Difference in critical behaviour of the phase transitions nematic-smectic A1 and nematic-smectic Ad

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Difference in critical behaviour of the phase transitions nematic-smectic A1 and nematic-smectic Ad

Wim Bouwman, Wim de Jeu

To cite this version:

Wim Bouwman, Wim de Jeu. Difference in critical behaviour of the phase transitions nematic- smectic A1 and nematic-smectic Ad. Journal de Physique II, EDP Sciences, 1994, 4 (5), pp.787-804.

�10.1051/jp2:1994165�. �jpa-00248001�

Classification Physic-s Abstiacts

64.70H 64.60F 61.30

Difference in critical behaviour of the phase transitions

nematic-smectic Ai ^and nematic-smectic A~

Wim G. Bouwman

(*)

and Wim H. de Jeu

(**)

FOM-Institute for Atomic and Molecular Physics, ^Kruislaan 407, 1098 SJ Amsterdam,

The Netherlands

(Receii,ed 21 June 1993, revised 7 February 1994, accepied 9 Febt.uary 1994)

R4sum4. A l'aide de la diffusion de rayons X h haute rdsolution, _{nous avons} dtudid les transitions de phase

n6matique-smectique

Aj dans le BOPCBOB _et

n6matique-smectique

A~ ^dans

un mdlange ^binaire de 7CB _et BCB (7,6CB). Les valeurs _connues de

l'exposant

critique de la chaleur

spdcifique

_{pour ces} deux

systkmes

sont en accord avec les prddictions _pour le modkle 3D XY. Pour le BOPCBOB _nous trouvons des valeurs de la

susceptibilitd

_yet de la

longueur

de corrdlation

longitudinale

_vi

qui

sont aussi _en accord _avec le modble 3D XY, _ce

qui

confirme les prddictions de de Gennes.

Cependant

les valeurs de yet de _vi obtenues pour le 7,6CB _sont plus grandes _que les valeurs du modble 3D XY. Ceci

suggkre

l'existence d'une diffdrence dans le comportement

critique

des transitions de phase

n6matique-smectique

Ai et n6matique-smectique Ad- Dans _tous les _{cas, vi est} sup6rieur h _v~. De plus, nous

analysons

la d6pendance ^{des valeurs}

obtenues pour les exposants

critiques

_avec la forme

sp6cifique

du facteur de structure utilis6e.

Abstract. A high resolution

X-ray study

^is

presented

^{of the nematic} to smectic Aj phase transition in BOPCBOB and of the nematic _to smectic Ad Phase transition in _a binary mixture of 7CB and BCB (7.6CB). The known critical exponents ^{of the heat}

capacity

of both these systems

agree with the predictions ^{for the 3D} XY model. For BOPCBOB

we find values for the

susceptibility

_y and the

parallel

correlation

length

_vi that also agree with the 3D XY model, thus

confirming

the predictions of de Gennes. However, the values _y and _vi of 7.6CB _are larger than the 3D XY values. This indicates _a possible ^{difference in} the critical behaviour of the nematic-

smectic Ai and the nematic-smectic Ad Phase transition. In all _{cases vi} is larger than v~, Furthermore it is shown how the values for the critical exponents depend _on the

specific

form

chosen for the _structure factor,

1. Introduction.

In the nematic

liquid crystalline phase,

orientational order is present ^but no

long

_range

positional

order

[1, 2].

The

anisotropic

molecules _are

aligned,

_{on average,} with their

long

axis

(*) Present address _: Ris~ National

Laboratory,

P-O, Box 49, Roskilde, DK-4000, Denmark, (**) Also Open

University,

P,O. Box 2960, 6401 DL Heerlen, The Netherlands.

parallel

to the director _n. In the

liquid crystalline

smectic A

phase

the orientational order of the

long

molecular axis is retained, while the centres of _{mass are, on} average, distributed in

equidistant layers perpendicular

_to the

long

_axes, thus

forming

a one-dimensional

crystal

of

liquid layers.

The

phase

transition between the nematic and smectic A

phase (NSA Phase transition)

_can be considered _as

freezing

in

just

_one dimension. This _seems therefore _to be _one of the _most

simple phase

transitions in _nature and _an ideal system to

study.

