Critical damping of first and second sound at a smectic A-nematic phase transition

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Critical damping of first and second sound at a smectic A-nematic phase transition

F. Jähnig

To cite this version:

F. Jähnig. Critical damping of first and second sound at a smectic A-nematic phase transition. Journal

de Physique, 1975, 36 (4), pp.315-324. �10.1051/jphys:01975003604031500�. �jpa-00208256�

315

CRITICAL DAMPING OF FIRST AND SECOND SOUND

AT A SMECTIC A-NEMATIC PHASE TRANSITION (*)

F. JÄHNIG

Max-Planck-Institut für biophysikalische ^Chemie

34 Göttingen-Nikolausberg, Germany (Reçu ^{le 15} juillet 1974, ^{révisé le 2 décembre} 1974)

Résumé.

On étudie à partir ^{d’une théorie} hydrodynamique généralisée ^les modes propres ^au

voisinage de la transition de phase smectique A-nématique. ^{Il est} nécessaire de tenir compte ^des

mouvements non hydrodynamiques ^du paramètre d’ordre du smectique ^{et du directeur} qui ^{se ralen-}

tissent à la transition de phase. ^Ceci conduit, ^{en accord avec} les résultats expérimentaux existants,

à une anisotropie spéciale de l’atténuation du premier ^{son et} du deuxième son. On discute la dépen-

dance en température de l’atténuation critique ^{en la} comparant ^{aux résultats} expérimentaux ^récents

sur les viscosités critiques.

Abstract.

Starting ^{from a} generalized hydrodynamic theory ^the eigenmodes ^{in the} vicinity

of a smectic A-nematic phase transition are derived. It is necessary ^to take account of the non-

hydrodynamic motions of the smectic order parameter and the director which slow down at the

phase transition. This leads to a special anisotropy of the critical damping of first and second sound which agrees with existing experimental results. The temperature dependence of the critical damping

is discussed in comparison ^{with recent} experimental ^{results on} the critical viscosities.

LE JOURNAL DE PHYSIQUE ^TOME 36, ^AVRIL 1975,

Classification Physics ^Abstracts

7.130

1. Introduction.

Considerable work has been done to investigate the smectic A-nematic phase

transition. Up ^{to now} mainly ^the pretransitional

effects due to fluctuations of the smectic order _para- meter in the nematic phase have been considered.

They ^show up statically ^{in the} divergence ^{of some of}

the Frank elastic constants at the phase transition TAN,

as predicted theoretically by de Gennes [1], ^and dynamically ^{in the} divergence ^{of some} of the Leslie

viscosities, ^as predicted by Jâhnig and Brochard [2]

[JB] ^{and McMillan} [3]. ^The dynamical ^behaviour

of a smectic A on both sides of the phase ^transition

was treated by ^Brochard [4] applying dynamical scaling theory. ^{She was} mainly interested in the temperature dependence of the critical eigenmodes

in the hydrodynamic regime ^{and their wave vector} dependence in the critical regime. ^{In the} present paper ^{we want to} investigate ^{in more} detail the aniso-

tropic properties ^{of the} eigenmodes ⁱⁿ the smectic

phase ^{in the} hydrodynamic regime. Special ^attention

will be given ^to the critical damping of first and second sound.

As a theoretical framework we use the unified (*) Supported by the Deutsche Forschungsgemeinschaft.

Presented partially ^{at the} Fifth International Conference on

Liquid Crystals, Stockholm, ^June 17-21, 1974.

hydrodynamic theory ^for crystals, liquid crystals ^and liquids ^of Jâhnig and Schmidt [5] [JS]. They gave

already ^the application ^{to nematics. For our} problem

this has to be generalized ^to include the complex

smectic order parameter t/J. ^Its magnitude §o ^{is a} nonhydrodynamic variable. Its phase ^{is a} nonhydro- dynamic variable in the nematic phase, but in the smectic phase ^{it is a} hydrodynamic variable, ^the displacement u, of the layers. ^The orientational

displacements ^{of the} molecules, the director displa-

cements «5n_b behave in the opposite ^sense being ^a hydrodynamic ^{variable in} the nematic phase ^{and a} nonhydrodynamic variable in the smectic phase (more exactly ^the transverse part ^of c5n .L). ^{As the} nonhydrodynamic variables slow down at TAN they

have to be included in the set of dynamical ^variables

for our problem. ^{As a} special case, the theory pre- sented includes also the hydrodynamics of smectics A which has already ^been given by ^{de Gennes} [6], ^and

more recently by Martin, Parodi and Pershan [7]

as one application of their unified hydrodynamic theory.

