Relaxation modes of the fluctuations of the order parameter in the vicinity of the uniaxial-to-biaxial nematic phase transition

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Relaxation modes of the fluctuations of the order parameter in the vicinity of the uniaxial-to-biaxial

nematic phase transition

Y. Galerne

To cite this version:

Y. Galerne. Relaxation modes of the fluctuations of the order parameter in the vicinity of the uniaxial-to-biaxial nematic phase transition. Journal de Physique, 1986, 47 (12), pp.2105-2115.

�10.1051/jphys:0198600470120210500�. �jpa-00210404�

Relaxation modes of the fluctuations of the order parameter ^{in the} vicinity

of the uniaxial-to-biaxial nematic phase ^transition

Y. Galerne

Laboratoire de Physique ^{des Solides, Bât.} 510, Université de Paris-Sud, ⁹¹⁴⁰⁵ Orsay Cedex, ^France

Résumé. 2014 Une analyse classique des modes de relaxation des fluctuations du paramètre d’ordre, ^est développée ^au voisinage ^{de la} transition de phase nématique uniaxe-nématique ^{biaxe. En} employant ^des

coordonnées polaires, ^on peut suivre les 5 modes de relaxation du paramètre ^{d’ordre à travers la} transition de

phase nématique uniaxe-nématique biaxe : 2 de ces modes sont des modes d’amplitude, ^{et les 3} autres sont de type orientationnel. On montre que les modes d’amplitude ^en phase biaxe, mélangent ^les composantes ^de biaxialité et de biréfringence ^du paramètre d’ordre, ^{le mode} critique (quasi-biaxe) présentant des variations en

température ^{deux fois} plus grandes qu’en phase uniaxe conformément à la théorie classique. ^{On étudie aussi,}

sur des cas particuliers, ^les effets des courants induits sur le comportement prétransitionnel ^{des modes} orientationnels, ^en phases nématiques ^{uniaxes et biaxes.}

Abstract.

A classical analysis ^{of the} relaxation modes of the fluctuations of the order parameter ^is developed ^{in the} vicinity ^{of the uniaxial to} biaxial nematic phase transition. Using polar coordinates, ^{the 5}

relaxation modes of the order parameter ^{are followed across} the uniaxial to biaxial nematic phase transition : 2 out of these modes are amplitude ^{modes, and the 3 others are} orientational modes. It is shown that the

amplitude modes in the biaxial phase ^{mix the} components ^of biaxiality ^and birefringence of the order parameter, ^{the critical} (quasi-biaxial) ^mode presenting temperature variations twice larger ^{than in} the uniaxial

phase, according ^{to classical} theory. The backflow effets on the pretransitional behaviour of the orientational modes are also studied in the uniaxial and biaxial nematic phases.

Classification

Physics ^Abstracts

61.30

64.70M

1. Introduction

The biaxial nematic phase Nb , ^{recently discove-}

red by ^{Yu and} Saupe [1] ^{in the} lyotropic ^{mixture of} potassium laurate, decanol and H20, ^{is an interme-}

diate phase between the two uniaxial nematic phases (indifferently ^noted Nu) ^{of disk} Nd and cylindric

( Nc ) ^{types, both} well known in the lyotropic

systems [2]. ^These three different nematic phases ^of

micellar aggregates dispersed ^{in water have} naturally

stimulated experimental and theoretical studies [3], especially around the Nu-Nb phase transitions. In

particular, it has been noticed by Jacobsen and Swift

[4] ^{that the} Nu-Nb phase transition has some analo-

gies ^{with the} isotropic ^{to nematic} phase transition in two dimensions (because ^the biaxial orientational order in the 2D-plane perpendicular ^{to the uniaxial}

director n, is isotropic-like ^{in the} Nu phase, ^and

nematic-like in the Nb phase). ^The Nu-Nb phase

transition may therefore be described by ^{a reduced}

order parameter ’aJJ’ ^and the de Gennes model [5]

of the isotropic-nematic phase transition may be extended to the Nu-Nb phase transition. This allowed Jacobsen and Swift to predict ^the pretransitional

biaxial modes which have recently been observed in the Nu phase using Rayleigh scattering [6]. ^{The 2D-} analogy ^{with the} isotropic ^{to nematic} phase ^transition

is a fruitful approach ^{which can} also be used to show that the Nu-Nb phase transition may be second order

[7]. ^The argument is that the ^same 2D-nematic state

corresponds ^{to the same} orientational order parame- ter, ^{but with} opposite signs depending ^{on whether}

the director is chosen along ^{m or I} (Fig. 1). ^This

Fig. 1. - The orientational order parameter

of a two-dimensional nematic phase, keeps ^{the same} amplitude, ^but changes sign ^{when chosing the symmetry}

axis I instead of m as the origin ^axis of the tilt angle 0 ^of

the molecules (sketched ^{here as} ellipses).

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198600470120210500

shows that the free energy of the 2D-nematic phase

is an even function of the order parameter, ^and consequently ^{that the} 2D-nematic-to-isotropic phase transition, and with the analogy, ^the Nu-Nb phase transition, may be second order.

The analogy ^{with the} 2D-nematic phase ^{and the}

reduced order parameter’ a{j’ though satisfactory

for most questions, ^{have to be left when} studying ^the

relaxation modes of fluctuations of the order para- meter Qa(j in ^{the Nb phase, because it} artificially decouples ^the components of the order parameter

Qa{j. ^In this paper, after recalling ^{the basic} equations

of the nematic phases (Sect. 2), ^and briefly ^presen- ting ^the 2D-analysis ^{of the critical} modes in the

Nu phase (Sect. 3), ^we analyse ^the relaxation modes of the Nb phase ^{on the more} general basis of the order parameter Qa{j (Sect. 4). ^The analysis ^{of the} Qa{j relaxation modes is performed ^{in the} vicinity ^of

the Nb-N. phase transition, ^within the framework of classical theory, ^and using ^{the «} one-constant

approximations ^» [8] ^{on both the elastic and} viscosity

coefficients to simplify ^the problem. ^{The relaxation}

modes are divided into two categories : ^{the modes of}

the fluctuations of the amplitude of the order parameter (amplitude modes), ^and the orientational modes. Their temperature ^behaviour is examined around the Nu-Nb phase transition, respectively ⁱⁿ

sections 5 and 6. In the remaining ^{section we}

summarize the results and make ^some concluding

remarks.

