopic
due to the rapid oscillation across the layers. To find macroscopic polarizations, we must take
into account the non-zero polarizability modulation in (4). Now ai;(r)E;(r) is given in the
reciprocal space by
Pi,k " £ aij,
k-k' E;, k' (8)
~,
For instance, &k, couples with E-k,+q to build pq macroscopic. Since both &k, and E-k,+q
are proportional to ~fi, pq is proportional to ~fi~. Further coupling can give rise to components at wave vectors ~ nk~ ~ mq of higher orders in
~fi(n, m integer). To derive the strain-polarizations corresponding to flexo-, piezo- and ordo-electricity, it is possible to calculate E~k, and E~k,~q explicitly, and to couple them at lowest order in
~§ with &~k, and &~k,+q according to the selection rules giving pq. Then, coming back to the direct space, one can derive the expression
of p(r) directly in function of the strains. Such a procedure, requiring independent calculations for the flexo-, piezo- and ordo-cases, is heavy. The same results can alternatively be obtained,
at lowest order in
~§ and q/ks, in a more transparent way using the Maxwell stress tensor as
we now show.
Consider again the general case with ~§(r) and u(r). Because a linear variation of u cor-
responds to a uniform compression, it cannot give rise to a polarization. It is then more convenient to use the variables k(r) and n(r) defined in (2) instead of u(r). Then ~§, k and n
will enter the formulae at the same order of derivation. Now we wish to derive the electrical
field. It is probably not simple to get it directly from (3) and (6), but it is straightforward to
check that
E(r)= ~po ~fi(r) sin(k(r) r) n(r) + o(1) (9)
where O(1) means terms of order in the derivatives of ~§, k, n, is solution of divE
=
41rp + O(1), with p given by (3) under the aproximations (2). At this order, E is simply the local field function of the local variables. The Maxwell stress tensor, defined as
M,j = ) (E~
bij 2 E, Ej (10)
1004 JOURNAL DE PHYSIQUE II N°5
can be calculated with E given by (9). It has a non-zero mean value (locally averaged over a large number of layers)
lmul (r)
=
~)~)()~~
lbij 2 ni(r) nj(r)I + O(i) (11)
Using the proportionality between da;> and p, we can write the macroscopic polarization as
Pi = iba~jE;) = ~)) IPE;) l~~~
Since the density of electrical forces derives from the Maxwell stress tensor, I-e- pE, = -am Mm;, we directly obtain a compact form for P :
Pi = P~~" am (Mm;) (13)
The macroscopic polarization is thus explicitly given by
-Pi = ~rpo &alp am (p~§2 lbm; 2
nmnj)) + O(2) 1i4)
Developing the divergence one directly obtains, the ordo-polarization via am (~fi~), the piezo- polarization via am (1/k~), and the flexo-polarization via am (nmn;). Using (2) and (5), (14)
can be written in a less transparent but more classical way [11:
~~ " ~ll
~ t~~ + ~~~ Ill
+ Ill
+ ~~~Iii
~~~ " ~~ ~ t~~ + ~~~t
fi (~§2) fi2~
~' ~~
by ~ ~~~ 0z0y ~~~~
where eg and ei are two new ordo-coefficients that we introduce, and Aei, Ae~, Ae3 are the smectic contributions of our model to the flexo-piezo coefficients. Aei and Ae3 renormalize in
~fi~ the usual nematic flexo coefficients ei and e3, respectively associated with splay and bend deformations. The piezo-coefficient associated with layer dilation is Ae2. Since ez is exactly
zero in the nematic phase, Ae2 is equal to the e2 coefficient introduced from a symmetry basis by de Gennes [Ii. The flexo-, piezo-, ordo- deformations are shown in figure 1. Calculation of
the coefficients gives:
~
7rPo~~o
II ~$ II
~s
~~ ~ $~~~
6ei " -Ae~ = 2~fi~eg
Ae3 " -4~§~e1 (16)
a~ b) c)
Fig. i. Slowly varying strains or gradients can give rise to macroscopic polarizations: a) flexo-
electricity associated with layer curvature (liei, lie3)I b) piezo-electricity associated with layer spacing gradient (lie2)I c) the new ordo-electricity associated with smectic order gradient jell, e1).
