Elasticity and behaviour under magnetic fields of cubic lyomesophases and smectic D liquid crystals

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Elasticity and behaviour under magnetic fields of cubic lyomesophases and smectic D liquid crystals

H.R. Brand, P.E. Cladis, Y. Couder

To cite this version:

H.R. Brand, P.E. Cladis, Y. Couder. Elasticity and behaviour under magnetic fields of cubic lyomesophases and smectic D liquid crystals. Journal de Physique, 1986, 47 (8), pp.1417-1422.

�10.1051/jphys:019860047080141700�. �jpa-00210336�

Elasticity ^{and behaviour under} magnetic ^{fields of cubic} lyomesophases

and smectic D liquid crystals

H. R. Brand (*)

ESPCI, ⁷⁵²³¹ Paris, ^France

P. E. Cladis

AT & T Bell Laboratories, Murray Hill, ^NJ 07974, ^U.S.A.

and Y. Couder

GPS, Ecole Normale Supérieure, ⁷⁵²³¹ Paris, ^France

(Reçu ^{le 30} septembre ^1985, révisé le 18 avril 1986, accepté ^{le 22 avril} 1986)

Résumé.

Nous donnons une estimation de la constante élastique de cisaillement des cristaux tels que les ^smec-

tiques ^{D et les} mésophases lyotropes cubiques. ^La valeur trouvée est de trois ordres de grandeur plus petite que les cristaux atomiques ^les plus ^mous (He) ^et de trois ordres de grandeur plus grande que les cristaux colloïdaux. Le comportement ^{sous un} champ magnétique ^{extérieur est} discuté, ^en particulier l’importance ^pour les cristaux

lyotropes ^{des termes de} couplage ^au paramètre ^{d’ordre est mise en évidence.}

Abstract.

We estimate the shear elastic constant in cubic liquid crystals ^{such as D} phases ^{and cubic} lyome- sophases ^{and find a} value, which is about three orders of magnitude ^less than that of _very soft atomic crystals (4He)

and three orders ^of magnitude higher ^when compared with colloidal crystals. The behaviour under an external

magnetic field is discussed and the importance ^of coupling ^{terms to the order} parameter, especially ^{for cubic} lyomesophases, ^is pointed ^out.

Classification

Physics ^Abstracts

64 . 70E - 61. 30G - 03 . 40D

In the present paper ^we analyse for the first time the

elasticity ^{of cubic} lyomesophases ^{and of D} phases ^and

we investigate ^the influence of external magnetic fields, ^a point ^{that has not} been addressed before.

The effect of electric fields has been discussed _pre-

viously by Saidachmetov [1]. ^{In addition to the terms}

found in reference [1] ^{which are} of fourth order in the electric field, ^we find that there are cross coupling

terms to the order parameter ^{and to} the strain that are

second order in the electric field

Lyotropic liquid crystals [2], liquid crystalline phases which appear in multicomponent systems ^and

change ^their phases predominantly by changing ^the

relative concentrations of their constituents, ^come mainly ^{in two} groups : layered phases, ^{which are}

similar to smectic liquid crystals and columnar phases,

which show long range positional ^{order in two dimen-}

sions. The transition between these two types ^is

frequently ^achieved [2, 3] ^{via an} intercalated cubic

lyomesophase (sometimes called viscous isotropic phase). ^These phases ^have been first identified in soaps [4] ^but subsequently they have been found in many different systems [3, 2], including ^also biological

systems ^{such as} lipids [5]. The lattice constant of these cubic phases varies between 60 and 120 A and they

show sharp Bragg peaks ⁱⁿ X-ray diffraction as

expected ^{from a} crystal [2, 5] ; ^{on the} length ^{scale of the}

molecular diameter (~ 4,5 A), ^{however, this} phase

shows complete fluidity. The unit cell contains between 200 and 1 200 molecules. Various models have been put ^forward to account for the microscopic ^structure

of these phases (cf. ^{Ref. [3] for a} review) ^including

bi-continuous structures [6]. Methods used to inves-

tigate the cubic lyomesophases ^so far include-aside from the original X-ray measurements and direct observations in the polarizing microscope-freeze

fracture electron microscopy [7] ^{and NMR} [3].

In the present ^{note we} suggest ^{that the use of more} macroscopic methods such as measurements of the shear elastic constant and investigation of the beha-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019860047080141700

1418

viour under an external magnetic field is also important

to understand these fascinating systems ^{which are} fluid on a length scale of less than about 100 A and solid like on longer ^distances.