For this _reason

much theoretical and

experimental

work has been done _to characterize this

phase

^transition

during

the last twenty years. The basis of the theoretical

understanding

of the

NSA Phase

transition is the

analogy

with the normal _to

superconducting phase

transition in metals, _as

shown

by

de Gennes

[3]. Taking

the director _n

parallel

to the _z

direction,

the

density

modulation around the average

density

_{p~ can} be described

by

P

(r)

=

poll

₊

Rej4i(r)exp(iqo z)j)

,

(i)

where the

complex

^order parameter #i has an

amplitude

#i and _a

phase

#. The _average of this order parameter

(#i)

is _zero in the nematic

phase

and has _a finite value in the smectic

phase.

One of the

interesting

features of

phase

transitions is that

apparently

^different transitions _can be described

by

the _same universal mathematics,

leading

to the _same critical exponents

[4].

A

determining

^{factor is} the

dimensionality

of the order parameter, ^{which is} two in the _case of the

NS~ phase

transition.

According

_to renormalization group

theory,

^this

phase

transition should

therefore fall into the 3D XY

universality

class

[5].

Some

special

features

complicate

the

phase

^transition in these systems

[6].

For systems ^with

a short nematic range the

NS~ phase

transition is first order. For

compound

with _a

larger

nematic range a tricritical

point

is

predicted [7, 8]

and found

[9-11].

Calorimetric data

(in particular

the associated critical exponent

a)

for

NS~ phase

transition in systems ^with a

large

nematic range can be well described

according

to the 3D XY

universality

class

[12].

When the nematic range is

reduced,

_{a cross-over} of the values for the critical exponents ^{is found} upon

approaching

the tricritical

point [13].

Another feature is the

algebraic decay

^{of the}

long-range

smectic order, due _to the presence of undulations of the smectic

layers.

This effect is related to the

anisotropy

observed for the critical exponents

describing

the correlation

lengths

of the

smectic fluctuations in the nematic

phase

_{vii ~ v~.}

The last feature

given

above _can be clarified

using

the Landau-de Gennes free energy. The free energy

describing

the

phase

transition _can be written _as

[14]

f(4i, 3n)=)A14il~+ C14il~+jE14'l~+Cilvi 4'l~+Ci((Vi -iqo3n)4'(~+

+

[Kj(V.3n)~+K~(n.Vx 3n)~+K~(nxvx 3n)~]. ₍₂₎

2

The first five terms

correspond

to the

Ginzburg-Landau formalism,

in which the

phase

transition is driven

by

a

change

in

sign

of A. In the fifth _term for the

perpendicular gradient,

director fluctuations have been included _to preserve invariance of the free energy with respect

to small uniform rotations of the director and the smectic

planes.

The last three terms describe the Frank distortion free energy for _a

nematic, Kj, K~

and

K~ being

the elastic _constants for

splay,

twist and

bend, respectively.

Below

T~~

in the smectic

phase

a

splay

deformation of the director

corresponds

to undulations of the

layers,

as shown in

figure

I.

Bending

the

layers

^does

not affect the correlation

length iii

^{in the} direction

parallel

to the director. However, these deformations do decrease the correlation

length i~ perpendicular

to the director. In the nematic

phase

^this

coupling

between the director and the smectic order parameter ^increases with the

strength

and size of the smectic fluctuations. Therefore upon

approaching

the

phase

transition the

perpendicular

correlation

length

becomes

increasingly perturbed by

the

splay

k

/li

Fig.

I. In the smectic phase ^{the director} n is perpendicular to the layers.

Splay

deformations of the _n

correspond

to undulations of the smectic

layers.

deformations, leading

to a value for the critical exponent _v~ ^{smaller than the 3D} XY value calculated without this

coupling [6].

The

quartic

term that appears in the structure factor _can also be attributed _to the

splay

term

[15].

Als-Nielsen et al.

[16]

have shown for the first time that such _a

quartic

term

significantly improves

the

fitting

of the data and

suggested

that it

originates

from the

splay-mode

director fluctuations. The full _structure factor is described

by

S(q)

= ~

~"

~ ~ ~ ~ ^,

(3)

' ₊

ill

(q= q=o) ⁺

ii

_qi +

is

qi

where q~

(parallel

n) and q~ =

(q(

₊

_q()~/~

_{represent the components of the}

wave vector

transfer. The critical behaviour associated with the smectic

susceptibility

_« is described

by

the exponent y.

So far, we have not discriminated between different smectic A

phases [17, 181.

As _we shall

see there _are indications that the difference in the _structure of the various types ^{of smectic A}

phase

could be

important

for the critical behaviour of the

phase

transition _to the nematic

phase.

