Modulated smectic C*-uniform smectic C* phase transition in an electric field

HAL Id: jpa-00209574

https://hal.archives-ouvertes.fr/jpa-00209574

Submitted on 1 Jan 1983

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.

Modulated smectic C*-uniform smectic C* phase transition in an electric field

O. Hudák

To cite this version:

O. Hudák. Modulated smectic C*-uniform smectic C* phase transition in an electric field. Journal de

Physique, 1983, 44 (1), pp.57-66. �10.1051/jphys:0198300440105700�. �jpa-00209574�

57

Modulated smectic C*-uniform smectic C* phase ^transition

in an electric field

O. Hudák

Institute of Physics ^{of the} Czechoslovak Academy ^{of Sciences, Na Slovance 2, Prague 8, 180} 40, Czechoslovakia (Reçu le 24 août 1981, révisé le 20 juillet ^1982, accepté ^{le 23} septembre 1982)

Résumé.

La transition de phase ^{entre le} smectique ^C chiral uniforme et le smectique ^{C chiral modulé est étudiée} analytiquement ^en prenant ^en compte ^le couplage ferroélectrique ^et diélectrique pour ^un smectique caractérisé par ^une anisotropie négative. Dans le cadre de la théorie de Landau on a trouvé les distorsions de la structure

hélicoïdale, ^la dépendance du pas de l’hélice par rapport ^au champ électrique, ainsi que les domaines d’existence des différentes phases ^{dans le} diagramme ^{E-T. La} transition considérée est continue. On décrit l’évolution avec

le champ ^{de la} polarisation ^{et de la} susceptibilité de même que celle des parois ^{dans la} phase uniforme. Dans le spectre des fluctuations on a trouvé non seulement les branches du phason ^{et du} phonon, mais aussi une nouvelle branche d’oscillations. Pour décrire la disparition ^des parois ^{on a} proposé ^{soit le} mécanisme de déroulement de la structure hélicoïdale, ^{soit un} processus de disparition ^à l’aide de défauts linéaires.

Abstract

A detailed analytic description ^{of the} phase transition in an electric field between the modulated smectic C chiral phase and the uniform smectic C chiral phase ^is given taking ^{into account} both ferroelectric and dielectric coupling ^{to the field in the case of} negative anisotropy. Within the Landau theory ^of phase transitions we

find helix distortions, ^and establish the field dependence ^{of the} pitch ^{and the} phase boundaries in the E-T phase diagram. ^The transition considered here is continuous. The evolution of the polarization ^and susceptibility ^{with the}

field is described as well as domain walls in the uniform phase. The fluctuation spectrum ^{for some values of the field} also contains, besides the usual phason ^and phonon branches, ^{a new} oscillatory ^{branch. The} disappearance ^{of the}

walls occurs through unwinding ^{in our model but we also} suggest ^a possible ^mechanism through ^{line defects.}

J. Physique ⁴⁴ (1983)57::66 ^JANVIER 1983, ¹

Classification

Physics ^Abstracts

61. 30G

1. Introduction.

Ferroelectric liquid crystalline phases ^are interesting physical systems ^{which have} been intensively ^studied [ 1 -3] during the few last years.

In a smectic C liquid crystal rod-like molecules are

arranged ⁱⁿ parallel layers, within which they ^have

the character of a two-dimensional liquid. ^{The macro-} scopic ^local symmetry corresponding ^{to the mono-}

clinic surroundings consists of symmetry operations belonging ^{to the} C2h point ^{group. A tilt} angle ⁰

between the molecular director and the layer ^normal

is nonzero and is the same within all layers.

If the molecules ^are chiral, ^{the mirror} plane ^{and the}

inversion centre are absent, ^{the axis} C2 ^{is the} only symmetry operation. ^{In this case} the smectic C phase

is called the chiral smectic C phase (SmC*). ^The

essential property ^{of the} ground ^{state of the SmC*}

phase is that it is modulated in the layer ^normal

direction in the form of a helix [2]. ^{This fact corres-} ponds with the presence of the Lifshitz invariant in the Landau free _energy expansion ^{in the case of the} SmA*(0 ⁼ 0)-SmC* phase transition [3].

Symmetry arguments ^{lead to} the conclusion [2, 4],

that liquid crystals composed of chiral molecules have a spontaneous polarization in the SmC* phase

as experimentally observed in [2, 5-7]. ^{The dielectric}

behaviour of these materials _possesses a number of

properties specific ^to ferroelectrics [3, 8]. ^{The macro-} scopic polarization in the SmC* phase ^{is zero. This}

is due to rotation of the nonzero in-layer polarization

vector in the direction normal to the layers. ^{A nonzero}

bulk polarization may be induced by applying ^an

external electric field parallel ^{to the} layers. ^{The mole-}

cular tilt direction becomes uniform in _space and the modulated structure disappears ⁱⁿ strong enough

fields [2, 5].