The cubic lyomesophases ^{have a close} analogue ⁱⁿ

the _range of thermotropic liquid crystals, namely

the so-called smectic D liquid crystal phases [8, 10]

which are not layered ^at all but also cubic as has been shown by polarizing microscopy [8, 10] ^{and in} X-ray

diffraction [11, 12]. As with the cubic lyomesophases,

the size of the smectic D unit cell is about 100 A and it contains of the order of 103 molecules. More recently

also a second class of compounds showing ^cubic thermotropic phases has been found in hydrazine

derivatives [13] ^and they have been shown to be immiscible with the D phases, ^but they ^{seem to have}

otherwise similar properties [13]. ^{There is a difference}

in the phase sequence observed, however. Whereas D

phases ^{occur between a smectic C} phase ^{at lower tem-} peratures ^{and a smectic A} phase ^{or the} isotropic liquid

at higher temperatures, ^the hydrazine derivatives are

sandwiched between a truly crystalline phase ^{and a C} phase ^at higher temperatures.

The size of the unit cell of both, ^cubic lyomesophases

and D phases, lies between that of typical ^atomic

or ionic crystals (a ^few Angstroms) and that of colloidal

crystals (0. 1, - - ., 10 im) and cholesteric blue phases ( ~ 3 ⁰⁰⁰ A). The latter occur in systems which have a

cholesteric phase ^{with a short} pitch [14] ^and they

intervene in the cholesteric to isotropic liquid ^transi- tion ; the unit cell of cholesteric blue phases ^{is cubic}

and contains about 10’ molecules.

As a consequence ^one might expect ^the macroscopic properties ^{of D and cubic} lyomesophases ^{also to be}

intermediate between atomic and colloidal crystals.

A last system ^which might ^show quite ^similar macroscopic properties ^as the cubic phases ⁱⁿ lyo- tropic ^and thermotropic systems ^{is the} tetragonal phase [ 15J discovered _very recently ^{in a} pure chiral [16]

compound. As with the cubic systems, ^the tetragonal

unit cell contains about 103 molecules, ^{is fluid on} length scales smaller than the lattice parameters (75

and 68 A) ^{and shows} Bragg peaks ⁱⁿ X-ray diffraction.

For the gradient ^free energy of a cubic crystal ^we

have [ 17]

taking ^{the axes} of the cubic phase parallel

to x, y and z

and for a tetragonal system with the four fold axis in the _xy plane

with the displacement ^{vector u.}

To get ^an order of magnitude estimate for the shear elastic constant C3 ^we take the data available for the deformations of nematic [18] and smectic C liquid crystals [19]. Equating the elastic _energy written down in equations (1) ^to the deformation _energy K(Vn)’

with n the director, using ^{values of K - 3 x 10-8...}

5 ^x 10-’ dyn/cm2 and lattice parameters ^{between 60} and 120 A, ^{we obtain for} _C3 ^values ranging ^from’

5 ^x 104 to 5 ^x 105 dyn/cm2. ^The procedure ^of evaluating C3 ^{should be correct} (1) 1) ^{since the cubic}

and tetragonal phases studied here are fluid inside the unit cell (as ^{fluid as a nematic or} the director in a C

phase) ^and 2) ^{since a} similar estimate has turned out to be correct previously for cholesteric blue phases [20].

Since the elastic constants in lyotropic ^{materials are}

of the same order of magnitude ^{as in} thermotropics [21] ]

the same analysis ^should apply ^{to cubic} lyomesophases.

It seems very interesting ^{to note} that the value

predicted here for the shear elasticity is three orders of magnitude smaller than that found for _very soft atomic crystals ^{such as He} [22, 23], ^{where one has}

~ 3 x 108 dyn/cm 2 ^{and about three orders} of magni-

tude bigger than that found for colloidal crystals [24, 25] (where ^one has, depending ^{on the} density ^of particules, 10, ..., 100 dyn/cm2) and cholesteric blue

phases [20] (C3 N ^{4 x} 102, ..., ^{2 x 103} dyn/cm2).

These numbers also show that the shear elastic constant scales as a- 2 with a the lattice parameter.