2. Free energy and dynamical equations ^{of the}

nematic phases.

The order parameter ^{of a uniaxial or} biaxial nematic

phase ^{is a} symmetric, traceless, second-rank tensor

Qa{j ^{[8]. With the order state variables} (temperature

and presure), it determines the physical ^{state and} consequently, ^the free energy and the dynamical equations of the nematic phase. Diagonalizing ^these dynamical equations yields ^the hydrodynamical

modes of the order parameter ^{in the uniaxial and} biaxial nematic phases.

2.1 FREE ENERGY OF THE NEMATIC PHASE NEAR THE UNIAXIAL-BIAXIAL NEMATIC PHASE TRANSI- TION.

2.1.1 Local free energy. - Probably depending ^on

the experimental conditions, ^{the uniaxial to biaxial}

nematic phase transition is observed to follow a

classical [6, 9] ^{or a critical} [10] behaviour. In the classical regime, ^{to which we restrict our attention} here, the free energy density ^{of a} uniformly ^oriented sample may be expanded in powers of the tensorial order parameter Qa{j ^{as :}

being the invariants of Qa{j. ^{To a} first order approxi- mation, ^the coefficients a and b may be supposed ^to

have linear variations with temperature, ^{while the}

other coefficients c, d, e ^and f are ^constant [9]. ^The

free energy expression (1) is valid for both the uniaxial and biaxial nematic phases. ^{In the} Nu phase,

the invariants are linked by the relation cr 3 ^{= 6} uj,

while they just obey ^{to the} inequality ui> ⁶ o- 3 2 ⁱⁿ

the Nb phase [9].

2.1.2 Elastic energy. - then fluctuations of the order parameter ^are considered, ^{elastic terms have}

to be added to the free _{energy of} the uniform sample (Eq. (1)). ^To the lowest order, these elastic terms are [5]: [5] : "2 L, ^{Ll 1} V.QOY V.QOY VaQ{j’Y VaQ{j’Y ^and and "2 1L2 ^Lz V-Q-P VyQ-yP’ ^VaQa{j

Making the usual one-constant approximation, ^we

may group them, and write the elastic energy in the

simplified Frank form [8] :

2.2 DYNAMICAL EQUATIONS OF THE UNIAXIAL AND BIAXIAL NEMATIC PHASES.

The dynamical equa- tions of the order parameter ^come from the tensorial relation between the force conjugated ^to Q a/3 ^and

the velocity ^of change ^of Qa/3. ^To the lowest order, this is a linear relation [11]. Making again ^{a one-}

constant approximation, ^we may ^assume that this linear relation is just given by ^{a scaler, i.e.} [5] :

Am

where F ⁼ Fo ⁺ Fel is the total free energy, and v is

a viscosity coefficient. Because the expression ^of Fo ^cannot readily be reduced to quadratic ^{terms, the} diagonalisation ^of equation (3) ^{is not} straight-

forward. The uniaxial and biaxial cases have to be considered separately. ^The only ^{evident feature on}

the form of equation (3) is that the hydrodynamical

modes of the order parameter ^{in the nematic} phases

are of relaxation type [8].

3. Relaxation modes of the reduced order parameter

in the uniaxial nematic phase.

3.1 FREE ENERGY IN THE UNIAXIAL NEMATIC

PHASE. - ^As suggested by Jacobsen ^{and Swift} [4], ^it is often sufficient, ^when studying the uniaxial to biaxial nematic phase transition, ^{to use the two-}

dimensional order parameter’ a{j’ obtained from the traceless projection of the order parameter Qa{j in

the plane perpendicular ^{to the uniaxial nematic}

director n. This reduced order parameter vanishes in

the uniaxial nematic (disordered) phase. It therefore allows no to expand the local free energy in the Landau form around the transition. To a first order

approximation (valid ^for describing small biaxial fluctuations in the uniaxial nematic phase), ^this expansion ^{is limited to one term} (§ 5.3) :

a being ^a coefficient varying linearly ^with tempera-

ture and vanishing ^{at the} Nb-Nu transition (Eq. (19)).

The elastic terms of equation (2) may also be written as a function of the reduced order parameter

’a¡3. ^{For a} fluctuation of wave-vector q, the elastic energy reads :

As equations (4) ^and (5) show, the total free energy of ^a q fluctuation in the Nu phase ^{is a} quadratic function of the components of the reduced order parameter ’af3. ^From equation (3), ^{we deduce}

then that each’ aJ3-component ^{relaxes to zero}

according ^{to its own} differential equation. ^However, because’ af3 ^{is a symmetric and traceless tensor, its}

diagonal ^and off-diagonal components ^are respecti- vely ^{linked, and} thus, aJ3 ^{has only two} independent

relaxation modes in the uniaxial nematic phase ^near

the biaxial transition: the diagonal and the off-

diagonal modes. These modes, ^because they ^concern

the order parameter C,,,, ^which vanishes in the N.

phase, ^{are the two} critical modes of the N.-Nb phase

transition.