There are two mechanisms in the influence of the electric field on the SmC* phase [9], namely ^{the linear}

ferroelectric coupling between the molecular dipole

moments and the field, ^{and the} dielectric coupling

which is quadratic in the field. Neglecting ^{one of these}

two coupling mechanisms then the other one can be described and treated in a similar _way as in the case

of the influence of the magnetic ^{field on the} cholesteric

phase [10]. Thus, ^{the case} of ferroelectric coupling ^is

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198300440105700

studied in [2, 11-13] ^{and the case of the dielectric} coupling ⁱⁿ [1, 14, 15]. ^{The first} attempts ^{to take into}

account both types ^of coupling simultaneously ^were

made in [16, 17]. ^{The basic} equation describing ^the

structure of the SmC* phase ^{in an} applied ^{field is} numerically integrated ⁱⁿ [16]. ^{It is} found that the

pitch of the helix increases its value with the increasing

value of the field. Above some critical value of the

field, E, ,, ^{the helix} disappears. ^The explicit analytic

calculations of E, ^can be found in both papers [ 16, 17].

For special ^{values of the material constants it is} possible ^{to find also an} explicit analytic description

of the helix pitch dependence ^{on the field} [16]. ^{In the}

case of positive dielectric anisotropy ^{it was shown}

in [ 161 that domains with two different spatial ^orienta-

tions of the molecular tilt directions occur in the smectic C* phase ^{with the unwinded helical texture.}

It is interesting ^{that these domains can be} joined by

domain walls of two different types [16].

Due to lack of the complete explicit analytical description of the helix structure for fields near the critical value Ec ^{we decided to} perform ^{such a} study.

We chose the case of the negative dielectric aniso- tropy, because the case of the positive ^dielectric

anisotropy ^was partially ^treated analytically ⁱⁿ [16].

Our calculations thus complement ^{those in} [16] ^and [1 7].

The _paper is organized ^{in the} following way. The basic equations and formulas for description ^{of the}

transition are in the section 2. Our main results are in sections 3, ^{4 and 5. The} domain wall state in the uni- form SmC* as well as in the modulated SmC* phase ^is

described in section 3. The static characteristics of the transition are found in section 4. The dynamics ^{of the}

modulated SmC* phase is described in section 5.

In the last section a discussion of the various aspects of the transition treated here _may be found.

2. Basic equations.

The Landau free energy

expansion describing ^the phase transition between the modulated SmC* phase (this phase ^{we denote} by SmC* following ^{the notation} introduced in [3]) ^and

the uniform SmC* phase (denoted by SmC* in the

following, ^{as in} [3]) ^{is exactly that} expansion ^which

is used in the description of the transition SmA*-

SmC* in [11, 18]. The free energy density expansion

has the form :

where fA ^{is that} part of the free _energy density ^which

is not connected with the transition, ç 1 = nz nx,

Ç2 = nz ny ^{are two} components ^{of the} primary ^two-

dimensional order parameter describing ^{the transi-}

tion SmA*-SmC*, ^{n =} (n.,, _ny, nz) ^{is the director}

vector (n)2 = l, k 3 3 ^{is an} elastic constant, ^{a =}

(T - To). ^ao is the usual temperature dependent

constant, ao > 0, To > 0, ^b > 0, xo > 0 are tempe-

rature independent constants,, pf > 0 and yp > 0

are flexoelectric and piezoelectric coupling constants,

E,, >, ^{0 is the} x-component ^{of the} external electric field vector E ⁼ (Ex, ^0, 0), ^{z is a} coordinate in the direction normal to the layers, x and y are ^{two coordi-}

nates within the plane parallel ^to layers, the coordi- nate system (x, y, z) ^is orthogonal, Px ^and Py ^{are two}

components ^{of the} secondary ^order parameter : ^the

in-layer ^electric polarization ^{vector P =} (Px, Py, ^0),

ea is the dielectric anisotropy constant, Ba ⁼ (8j) ^{jj I}

^{8 1 )} (see [1]), ^{this constant is assumed to be} only negative

in the following.

We assume that the order parameters Çl’ Ç2’ Px

and Py depend only ^{on the} coordinate z. This _assump- tion follows from the fact that the modulation in the

z-direction is preferred by ^{the Lifshitz term}

SI ^{dz - e;2} -UZ-) dl ^{in (1). It} is convenient to eliminate

( ^{1 dZ 2} z )

the polarization ^vector components Px ^and Py ^{from (1)} using ^the Lagrange-Euler equations ^{for them :}

Moreover it is convenient to use spherical coordinates in the director vector space :

where the tilt angle 0 takes values from the interval

( 0, n > and the tilt angle direction within the layers ^is

determined by ^the phase angle lp.