The introduction of the displacement u., yields

an elastic coupling between translational and orien- tational displacements, ^and anisotropy in compres- sional behaviour, ^as shown in section 2. Dynamically

it implies ^{a further} dissipative process, the _permea-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01975003604031500

tion. This effect is introduced together ^{with the set}

of equations ^of motion for all dynamical variables in section 3.

The influence of the thermodynamic fluctuations of the modulus of the order parameter ^{on the} dyna-

mics of a smectic A will be taken into account by

the critical contributions to the elastic constants and viscosities. The critical contributions calculated

by ^{JB for T} > TAN ^{have the same} critical behaviour at T TAN. ^An additional contribution exists below

TAN which is shown to diverge also in the same way.

These effects will be discussed together ^{with the} generalization of the work of JB to compressible

systems ^{in Section 4.}

The eigenmodes ^are presented and discussed as to their critical behaviour in section 5. Finally,

in section 6 the results for the damping of first and second sound are compared ^with experimental ^results.

In this connection we also discuss some recent expe- rimental results on the critical behaviour of the twist viscosity y1.

2. The elastic _energy.

To derive the elastic energy Fel ^{of a smectic A} phase ^{we define the dila-}

tion 0 (in volume), ^the displacement u. of the layers lying ^{in the xy} plane, ^{and the} angular displacements [5]

CPx and CPy ^{of the axes} of the molecules. In the general expression for the elastic _energy Fei ^{as a} quadratic

function of the strains

only ^{the terms} including 0, Uz, CPx ^or (py ^{have to be}

kept ^as finite contributions. Because of the _appea-

rance of the strain r which is typical for smectics [8]

we show the derivation of the elastic _energy in some

detail.

In general the elastic _energy can be written as

where Gijk is the Levi-Civita tensor. P, nô, IIaJ, ^and Tij

are defined as conjugate variables and can be _express- ed as

For the uniaxial symmetry ^{of a smectic A} phase ^the nonvanishing components ^{of the} elasticity ^tensors C, E, G, ^{and H are} [8]

The components ^of Kijkl ^are the Frank elastic cons- tants [5]. ^A further condition on the elastic constants is imposed by the invariance of the elastic _energy under a homogeneous rotation of the layers ^{and the}

molecules (staying ^{normal to the} layers ⁱⁿ equili- brium)

The constants Ci, El, E2, E3 ^and H2 ^{have to be set} equal ^{to zero as} they give ^{rise to terms} containing

variables other than 0, uZ, and p y. Eqs. (2.2) ^then yield

where we have substituted C2 ⁼ C, E5 = B, ^and Hl ^{= H. As} 1Ij ^{in eq. (2. 5) has no diagonal part,}

P can be identified with the _pressure, and the stress

acting ^{on the centers of mass} of the molecules is

given by

The result for the elastic _energy is, ^{if we further-}

more introduce the director ^{n =} _no + ôn, ^with bnx ⁼ (py ^and ôny ^{= - PX,}

with

The elastic constants Hi ⁼ G describe the elastic

coupling between the centers of mass and the axes

of the molecules, by which smectics differ from ^nema- tics. In nematics, ^this coupling ^is purely dissipative

and the viscosities _y, and ₇₂ are the dissipative ^coun- terparts ^to Hl ^{and G,} respectively.

JS set up ^a unified hydrodynamic theory ^without introducing displacements. ^{For a definite} phase, meaningful displacements ^can be defined and used to express the elastic energy. This is shown for a

smectic A phase ⁱⁿ Appendix ^A, the result being

eq.(2.6).