3.2 DIAGONAL MODES. - A pretransitional (or ^cri- tical) fluctuation of ’aJ3 ^{in the Nu phase} is sketched in figure 2a, ’af3 being represented by ^an ellipsis ^of non-zero ^trace. This fluctuation which belongs ^{to the} diagonal mode, is drawn in the simple ^{case where q} lays ^{in the} plane perpendicular ^{to the uniaxial}

director n. The mean-squared amplitude ^{of the} diagonal mode q is obtained by writing ^the equiparti-

Fig. ^2.

Pretransitional biaxial fluctuations of the order parameter ⁱⁿ the uniaxial phase, the uniaxial director ⁿ

being perpendicular ^{to the} figure, ^{and the} wave-vector q horizontal. The ellipses and circles represent locally ^the

order parameter’ afJ. a) Diagonal ^mode. b) Off-diagonal

mode. The arrows sketch the hydrodynamical ^{flow which} helps the order parameter fluctuation to relax back to

equilibrium.

tion of energy: (A 2) = - kB T 2’ and its relaxation

B + Kq

time constant T is given by ^the dynamical equation (3) :

The coefficient if varying proportionally ^{to the}

temperature difference to the transition (Eq. (19)),

this equation expresses the critical slowing-down ^of

the diagonal ^{mode at the} Nb-N. transition. Such a

critical behaviour, depicted ⁱⁿ figure ^{3a for two}

different wave-vectors

(qf ^{= §q) , has recently}

been observed by Lacerda ^etal. [6 .

Fig. ^3.

Relaxation frequency ^{of the} pretransitional

modes in the Nu ^{phase versus} temperature ^{for two wave-} vectors q1 (heavy line) and q2 ⁼ J2 qt. a) Diagonal

modes. b) Off-diagonal modes. Because of back-flow,

these modes ^are faster than the corresponding diagonal

modes. Let ^us note that they ^both extrapolate ^{from the}

same zero-frequency point.

3.3 OFF-DIAGONAL MODES.

In figure 2b is sket- ched a critical fluctuation of ’al3 ^{of the} off-diagonal type ^{in the} Nu phase, again ^{with q} laying ^{in the} plane perpendicular ^{to n and} using ^{the same} representation

as for the diagonal ^{mode in} figure ^{2a. The mean-} squared amplitude ^{of the} off-diagonal ^mode q is

again (A 2) = - kB T 2’ ^and its relaxation time T is

a+Kq given by :

It has to be noted that the viscQsity coefficient v’ implied in the relaxation of the off-diagonal ^mode

is different from that of the diagonal ^{mode v. The}

reason is the hydrodynamical coupling [4, 5] ^{of the} off-diagonal ^{terms of} ’aJ3 ^to the shear flow. This back-flow effect (arrows ⁱⁿ Fig. 2b) helps the relaxa- tion process and makes ^v’ smaller than ^v. Equa-

tion (7) expresses the critical slowing-down ^{of the} off-diagonal mode at the transition. This critical

behaviour, depicted ⁱⁿ figure 3b, ^{has also been}

experimentally ^verified [6].

4. Relaxation modes of the order parameter ^{in the} biaxial nematic phase ^near the uniaxial transition.

In order to get the behaviour of the relaxation modes of the order parameter in the biaxial nematic

phase, we now use the complete ^order parameter

Qaø’ ^{instead of the reduced order} parameter

which artificially decouples ^the components ^of

Qaf3 .

4.1 POLAR COORDINATES. - It is often more

convenient to use polar coordinates than Cartesian coordinates to describe the order parameter Qa/3 ⁱⁿ

the biaxial nematic phase, because this representa- tion directly explicits ^the eigen-values ^{and the} eigen-

directions of Qaf3. ^{In order to} illustrate this point, ^let

us come back again ^{to the} (symmetric traceless) ^two-

dimensional order parameter l af38 ^{Its components}

may be written in both Cartesian and polar systems :

’0 ^{and -} ’0 being ^the eigen-values of , a3’ ^{and 9 and}

0 + ; ^{the angles of its} eigen-directions ^{with the} laboratory ^axes. The fluctuations of ’a3 ^{from its}

average ^state (the eigen-directions ^{of which} defining

the reference axes) are.:

-1y -1y

This-shows that the fluctuations of the diagonal ^and off-diagonal components ^of ’a¡3 ^{in the Nb phase} respectively correspond ^to fluctuations of the polar

and angular variables, provided that the reference

axes are chosen along the average eigen-directions.

Such a use of the polar variables, however, ^is acceptable ^as long ^{as the} angular fluctuations of

03BEa3 ^{are small} ( d e 1 ) , ^which implies (when estimating d’11 - d’12) ^that dCo .r. ’0’ i.e. that the fluctuations of the amplitude of the order parameter

are small if compared ^{to the order} parameter ^itself.

If one just considers the thermally ^{excited fluctua-}

tions of ’a/3’ this condition is equivalent ^{to the} Ginzburg ^criterium [12]. It indicates that the variable

change is valid in the Nb phase ^far enough ^{from the}

uniaxial transition to follow the classical regime.

Similar considerations readily ^{extend to three}

dimensions. The six components ^{of the} symmetric

order parameter Qa¡3 ^{(only five} components ^are

independent ^because Qa¡3 ^is traceless) may be converted to the three polar and the three angular

variables (the eigen-values and the Euler angles ^of

the eigen-directions with the reference-frame of

Qa/3) under the condition that the Nb phase ^is

considered in the classical regime.

Let us now express the free energy and the

dynamical equations ^{of the} Nb phase, using the polar

and angular ^variables.

4.2 FREE ENERGY AND ELASTIC CONSTANTS IN POLAR COORDINATES. - The free energy of ^a uni- form sample, being ^{a function} of the invariants of

Qaf3 (Eq. (1)), ^{is a} function of the eigen-values ^of Qaf3 ^{only, i.e. a} function of the polar ^variables only.