The

simplest

version is the smectic

A~ phase

in which the molecules _are either

symmetric

or

effectively

so because of _a random

up-down

orientation. When strong

end-dipoles

are present,

dimers may be formed with

partial overlap.

In the

resulting

smectic

Ad Phase,

the

layer

distance d is

larger

than the molecular

length

^f

(d

_~ ^f and smaller than twice the molecular

length (d~

2

f).

Other

possible phases

occur for

compounds

^with strong

end~dipoles

and

usually

at least three aromatic

rings.

The smectic

A~ phase

has antiferroelectric double

layers.

The repeat ^{distance here is twice the molecular}

length (d

m

2

f

_{). In} these types of

compound

also the smectic

Aj phase

_may be found. The difference between the smectic

A~

and the

smectic

Aj phase

^{is subtle.} In the smectic

Aj phase

there _are strong ^smectic

A2

fluctuations, while the

layer

^thickness

corresponds

to the molecular

length (dw1) [19].

Earlier _we have

presented preliminary

results for the nematic _to smectic

Aj (NSAj) ^Phase

transition in BOPCBOB

(see

Tab.

I)

which _was the first

compound

to show 3D XY values for _y and vii

(though

still _vii

/v~

_~

l),

in addition _to the

already reported

3D XY value for _a

[15].

Recently

more systems ^with

NS~~ phase

transitions

exhibiting

^{similar critical} behaviour have been found

[20].

However, for _some

NA~ phase transitions,

different critical exponents ^have been

reported.

For two systems ^with a 3D XY value for the exponent a, indicated

by

the

acronyms 40.7

[21]

and

iS5 _{[13, 22],}

_{the values found for}

y and _{vii are}

significantly larger

than the 3D XY values. The

objective

of the work to be

presented

in this paper is

(I)

_to

provide

full information _on the

BOPCBOB-system [15]

_;

(it)

to

expand

these data _{to a} system ^with a

NS~~ ^phase

^{transition with XY character for the} calorimetric data. The natural choice for such _a

Table I. Chemical _structure

of

the

compounds

studied in this _{paper. ~b} stands

for

_a

pheny/

group,

Compound

Chemical _structure

nOPCBOB

C~H~

~ ~

j-O-j -OOC-~b-O-CH~-~b-CN

nCB

C~H2

n + i fb fb ^-CN

study

is the series of nCB

compounds. They

^{differ from BOPCBOB}

by having

a shorter central

core (two aromatic

rings

instead of

three,

_see Tab.

I) leading

_{to an S~~}

phase.

Thoen et al.

[10]

characterized the _{cross over} from tricritical _to XY behaviour of the

NS~~ phase

^{transition in this}

series with adiabatic

calorimetry.

The results indicate that in

spite

of the fact that for both

BOPCBOB and 7,6CB the exponent _a agrees with the

predictions

for the 3D XY model, for

7.6CB the values _y and _{vii are}

larger

than the 3D XY values. This could indicate that the critical

behaviour of the

NA~

(and the

NA~) phase

transition differs from that of the

NAj phase

transition.

The paper is

organized

_as follows the _next section describes the

experimental

_apparatus.

Section 3 presents the results and data

analysis

and shows the

importance

of the form chosen

for the structure factor. Section 4

gives

_a discussion

including

the theoretical

descriptions

available _to

interpret

the

experimental

situation. Section 5

gives

conclusions. In _a

appendix

_we present ^{details of the line}

shape analysis.

2.

Experimental.

The 3D XY critical behaviour of _a

NS~ phase

transition is studied for

octyloxyphenylcyanoben- zyloxybenzoate (BOPCBOB) [23].

The

sample

of

BOPCBOB,

shown in table I, was obtained from

professor

G.

Heppke (Technical University

of

Berlin, Germany). High

resolution _ac

calorimetry

showed non-inverted XY critical behaviour

[24].

The critical behaviour of _a

NS~~ phase

^transition has been studied in mixture of

alkyl-cyanobiphenyls,

shown in table1

(7CB

and

BCB).

These

compounds

were used _as obtained from BDH

(Poole,

Dorset,

UK).

Thoen et al.

[10]

found for the mixture 7.6CB with adiabatic

scanning calorimetry

an XY value for _a.

Following

their

recipe

^7.6CB was

prepared

as a mixture of-7CB with BCB with _a

concentration of xsc~ = 0.5874.

