Critical slowing down near the smectic-A-hexatic-B transition

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Critical slowing down near the smectic-A-hexatic-B

transition

R. Geer, H.Y. Liu, E.K. Hobbie, G. Nounesis, C.C. Huang, J.W. Goodby

To cite this version:

Critical

_slowing

down

near

the

smectic-A-hexatic-B transition

R. Geer

_(1),

H. Y. Liu

_(1),

E. K. Hobbie

_(1),

G. Nounesis

_{(1, *),}

C. C.

_Huang

₍₁₎

and J. W.

_Goodby

_{(2, **)}

(1)

School of

_Physics

and

Astronomy, University

of Minnesota,

Minneapolis,

Minnesota 55455,

U.S.A.

(2)

AT & T Bell Laboratories,

Murray

Hill, New

_Jersey

07974, U.S.A.

(Reçu

le 27

_février

1989,

accepté

sous

_{forme définitive}

le 10 mai

₁₉₈₉₎

Résumé. 2014 Nous

avons fait des mesures de

_{capacité calorifique}

et conductivité

_thermique,

avec une

_grande

résolution, sur un échantillon de

_{n-propyl-4’-n-heptyloxybiphényl-4’-carboxylate}

(370BC)

au

_voisinage

de la transition

smectique A-hexatique

B. La conductivité

thermique

a une

forte décroissance

_près

de la transition, ce

_qui

montre clairement et confirme le ralentissement

critique

des fluctuations

thermiques, déjà

observé sur des échantillons de séries

_homologues.

On

vérifie que l’anomalie est en accord

_quantitatif

avec la théorie

classique

de la

dynamique

des

phénomènes critiques.

Abstract. 2014 We have

performed high-resolution heat-capacity

and

_{thermal-conductivity}

investi-gations

on the same

_sample

in the

_vicinity

of the smectic-A-hexatic-B transition of

n-propyl-4’-n-heptyloxybiphenyl-4-carboxylate

(370BC).

The thermal

_diffusivity

shows a

_{pronounced drop}

near the transition. This

clearly

demonstrates and confirms the critical

slowing

down of the thermal fluctuations observed in the other member of the

_homologous

series. The

_anomaly

is found to be in

_quantitative

_agreement with the conventional

theory

of critical

dynamics.

Classification

Physics

Abstracts 61.30 -

64.70 - 64.70M

Halperin

and Nelson

_[1]

formulated a

_two-stage

dislocation mediated

_{melting theory}

which would take a two-dimensional

_(2D)

solid to a 2D

_liquid

via two

_phase

transitions instead of

one. This is a dislocation

_unbinding

transition,

first

_{proposed by}

Kosterlitz and Thouless

_[2]

followed

_by

a disclination

_unbinding

transition. These transitions can be

_continuous,

unlike

the first order

_melting

transition of bulk materials.

Intermediate between the transitions a hexatic

_phase

is

_predicted

which has short range

positional

order and

_{algebraically decaying}

bond-orientational order

_(order

in the local lattice

orientation).

Birgeneau

and Litster

_[3]

_argued

that the

_stacking

of such hexatic

_layers

would

(*)

Present address : Center for Material Science and

Engineering,

M.I.T.,

Cambridge,

MA _02139,

U.S.A.

(**)

Present address : School of

Chemistry.

The

_University,

Hull HU67RX,

England.

3168

produce

a kind of smectic

_{liquid-crystal}

_mesophase

which would have three-dimensional

_long

range bond-orientational order due to weak

_{intra-layer coupling,}

while both inter- and

intra-layer positional

orders remain short range. Without

long

range

positional

order,

the

_layer

structures in both the

_{smectic-A(SmA)}

and this intermediate

_phase

_{(the hexatic-B(HexB)}

phase)

undergo large

fluctuations. The HexB

_phase

was first detected

_{through X-ray}

diffraction studies

_by

Pindak et al.

_[4]

in 650BC

(n-hexyl-4’-n-pentyloxybiphenyl-4-carboxy-late).

High-resolution calorimetry investigations by Huang et

al.

_[5]

show that the SmA-HexB transition in 650BC is continuous with a

_{heat-capacity anomaly}

described

_by

the critical

exponent « -

0.60.

_Employing

an

experimental technique

which is an extension of the ac

calorimetry

method

_[6],

Nounesis and coworkers

_reported

a

_strong

_anomaly

associated with the thermal

_conductivity

of 650BC in the

_vicinity

of the SmA-HexB transition

_[7].

This is described

_by

a critical

_{exponent a}

= 0.51 ± 0.04 and a critical

_amplitude

ratio

_{A’ /A-}

= 0.93.