59

It is also convenient to use the constant tilt angle ⁰ approximation, ^{0 = constant. This} approximation ^is asymptotically correct at the transition point E

Ee

because the crucial term in the free _energy that favours

a modulated ordering, the Lifshitz term, involves the derivative of the phase angle (fJ. In analogy ^{to the}

similar problems in incommensurate-commensurate transitions (see ⁱⁿ [19]), this form of the Lifshitz term

suggests that close to the transition to the uniform

phase, ^the ground ^{state of the modulated} phase ^will

differ from that of the uniform phase principally through modulation of the phase angle ^{(p. Thus the}

value of the constant tilt 0

constant near the tran- sition point ^{is in the} following ^{assumed to} be ^identical

with that corresponding ^{to the} uniform SmC* phase :

where a = & - xo. p§ ⁼ ao(T - To), To ⁼ To ⁺

Xo. J.lp . ^The dependence (4) of the tilt angle ^{0 on the}

ao

electric field Ex is restricted to those fields and tempe-

ratures for which :

The final result for the Landau free _energy density expansion (1) ^{is :}

where f’ ^{is that} part ^of f ^{which is not related to the}

transition at E ⁼ E. and where :

The Landau free _energy density expansion (6) may be transformed to the form_used ⁱⁿ [16, 17] using (7).

Note that the uniform SmC* phase ^is characterized by :

The modulated SmC* phase is described by ^{a stable}

modulated solution of the Lagrange-Euler equation

for the variable 11 :

The same equation ^{enables us} to describe ^{a state with}

domain walls in the uniform SmC* phase.

3. The domain wall states in the SmC* _phase.

The order parameter ’1, describing ^{the state of the}

system, ^{is a} phase ^{variable. Thus,} in accordance with the topological theory of defects in ordered media [20],

we expect ^{that domain walls} exist in the uniform

SmC* phase. They ^{are described as solutions of the}

double sine-Gordon equation (9) satisfying ^{the boun-.}

dary conditions :

Such solutions were found in [21, 22] :

11 ⁼ 2 arctg [exp((2 ⁺ À)I/2. (r - ro) + 4)] ⁺

+ 2 arctg exp((2 ⁺ À)I/2.-(r - ro) " A)] ⁽¹¹⁾ where ro . is ^an arbitrary number and the quantity ^A

is defined by ^the equation :

The configuration (11) ^{is stable} against ^small pertur- bations, ^see [22]. ^{The free} energy of this configuration

may be easily ^found using the results of [22] ^and

equations (11), (7) ^and (6) :

Here F’ is the free _energy of the uniform SmC* phase.

It is evident from (13) that the modulated configura- tion_(ll) ^{becomes more} favourable than the uniform SmC* state q ^{= 0} (mod. ² yc) for those values of the field Ex, ^of temperature ^T and of the material cons-

tants, for which 6 > ðe(Å). ^{In fact the} equality

6 ⁼ 6c(A) defines ^{the phase} boundaries between the uniform SmC* phase ^{and the modulated SmC*}

phase. ^{It is} easy ^to show that the critical field value Ee,

i.e. that value Ex for which 6 ⁼ ðe(Å), ^is exactly ^that

found in [ 16, 17].

Let us now discuss the form of the configuration (11).

As we already discussed elsewhere [23], ^{there are two}

different interpretations of the form of (11). ^{For those}

values of the electric field Ex for which 0 A 2, i.e. for E,,o Ex, ^{there is a domain} around the point

ro ^at which the state (11) ^{is almost} constant, r N ^n.

For smaller fields Ex Exo this domain disappears.

Thus for Ex > Exo ^{there are} three domains (0, n, ² n).

Neighbouring ^{domains are} separated by the n-kinks.

There are two n-kinks in the state (11) ^{in this case.}

On the other hand for smaller fields Ex Exo ^there

are only ^{two domains} (0 ^{and 2} 7r) separated by ^a single

2 n-kink. The origin ^{of this} phenomenon (two ^n-walls

become coupled ^{into a} single ² n-wall) is in the exis- tence of a local minimum of the nonlinear potential

in (6) ^for Ex > Exo at q ^{= n.} This minimum disappear

for smaller fields, Ex Exo, ^In this later case the ferro- electric coupling ^{term -} Å(1 - ^cos q) prevails ⁱⁿ (6).

For higher fields, Ex > Exo, the dielectric coupling

becomes comparable with the ferroelectric coupling.

Mutual competition ^{of both} types ^of coupling ^now

leads to decomposition ^{of the 2} n-wall into a pair ^of n-walls, ^see figure ^1.

Note that the domains q

⁰ (mod. ² n) correspond

to the tilt angle direction q = - (7r/2). Because the electric field is oriented in the direction of positive

values of the coordinate _x, the molecular directors are

Fig. ^1.

^Two forms of the 2 n-kink solution (11) : a) ^for

fields Ex below the value Exo ^{this solution} represents ^a single ² n-domain-wall; b) ^for fields Ex ^above Exo ^this

solution represents ^a pair ^{of two} Tc-domain-walls bounded into one 2 n-kink.

oriented perpendicular ^{to the field E, see} figure ^2.

On the other hand domains il ^{= a} (mod. ² 7c) ^corres- pond ^{to the} phase angle ^qJ

⁺ (Tr/2) and the orien- tation of the director changes by ^the angle A(p

ⁿ

with respect ^{to the} previous ^case.