The elastic energy Fel may also be written ^{as a} function of the polar ^and angular variables. It is a

quadratic function of the gradients ^{of the} polar ^and angular variables, without crossed terms because, containing angular variables, the crossed terms are not invariant in a mirror symmetry ^while Fel ^{is. In}

the two-dimensional example, the elastic energy may be written :

which illustrates the decoupling between the polar

and angular variables. From this expression, ^we

deduce the elastic constant associated to the angular

fluctuations :

while K is the elastic constant for the polar ^fluctua-

tions.

This result readily ^{extends to} three dimensions.

The reference axes being ^chosen along the average

eigen-directions of the order parameter Qafj (§ 4.1),

the gradients ^{of the} diagonal Qaa ^{and off-diago-}

nal (Qafj ^{with a # /3) elements} respectively ^corres- pond ^to polar (or amplitude) fluctuations of the order parameter, ^{and to} angular (or orientatio-

nal) fluctuations around the eigen-direction ^y ( # a, f3 ). The extension of the two-dimensional

example ^to three dimensions shows that the elastic constants of these elementary deformations are K and ky ⁼ 2 K Qaa - QfJfJ ) 2 respectively. ^The Nb phase has therefore 3 different orientational elastic constants in the ^« one-constant approximation » model, ^{instead of 12 in the} general ^case [13], ^{and its}

elastic energy expresses ^{as :}

1 1

Let us recall that this result is valid for small fluctuations only. This condition is also needed for the rotations around the three orthogonal eigen-

directions to commute in equation (2).

At the uniaxial-to-biaxial nematic phase transition,

two eigen-values becoming equal, ^{the two corres-} ponding orientational elastic constants K become

equal, while the third one vanishes. In the Landau

approximation (§2.1.1), this last elastic constant goes ^{to zero as} AT, ^the temperature difference to the transition : ky - K ^AT.

4.3 EQUATIONS ^{OF DYNAMICS AND} VISCOSITY COEF-

FICIENTS.

The dynamical equations of the biaxial

nematic phase (Eq. (3)) may be written using ^the polar ^and angular variables, ^under the condition of small fluctuations, ^as for the free energy. The reference axes being ^chosen along the average eigen-

directions of the order-parameter, ^the polar ^and angular ^{variables are} decoupled ⁱⁿ the free energy

expression (§ 4.2). They ^are therefore also separated

in the dynamical equations. It results that they

describe independent relaxation modes of the order parameter, ^the amplitude and the orientational modes. The amplitude modes, ^which correspond ^to

fluctuations of the diagonal ^elements Qaa, undergo

the same viscosity coefficient v. The orientational modes (which ^are fluctuations of the off-diagonal

elements Qaf3’ ^i.e. fluctuations of rotation around the eigen-directions ^y (=F ^a, 8 ^{are submitted to}

the rotational viscosities v ’Y ^{= 2 v’} Qaa Q fJ fJ ) 2.

In this expression, v’ stands for v corrected from the back-flow effects (which depend ^{on the q} direction),

as for the off-diagonal mode in the Nu phase (§ 3.3).

Except ^{for the back-flow} corrections, ^the Nb phase presents therefore 3 different orientational viscosities in this simplified model, ^{instead of 12 in the} general

case [14]. ^{Let us notice that the} problem ^of viscosities remains simple, ^{as the} problem of the elastic coeffi-

cients, because the wave-vector q is decoupled ^from

the eigen-directions ^of QafJ ^{in the «} one-constant » model (Eq. (10)). ^{Let us note also that} similarly ^to

the orientational elastic coefficients, the rotational viscosities exhibit a pretransitional ^behaviour

( V ’Y - V ’ AT ^{until they vanish at} the correspon-

ding Nb-Nu p ase transition.

5. Relaxation modes of the amplitude of the order parameter ^{in the} vicinity of the uniaxial to biaxial nematic phase transition.

As discussed in § 4.3, the reference axes being

chosen along the average eigen-directions ^{of the}

order parameter Qaf3’ ^{the amplitude} modes of the order parameter ^{in the} Nb phase correspond ^to

fluctuations of the diagonal ^elements Qaa. They

appear therefore ^to continue the diagonal ^{modes of}

the uniaxial nematic phase already ^{discussed in}

§ 3.2. This continuation of the diagonal ^modes through ^the Nu ^and Nb phases ^{allows us to} apply ^the

same treatment from both sides of the transition.

5.1 BIREFRINGENCE AND BIAXIALITY VARIA- BLES.

Let us first consider a perfectly ^oriented

sample ^{free of} orientational fluctuations. Its Nb physical ^state may be represented by ^{the vector} Q ⁼ 611 i ⁺ Q22 j ⁺ Q33 k, {i, j, k} being ^{unit vec-}

tors parallel ^{to the} eigen-directions ^of Q.0. ^The

problem being restricted to the vicinity ^{of the}

uniaxial-to-biaxial nematic phase transition, ^{it is}

convenient to use the orthogonal ^basis of the unit vectors :

This new basis has the advantage ^to evidence the pure isotropic, uniaxial and biaxial states, respecti-

vely (i being parallel ^to the uniaxial director).

Calling ^{z, x} and y the components ^{of the state}

vector Q ^{in this new} basis, the invariants of

Qa/3 ^{may be written :}

z ⁼ 0 expresses that ^no isotropic fluctuations of QafJ

are considered here (QafJ ^{being a traceless tensor).}

The problem is therefore restricted to the two- dimensional xy-fluctuations ^of birefringence ^and biaxiality.