317

In order to investigate ^our system ^{in the} vicinity

of the smectic A-nematic phase transition, ^{we must} generalize the elastic energy ^to include the smectic order parameter t/J. ^This generalization ^has already

been given by ^{de Gennes} [1] ^{in form of a} Ginzburg-

Landau expression

where qS ^{= 2} nld, ^{and d the} interlayer ^distance.

In the ordered phase ^the complex ^order parameter

can be decomposed ^{into its} magnitude t/lo, ^{and its} phase determined by ^the displacement u.,

Inserting ^this equation ^into eq. (2. 8) ^{it is} easily ^seen

that the _energy connected with the variables 0, Uz, and c5n.l ^{is identical to the} above result _eq. (2.6),

under the substitutions

3. The équations ^{of motion.}

^For the determina- tion of the eigenmodes ^{we need the} equations ^of

motion for the dynamic ^variables by ^{which we}

describe our system. ^{These are the} particle density ^p

or the dilation 0 related by p ⁼ po(1 - 0), ^the

momentum p, the angular displacements qJx and qJy, the displacement Uz, and the magnitude t/J 0 ^{of the}

order parameter. ^{We shall} neglect thermal variables.

The equations of motion for the variables 0, p, qJx, and qJy ^are known from the hydrodynamics of nematics which we will use in the formulation of JS.

The motion of _uz, being hydrodynamic ^below TAN,

was treated within the hydrodynamic theory ^of Martin, Parodi, and Pershan [7]. With this motion

a further dissipative process is connected, ^the per- meation. It has not been taken into account by ^JS

but can easily ^be treated within their framework

as shown in Appendix ^{B. The result is} given by ^the equation of motion for _Uz

where v is the velocity ^and Âp ^{is the} permeation ^cons-

tant.

The dynamic behaviour of the order parameter t/1 0

is usually ^{assumed to follow a} simple relaxation law

The relaxation time i has been discussed by ^Bro-

chard [4].

Far below TAN ^{one can} eliminate the fast non-

hydrodynamic ^{motions of} the variables g/o ^and bOi by the conditions

The latter condition states that the molecules are

fixed normal to the layers. ^It implies ^{that the trans-}

verse part ^of l5n.l vanishes ; ^the longitudinal part ^is still a hydrodynamic variable. Under the conditions eq. (3.3) ^our theory ^{reduces to the} hydrodynamic theory of smectics A.

From the work of JB for T > TAN ^{we know that}

the motion of t/J 0’ ^its dynamical fluctuations, yields

contributions to some of the Frank elastic constants and viscosities which diverge ^at TAN. ^{For T} TAN, essentially ^{the same} critical contributions arise from the fluctuations of t/J o. ^{Therefore we will take account}

for the motion of t/Jo by keeping these critical contri- butions which will be investigated ^{in more detail}

in the next section.

The complete ^{set of} equations of motion is then

given by

where the dissipative stress tensor TCij = rc1j + n’ij

was split up into its symmetric ^and antisymmetric

parts. Explicit expressions for them define the vis- cosities (presented without restriction ^to critical

ones)

with

(In ^JS Yi and _y2 were called (j ^{and - v,} respectively.)

These viscosities are related to the ones used by ^Mar- tin, Parodi, and Pershan [7] by

4. The critical elastic constants and viscosities.

Above the phase transition the critical contributions to the Frank elastic constants and the viscosities have been calculated for the incompressible system by JB

where jjj ^Il = (2 aM v) - 1/2 ^{is the coherence} length

in the z direction, ^z the relaxation time of the order parameter ^{and T the} temperature with Boltzmann’s constant KB ^{= 1. Just those} viscosities, ^{for which}

an elastic counterpart, E4 ^{= G} = Hl ^and E5, has

been introduced in section 2 for the description ^of

a smetic A, ^{show a} critical behaviour.

We don’t want to give ^a rigorous derivation of the critical effects for T TAN, but rather discuss the

qualitative features. As ^an example ^{we treat the}

critical contributions to y,. Following ^{JB we start}

with the expression ^{for the stress tensor} compo- nent IIZx, eq. (2. 5c) with the substitution _eq. (2.10c).