The polarization ^{vector P, see} (2), has, in the uniform SmC* phase, ^the components :

The state (11) has in the case Ex > Exo ^{within the} domain I ^{= n} different from (14) ^the polarization

components :

Dynamic properties ^{of the} state(ll) may be obtained from the Landau-Khalatnikov equation for the order parameter [24]. ^{From the} free energy density expan- sion (6) ^{and from} (11) ^{we obtain} with q ⁼ (i i) + ð r

and linearizing ⁱⁿ 6 il :

Assuming ^that the solution of (16) ^{has the form}

=

6 q(r). exp( - tIT) ^we obtain the eigenvalue pro- blem :

Fig. ^2.

Macroscopic orientations of molecules (=)

within domains of 2 n-kink configurations : ^{a) domains}

with this position ^are realized in both cases Ex > Exo ^and Ex Exo ; b) ^{a domain with this} position is realized only

in the cases Ex > Exo,

61

This problem ^was discussed in [23]. ^The potential

Acos ?1(11) ^{+ 2 cos 2} ’1(11) ^seen by the excitation bil

is too complicated ^{and the} problem (17) probably

cannot be solved explicitly. Nevertheless some pro-

perties ^{of solutions of} (17) ^{are known} [23]. ^{It is} easy to find the Goldstone mode :

This mode corresponds ^to the broken translational symmetry ^{of the} system by ^the presence of the loca- lized state (11). The relaxation time for phonon-like

excitations with the asymptotic ^behaviour

where k is given in the reduced units (7). ^From (19)

it follows that the relaxation time is finite for long- wavelength excitations of the phonon type. ^{It was} established by various methods see [23], that there exist another mode besides the Goldstone mode and phonon-like modes. The relaxation time _T, of this mode is, from the discrete part ^{of the} frequency spec- trum :

The whole _range of Df(À.) dependence has the form

qualitatively ^{shown on} figure ^{3. The} interpretation ^of

the excitation with relaxation time T, ^{is based on the}

fact that the configuration (11) represents ^{two n-walls} bounded together. ^{When the centres of these two}

n-walls are symmetrically shifted from their equili-

i positions (given by (11)) ^{then a} restoring ^force

les active, ^{the walls are forced to return back}

-

The relaxation frequency 1/(T,) dependence ^on

,

see (20).

to the starting equilibrium positions. The relaxation

time T1 corresponds exactly ^to this motion.

4. The SmC*-SmC* transition.

The transition bet-

ween the uniform SmC* _phase and the modulated

SmC* phase ^was numerically described in [16].

Unfortunately ^{numerical results} give ^an uncomplete description ^of the transition within the region ^{of the}

critical behaviour of the system. ^{It is} always ^conve-

nient to investigate ^an analytical description ^{of the} phase transformation. The aim of this section is to find the critical dependences of the various quantities

’ characterizing ^our system.

We start our analysis with the free _energy density expansion (6). ^The ground ^state describing ^{the SmC*}

or SmC* phase is that solution of the double sine- Gordon equation (9) which minimizes absolutely ^the

free _energy. Any solution q ^{of the} equation (9), ^which monotonously increases its value with increasing r, has the form :

where _ro is an arbitrary constant, a ^{is an} integration

constant such that

am is the amplitude elliptic function, ^{see in} [25], ^with

the modulus q2 ^{= 1}

(B2/A 2) ^{where the constants}

A and B are defined by :

.

The form of the solution (21) for q ^near the value 1 and for two different cases is shown on the figure ^4.

Fig. ^4.

Two different forms of the SmC* phase ground

state : a) ^for Ex > Exp ^{and b) for} Ex E.,,,. ^{The second}

case is realized only ^if Ee > Exo, ^{see section 6.}

We see that in both cases, Ex Exo ^and Exo Ex,

the state (21) ^{is a} periodic array of domain walls (11).

In fact, for q

^{1 we} obtain from (21) ^{the state} (11) exactly. ^The periodicity length ^L of the modulated state (21) ^{is that} length ^which equals ^to the distance between the centres of two neighbouring ^{domains 0} (mod. ² n) ^{and 2 n} (mod. ² n) :

Here K(q) ^{is the} complete elliptic integral of the first

kind, ^{see in} [25]. The second equation ⁱⁿ (23) repre- sents a one-to-one relation between the length ^{L and}

the integration constant a for all a !! 0. The value of the constant a indexes all possible ^states (21).

From the condition of the thermodynamic stability

of the Landau free energy ^we obtain which of all the

possible ^states (21) is realized for a given temperature and field. The equilibrium value of the constant a is that which minimizes the averaged free energy den-

sity f ^defined by ^the equation (6) and the relation :

Finding ^the equilibrium ^{value a we can} then find the

equilibrium ^{value L of the} periodicity length ^L

from (23). ^In fact, ^the equilibrium ^{value L is identical}

with the pitch ^{of the modulated structure.}

The averaged free energy density 7 of the modulated state (21) with a >_ 0, ^an arbitrary ^{number, has the}

form :

where E(q) ^{is the} complete elliptic integral ^{of the}

second kind, Z(s, q) is the Jacobian Zeta-function

with sin2 E --_ ( 1 - B 2). q- 2. ^The ground ^state ’of

the uniform SmC* phase ^{has its} averaged free energy

density exactly equal ^{to the value} f ’ ⁱⁿ (25).