In _{this paper} we

_report

on thermal

conductivity

studies for the SmA-HexB transition in

370BC,

another nmOBC

_compound

_{(n-alkyl-4’-n-alkoxybiphenyl-4-carboxylate).}

The SmA-HexB transition of 370BC has a similar

_anomaly

in the thermal

_{conductivity.}

The heat

capacity

of 370BC over the same

_temperature

_region

is consistent with that found for other

nmOBC

_compounds

with critical

_{exponent a =}

0.60.

_Fitting

the thermal

_diffusivity

data to a

simple scaling

model

_[8]

allows us to determine another critical

_exponent

_(~)

associated with the SmA-HexB transition.

_Furthermore,

we demonstrate that three

_experimental

available

critical

_exponents

_{(a, q}

and

₍₃₎

for the SmA-HexB transition

_satisfy

the

_scaling

relations. Here critical

_{exponent 03B2}

characterizes the

_temperature

variation of the order

parameter.

Details of the

_experimental

_technique

and its motivation have been

_reported

_[7].

Due to the

relatively

poor thermal

conductivity

of

_{liquid-crystal compounds}

_(about

one-fifth that of

glass)

a

_{special sample}

cell has been

_designed

to

_{significantly}

reduce heat loss

_by

conduction

through

the

_sample

container. The cell consists of a thin

sample

of

liquid-crystal

compound

housed between two

_chemically

etched

_glass

slides

_[6].

To determine thermal

_{conductivity,}

A?A

the

_apparent

heat

_capacity

_{per unit} area,

CA

APA

is measured as a function of

w AT

temperature

throughout

the transition

_{region. OP A is}

the ac _{power oscillation}

amplitude

_per

unit area

_applied

to the bottom side of the

_sample

cell, w

the

_{angular frequency}

of this

oscillation,

and AT the

_resulting

_temperature

oscillation

_amplitude

detected on the

_top

side of

the

_sample

cell. This variation of the

_apparent

heat

_capacity

was measured with

_high

data

density

( - 1200

data for each

_frequency)

for eleven different

_{frequencies. Through}

interpolation

of the

_data,

the

_frequency

_response curve, i.e.

C-1 =

wAT

vs.

f ( =

2 (J)7T ),

is

APA

2 ir

obtained for each of the 106 different

temperatures.

A linear

_fitting

_[7]

was done for

This determines _WH, the

_high

cutoff

_frequency

which is

_directly

related to the effective

thermal

_diffusivity

of the entire

_sample

cell,

and can be used to determine the thermal

conductivity

of the

_{liquid-crystal}

_{sample through}

the

_expression

where

_{C A}

= 2

hG Ce

₊

hL C L

is the total heat

capacity

per unit area. The

_quantity

h refers to

thickness,

C to heat

_capacity,

and K to thermal

_{conductivity.}

The

_subscripts

G and L refer to

measurements were done for the

_glass

slides used to hold the

_sample

over the relevant

temperature

region

to determine the

_glass

contribution to

_{equation (2).}

Other methods used to measure thermal

_conductivity

_{of liquid-crystal}

_mesophases,

_{e.g., the}

steady

state method

_[9],

thermal lens effect

_[10],

and forced

_{Rayleigh light scattering}

technique [11, 12],

require

a

_{fairly large}

_temperature

_gradient

inside the

_sample

to obtain a

measurable

_signal.

Our

_{technique requires only}

a small amount of

_sample

_(-

15

_mg)

with a

temperature

oscillation of about 3 mK rms over the

_sample.

Both are

_important

factors in

studying

critical

_phenomena.

The measurements of Nounesis and coworkers

_[7]

showed a

_peak

in the thermal

conductivity

and a

_drop

in the thermal

diffusivity, DT

=

KICP,

of 650BC near the transition

temperature

(Tc).

Our results on the heat

capacity,

thermal

conductivity

and thermal

diffusivity

near the SmA-HexB transition of 370BC show the same behavior as that of

650BC. The heat

_capacity

data have been

_published

in reference

_[13].

The transition

temperature

for our

_sample

was

_{70.960 °C,}

and remained

_stable,

with a

_Tc

shift less than

5 mK/d

_during

the course of the measurements.

The

_drop

in thermal

_diffusivity

in both 370BC and 650BC illustrates critical

_slowing

down of the thermal fluctuations. For smectic

_{liquid crystals,}

thermal studies

_by

Rondelez and coworkers

[11]

show that heat

_transport

is

_{essentially governed by}

orientational

_ordering

and molecular

_properties,

but not

_{by long}

_range

_{positional ordering,}

or the smectic

_layer

order. This

_suggests

a direct

coupling

between thermal and hexatic order fluctuations near the

transition in our

_system.