Though

the set-up ^and measurement

technique [161

_are in

principle

standard for this type ^of measurements, there _are many subtleties that

justify

a somewhat detailed discussion. Chan et al. have

given

an excellent

description

of _a similar

experiment [25]. During

a measurement the

compound

_was

kept

ⁱⁿ _a

beryllium

container which _was 1.5 _mm thick, 3 _mm wide and 8 _mm

high.

The thickness is

roughly equal

to the lie

absorption length

^of

organic compounds

for

X-rays,

thus

assuring

that the scattered

intensity

is maximized. The container is closed with

a teflon cap which is

tightly

fit in order to prevent

leaking.

The

beryllium

cell is contained in _a copper block. This inner _oven is

gold plated

to reduce the heat losses to the outer oven

by

radiation. A 6 _mm diameter hole allows for the

X-rays

_{an access}

angle

of 35° with the normal to the

sample.

At both windows _a 50 _~Lm

mylar

foil with _a lo _~Lm aluminium

coating

is

glued

upon the inner oven as a radiation shield. This radiation shield

helps

to distribute the temperature more

evenly.

The inner _oven is heated

electrically by

a nickel oxide resistor wire.

The temperature ^{is measured}

by

a

4-point

measurement of the resistance of _a Pt100

platinum

resistor and _set with _a

stability

better than 3 mK

by

the combination of _an AZA F26 resistance

bridge

and _an AZA 300 PID temperature controller.

Two stainless steel tubes

keep

the inner _oven in

place

in the _{outer oven.} Nickel oxide resistance wires _are used _to heat the _{outer oven}

electrically.

A Eurotherm 825 PID temperature controller stabilizes its temperature ^{0.5 K below the}

phase

transition

temperature

^with an

accuracy of 0.I K. In this _way

gradients

_are

kept

^{smaller than 3} mK _over the illuminated

sample height

of 3.0 _mm. The

gradients

were measured

by locating

the

phase

transition temperature ^{of different}

parts

^{of the}

sample,

selected

by setting

_narrow horizontal slits

[22].

Inside the outer oven two

permanent

cobalt-samarium

magnets

are

placed

_on either side of the inner _oven, which

provides

_{over a gap} of 13 mm a field of 0.4 T _to

align

the director of the

sample.

With the

experiment

with

7.6CB,

the _corners of the magnets were cut off in order _to reach the full _access

angles

_up to 35° allowed

by

the _oven.

Beryllium

windows

provide

an

entrance and exit for the

X-rays.

The outer oven is in _tum insulated with

glass

wool and

aluminium foil.

The

X-ray

_source is _an Enraf-Nonius GX-21

rotating

anode generator ^with a copper

anode, operated

at 7 kW with _an effective spot ^{size of 0.5} x 0.5 mm2. The

K~j

line (A ₌

1.54051)

used is broadened

owing

_{to a} finite life time of the excited state, which _means that it has _a

Lorentzian

spectral shape.

Its relative half-width-at-half-maximum

(HWHM)

is

given by

AA/A

= 1.6 _x

10~~

The

triple

axis

spectrometer,

^built at the Ris~l National

Laboratory

in

Roskilde, Denmark,

has been

designed

to obtain _a

high

resolution in

reciprocal

_space

(in angles)

and _a low

background. Figure

2 shows _a schematic top ^{view of the} spectrometer. ^Next to the

rotating

anode

(A),

the first slits made of lead

(B)

select the part ^{of the}

X-rays

which _can

eventually

reach the

sample,

thus

already removing

much of the

background.

At 385 _mm from the

electron focus at the anode _a

single Si(I

I I

) crystal (C) serving

as a monochromator is fixed _{to a} rotation table. The monochromator

crystal

is

positioned

to reflect the

K~

lines in this situation the

K~

^{line is} not

accepted. Bremsstrahlung

with _an

integral

number times

higher

_energy

(A/2,

A/3, is reflected under the _same

angle

and has _to be removed later. To _remove the

K~~

^{line 0.5} mm wide slits

(E) (Huber,

3 000

series)

are

placed

460 _mm away from the

monochromator, just

before the

sample.

The

K~

lines _are

spatially separated

at this

position.

z

y~~+

x

c ~

D

A G

Fig. 2. Schematic top ^{view of the} triple ^axis spectrometer ^{and the definition of} the _x= scattering plane.

A

rotating

copper anode, B slits, C monochromator

crystal,

D capton ^foil to scatter part ^{of the direct} beam to a monitor above it, E _: xy

adjustable pre~sample

slits, F

sample

rotation table, G

analyser

slits, H _: analyser crystal (the same as C), and I NaI scintillation counter.