To find which of two phases is realized for a given Ex ^{and T we must compare} f ^and f ’ using (25).

The difference Af =- ( f - f ’) ^is negative ^{for all} positive values of the constant a and becomes zero

for a ⁼ 0. Thus the value a ⁼ 0 (q ⁼ 1) corresponds

to the transition point between the modulated and uniform states. We can then expand (25) ⁱⁿ powers of the small quantity a ? 0 and find the difference åf

near the transition point ^{in a more} convenient form for the interpretation purposes. Moreover it is conve-

nient to write this expansion ^{in terms of the} periodicity length ^L using (23). ^From equation (23) ^{we see that}

for a -+ 0 (from above) ^the length ^{L -+ oo. The} averag- ed free energy density expansion ^{in powers of small} quantities has the form :

where the constant ðc(À.) ^{is defined in} (13). ^The region

of validity ^{of the} expansion (26) is confined to those values of the length ^L for which the inequality :

holds. This inequality ^can be satisfied for _every value of the field Ex ^{if L is large} enough.

We can interpret ^the equation (26) ^{in terms of}

domain walls in a similar _way as in the theory ^of

incommensurate-commensurate phase transitions in

improper ferroelectrics of the K2SeO 4 ^and (NH4)2BeF4 types, ^{see in} [26]. Noting ^that (2 nIL) ^is proportional

to the number of regions in the z-direction in which the configuration of the order parameter is almost the

same as in the state (11), ^we may interpret the expan- sion (26) ^{in the} following ^{way. The} coefficient of (2 n/ L)

is the _energy required ^{to create a 2 n-kink state} (M).

Such kinks are distributed periodically in the z-direc-

tion, ^with separation ^{L between} neighbouring ^kinks.

The interaction of the neighbouring ² n-kinks of the type (11) ^is given by ^{the last term in} (26). ^{This term} represents ^a repulsive interaction potential : by ^increas- ing the distance L between neighbouring ^{2 n-kinks,}

this term decreases. Note that the form of 2 n-kinks was

discussed in the previous section. The term « 2 n-

kink » is equivalent ^{to the term « 2 n-domain wall » in}

the case Ex __ Exo only. ^{In the case} Ex > Exo ^{this term}

is equivalent ^{with the term o} pair ^{of two} distinguishable

n-walls ».

_

The equilibrium ^value L of the periodicity length

and the equilibrium ^{value of the} averaged free energy

density ^{are :}

63

where the critical value Ee is defined by ^the equality 6(E,) = 6,(E,) and, in fact, it ^was explicitly ^calculated

in [J6, 17]. ^{For the} fields ^{above the critical value we}

obtain the uniform SmC* phase ^{values :}

The width s of the region ⁱⁿ the z-direction where the modulated state (21) ^with a(L) given by (29) ^and (23) appreciably changes is defined by the coefficient of ^z in (21) :

This width is of the order of the domain ^{wall width} (n-walls ^{or 2} n-walls). ^Near the transition to the uni-

form ^{phase it is} negligible ^with respect ^to the distance

L - aJ ^between two neighbouring ^{2 n-kinks. This}

fact enables ^{us to} consider the regions ^{where the} phase angle ^qJ appreciably changes ⁱⁿ analogy ^with

domain walls and the regions ^{where the} configuration

is almost homogeneous ⁱⁿ analogy ^{with domains.}

Far from the transition boundaries between the modulated and uniform phases the values of the width s and the periodicity length L ^become comparable.

In this case it is more appropriate ^to speak ^{about the}

helix structure instead of the domain wall structure of the modulated phase.

The polarization components Px ^and Py ^{may be}

calculated from (2), (7) ^and (21). ^Their averaged ^values

are given by (for Ex E,,) :

where the constant ç is defined by sin2 ç ⁼ (A 2 - 1).(A 2 - BZ)-1. ^From (32) ^{we calculate} the dielectric.sus-

ceptibility of the modulated phase. Near the transition point ^{we obtain :}

for 6 -+ 6,(A) (i.e. E.,, -+ E,,,) from above (resp. below),

where C and C’ are constants depending ^{on the}

material constants. The singular ^{term in} (33) represents the most singular part ⁱⁿ xxx. From (33) it is clear that

as the field value Ex ^{increases to} the critical value E,,

the susceptibility xxx diverges. ^The divergence ^{is of}

the Curie-Weiss type ^{with the} logarithmic correction.

The susceptibility Xxx of the uniform SmC* phase

remains constant at the transition point :

For those fields and materials, for which the relation (4)

holds near the transition point, ^{we obtain :}

which is a finite number at the transition point

JOURNAL DE PHYSIQUE.