Splitting up the order parameter t/J 0 ^{in its mean}

value t/Joo ^{and the} fluctuating part liÍo,

11, "Il, r w , 1.. -,

we find with

The first term 1Ii represents ^the ordinary hydrody-

namic stress, ^whereas llzx ^and 1I.-X ^{are due to fluctua-} tions tÍ o.

The last term ÎIZx corresponds ^{to the} stress tensor

component ^used by ^{JB for the} calculation of Y1

at T > TAN. ^Below TAN ^{it will} yield ^a contribution y1 with the same critical behaviour

This symmetry of the fluctuation effects, ^with _respect

to TAN, ^is already known from the analogous pro- blem of superconductors [9, 10].

The second fluctuating ^term Îlzx ^exists only ^for

T TAN, ^{as it is} proportional ^to yGoo, ^and gives

rise to an additional critical contribution yi. ^The

Kubo formula for Yi ^is given by

This thermal average ^{can most} simply be evaluated

on working ^{with the} nonhydrodynamic ^variable tPx ⁼ Oxuz ⁺ c5nx instead of the variables Uz and c5nx separately. Dynamically px ^{describes a relaxation} with

and c5Yl = Yi + 00FF 1, ^{as can be seen from} eq. (5. 6).

The static fluctuations of 4>x ^are given approximately by

Following the lines of JB we get ^from eq. (4.6)

and with 1 tfro(K) 12 ) ~ (K2 + C-2)- ’ and eq. (4. 8)

Inserting eq. (4.7) this relation is fulfilled for

Critical contributions of this type ^were introduced first by Landau and Khalatnikov [11]. î, ^and yi

show the same behaviour, ^{as discussed} already ^for

the Â-transition in He by Hohenberg [12]. ^The complete

critical contribution at T TAN’

319

therefore has the same critical temperature divergence

as i ^{for T} > TAN. ^This symmetry would also follow from dynamical scaling [4].

The term 1IÀ, ^{eq. (4.4a), and the} corresponding

terms of the other stress tensor components ^are independent of the fluctuations §o and linear in the

displacements. They represent ^the ordinary hydro- dynamic ^stresses and will be used in Section 5 for the calculation of the eigenmodes.

If we wanted to calculate the director _response function QafJ ^{defined by [2] ]}

the fluctuations of _Uz arising from the first term in _eq. (4.4a) would contribute to Qap. ^{We only}

mention that these fluctuations would serve to make the static director _response function purely ^transverse

in the limit q --+ ⁰

(We introduced the assumption MT ⁼ Mv ^which

can be avoided by ^a scale transformation in momen- tum space.)

As spatial symmetry ^{does not} change ^at T AN,

the other critical viscosities and elastic constants, eq. (4.1), diverge ^{for T} TAN ^{in the same} way ^as for T > TAN.

For a generalization ^{of our results to a} compressible

system, it is sufficient to continue the work of JB at T > TAN because of the symmetry of the fluctua- tion effects. The two additional viscosities by ^which

a compressible system is described in comparison

to an incompressible system ^{are critical}

as shown explicitely ⁱⁿ Appendix ^C. ôtll ^and 8r3 depend ^on the existence of the coupling ^{constant C.}

The critical behaviour of the permeation ^constant

we take from dynamical scaling [4]

5. The eigenmodes.

The critical eigenmodes ⁱⁿ

the vicinity of the smectic A-nematic phase ^transition

are the solutions of the equations ^of motion, eq. (3.4), restricting ^{ourselves to} the critical parts of the elastic constants Ki ^{and the} viscosities fli ^and y,. Without loss of generality ^{we can} choose the wave vector q to lie in the xz plane, ^{with 0} = (z, q). ^{Then the} eigenmodes ^with q in the xz plane (case 1) ^{and the}

others with _q perpendicular ^{to it} (case 2) ^{are decou-} pled.