The

height

of the

part

of the

sample

with the

required homogeneity

in temperature and orientation is selected

by setting

the horizontal slits at 3.0 _mm. A

capton

^{sheet is}

placed

115 _mm behind the monochromator _{to scatter} about 0. I fb of the direct beam into _a detector

(D) perpendicular

to the

scattering plane

in order to monitor the

incoming

beam

intensity.

The heart of the spectrometer is _a

heavy

column _on a

rail, supporting

the

sample

table

(second axis)

and the detector _arm. The rail makes it

possible

to centre the column into the X- ray beam. The

sample

table

(F)

is situated _on the centre of the

column,

550 _mm behind the monochromator. An

xy-translation

table allows to put the centre of the

sample

in the _centre of the beam. A detector _arm is connected to the _same column with the column axis _{as centre} of rotation.

Analyser

slits

(G),

525 _mm away from the

sample,

cut off

background

and _more of the tails of the direct beam and narrow down the vertical resolution. Its

settings

_are 0.7 _mm

wide and 5.0 _mm

high.

Just behind this slit is the third

axis,

_an

analyzer crystal (H),

at a

distance of 585 _mm from the centre of the

sample

table. This is _a

Si(I

I

I) crystal, just

as used also for the monochromator. The combination of the two

crystals,

set in the

non-dispersive

mode

[25],

determines the resolution of the direct beam which is I.I

mdegree

HWHM. The detector

(I)

is

placed just

behind the

analyser.

Evacuated

flight

_passes are

positioned

between the generator ^{and the} monochromator, between the monochromator and the

sample,

and

between the

sample

and the

analyser

to reduce air

scattering.

All tumtables and translation stages are driven

by Slow-syn

stepper motors, in combination with _a Huber stepper motor

control.

By

the _use of additional lo gears this allows for steps ^{of either I}

mdegree

_or I ~Lm.

Both the monitor

(D)

and the detector

(I)

consist of _a NaI scintillation

crystal directly

attached _{to a}

photomultiplier.

A NaI

crystal

^emits

light,

_or scintillates, when

X-rays

_pass

through

it, which is detected

by

the

photomultiplier.

The output ^{of each}

photomultiplier

is

amplified

and fed into _a

single

channel

analyser.

Here _an energy window is _{set to} cut off the thermal

background

_on the low side and the

Bremsstrahlung

at A

/2, /3,

etc. also reflected

by

the monochromator and

analyser crystal,

_on the

high

side. A dual _{counter counts}

continuously

and transports ^the output ^{of the}

single

channel

analysers

to an Olivetti M24 computer. ^The detector output ^{is normalized}

by

the monitor output to correct for drifts in the

X-ray intensity

from the generator. ^The stepper motors, the counter and the temperature controllers _are all interfaced with _an IEEE-488 system. The Eurotherm temperature controller is interfaced with _a modified RS232 interface.

At the start of _a measurement

series,

first the director of the

sample

is

uniformly aligned

in

the nematic

phase

with the

magnetic

field. To determine the

phase

transition temperature

T~~

the

sample

is cooled down

slowly (±

I

K/h)

from the

magnetically aligned

nematic

phase

into the smectic

phase.

In this way a

mosaicity

^with a HWHM of

typically

0.3° is reached.

Then with fast, short _{scans at}

increasing

temperatures

T~~

is determined within 3 mK.

T~~

is

easily recognized by

the

change

in the

peak intensity,

_as well _as in the

longitudinal

and

transverse line width

(Fig. 3).

With

increasing

temperature, ^{in the smectic}

phase

the

intensity

decreases and the

longitudinal peak

width is still

nearly

resolution limited,

though slowly increasing.

The transversal line width is determined

by

the

mosaicity,

^{which becomes smaller,}

Above

T~~,

in the nematic

phase,

^the

intensity

decreases

abruptly

and the

rapidly decreasing

correlation

lengths

lead _to

broadening

of the

peak.

The

change

in the transversal line width shows up as a

dip

^which is the clearest indication for

T~~.

^{In the} _range ^of

reciprocal

space where the measurements were taken

(up

to q= m 0.2

l~ '),

the

experimental background

was

mainly

due _to the dark current of 0.05 cts/s of the detector. The

intensity

of the

quasi Bragg

peak

decreased from l 000cts/s close _to the

phase

transition to l ct/s _at the

highest

temperatures ^measured.