T. 44, No 1, ^{JANVIER 1983}

The fact that the susceptibility xxx diverges going

from the modulated phase SmC* ^{to the} uniform SmC*

phase while it remains a finite number going ^{in the} opposite ^{direction to the} transition point is connected with the peculiar character of the transition under consideration. This peculiarity ^{is of the same} type ^as that considered by ^{de Gennes} [1], ^{in the case of the}

cholesterics in external fields. So we will not consider further details of the transition but give only ^the following remarks. It is easy ^to show that the uniform

SmC* _phase is a stable phase ^{on both sides} of ^the

transition point Ex ⁼ E,,. ^{This statement is} equivalent

with our result that the susceptibility Xxx (34) ^{of the}

uniform phase ^{does not} possess any singularity ^{at the}

critical point On the other hand the modulated phase

becomes unstable for fields Ex > E,, and the diver-

gence of the susceptibility xxx of this SmC* phase ^is

intimately ^connected with this instability. ^{The ins-}

tability ^follows here from the fact that the modulated

state (21) develops ^{into an unstable state when the}

integration ^{constant a becomes} negative (i.e. Ex

becomes higher ^than Ec) : ^the eigenvalue problem (17)

with q ^{from (21 ) instead ?1(11) has} negative eigenvalues Q2 0 for those fluctuations ð’1 which drive the state (21) ^{with a 0 to the uniform} state ^{= 0}

or to the state (11)..

The peculiar behaviour of the system ^{around the} transition point Ex = E,, should be experimentally

observable. In fact, ^{it was} probably ^{observed. In} experiments by Hoffmann et al. [8], ^{it was found that}

the conoscopic figure ^{shifted as the field} strength

increased to a value E,,,, ^the figure became that of a

biaxial crystal pointing ^to complete unwinding ^{of the}

helix. As the field strength decreased, ^the figure ^{of the}

uniaxial crystal reappeared ^{at a} field about 20 %

weaker than Ec. This observation is consistent with the above predicted peculiar character of the transi- tion at Ec’ Note that the uniform SmC* _{phase may}

be shown to become unstable at fields below the value

Ec. This follows from two facts : ¹⁾ for fields slightly below Ec the uniform SmC* _phase is stable and 2)

at the zero field Ex ^{= 0} the uniform phase qJ ⁼ -(n/2)

is not a stable phase : ^{there is no} preferred position

of molecules for Ex ^{= 0,} their molecular tilt angles change ^{from a} layer ^{to the} following layer according

to (p ⁼ (z - zo) qo, where _zo is an arbitrary ^constant

and _qo is the zero field helix wavevector. It is then clear that the uniform phase ^{has to lose its} stability

for field values below Ec ^{but above} Ex ^{= 0.}

5. Dynamics ^{of the} SmC* phase ^{near the transition}

at Be.

^To describe the dynamics ^{of the} system ^near the transition point ^{we use the} Landau-Khalatnikov

type ^of equation, ^{as done in} [27-31] ⁱⁿ describing ^the

dielectric relaxation in ferroelectric liquid crystals.

The form of the dynamic equation ^is given by (16)

with ’1(11) replaced by q ^from equation (21). Using

the same procedure ^{as the} step ^from equation (16)

to equation (17), ^{we obtain an} eigenvalue problem.

This problem ^is formally identical with the eigenvalue problem ^treated by ^us [32], ⁱⁿ connection with the incommensurate phase ⁱⁿ K2SeO4 ⁱⁿ electric fields.

Our calculations are of approximative ^{nature and}

follow the procedure developed ⁱⁿ [33] ^for description

of the fluctuation spectrum in incommensurate phases.

Thus, translating the results of the _paper [32] ^into

the language ^of purely relaxational dynamics ^of

ferroelectric liquid crystals ^we obtain the following

results. There are three branches of the relaxational

frequencies corresponding ^to three various types ^of motion of the modulated structure SmC. for fields

slightly below the critical field, Ex Ec. ^{The modulat-}

ed structure within this field region ^{consists of a} periodic array of 2 n-kinks of the type (11) separated by the distance L, ^see section 4. The kinks interact Due to this interaction the motion of _every single

2 n-kink becomes correlated with the motion of

neighbouring ^kinks. Consequently, three branches of relaxation frequencies ^{are found} corresponding ^with

the phonon-like excitations and with two modes _iso

and T, ^{which were} found in the case of the single ^{2 a-}

kink dynamics, ^see section 3. The explicit form of the relaxation frequency Ti(k) dependence ^{on the wave-}

vector k of the excitation of the i-th type (i ^{= 0.1 and} phonon-like) ^is given by :

and

The figure ^{5 shows} qualitatively ^{the behaviour of the}

relaxation frequencies.

We see that longwavelength excitations of the domain wall lattice are relaxed within _very large ^time

intervals if they ^{are of the i = 0} type. This fact follows from the observation that the Goldstone mode be-

longs ^to this branch.

While the two lowest branches on the figure ⁵

describe the relaxation frequencies ^of the excitations of the domain-wall lattice, ^the highest branch des- cribes the relaxation of the excitations within domains.

This fact becomes clear considering figure ^{5 which}

shows the dependence ^of relaxation frequencies ^{on the}

Fig. ^5.

Relaxation frequencies ^{of the} domain wall lattice excitations (1/(il) ^and 1/(T,)) ^{and of the} phonon-like ^excita-

tions 1/(ipb.) within the extended zone scheme.