Recently

Hobbie and

_Huang

_[8]

_proposed

a

_{simple scaling}

model for smectic

_{liquid crystals}

that

_predicts

the thermal

_diffusivity

to be

_{given by}

where y

is the order

parameter

susceptibility, e - X

1/ (2 - ~) the « bulk » correlation

_{length, t}

the reduced

_temperature

and

_F,,

the kinetic coefficient associated with the order

_parameter.

B is a

_background

term that should be a

_relatively

smooth function of

_temperature.

For the

conventional

_theory

of the critical

_dynamics

of nonconserved order

_parameter

fluctuations

(z

= 2 - _q

),

Fe

will be

_independent

of the correlation

_length.

_{Taking X -}

_{1 t 1- ’Y}

and

_using

static

_scaling,

the above

_expression

for

_DT

1

_predicts

_{CPIDT = A 1 t 1 - "’ + "n}

_{+ B,}

where

Fig. 1.

- The

3170

Fig.

2. -

Temperature

variation of thermal

_{conductivity (a)}

and thermal

_{diffusivity (b)}

in the

vicinity

of the SmA-HexB transition of 370BC. The dots and line represent the measured and calculated behavior,

respectively.

A +

_{/A - = (X -}

_{lx + )(3}

+ 11 )/(2 - 11).

A

_power-law

fit of the ratio

_CplDT

was carried out with a

weakly

linear

_background

term. The results are shown in

_figure

1. From this

_exponent

and the

previously

determined value of a we find v~ = - 0.10 ± 0.03 and

X + /X -

= 1.13,

in very

good

agreement

with similar fits of the 650BC data

_[8].

_Employing

the

_scaling

relations

(dv = 2 - a, 2 v - TI v = 2 - 2 f3 - a),

we obtained

03B2

=

0.18,

which is consistent with the

value measured

_by

Ho and Rosenblatt

_[14]

for 650BC. The fits of the measured thermal

diffusivity

and

_conductivity

are constructed

_according

to

_DT

=

C pl (A :t 1 t 1- a + 11 v

+ B )

and K =

_{C P/ (A± t - « + ~v}

_{+ B )}

_using

the fitted

_expression

for

_CP.

These fits are shown in

figure

2a and

2b,

respectively.

The fact that the heat

_capacity

_{(C )}

has a

_sharper

rise than the

drop

in the thermal

_diffusivity

_{(DT )}

near the transition

_temperature

leads to a

_divergence

in

the thermal

_conductivity

_(K

= C x

_DT).

_Although

we know that the

_drop

in

_DT

is related to

the critical

_slowing

down of thermal

_{fluctuations,}

we do not have

_{microscopic explanations}

for the

_divergence

in thermal

_{conductivity.}

This

_agreement

of the data with the conventional

theory

of critical

_dynamics

_(z

=

2 - q )

depends strongly

_on the

validity

of the inferred

f3.

Although

the value of q is

_{unusually large}

and

_negative

(q

= - 0.21 _± 0.03

),

it should be

noted that the

_exponents

y and v are

_{only slightly}

removed from their mean field values

( y

= 1.04 ± 0.05 and v = 0.47 _±

0.01).

In

_light

of this

_strange

value of ~, it is worthwhile to consider some other

_{possible dynamic}

schemes. For z

_unknown,

_{dynamic scaling gives}

F e2 -,l -z

_which,

when

_substituting

into the above

_expression

for

_DT-1,

_gives

the

_singular

_part

of

_CP/DT

_proportional

to

1 t 1- a + v (Z - z). This

implies

DT - 03BE2-z

which is what one

_gets

_{by extending}

the

_dynamic

scaling assumption

to the thermal diffusion mode

_{úJ q}

=

DT

k2.

The fact that

a) qsoftens

weakly

near

_{T, implies}

that z is

_{slightly larger}

than 2. Renormalization group calculations of z for an

auxiliary

conserved

_density

that

_couples

to the order

_parameter

fluctuations like the

temperature

have been carried out

_{by Hohenberg}

and

_Halperin

_[16].

For a one

_component

order

_parameter,

which would be an

_{incomplete description}

of hexatic

ordering, they

find that

z = 2 ₊

a 1 v,

which would

imply

a

_softening

of

_DT

with an

_exponent

a. In that case the thermal

_conductivity

is

_predicted

to be smooth

_through

the transition. For a two

_component

order

_parameter,

which in this case would be more

_realistic,

_they

find that there are

value of z is not clear. The situation can be made more

_{complicated by}

_{nondissipative}

couplings

between the order

_parameter

and other

_hydrodynamic

modes of the

_system

that arise from the Poisson bracket relations between these

_quantities.