The resolution in the q~ direction is determined

by

^the

spectral

line width of the

K~j

line and

by

the

divergence

allowed

by

the monochromator

crystals (Darwin-Prins curve).

1600

~

oo

~ E

~

#

o

i

~

~

. ~

4~

j D

Q ~ ^4j

(

E

"

D°

.

~

~

~

D

h )

. Gl

11 Ob

D

~

OD ^Q

0

119.43 119.44

sample

and the anode 935 _mm. The

X-ray

_{source can} be considered _{as a}

point

_{source, so} that

with these dimensions _a

trapezium shaped

resolution function is obtained, which _can be

described

by

a Gauss function

R~(q=)

= exp

[- (q~/«,)~]

with _a HWHM

«~

=

2.0 _x 10~ ^~

l~

'

The effective resolution function is the

product

of the resolution functions in the _two relevant directions _:

expi- (qv/«y)~i

~ ~~~

~

~"~~~~ ~~~~~~

_~

ii

+

(q=/«-)~i

^~~~

The additional effect of the

mosaicity

of the

sample

could

complicate

the resolution. The

mosaicity

is

certainly important

in the smectic

phase

_; its

possible importance

in the nematic

phase

will be discussed later. In the smectic

phase

the instrumental resolution function convoluted with the

mosaicity

results in _an effective resolution function

R~ (q

). In _{our case} the

mosaicity gives

the HWHM of the solid

angle

that determines the _arc of the

sphere

_over which the

Bragg peak

is smeared _out. Over _{a range} of at least 15 times the HWHM the

mosaicity

_can be well described

by

a Lorentz function. In

principle,

the

mosaicity couples

in all directions

with the resolution function if both have similar dimensions. In

practice,

this

coupling

is

only

relevant in the q,

direction,

where the

mosaicity completely

determines the effective

resolution. A

typical

value for the

mosaicity

of 0.3° HWHM

gives

rise _{to a} Lorentzian

shaped

resolution function

R~(q_,

₌

[I

+

(q,J«, )~]~

^{' with} a HWHM _«,

=

9 _x

10~~ l~

' The effec- tive resolution function with the

mosaicity

correction included is then

given by

R~(q

=

R,(q~

R

(q)

₌ ^~~~

~~

~~~~"~~~~

~

(5) Ii

+

(q/~r,) j ii

₊

(q~/~r~)

3. Results and data

analysis.

The

NS~ phase

transition temperature

T~~

=

121.4 °C of BOPCBOB [271 could be determined

by

a

sharp

_cusp in the line widths with _a relative accuracy of 2 mK (see

Fig. 3)

and _was found

to decrease

by

0.6

mK/h.

The

mosaicity

of the

sample

in the smectic

phase just

below the

phase

transition temperature was 0.2°. The _wave vector

corresponding

to the modulation

period, qo=2

grid _was found at

0.21451~'

_and

_hardly

_{shifted with} temperature I _x 10~ ^~

l~ '/K).

_Some

_typical experimental

data _are shown with their fits in

figure

4. Values

1.25

o 1

~

0.75 O-S

0.25 o

o

0.9 0.95 1 1.05 1-1 O-S 0.25 0 0,25 O-S

qJ/qo q~/qo

Fig.

4. Fits _{to a}

longitudinal

(left) and transversal (right) scan of the smectic fluctuations of 7,6CB _at

r = x 10~~ using equation (A.8). The drawn line represents a fit with _a quartic term included, the dashed line represents a fit without _a

quartic

term,

for _«,

fj, f~,

^and

f,,

^{shown in}

figure 5,

have been extracted from these type ^of

X-ray scattering

data with the line

shape analysis

described in the

appendix.

The data for

« and

iii ^{have been fitted} to a

single

_power law _over the reduced temperature range of

lx10~~~r~5 x10~~.

_{Correction to}

scaling

terms for the data further away from

r =

0 _{were not} included in the

analysis.

Data for

f~

were

only

reliable for reduced

temperatures up ^to r =

I _x 10~ ~, since at

higher

values the

peak

became wider than the _access

angles.

Since the tails of the

peak

were

already

cut off at lower reduced temperatures,

f~

was

only

fit up to _{r =} 3 _x 10~ ^~

Extrapolated

values for

f~

^and

f~

_were used in the fits to

obtain wand

iii.

^The

stability

of the fit parameters was checked

by

_range

shrinking.

j~7

l~

~

~~

8 a

fl

^io~

_q~j

°

8 lo

)

~~~

qoli

~

lot

qols

a

io°

10~~ 10'~ 10'~ 1o'~ IO'~

reducer ~mperatule

Fig.