65

wavevector k for normal excitations of the modulated

structure. ^{If this} wavevector I k I is in the interval

(0, 1t/L) ^then the corresponding wavelength ^{is in}

the interval (L, oo). These normal modes describe oscillations of the centres of 2 n-kinks around their

equilibrium positions. The normal modes with their

wavelength from the interval (Lfi L) ^{describe oscil-}

lations of the centres of two n-walls bounded to single

2 n-kinks in the whole domain wall lattice. The last type ^{of normal} modes, phonon-like modes, ^{are those}

excitations which are bounded within the region ^of

domains.

_

The dynamics of the uniform SmC* _phase is described by ^a single relaxation frequency ^branch.

Its form _may be found using (16) ^and (17) ^with _{il(l 1)} replaced by il ^{= 0. We obtain :}

Comparing (35) ^and (36) ^{we see that} three branches of the relaxational frequencies ^{of the} SmC* ^phase develop ^{into a} single branch of the SmC* phase.

From (35) it is clear that the branches Tõ l(k) ^and

’tl1(k) disappear ^at E, ^{and the} phonon-like ^branch T ’ ^develops into the branch (36).

6. Discussion.

The SmC* phase ^{has a helix}

texture in zero field For small fields 0 E E,

the pitch of the helix is not affected but its form is

distorted Near the transition to the uniform SmC*

phase the modulated SmC* phase ^{consists of almost}

uniform regions ^{of molecular} ordering, ^{which are} separated by domain walls. For fields with their value within the region Ex Exo ^{these walls are of the 2 n-} type ^and neighbouring domains differ by the rotation of the phase angle ^{of the director} by ^{2 n.} It however,

the critical field Ec ^is higher than the value Exo,

i.e. if Ec > Exo, ^{then the} neighbouring domains of the uniform molecular ordering ^differ by ^the angle n

and they ^are separated by domain walls of the Tc-type.

It is possible ^to show that the length ^{of domains}

’1 ⁼ 0 (mod ² 7c) ⁱⁿ any ^case always ^{increases to} infinity ^{as the field} Ex ^{increases to} E,. On the other hand the length ^of domains ’1 ^{= n} (mod ² a), ^if they

are realized, ^remains always finite, ^{even at} Ex ⁼ E,.

Above the critical value of the field the uniform phase

’1 ⁼ 0 (mod ² n) is realized as the ground ^{state. The}

character of the transition is peculiar ^{and it was discus-}

sed in previous sections. The form of the 2 n-kink (11)

in the uniform phase depends ^on whether Ec > Exo

or E, Exo. ^{In the first case there are} only ^{2 n-walls} above E,. ^In the second case for fields Exo >__ E, > Ec

the 2 n-wall configuration may be realized but above

Exo, ^{i.e. for} Ex >__ Exo > E,, ^this configuration splits

into two distinguishable ^n-walls.

The modulated SmC* ^{phase near} the transition to the uniform SmC* _phase represents ^an equilibrium

state between two processes, the creation of 2 ^n- kinks and their mutual repulsion, ^see (26). ^{The tran-}

sition point corresponds ^to the field at which it is unfavourable to create 2 n-kinks because theiiforma- tion energy becomes nonnegative ^and repulsive ^inter-

wall interactions then lead to disappearance ^{of walls}

from crystal samples. ^This process consists of the motion of walls out of the sample, the modulated structure unwinds. The unwinding is, however, ^an unphysical ^{process in} large samples ^{as the} velocity

of molecular rotation around the local z-axis may be-

come unrealistically large. Nevertheless, ^our theory

enables us to find another _process of wall disappearan-

ce. Let us assume that closed or open line defects

lying within the layer identical with the centre of the domain wall _may exist These line defects separate the configuration of molecules corresponding ^{to the}

centre of the wall from that corresponding ^{to the}

uniform phase. ^{Let S} be the total surface of the layer

enclosed by the line defect and corresponding ^with

the centre of the wall. The total free _energy of a

single ² n-kink with respect ^{to the uniform} phase ^near

the transition to the uniform phase ^{has the} form,

see (13) :

Thus the three-dimensional character of crystals ^was

taken here into account The repulsive ^inter-wall

energy is neglected ⁱⁿ (37) assuming ^{that we are almost}

at the transition point ^{The last term in} (37) represents the contribution by ^{the line defect with its} length ^/

and the length density ^of the creation defect free energy F _linedef. to the total free energy: Note that for closed line defects S = 11. Within the modulated phase ^we

have 6,(A) 6 and thus the maximal possible ^sur-

face S is realized. The term 6,r (A) - 6 changes ^its sign

at the transition point Thus, ^{it becomes} preferable

for the wall to disappear ^{as a whole} shrinking ^{of its}

surface S to zero with the aid of line defects.

The results of this _paper are in agreement ^{with the}

experimental ^data from papers cited in the introduc- tion. Nevertheless further quantitative experiments

are necessary ^to verify explicit analytical ^formulae

found in this _paper. We hope ^{that our} results will stimulate such quantitative measurements.

Acknowledgments.

The author is indebted to

Dr. ^V. Janovec, ^{Dr. M.} Glogarova ^{and to Dr. V.}

Dvorak for stimulating discussions.