5.2 EQUATIONS OF BIREFRINGENCE AND BIAXIA- LITY.

Let us now consider a homogenous sample,

i.e. perfectly ^{oriented, with a constant} order parame- ter ( q ⁼ 0 ) . The free energy given by equation (1)

may be rewritten:

where A, B and C ^are independent polynoms ^{in x, A}

and B varying linearly ^with temperature through ^the

coefficients a and b :

The equilibrium ^{state is} given by the minimum of energy :

This system ^{has two} solutions :

-

and

which correspond ^{to the uniaxial nematic state} (biaxiality y ⁼ 0), ^{and to the} biaxial nematic state

(y :0 0 ) respectively. ^{Let us note that the} N.-Nb phase transition being second order (Sect. 1), ^the

biaxial solution (Eqs. (16)) joins continuously ^the

uniaxial one (Eqs. (15)) ^{at the} transition. It thus results that y2 ^and consequently, ^{B and} Ax, ^{vanish at}

the transition as AT, ^the temperature difference to the biaxial transition.

5.3 EIGEN-VALUES ^{OF THE FREE ENERGY. - The} second derivatives of the free energy give ^the

coefficient of the restoring ^{force to the} equilibrium

consecutive to a small fluctuation. To the first-order in AT (or y2) ^{at the} transition, they ^{are :}

On the uniaxial side of the transition (y ⁼ 0 ) , ^the coupling ^{factor C between} birefringence (x ) ^and biaxiality ( y ) ^{is zero, i.e.} the matrix of the second derivatives of the free energy is diagonal ^{in the} { ^{I, U,} B} ^{reference frame.} It results that the

amplitude ^{modes are of} purely birefringent ^and

biaxial types ^{in the uniaxial nematic} phases, ^{with the} respective eigen-values :

This last eigen-value g. allows a comparison ^{to the} Nu free-energy expansion (Eq. (4)) ^{used in} (Chap. 3) :

Calculating ^{we deduce:}

The coefficient ii undergoes therefore linear varia- tions with temperature, and vanishes at the Nb-Nu

transition (Eqs. (16)), ^as announced in § ^3.1.

In the Nb phase (y2 = - ^B/2 C ), the matrix of the potentials (Eqs. (17)) ^{is no} longer diagonal (C # 0). Consequently, ^the amplitude ^{modes are}

no longer purely birefringent ^{or biaxial. However, C} vanishing ^at the transition as AT, ^the mixing ^{of the} birefringent ^{and biaxial} components begins slowly ^at

the transition, ^{and the} amplitude ^modes keep ^a

dominent character, birefringent ^{or biaxial near the} N.-Nb transition. To first-order in AT, ^their eigen-

values are respectively :

with ^a

A comparison ^with equations (18) immediately

shows that Àb and J.Lb ^are different from the values

A. ^{and -} u./2 ^{that one} would have expected ^{at first} sight. ^This is the consequence of the coupling ^factor

C. Let us notice also that, ^for vanishing AT, ^which makes B - 0, both J.Lu and J.Lb tend ^to zero, and that

Àb ^{tends towards} Àu =F- o. ^{This confirms that the} quasi-biaxial ^{mode is critical at} the transition whe-

reas the quasi-birefringent ^{mode is not.}

5.4 AMPLITUDES AND TIME CONSTANTS OF THE AMPLITUDE MODES AT q ^{= 0.}

^The average squar- ed amplitudes ^{of the} dominently birefringent ^and

biaxial modes ^are given by ^the equipartition ^of

energy ^to be respectively :

As a consequence ^{of the «} one-viscosity approxi-

mation ^» (§ 4.3), any fluctuation of the amplitude ^of

the order parameter Qa/3 ^{relaxes with the same} viscosity coefficient v. In other words, v ^{is like an} isotropic viscosity ^{in the} three-dimensional _{space of} the amplitude fluctuations. This property ^remains valid when going ^{from the} orthonormal basis of this space {i, ^j, k} ^{to another one :} {I, ^U, B }, ^or equivalently ^to the orthonormal basis of the eigen-

vectors of the free-energy. v is therefore the viscosity

coefficient of both the amplitude ^modes (the ^birefrin-

gent ^and biaxial-like ones). Their relaxation time constants T are given respectively by :

5.5 TEMPERATURE DEPENDENCE OF THE AMPLI-

TUDE MODES. - In the framework of classical

theory, ^the temperature variations of the viscosity

coefficient v _{may be} supposed ^{to be} regular ^{at the} N.-Nb phase transition. The temperature depen-

dence of the amplitudes ^{and time constants of the} amplitude modes around the N.-Nb phase ^transition

comes therefore mainly ^{from the} temperature depen-

dence of the eigen-values ^A and A ^of the free energy.

In the Nu ^and Nb phases, ^{and to} zero-order in A T,

the temperature variations of the (quasi-biaxial)

critical mode are respectively given by :

temperature derivative of the birefringence ^{in the} N. phase ^{at the} transition, ^and by :

Eq. (16)) is ^the temperature derivative of the bire-

fringence ^{in the} Nb phase ^{at the} transition to the N.

phase. ^{After a} straightforward calculation, ^{it reduces}

to :

So, though /Lu and A b ^{are not in} the usual ratio - 2,

their temperature derivatives are (the compensation being exactly ^realized by ^the difference between the temperature derivatives of the birefringence ^{in the} Nu ^and N phases: dx I ^{u and dx The quasi-}

biaxial (critical) amplitude ^mode undergoes ^there-

fore temperature variations twice larger ^{in the}

biaxial nematic phase (ordered phase) than in the uniaxial (disordered) ^one (Fig. 4). ^{This result is}

classical [15]. ^{It has} already ^been given ^{in refe-}

rence [6] ^{where the reduced order} parameter C., ^is

Fig. ^4.

Relaxation frequency of the critical quasi-

biaxial amplitude-mode ^versus temperature ^{for two wave-} vectors q1 (heavy line) and q2 ⁼ J2 q1. The Nu ^{part is the}

same as in figure 3a. The dashed lines give ^{the non-} hydrodynamical ^limit.

directly chosen for describing ^the N.-Nb transition,

and in reference [16] ^{where the} coupling ^{factor C}

between the purely birefringent ^{and the} purely

biaxial deformations seems to have been neglected.