Such a

_coupling

between

the

_phase

of the order

_parameter

_{and the energy}

_{density gives}

rise to the anomalous

_transport

of heat above the lambda transition of

4He.

In smectic

_liquid

_crystals,

however,

none of these

couplings

involve thermal fluctuations

_directly,

and the thermal diffusion mode is

_purely

diffusive in both the SmA and HexB

_phases.

In

addition,

it is most

_likely

fluctuations in the modulus of the order

_parameter

below

_T,

that are relevant in

_determining

z, as

_opposed

to the

case in

_superfluid

4He

where

_amplitude

fluctuations are frozen out in the ordered state. There has been

_speculation

that the SmA-HexB transition is

_tricritical,

so it is worth

_mentioning

that the critical

_exponent

z has been calculated to be 2 for relaxational models at a tricritical

_point

[16].

Hence extended

_{dynamic scaling predicts}

that the thermal

_diffusivity

would not soften at

Te.

Whether or not the

_softening

that is observed would arise a finite distance away from such a

_point

is not _{clear. At any rate,} the value

of f3

that has been measured for 650BC

_along

with the

_similarity

between the behavior observed for both 650BC and 370BC

_suggest

that the conventional

_theory

is

_adequate.

A direct

_experimental

determination of the

_exponent

q

would shed more

_light

on this.

For 370BC the

_fitting

_{range for the heat}

_capacity

extended to

_about

_{T - Tc}

₁

$ 20

_mK,

showing

a

_power-law

behavior closer to the transition

_temperature

than the ratio

CP/DT,

which becomes rounded

_for

_{T - Tc 1}

30 mK. Similar behavior was observed for

the 650BC. _{One may}

_speculate

that the

_rounding

in _wH

_(and

hence

_CPIDT)

is due to our

unique

measurement

_technique

_[17].

_Recently,

Hobbie and

_Huang

have

_investigated

the

temperature

variation of thermal

_conductivity

in the

_vicinity

of the SmA-smectic-C

_(SmC)

transition of one

_{liquid-crystal}

_compound

[18].

_They

have found that both heat

capacity

and

thermal

_conductivity

show

_sharp

_jumps

at the mean-field SmA-SmC transition.

Furthermore,

judging

from the 10 %-90 % width of the

_{jump, they}

obtained a

_sharper

_jump

for the thermal

conductivity

than that for the heat

_capacity.

Thus,

this result indicates that our measurement

technique

will not increase the

_{rounding region}

of the thermal

_conductivity

in

_comparison

with that for the

_{heat-capacity anomaly.}

In

_conclusion,

we have

_{performed high}

resolution thermal

_conductivity

studies on the

SmA-HexB transition of

_370BC,

_confirming

the results of Nounesis et al.

_[7]

with 650BC. Our results once

_again

show a

_dip

in the thermal

_{diffusivity, demonstrating}

the critical

_slowing

down of the thermal fluctuations consistent with a conventional

_theory

of the

_dynamics

of the

hexatic order

_parameter

fluctuations. Another critical

_exponent

_(i.e.

_q)

is obtained. Its value satisfies the

_scaling

relation with the other two critical

_exponents,

_namely,

a and

_/3.

A clear

picture

of the fundamental nature of the SmA-HexB transition is still

_lacking.

Acknowledgment.

3172

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[15]

The thermal

_{diffusivity, DT,}

can be related to the thermal

conductivity,

K,

by DT

=

K/(03C1CP),

where p is the

_{specific gravity.}

Both the smectic-A and hexatic-B

phase

do not have

_long-range

positional

order, so the _temperature variation of the nearest

_{neighbor spacing}

can

_give

us

approximately

the

_{density change through}

the transition.

According

to the

_X-ray

measure-ments

_(DAVEY

S. C., BUDAI _J., GOODBY J. W., PINDAK R. and MONCTON D. E.,

Phys.

Rev.

Lett. 53

(1984) 53),

there exists no

_anomaly

in the

_{nearest-neighbor spacing through}

this transition. Thus we _expect no

_anomaly

in p

through

this

phase

transition.

[16]

HOHENBERG P. C., HALPERIN B. I., Rev. Mod.

_Phys.

49

₍₁₉₇₇₎

435.

[17]

The

_equivalent

circuits model

_leading

to

_equation

₍₂₎

is valid near

_Tc

and does not contribute to

rounding.