References

[1] ^DE GENNES, ^P. G., ^The Physics of Liquid Crystals (Clarendon Press, Oxford) ^1974.

[2] MEYER, ^R. B., LITBERT, L., STRZELECKI, ^{L. and} KELLER, P., ^J. Physique-Lett. ³⁶ (1975) ^L-69.

[3] PIKIN, ^S. A., Structural Transformations ⁱⁿ Liquid Crystals (Nauka, Moscow) 1981, in Russian.

[4] MICHELSON, A., CABIB, ^{D. and} BENGUIGUI, L., J.

Physique ³⁸ (1977) ^961.

[5] OSTROVSKU, ^B. I., RABINOVICH, ^A. Z., SONIN, ^A. S., STRUKOV, ^{B. A. and} CHERNOVA, ^N. I., ^Pis’ma

Zh. Eksp. Teor. Fiz. 25 (1977) ^80.

[6] MARTINOT-LAGARDE, Ph., ^J. Physique-Lett. ³⁸ (1977)

L-17.

[7] NERANSKI, P., GUYON, E., KELLER, P., ^J. Physique

36 (1975) ^1005.

[8] HOFFMAN, J., KUCZY0144SKI, ^{W. and} MALECKI, J., Mol. Cryst. Liq. Cryst. ⁴⁴ (1978) ^287.

[9] DURAND, G., MARTINOT-LAGARDE, Ph., Ferroelectrics 24 (1980) ^89.

[10] ^DE GENNES, ^P. G., Solid State Commun. 6 (1968) ^163.

[11] PIKIN, ^{S. A. and} INDEBOM, ^V. L., ^{UFN 125} (1978) ^251.

[12] MICHELSON, A., BENGUIGUI, L. and CABIB, D., Phys.

Rev. A 16 (1977) ^394.

[13] OSTROVSKIJ, ^B. I., PIKIN, ^{S. A. and} CHIGRINOV, ^V. G., Zh. Eksp. Teor. Fiz. 77 (1979) ^1615.

[14] TRIMPER, S., Phys. ^{Lett. A 82} (1981) ^131.

[15] BLINOV, ^L. M., Electro- and magnetooptics of liquid crystals (Nauka, Moscow) ^1978.

[16] DMITRIENKO, ^{V. E. and} BELYAKOV, ^V. A., ^Zh. Eksp.

Teor. Fiz. 78 (1980) ^1569.

[17] MARTINOT-LAGARDE, Ph., ^Mol. Cryst. Liq. Cryst. ⁶⁶ (1981) 61.

[18] BLINC, R., ^{DE SA} BARRETO, ^F. C., Phys. Status Solidi b 87 (1978) ^K-105.

[19] ^{BRUCE, A. D.,} COWLEY, ^{R. A. and} MURRAY, ^A. F., J. Phys. ^{C 11} (1978) ^3591.

[20] MERMIN, ^{N. D., Rev. Mod.} Phys. ⁵¹ (1979) ^591.

[21] BULLOUGH, ^{R. K. and} CAUDREY, P., ⁱⁿ Solitons, Topics ^{in Current} Physics, Vol.17, ^{eds. R. K. Bul-} lough ^{and P.} Caudrey (Springer, Heidelberg) 1980, p. 107.

[22] HUDÁK, O. , Phys. ^{Lett. A 82} (1981) ^95.

[23] ^HUDÁK, O., Phys. ^{Lett. A 86} (1981) ^208.

[24] LIFSHITZ, ^{E. M. and} PITAEVSKII, ^L. P., Physical ^Kinetics (Nauka, Moscow) 1979, in Russian.

[25] ABRAMOWITZ, ^{M. and} STEGUN, ^I. A., ^Handbook of

Mathematical Functions (National Bureau of Stan-

dards, U.S.A.) ^1964.

[26] DVO0159áK, V., ⁱⁿ Proceedings of ^the Karpacz ^Winter School of Theoretical Physics, ^{eds. A. Pekalski}

and J. Przystawa (Springer, Heidelberg) ^1979.

[27] ^{BLINC, R. and} 017DEK0161, B., Phys. ^{Rev. A 18} (1978) ^740.

[28] ^LEVSTIK, A., 017DEK0161, B., LEVSTIK, I., BLINC, ^{R. and} FILIPIC, C., J. Physique-Colloq. ⁴⁰ (1979) ^C3-303.

[29] 017DEK0161, B., LEVSTIK, ^{A. and} BLINC, R., J. Physique- Colloq. ⁴⁰ (1979) ^C3-409.

[30] MARTINOT-LAGARDE, ^{Ph. and} DURAND, G., ^J. Phy- sique-Lett. ⁴¹ (1980) ^L-43.

[31] MARTINOT-LAGARDE, ^{Ph. and} DURAND, G., J. Phy- sique ⁴² (1981) ^269.

[32] HUDáK, O., ^{to be} published ^{in J.} Phys. ^{C : Solid}

State Phys.

[33] BRUCE, ^{A. D. and} COWLEY, ^R. A., ^J. Phys. ^{C 11} (1978)

3609.