In case 2 the equations of motion reduce to (the coupling ^{due to} the elastic constant H _may simply

be taken into account substituting - iWY1’ - iroY2,

-

i(0114 ^{in JS} by ^{H -} iwc5Yl)

The first eigenmode, eq. (5. la), ^{has zero} frequency

which means that this mode is noncritical. It corres-

ponds ^to the fast shear diffusion mode _wF2 of nema-

tics. _{In eq.} (5 .1 b) the elastic terms are of the relative

order Êi q2jH ~ (çq)2. ^{In the} hydrodynamic ^regime jq « ^{1 the} Ki ^{terms can be} neglected. Then eq. (5. 1b) yields ^a relaxational eigenmode

This mode corresponds ^to the slow director mode WS2 of nematics.

In case 1 the equations ^{of motion are}

To simplify the discussion of the eigenmodes ^we

introduce the assumptions i) A > B, ^{C and} ii) ^{B » C.}

The first assumption ^is justified experimentally [16]

and implies ^the neglect ^{of the constant C} ( ~ B).

The second assumption implies ^the neglect ^of c5r¡ 1

and c5r¡3. The results in the absence of assumption ii)

will be given ^later.

As discussed in case 2, ^the K3 ^{term in} eq. (5. 4d)

can be neglected against ^{the H term. Then this} equa- tion for the orientational motion decouples ^{from the} equations for the translational motion. Substituting

vZ ⁼ icz ^{in the} dissipative ^term it reduces to

This equation ^{describes a} relaxational mode

in the variable 4>x. ^It corresponds ^{to the director}

mode _ws1 of nematics.

We furthermore show that the H term describes

a purely transverse static director response. ^In the

static limit _eq. (5 .4b) yields ^{0 =} 0, ^and eq. (5 .4c)

and (5.4d) ^{reduce to}

Assuming MT ⁼ Mv ^as in eq. (4..14), ^{which means}

B ⁼ H, ^{we find on} eliminating ^{u from} eq. (5. 7b)

This result corresponds ^{to the} transverse static response function, eq. (4.14), ^{mentioned in connec-}

tion with the fluctuations of UZ.

The equations ^of motion, eq. (5.4), for the varia- bles 0, v,,, vz, and Uz ^{lead to} the secular equation

We make the insertion W2 ⁼ s2 q2 - iDwq2. ^For

the velocities we find the well-known equation [6]

Using ^the assumption A » ^{B the} velocities of first and second sound result as

For the damping ^{constant D we} obtain the equation (using A > B)

. n v i . i

yielding

This is our main result. It should be mentioned that

starting ^{from the} pure hydrodynamic theory ^of

smectics A and using only critical viscosities for the calculation of the critical eigenmodes ^would yield

a wrong contribution of c5r¡4 ^{in the} damping expres- sions. On the other hand, ^the hydrodynamic theory

of nematics _may be used as a check of our result eq. (5.13). The second sound corresponds ^{to the}

shear mode _(wF1 ^{= -} iDFl q2 of ^{nematics, and indeed}

one finds D2 ⁼ DF,, neglecting permeation, ^and

also Dl ⁼ D1,nem ^{(using only critical} viscosities in the expressions ^{of JS for} DFl ^and D1,nem). ^{We mention}

that permeation contributes to the damping ^{of first}

sound in a term of higher ^order bÀ.p(B2/A) ^{cos6 0.}

The result of Brochard [4], D2 ~ by ^sin2 0, ^was derived on approximating inconsistently the viscous terms in the equations ^{of motion.}

Releasing ^{the condition C B} used above the viscositites c5r¡1 ^and c5r¡3 change ^the anisotropy ^of

the damping of first sound

whereas the anisotropy ^{of the} damping ^{of second}

sound is not changed

The cases 0=0 and 0 ⁼ n/2 ^deserve special ^consi-

deration since the velocity of second sound _goes to

zero. For 0=0 the second sound mode decomposes

into two diffusional modes, ^{the critical} permeation

mode and a noncritical shear mode. For 0 ⁼ n/2

one finds also two diffusional modes, the undulation mode and a shear mode, both noncritical.