A comparison between the temperature ^variations

of the quasi-birefringent (non-critical) ^{mode from}

both sides of the transition, may also be calculated :

In general, ^{as verified from the} expressions (12),

none of these factors are zero. The temperature variations of the quasi-birefringent (non-critical) amplitude mode should therefore change ^{at the} transition, ^as depicted ⁱⁿ figure ^{5. This} point ^{is new.}

It could not be predicted ^if using the reduced order-

parameter C.,, ^{which measures the biaxial order} only [6], ^{or if} crudely decoupling ^{the two dimensions} (birefringence ^and biaxiality) ^{of the order} parameter

(Eq. (33) ^{of Ref.} [16]).

Fig. ^5.

Relaxation frequency of the non-critical quasi- birefringent amplitude-mode ^versus temperature ^{with the}

same representation ^{as in} figure ^4.

5.6 AMPLITUDE MODES OF NON-ZERO WAVE-VEC- TOR. - The sample ^{is now} supposed ^to undergo

fluctuations of non-zero wave-vector q. The ^corres-

ponding ^elastic energy has ^to be added to the free energy (Eq. (11)) :

This isotropic ^term in the xy (birefringence-biaxia- lity) two-dimensional space, does ^not change ^the eigen-vectors of the free energy. It adds Kq2 ^{to the} eigen-values, which become A + Kq2 and u ⁺ Kq2,

and adds conse uentl Kq ²

to the relaxation fre uen-

and adds consequently q to the relaxation frequen-

v q

cies (Eqs. (22)) ^{of the two} amplitude ^modes (Figs. ⁴

and 5). ^{Let us} notice that the effect of a non-zero

wave-vector is more sensitive on the quasi-biaxial mode, ^{because it} introduces a limitation to its critical

slowing-down ^{at the} transition, ^{than on the} quasi- birefringent mode which keeps relaxation frequen-

cies of comparable ^{order of} magnitude (Fig. 5).

5.7 LIMITATIONS ^{TO THE} CLASSICAL MODEL. - The above calculations ^are developped within the frame work of classical theory (§ 2.1.1 and § 4.1). ^The

central parts ^of figures 4 and 5 around the transition have therefore to be rejected. First, ^is suppressed

the part (not ^{indicated on the} figures) ^{which does}

not obey ^{to the} Ginzburg criterium, ^{i.e. such that}

1 AT 1 åTG, Ate being ^the temperature ^difference between the Ginzburg ^{crossover and the} transition.

Second, ^is eliminated the non-hydrodynamical region ^defined by q03BE > 1, where 03BE = 03BE0 t 2013 !

region ^defined by q03BE > 1, where 03BE ⁼ 03BE0 T

is the correlation length ^{of the order} parameter.

The correlation length ^{in the} Nu phase being given by Ke -2 =,U, ^the non-hydrodynamical limit reads

Kq2 ^{= .L. The} non-hydrodynamical ^{limit to the}

critical amplitude ^{mode in the} Nu phase

(¿ = Kq2 + JL) corresponds ^{therefore to :}

T v

Its temperature variations are linear, ^twice larger

than the modes (dashed ^{line in} Fig. 4). ^{In the} Nb phase, ^the non-hydrodynamical limit of the critical mode (¿ = 2 :-) undergoes also linear

T v

temperature variations, ^{but twice} larger than in the

Nu phase, ^as the critical amplitude modes, ^because

the inverse susceptibility tk ^{has a twice} larger tempe-

rature dependence in the ordered phase than in the disordered one (Eq. (23)).

The non-hydrodynamical ^{limits to} the non-critical

amplitude modes in the Nu ^and Nb phases, ^is similarly given by :

It makes again linear variations with temperature, but not going ^{to the} origin (dashed ^{lines in} Fig. 5).

6. Orientational modes in the biaxial nematic phase.

6.1 TEMPERATURE VARIATIONS OF THE ORIENTA- TIONAL MODES. - As noticed in § 4.3, the orienta- tional modes in the Nb phase ^are fluctuations of rotation around the eigen-directions a, /3 ^{and y.}

They ^{are of two kinds} depending ^{on whether the axis}

of rotation is perpendicular ^or parallel ^{to the}

uniaxial director n of the considered Nu-Nb ^transi-

tion. The former case is trivial and corresporids ^to

the two uniaxial orientational modes which are not

concerned with the transition. They ^{should therefore}

exhibit regular variations with temperature ^across

the transition.

The other case where the axis of rotation _y of the fluctuations is parallel ^{to n, is more} interesting,

because critical. The reference ^{frame being chosen}

along the average eigen-directions ^of Qap’ ^this

critical orientational mode corresponds ^{to the off-} diagonal fluctuations of C ao (a, f3 =F 1’). ^It

appears therefore ^to be the extension from the

Nu phase, ^{of the} off-diagonal ^mode already ^described

in § 3.3 (Fig. 2b). Its average squared amplitude diverges ^{at the} transition (from ^{now on, the} subscript

y is omitted in Ky ^and v -y) :

but its damping time ? follows less evident tempera-

ture variations (§ 4.3) :

v’ being ^the viscosity coefficient v corrected from the back-flow effects (§ 4.3). ^{Let us} examine the temperature variations of equation (25) ^{in two} typi-

cal examples.