Only two independent coefficients are in fact required. For instance, a direct measurement of the smectic-A ordo-electricity would allow us to separate the smectic and nematic contributions to flexo-electricity. To estimate the modulation ba°,
we use the relation e
= 1+ 41ra defining
the low frequency dielectric constant of the material. Neglecting the polarizability of the
aliphatic tails (for standard molecules), we can estimatc that ba° compares with the mean
polarizability a. lvith e
r- 10 for usual compounds, we find a r-1 and therefore ba°
r- 1.
With ~fi r- 0.6, po
r- 3 x 10~° cgs and 21r/k r- 30 I, the coefficients are estimated
r-
10~~ cgs.
This is indeed the usual order of magnitude of the neinatic flexo- and ordo-coefficients. The
polarizations we obtain are
r- ~fi~, the self-energy associated is then only
~- ~fi~, irrelevant close to second-order transitions. It seems quite obx,ious that in the absence of charge distributions
along the molecules, no spontaneous strain-polarization can appear. In principle, one should consider higher order harmonics of the charge distribution, linked to the existence of very localized dipolcs or quadrupoles. lIowex~er thesc components are strongly washed out by the
positional disorder inside the layers. Such contributions must only be considered for
~fi very close to unity. Comparing smectic with nematic ordo-electricity, the smectic "magic angle" km
(director tilt angle giving a polarization parallel to the interface which produces the gradient [4]) is defined according to formula (13) as cos~0m
~- 1/2 instead of1/3 for nematics. This reflects the anisotropy of the Alaxwell stress tensor and could in principle be used to separate smectic from nematic order-electric contributions. Finally considering the smectic fluctuations in a nematic phase, our model predicts a contribution ~-< ~fi~ > to the difference ei e3,
analogous to the quadrupolar "surfing" contribution recently introduced [7].
In summary, we have demonstrated the existence of a pure smectic-A contribution to the
flexo-, piezo- and ordo-elcctricity. Our model requires the presence of inhomogenous charge
distributions along the molecules and different polarizabilities between the center and ends of the mesogenic molecules. This is indeed the general case. In the presence of slow spatial strains, the inversion symmetry of the hoiuogenous smectic phase is broken. The strain-induced
polarizations come from the non-linear coupling betP>een two rapidly oscillating smectic waves, the electrical field wave and the polarizability wave. A macroscopic distortion at q builds smectic wave components at ~k ~ q. Their coupling, for instance at k + q and -k, gives
a macroscopic polarization at q. This is a kind of "static rectifying~' indeed very similar to the so-called "optical rectifying" in non-linear optics, where an R-F- n,ave at pulsation w w' if generated from two optical waves at w and w'. Practically, these polarizations can couple directly to external fields. This is probably the best way to test their existence. In principle it
1006 JOURNAL DE PHYSIQUE II N°5
is possible (although difficult) to separate nematic from smectic contributions and bulk from surface effects.
Acknowledgements.
We acknowledge stimulating discussions with Pr I. Dozov.
References
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[2] MEYER R.B., Phys. Rev. Lett. 22 (1969) 319.
[3] PROST J. and MARCEROU J-P-, J. Phys. France 38 (1977) 315.
[4] BARBERO G., DOzOV I., PALIERNE J-F- and DURAND G., Phys. Rev. Lent. 56 (1986) 2056.
[5] BARBERO G. and DURAND G., Mol. Cryst. Liq. Cryst. 179 (1990) 57.
[6] DOZOV I., MARTINOT-LAGARDE Ph. and DURAND G., J. Phys. Lent. France 44 (1983) 817.
[7] PALIERNE J-F-, presented at OLC'91, Cocoa Beach, Florida, (October 7-11 1991).