In table 1 the critical behaviour of the eigenmodes

below and above TAN is summarized omitting ^the complications ^{for 0 = 0 and 0 =} n/2. The relaxation of the phase of the order parameter ^{for T} > T AN

is denoted by WRO. Passing through ^the phase ^transition

from above to below the number of hydrodynamic

modes decreases by ^one. All critical modes show the scaling ^{behaviour discussed} by ^Brochard [4].

TABLE 1

The critical behaviour of ^the eigenmodes. ^The

number in brackets give the number of modes, ^the hydrodynamic ^ones being underlined.

6. Discussion of the critical damping.

The critical

damping ^{of first and} second sound depends ^{on the} angle 0, ^the frequency f ⁼ co/2 ^{n, and the} temperature difference AT ⁼ T - TAN. ^We will discuss the theore- tical results for these effects and _compare them with

experimental ^results.

321

The angular dependence ^{of the} damping ^{of first}

sound in a smectic A phase ^has been measured by

Lord [13]. ^The frequency ^was f = ³ MHz, ^{and the} temperature ^{was 7 OC} below the transition to the

isotropic phase. ^{As shown in} figure 1, the observed

anisotropy ^{of the} damping follows well a cos4 0 law as given by eq. (5.8a). ^{This differs} markedly

from the usual damping ^{in a} nematic which obeys

a cos’ 0 law (at comparable frequencies) and is much less anisotropic, ^as shown also in figure ^{1. In an} experiment ^on CBOOA, however, Miyano ^{and Ket-}

terson [14] ^{observed no} significant change ⁱⁿ

at f ^{= 12 MHz on} passing from the nematic to the smectic A phase. These results indicate that the smectic order is much softer below a smectic

A-isotropic transition than below TAN. ^{A critical} damping ^at the smectic A-nematic phase ^transition

in CBOOA has been observed, ^{on the other} hand,

at a higher frequency f = ^{560 MHz} by ^Bacri [15].

He found a critical divergence ^of Dl ^for 0=0, but no divergence ^{for 0 =} 90°, ⁱⁿ agreement ^with

our result _eq. (5.8a).

FIG. 1.

The damping ^of first sound in a smectic A phase. ^The experimental ^values (+) ^were determined by Lord. The theoretical

curve (-) ^{is a} fit of the anisotropy ^{to a cos’} 0 law. The damping ^{in a}

nematic phase (- -) is shown for comparison.

The anisotropy ^{of the} damping of second sound’

has not yet been measured accurately. ^But Liao, Clark, and Pershan [16] found, by ^{means of Bril-}

louin scattering, ^a strong ^{increase of this} damping

as 0 approached 45°. This behaviour _agrees qualita- tively ^{with our result} eq. (5.13b) neglecting permea- tion. For larger angles the second sound mode could not be observed. This cannot be explained ^{within our}

theory ^and may be due to the noncritical KI ^term

which in the smectic phase destroys ^the long-range

order in the order parameter [26].

The frequency dependence ^{of the} critical viscosities has not yet been calculated. Some insight may however be gained using ^the analogy ^{between our} problem

and the case of superconductors. Yi corresponds ^to

the fluctuation conductivity ^whose frequency ^dis- persion has been calculated within time dependent Ginzburg-Landau theory by ^{H. Schmidt} [10]. ^Its

behaviour is more complicated ^{than a} single ^Lorent-

zian. As this dispersion depends ^{on mz it shows} up also in the critical temperature dependence ^{of the} damping of first and second sound. For first sound this might have been observed by ^Bacri [15].

The temperature dependence ^of c5r¡s ^is given by

eq. (4.1e) ^and (4.15)

with e = Ar !/7BN’ ^{In the classical} region, ^where

mean field theory holds, ^this yields btl _, - ’e-0.5,

whereas in the non-classical region, ^{where the smectic}

A-nematic phase transition is assumed to obey scaling theory ^{as for the} Â-transition in He, this would give âtl.5 - ^{B- 0.33.} Evaluating ^{his data on the} damping

of first sound Bacri obtained the critical exponent ^o.33.

This result can however not be taken too seriously

as his assumptions for the evaluation are -open ^to criticism.