5. The X-ray results for BOPCBOB. The smectic susceptibility (arbitrary units) ^and the reduced correlation lengths have been

plotted

as a function of reduced temperature. ^{The solid lines} present ^fits by single _power laws.

For the 7.6CB mixture the

phase

transition temperature was found _at 25.7 °C with _a relative accuracy of 5 mK. The

phase

transition temperature increased

during

the

experiment

at a rate of about I mK/h. The lowest value found for the

mosaicity

was 0.28°. The smectic

peak

_was

found _at

qo=0.2021~'

_{Data, shown} in

figure6,

were taken in _a range from

3 _x

10~~

~ r ~ 2 _x

10~~

iQ6

j~5 ^a

eif

10~

q~j~

j

~°~

qoli

~ _10~

I

j~i ~ols

lo°

10-~ 10'~ 10'3 10-~

reduoedtemperatum

Fig. 6. The X-ray result~ for 7.6CB. The smectic susceptibility (arbitrary units) and the reduced correlation

lengths

have been

plotted

_{as a} function of reduced temperature, The solid lines present ^fits

by

single _power laws.

Whether corrections for the

mosaicity

to the data _are made _or not, has _a strong ^influence on

the extracted parameters

describing

the critical fluctuations. The

mosaicity

in the smectic

phase

is most

likely

caused

by

effects of the boundaries

penetrating

into the bulk. In that

phase

effects of the walls _can

penetrate

^for at least millimetres into the bulk.

However,

the

penetration length

for defects in the nematic

phase

^{will be} at most

equal

to the correlation

length,

which is of the order of microns. Hence the

mosaicity

in the nematic

phase

^will

only depend

on the

divergence

of the

magnetic

field lines _over the

sample.

In _a transversal scan the linewidth is determined in the smectic

phase by

the

mosaicity

and in the nematic

phase by

the

reciprocal

of the correlation

length I/f~.

In the ideal _case at the

phase

transition the linewidth

should become close to zero.

Temperature gradients

^will smear out this effect. A

larger

temperature

gradient

will thus result in _a

larger

^observed

mosaicity

and _a less

pronounced dip

at the

phase

transition temperature

(see Fig. 3).

This _means that the smallest

mosaicity

observed

during

_an

experiment

is still _an over-estimation of the

right

correction. Hence in the nematic

phase

it is

probably

^better not to attempt to correct for any

possible mosaicity

and to use

equation (4).

The effect of the

mosaicity

correction is demonstrated in table II. In the first line _our present

analysis

of the data for BOPCBOB is

compared

with the

previous

one in which

a

mosaicity

correction _was included

[lsl.

These earlier values for _y and _{v~ are} about lo fb

higher.

This difference does not affect the conclusions from the earlier

analysis.

The _same type

of effect is shown in the _same table for the data for 7.6CB. The

complete

set of

fitting

parameters

describing

the data without the

mosaicity

correction _are shown in table III.

Table II. Values

for

the critical exponents as obtained

by dijfierent

methods

of analysin

_g the

X-ray scattering

data

OfBOPCBOB

and the 7.6CB mixture. Leai,in g c

free

is

equivalent

to a

fit including

the

splay length is.

^This leads to the

boldly printed

data that _are considered _to be the

most reliable _ones.

Sample Quartic

_term No

mosaicity

correction

Mosaicity

corrected

? _VII Vi Y VI ~i

BOPCBOB _c

=

free 1.26 0.71 0.50 1.39 0.71 0.56

7.6CB _c

=

free 1.38 0.82 0.58 1.62 0.91 0.72

7.6CB _c

=

0 1.64 0.93 0.74 2.07 1.06 0.93

7.6CB _c

=

0.03 1.32 0.79 0.49 1.55 0.87 0.62

7.6CB _c

= 0.I 1,28 0.76 0.49 1.48 0.84 0.58

Table III. Critical exponents ^{and bare} correlation

lengths for

BOPCBOB and 7.6CB _as

obtained

from fits

with

single

_power laws without

mosaicity

correction.

Y vi vi v~ Qo

ii

o Qo

ii

o Qn

f~o

BOPCBOB 1.26 0.71 0.50 0.43 1.34 0.46 0.47

7.6CB 1.38 0.82 0.58 0.49 0.61 0.38 0.36

The critical

exponents

of both

systems

are

compared

with the theoretical values in table IV.