To determine C3 ^{(or an average} shear elastic constant in the tetragonal phase) experimentally in the cubic and tetragonal phases ^{discussed here, a} technique using

a torsional oscillator seems appropriate ^{as it has} already ^{been the case} for the shear modulus of the cholesteric blue phases [20]. Using e. g. the data of the

experimental configuration described in reference [20b]

a resonance frequency of about 200 Hz can be expected

Two alternative techniques ^{to determine} C3 ^also

come into mind immediately. As D- and cubic lyo- mesophases ^have properties intermediate between those of atomic and colloidal crystals, ^the measurement of the sound velocity ^as in solid helium [22, 23] ^seems possible. ^One could also measure the oscillations of a

freely suspended ^{film of a D} phase (as ^{in a thin} plate [26]) and infer from the frequency ^observed the elastic

properties. ^{If the} samples used consist of _many crystal- lites, ^{all the} experiments proposed ^will give ^an average shear elastic constant. In table I we have summarized for the various systems of interest here (cubic lyo- mesophases, ^D phases, ^colloidal crystals, cholesteric blue phases soft atomic crystals ^and typical crystals)

(1) ^{We assume that the} density modulation is sufficiently

small.

Table I.

data for the unit cell size, the number of atoms or

molecules _per unit cell, the actual (or expected)

values for the shear moduli, the critical shear rate for shear flow induced melting (ct also the discussion

below) ^{and the} question ^{whether or not} chirality ^{of the}

constituents is important ^{for the} phenomena ^observed

From the considerations presented ^{above we can}

also get ^{an order of} magnitude estimate for the critical thickness of a freely suspended ^{film of D} phases, ^i.e.

for the thickness below which the D phase ^is suppressed

in the film geometry. ^{If we} equate the elastic _energy

(Eq. (lb)) with ^{the gradient} energy of the cubic order parameter K(VQ)’ ^{we obtain}

assuming ^{for K a}

value of the same magnitude ^as for the elastic cons- tants of the director

for the critical thickness which leads to Ac ~ ^{0.05 um. Below}

this thickness D phases ^{should not exist and be} replaced by ^{a smectic A or C} phase. ^{A similar} analysis

can be performed for cholesteric blue phases. Taking typical values for C in the blue phase ^{and for K in the}

cholesteric phase ^we obtain for freely suspended ^films

of blue phases ^a critical thickness of about _{1 um. For} thinner films an isotropic ^{or a} cholesteric phase

should occur.

A second class of phenomena ^{which can} help ^to

understand the properties ^{of cubic} phases ^{is the} study

of the influence of an external magnetic ^{held This} technique ^{will be} particularly useful for lyotropic ^cubic

phases since for those the application of electric fields will lead (due ^{to their} conductivity ^{and due to} space

charge effects) ^to very complex ^behaviour.

The symmetry arguments parallel ^those given by

Saidachmetov [1] ^{for an} electric held

As it is clear from the cubic symmetry ^{the second}

i

order magnetic susceptibility

is isotropic ^{and can therefore not lift the} degeneracy.

To _preserve time reversal symmetry ^{cubic terms have}

to be ruled out and we have for the quartic ^terms

a result isomorphic ^{to that obtained} by ^Saidach-

metov [1] if the electric field is replaced by ^the magne- tic field To estimate Xi (4) ^we proceed ^{as follows.}

Equating ^elastic energy (K(Vn)2) ^{and magnetic energy}

(xa H2) in the smectic C phase ^we find for the threshold of the Frederik’s transition of the director inside the

plane ^{of the} layers ^{in a 1 mm thick} sample He ’" ^{50 G.}

If we assume that this value does not change drastically

in the D phase, ^{we obtain for} X (4) ^{as an order} of magni-

tude estimate X (4) 1 - ^{5 x lO-11} c.g.s. for a 1 mm

thick sample.

For cubic lyomesophases this estimate has to be taken with caution and should rather be taken as as

rough estimate than as a precise prediction. "

1420

If one allows for the simultaneous _presence of strain and a magnetic ^{field, cross} coupling ^{terms arise and we}

have an additional contribution in the generalized

free _energy Fg ^{(with the unit vectors h, mi, ni) with o having the same structure as} the elastic moduli C :

This contribution is completely independent ^of

the question ^{whether one is close to a} phase ^transition

or not and exists for all cubic phases discussed here and also for colloidal crystals and cholesteric blue

phases. ^An isomorphic ^term exists for electric field

effects, ^if Hi ^is replaced by Ei. ^{Such a term can be} expected ^{to be of} importance especially ^{for D} phases

and cholesteric blue phases since it is a contribution in the gradient free energy which is quadratic ^{in the}

external field and not quartic. ^It corresponds ^to

electrostrictive and magnetostrictive ^terms respectively

and we thus find three coefficients [17].