Lacking ^further measurements on the temperature dependence ^{of the} damping ^of first and second sound

we will discuss in place ^{of that the critical} divergence

of Yi ^{at T} > TAN which has been studied experimen- tally ^most extensively. ^In scaling theory eq. (4. 1 c) gives

An order of magnitude ^estimate yields (with ç /1 0 ⁼ 10 A, ^{a numerical error in JB was} already noted’by Huang ^{et al.} [18])

This leads to a small effect, ^{at AT = 0.5 OC we} get

Hardouin et al. [17] derived from their experimental

results on CBOOA a critical exponent 1.07, ^with

a value lo N ^0.2 poise ^{at AT =} 0.5 °C. Both results

are in contradiction with the theoretical scaling ^result.

Recently, ^critical exponents 0.37 for CBOOA [18]

and 0.36 for HAB [19] ^{with the same order of} magni-

tude of the effect, yio = ^0.4 poise ^{at AT = 0.5} OC, have been published.

In an attempt ^to interprete the results of Hardouin

et al. [17] ^we may profit ^{from the} expérimental ^results

of Cladis [20] ^{on the critical} divergence ^of K3 ⁱⁿ

CBOOA. She investigated the influence of impurities

on the critical exponent ^of K3 ~ Çll ~ ^{E v. For the}

pure system she found (evaluating the data with a

least-squares fit) the classical value v ⁼ 0.5, ^whereas for impurity concentrations higher ^{than ~ 1} % ^she

found v ⁼ 1. The latter value we propose ^to be inter-

preted ^as the classical value for a two-dimensional system, two-dimensionality being ^evoked by ^the impurities. Assuming ^that they ^{do not like to enter}

the ordered regions, ^these regions ^{extend no more}

over the coherence length jjj Il but over a distance

t ç Il ’ ^{as shown in figure 2. This} represents ^the condition for two-dimensional behaviour. It would be interesting ^{to test this} suggestion by X-ray ^stu-

dies [21] ^on doped systems.

Recently, ^accurate specific ^heat measurements by Djurek ^{et al.} [22] ^{on CBOOA} gave ^a nonclassical critical exponent ^{a, for AT} 1.5°C. Their sample

was of still higher purity ^{than the} samples ^without doped impurities ^{of Cladis} (because ^{of the} higher

transition temperature) for which she found a classical critical exponent v ^{in the same} temperature range.

Interpreting this difference as an effect of impurities

would imply ^{that the} impurities reduce the nonclassical

region. ^{For a} qualitative understanding of this effect

we discuss the temperature range of the fluctuation effects given by ^the Ginzburg ^criterium [23]

where j jj o ^{and çJ.o are the zero} temperature ^coherence

lengths, ^{and AC is the} jump ^{in the} specific ^{heat at}

the phase transition (without fluctuations). ^{In our}

model for the effect of impurities, ^the C;o’s ^{are not} changed by ^the impurities (in contrast to the ^{case of} superconductors). On the other hand, ^the impurities

may increase ^AC due to the entropy ^{decrease asso-} éiated with the aggregation outside the ordered

regions. ^{In this} way 8c would be reduced by ^the impurities. ^{The same} mechanism should serve not to

enlarge 8c ^on passing from three to two dimensions.

The result for the critical contribution yB2) ⁱⁿ

the two-dimensional case for classical behaviour

can be taken from the analogous result for _super- conductors [9]

with

Inserting t ^{= 350 A} (corresponding ^{to t = 10 d}

for CBOOA) ^{we find at} AT= 0.5 -C yb( = ^0.02 poise.

The critical exponent and the order of magnitude ^of

the effect in this case agree better with the experimental

results of Hardouin et al. on CBOOA than in the three-dimensional case discussed above. The _assump- tion of classical theory ^{for CBOOA over the} experi- mentally accessible temperature range is also strongly supported by the results of Salin et al. [24] ^{who found}

no critical divergence ^{for the} width of the twist mode

K2 N

WS2 ^{= -} 1 -,Z- q. ^{in CBOOA.}

Yi

Summarizing the discussion of pretransitional

effects it seems difficult to reach the nonclassical

region in CBOOA due to the still relatively high

transition enthalpy [19] ^at the weak first order tran- sition [22, 25, 26] and the influence of impurities.