6.2 PURE BIAXIAL TWIST MODE. - In this example,

q and the rotational axis y are chosen along ^the

uniaxial director n. This corresponds ^to the pure twist mode relative to the biaxial director m (as

defined in Ref. [17]) (Fig. 6a). ^{In this} configuration,

no back-flow can help ^{to return back to} equilibrium [8]. Consequently, in the twist mode, ^the viscosity

v’ is equal ^{to v. v’} should therefore not depend critically ^on temperature, ^nor the relaxation time ^T of this pure biaxial twist mode (Fig. 6b), given by :

Fig. ^6.

a) Twist mode relative to the biaxial director m

in the Nb phase. ^{The order} parameter, ^{sketched as an} ellipsis perpendicular ^to the uniaxial director n, oscillates around n. b) Relaxation frequency of the biaxial twist mode versus temperature. The critical orientational mode in the Nb ^{phase is continued by the} off-diagonal ^{mode in}

the Nu phase (Fig. 3b). ^{Because no back-flow} couples ^to

the twist mode, the associated viscosity ^v’ equals v ^{in the} Nu ^and Nb phases. The continued off-diagonal mode in the

Nu phase has therefore exceptionally ^{the same relaxation}

time as the diagonal ^mode (Fig. 3a).

6.3 PURE BIAXIAL m-BEND MODE. - In this ^exam-

ple, q is chosen perpendicular ^{to the common}

direction of the rotational axis _y and of the uniaxial director n, but parallel ^to the average direction of the biaxial director m. This corresponds ^to the pure biaxial bend mode relative to the director m

(Fig. 7a). As mentioned above, ^{back-flow effects}

can reduce the viscosity coefficient v’ of this bend mode. Using ^{the notations} of reference [8], ^and taking ^the back-flow into account, ^the effective bend rotational viscosity vb is calculated to be [8] :

Fig. ^7.

a) ^{Bend mode relative to} the biaxial director m

in the Nb ^{phase. m} oscillates about its average direction

parallel ^to q and perpendicular ^to the uniaxial director ^n.

b) Relaxation frequency of the biaxial bend mode versus

temperature. The orientational mode in the Nb ^{phase is}

continued by ^the off-diagonal mode in the Nu ^phase.

Because of the back-flow effects, ^the frequency ^relaxation

of this bend mode varies linearly ^with temperature ^{in the}

Nb ^{phase, while its} non-hydrodynamical limit varies qua-

dratically.

Let us estimate the bend rotational viscosity vb ^{in the case of a} strong coupling between the director m and the shear flow. In this frozen-in

situation, the Miesowicz viscosities become simply [18] l1b - a3 and l1c - - a2. ^{From the relation}

y1= a3 - a2, ^we then estimate vb ~ l1b’ i. e. the bend rotational viscosity corrected from the back- flow effects, is of the order of magnitude ^{of the}

shear-flow viscosity. vb should therefore not criti-

cally depend ^on temperature ^{in the} strong coupling

case (Fig. 8).

The opposite ^{case -of a} complete decoupling

between m and the shear flow corresponds ^{to a case}

where no back-flow can help ^the relaxation of the bend mode. The rotational viscosity vb for ^{the bend}

mode remains then uncorrected, ^and equal ^{to the}

twist rotational viscosity v(y1 in the usual notation).

It depends ^now critically ^on temperature (§ 4.3) : vb = V - v I1T (Fig. 8).

The general ^{case of the} intermediate coupling

between m and the shear flow corresponds ^{to a case}

where the two above relaxation mechanisms do exist in parallel, ^and compete. The effective bend rotatio- nal viscosity ^{is then smaller than} both TJb ^{and v, and}

Fig. ^8.

Temperature variations of the bend rotational

viscosity vb ^{from the} decoupling (Vb ⁼ 9 ) ^to the frozen- in (Vb = 1Jb) ^limits.

joins these values respectively ^{far from and near to}

the N.-Nb transition, ^{where the} two extreme relaxa- tion mechanisms dominate. In between, the effective bend rotational viscosity vb ^{follows a crossover} regime schematically represented ⁱⁿ figure ⁸ by ^the

dashed line. The temperature ^{behaviour of the} relaxation frequency ^of the pure bend mode is

deduced, ^and represented ⁱⁿ figure 7b. Far from the transition in the Nb phase, ^{the frozen-in case domina-}

tes, ^and the temperature variations of the relaxation

frequency may be approximated by :

This linear temperature dependence of the relaxation

frequency ^{of the biaxial bend mode has been} experi- mentally ^observed by Lacerda and Durand using light scattering spectroscopy [16]. ^{Let us} notice that its extrapolation ^{to zero} frequency gives ^an easy determination of the transition temperature Tc. ^On

the other side of the transition, ^{in the} Nu phase, ^the

biaxial bend mode becomes the off-diagonal ^mode

discussed in section 3. As shown in figure 7b, ^the

two modes do not coincide at the transition because

they undergo different viscosities in the Nu ^and Nb phases. On the biaxial side of the transition, ^the back-flow coupling vanishes, ^and vb - v (Fig. 8) ^so

that the bend mode joins the twist mode (Fig. 6b),

while in the Nu phase, ^the off-diagonal mode, keeping coupled ^{to the} back-flow, undergoes ^{a lower} viscosity ( v’ v ). However, ^{there is no disconti-} nuity ^at the transition in reality, because the above calculations, being developed within,ctassical theory,

are not valid _very ^near to the transition, making ^the

central part ^of figure ^{7b to be discarded.}

6.4 LIMITATIONS TO THE CLASSICAL MODEL. - The calculations of the relaxation modes of the order parameter ^{at the} Nu-Nb phase transition are develop-

ed within the classical theory (§ ^{2.1.1 and §} 4.1). ^As

in § 5.7, ^the part around the transition in figures ^6b

and 7b has to be eliminated from this study. ^{A first}

part, ^such that I åT I TG, ^{has to be} suppressed

because it does not satisfy ^the Ginzburg ^criterion.

The second part ^{to be} rejected ^{is the} non-hydrody-

namical region ^where qg > 1.