To discuss an application ^{of the terms} presented

in equations (3, 4) ^{we consider a} simple example. ^We

assume that a field is applied along ^{the z}

direction,

i.e. E

(0, 0, E). ^{In this case} equations (3, 4) predict

a change ⁱⁿ birefringence ^{due to} the deformation of the unit cell which can be detected experimentally provid-

ed qll ^and t/J 2 ^{are not too small} (we have for the contribution to the free _energy

, Conversely, ^a uniaxial pressure applied in y

or z direction will result in the appearance of an

electric field in the x - direction.

To get ^{an order of} magnitude ^{estimate for} qli ^we

make use of the inequalities obeyed by ^the transport coefficients to guarantee positivity ^{of the} entropy production. ^{We thus obtain the} upper bound

where C is a typical ^{elastic constant.}

But this class of terms is not the only ^{one which is} quadratic ^{in the} strength of the external field If one is close to a phase transition- and this is always ^{true for}

cubic lyotropic phases which exist only over a narrow

range of concentrations and also for cholesteric blue

phases-cross couplings ^{to the order} parameter ^{are also}

important ^{and we have}

with Qi jkl ^{the order parameter for cubic} phases [1].

’iklmn contains six independent parameters

where h, mi, ni ^are the three unit vectors along ^the edges of the cubic structure.

In the case of a magnetic ^{field H in z - direction} FQ simplifies considerably ^{and reads}

As for the case of the cross couplings between the

magnetic ^{field and the} strains, equations (5), (6) ^lead

to a change ⁱⁿ birefringence ^and

^{if the D to isotro-} pic transition is only weakly first order in some mate-

rials

to a finite value of the order parameter ^{of the} cubic phase ^{in the} initially isotropic phase, ^{i.e. to a}

field induced D phase.

The order of magnitude ^{of the} coefficients ^can

be estimated as for the 4/i ^{and we obtain}

That is external magnetic (or equivalently electric)

fields are coupled ^to fluctuations of the order _para- meter. This effect is well known [27], e.g. close to the

isotropic-uniaxial nematic transition. In D phases ^one

can expect ^{those terms to be} important mainly ^close

to the phase transitions. We mention in passing ^that

the relevance of these terms for both, electric and

magnetic fields, has been overlooked so far by ^workers

in the field of cholesteric blue phases.

It is clear that similar cross coupling ^{terms between}

external fields and the displacement ^{vector or the}

order parameter also exist for the tetragonal M2 phase, ^but they ^are probably ^less important (except possibly ^the coupling between order parameter ^and external fields) ^{for this} phase, ^since tetragonal sym- metry allows for an anisotropy ^of magnetic ^and

electric susceptibilities.

In conclusion, ^{we have} pointed ^out that the shear elastic constant of cubic lyomesophases, ^D phases

and the tetragonal M2 phase ^{has a} value which lies between that of _very soft atomic crystals like He and colloidal crystals. This also implies that the shear induced melting observed for colloidal crystals [28]

will take place ^{at much} higher frequencies ( ~ 104 Hz)

and thus seems to be out of reach experimentally,

whereas there is ^a better chance to observe it in cholesteric blue phases ( f ~ ¹⁰⁰ Hz) provided ^the single crystals ^are big enough. Polycrystalline samples

are not useful for this purpose since ^{one wants to}

study the actual melting process and ^not the motion of grain boundaries and other defects; therefore the whole sample ^{must be a} single crystal.

New cross coupling ^terms between external _magne- tic (and electric) fields and the distortion or the order parameter ^{have been} pointed ^out. Their influence can

also be measured experimentally, e.g. by measuring

the torque ^exerted by ^a magnetic ^{field on a} sufficiently big ^{smectic D} single crystal floating ^{in an} isotropic background ^We predict ^{that the} torque ^{has two} contributions : one proportional ^{to H2 and one} proportional ^{to H4. As a function of} temperature ^the latter can be expected ^{to become more} important

relative to the former.

The chances to observe the effects predicted ^here

are best for well aligned samples, ^{which can} be obtained either by using ^a suitable surfactant or by studying freely suspended thick films.

Acknowledgments.

It is a pleasure ^{to thank Jean} Charvolin for a stimulat-

ing discussion on cubic lyomesophases. ^{H.R.B. wishes}

to thank the Deutsche Forschungsgemeinschaft ^for

support of his work. The work of P.E.C. was partially supported by ^NSF grant DMR-8404942.

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