For other systems the situation _may be more favo- rable.

The above discussion of pretransitional effects,

which is based on the suggested influence of impurities

should be regarded ^{as a first} attempt ^to bring together

some of the contradictory experimental ^results. Surely,

further experimental investigations ^{are needed to}

clear _up the controversial situation. The critical effects on the low temperature side of the phase

transition will certainly ^{show the same} problems.

Acknowledgments.

^Most of this work was done

during my stay ^{at the} Collège de France. 1 would like to thank all the members of the Physique ^{de la}

Matière Condensée for the pleasant ^{time 1} spent ^there.

Furthermore 1 thank the referee for pointing ^{out two}

errors in the first version of the manuscript.

APPENDIX A In their unified theory JS avoided introducing ^dis-

placements ^or strains. Instead they ^{use the stresses} Hj ^and Tu ^{as dynamic} variables, ^whose equations

of motion were found for a system with translational

and orientational degrees of freedom to be [8]

323

v and il are the drift and angular velocities, respec-

tively. ^In contrast to section 2, ^the diagonal part ^of the stress tensor has not been written down separately,

but was included in the complete symmetric part 77y.

For the symmetry ^{of a smectic A} phase ^{the nonva-} nishing components ^{of the} elasticity ^tensors E, G, and H are given by eq. (2. 3). ^{In addition to} symmetry,

a smectic A phase is characterized by ^{the invariance}

of the elastic _energy or stresses under i) ^a homogeneous

rotation of the layers ^{and the} molecules, ii) ^{a strain}

in the plane ^{of the} layers, ^and iii) ^a rotation of the molecules around their long ^axes. This leads to the conditions

The r.h.s. of eqs. (A.1) ^{can now be} expressed ^{in terms}

of ViVi’ ViVz, Q,, ^and Dy. ^{If we introduce displace-}

ments 0, u,-, cpx, and çoy by

these displacements ^are meaningful quantities ^{in a}

smectic A phase ^{in the sense} that their static fluctua- tions stay finite. On linearizing eqs. (A .1) ^and integrat- ing them, ^the components ^{of the stress} tensors, with

IIscij ⁼ Pbij ⁺ IIsij, ^{agree with eq. (2.5), under the}

substitutions

APPENDIX B The permeation effect is already ^covered by ^the

hydrodynamic theory of smectics A where the mole- cules are fixed to the normal of the layers, eq. (3. 3).

For simplicity ^{we will discuss it} within this theory neglecting the second order elastic constant Kl.

In analogy ^{to JS we start with the} expression ^{for the}

conserved thermodynamic potential cpN per particle

where s’ is the entropy per particle. This relation has to be completed by ^the equations of motion for the

quantities p, p, and _ç, the nonconserved quantity s, and the quasi-conserved quantity uz (or V zUz)

Applying the standard procedure ^{to derive} hydrody-

namic equations the convective currents result as

In the case with dissipation ^{one finds}

As under time reversal Aii ^{is odd, whereas ViT and}

vin:z ^are even, ^no dissipative coupling exists between them. Therefore the constitutive equations ^are

Spatial symmetry allows for five independent ^visco-

sities and two independent components of each the thermal conductivity _Kij, ^the coupling ^constant pj, and the permeation ^constant Âii. Neglecting ^the coupling _Jlij the linearized equation of motion for _Uzz reads

From the case q,- ^{= 0 we} find Â. 1. ^{= 0. After} integration

of this equation ^we find the result _eq. (3. 1), ^with

Àz ⁼ 4.

APPENDIX C The Kubo relations for the viscosities defined in

eq. (3. ) ^{have been} given by ^{JS. For the} compressional

viscosities they ^are

The fluctuating parts ^{of the stresses} follow from eq. (2.5a) ^and (2.5b)

Fluctuations of 0 will be neglected. Comparing ^with

the result of JB for the case Co ⁼ 0, eq. (4. le), ^{we find}

and analogously

As n23 ⁼ ifl ifs the condition for a positive entropy production is fulfilled.

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