The non-hydrodynamical ^limit being characterized

by the relation Kq2 ^{= ’a} (§ 5.7), the limit of the critical orientational mode in the Nu phase (off- diagonal ^mode of ’aP) ^{is given} bY : / = 2 03BC . ^In

general, this limit is different from the limit of the critical amplitude ^mode (diagonal ^{mode of} but

in the particular ^case of the twist mode ( q//n ) , ^{it is}

the same : 1 - 2 A , because of the absence of the

T v

back-flow effects in this geometry (Fig. 6b).

The non-hydrodynamical ^{limit of} the critical orien- tational modes in the Nb phase depends ^more clearly

on the back-flow effects. The biaxial twist mode,

which involves ^no back-flows, relaxes with the

frequency 1 = q ^Kq2 (Eq. (26)), ^{and has therefore its}

T v ( ))

non-hydrodynamical ^limit at .! = IL. This limit

T v

undergoes linear variations with temperature, ^and because ii ^{has twice} larger temperature variations in the Nb phase than in the Nu phase, ^{the two non-} hydrodynamical limits in the Nu ^and Nb phases ^have

the same slope ⁱⁿ figure 6b. The biaxial bend mode,

which on the other hand, involves back-flow effects,

relaxes with the frequency - 1 = ^{Kq2 åT} (Eq. (27)),

T 71 b

and has therefore a non-hydrodynamical ^limit:

which varies quadratically ^with temperature

(Fig. 7b).

7. Summary.

Let us now summarize the main results of this classical approach ^{of the} relaxation modes of the fluctuations of the order parameter ^{in the} vicinity ^of

the uniaxial to biaxial nematic phase transition.

Using polar coordinates in the appropriate ^reference frame, the 5 relaxation modes of the order parameter

Qa03B2 ^{(for a given q} wave-vector) ^{are followed across}

the Nu-Nb ^phase transition: 2 modes are amplitude (or diagonal) ^{modes, and the 3 others are} orientatio- nal (or off-diagonal) modes. As could be predicated

when using the reduced order parameter C.,, ^{[6], 2}

out of these 5 modes ^are critical : one amplitude

mode and one orientational mode. However, ^{the use}

of the reduced order parameter ^{should be limited to} the Nu phase ^where equation 19 is valid. In the Nb phase, ^{the use of} C,,,, ^is misleading, and masks the

mixing ^{of the} biaxiality ^and birefringence compo- nents of Q,,o ^{in the amplitude} modes. The critical

amplitude ^mode (Fig. 4) is therefore quasi-biaxial

instead of being purely ^biaxial. Though ^the problem

is greatly simplified by using ^{the «} one-constant

approximation » ^{on both} the elastic and viscosity

constants, ^the temperature behaviour of the critical orientational mode is complicated by ^{the occurrence}

of back-flow effects which depend ^{on the} q-direction (Figs. ^{6 and} 7). Among ^{the 3} non-critical modes, ^one

is the quasi-birefringent amplitude ^mode which,

because of its interaction to the quasi-biaxial ampli-

tude mode, ^should present ^a break in its temperature

variations (Fig. 5) ; ^{the two} other modes are the two non-critical orientational modes. The relaxation fre-

quencies ^of these non-critical orientational modes do not vary ^at the Nu-Nb phase transition. They ^{are :}

where v’ stands for v corrected for non-critical back- flow effects.

As discussed in § ^5.7 and § 6.4, ^{the classical} approach ^{is limited} by ^the Ginzburg crossover, i.e. is restricted to temperatures farther from the transition

than A To. Specific ^heat measurements are lacking ^to

estimate ATG, ^{but from the} analysis ^{of the} tempera-

ture variations of the order parameter [9] ^{and of the}

critical slowing-down ^{of the} biaxiality fluctuations

[6] ^{at the} Nd-Nb phase transition in the typical system of K-laurate, decanol, D20, it appears that åTG ^is

smaller than the temperature resolution of these

experiments, i.e. åTG 0.05 K. On the other hand,

this system is shown in reference [10] ^{to behave} critically below 0.03 K from the transition. We may therefore estimate 4TG in the range 0.03-0.05 K,

may be slightly depending ^{on the} sample composi-

tion.

The non-hydrodynamical region ( q > 1) ^cannot

also be described within the classical model. For a

typical q - ^1W cm-1, ^and taking 60 ^{= 10-6 cm from}

the light scattering measurements at the Nd-Nb phase

transition [6], ^we estimate the temperature range of the non-hydrodynamical region ^to be AT -- 0.03 K.

This value, ^{as the} Ginzburg temperature, ^{falls inside}

the temperature resolution of the light scattering experiments, ^so that the classical model developed

here is adapted ^for describing the available q-depen-

dent data at the Nu-Nb ^phase transition [6, 18].

The comparison ^{to the} preliminary experimental

results,is interesting. ^The temperature behaviour of the biaxial bend mode around the Nb-Nu phase

transition as depicted ⁱⁿ figure 7b, ^{is in} good agree-

ment with .the experimental ^results reported ^{in the} figure ^{5 of reference} [16], except ^{for the transition} temperature which for unclear reasons seems to have been chosen at the intercept ^{of the uniaxial}

mode to the temperature axis. The existence of the

crossover between the frozen-in and the decoupled

back-flow regimes, which deviates the biaxial modes from going ^{to the} origin ⁱⁿ figure 7b, ^{has also not} yet been observed. More experiments ^{would be neces-}

sary ^to confirm this point. ^{At the same time, it would}

be interesting ^to study ^the temperature behaviour of

the biaxial twist mode, ^{and to} compare it with

figure ^{6b. These} two measurements would provide ^a good ^test of the back-flow effects in the Nb phase.

We are very grateful ^{to G. Durand for} helpful

discussions.

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