The _errors in the exponents are

mainly

determined

by

the

uncertainty

of 2-5 mK in the

determination of the

phase

transition

temperature.

^This

gives

rise to

systematic

_errors. These

were estimated

by repeating

^the final fits with

T~~

^fixed at the

limiting

values of the

Table IV. Theoretical values

for

the 3D XY critical exponents

(in

the

superconductor

gauge

),

the

modified predictions for

the

liquid crystal

_gauge, and the

experimental

values

for

BOPCBOB and 7.6CB.

Exponent

3D XY

(a) ~~~~~~

~~(~~~~ ^{BOPCBOB 7.6CB}

gauge

( )

a 0.007 0.007 0.007

(d)

0.03

(e)

y 1.32 _w 1.32 1.26 _± 0.06 1.38 _± 0.06

vii 0.67 0.67 0.71 _± 0.03 0.82 _± 0.03

v~ 0.67

~ 0.67 0.50 _± 0.03 0.58 _± 0.03

v, ~

0.33(C)

0.43 _± 0.04 0.49 _± 0.04

(a) reference [5], ₁₆₎ reference [14], (C) references [6, 15], id) reference [24], (C) reference [10].

experimentally

determined range. Furthermore, the

uncertainty

in the parameters ^needed to model the resolution function

give

rise _{to an extra} increase of the _error bars. The statistical

errors are in

comparison

so small that

they

can be

neglected.

All parameters

describing

the structure factor _are

coupled

to

f~

as discussed in the

appendix.

In the literature the critical behaviour of

f~

^{has been connected} to that of

f~ by writing

the

quartic

term in the structure factor _as cf

(.

_When

c is left free this is

mathematically equivalent

to our

procedure.

The situation is different when _c is fixed _{at an} average

value,

_as has been done in _some

experiments [22, 25].

Some authors report the same values for the critical

exponents ^with a free

quartic

term and when _c is fixed _{at zero.} This

certainly

is _not the _case for the

experiments

described in this article.

Qualitatively,

the influence of

fixing

_c instead of

leaving

_c free is easy ^to understand. A value of _c

larger

^{than the}

optimum

_one will be

compensated by

a smaller value for

f~.

The effective smaller

integration

_range will then result in _a smaller value for

ii,

as discussed in the

previous

section. The smaller values for the

correlation

lengths

will result in _a

larger integrated intensity,

thus

giving

rise _to smaller values for the

susceptibility

«. These effects _are strongest ^close to the

phase transition,

where the

resolution function and _structure factor _are of

comparable

size. It

gives

rise _to smaller values for the critical exponents. ^{The last three lines in table II show this effect of}

fixing

the

quartic

term at a constant value. The _same effects _were found for BOPCBOB.

Figure

7 illustrates how

jo3

qo~i

jo2

jot

. cd

o c=0,03

. c=o.i loo

10-~ 10.4 10.3 10.2

Reduced

Temperature

Fig. 7. Results for the correlation

length

f~ of 7.6CB with the

quartic

term fixed _at

f)

_cf

(

_{for different constant} values of _c.

various ways of

fixing

the

quartic

behaviour affect the values obtained for the correlation

length f~

^The

larger

scatter in the values for the correlation

length

for the fits without _a

quartic

term is _a consequence of _worse fits of the _structure factor _to the data.

Evidently,

the details of the form chosen for the _structure form _are

highly

relevant for the values of the critical

exponents ^that are obtained.

4. Discussion.

The results for BOPCBOB _can be

interpreted

with the so-called gauge

theory

of the

NS~ phase

transition

by Lubensky

and co-workers

[14].

As mentioned in the introduction smectic

liquid crystals

_{possess no} true

long

_range order. The

long

_range ^order can be restored

by appropriate

order parameter transformations that map the

liquid crystal

_gauge on the

superconductor

_gauge. ^Elastic constants and

thermodynamic quantities,

such _as

specific

heat and internal energy, are

independent

of the gauge

used,

in _{contrast to} the correlation function.

In the latter _case the

superconductor

results have to be transformed back to the

liquid crystal

gauge. For the critical behaviour of the smectic

susceptibility

and the

longitudinal

correlation

length,

^this was found to have little effect.

However,

for the critical behaviour of the

perpendicular

correlation

length f~,

a cross-over is

predicted

^from _{v~ = vi /2} close to the

phase

transition temperature to v~ = vii for

higher

temperatures. ^{Thus in}

experiments

_an

effective exponent ^{will be observed with} a value in between these _two limits. As _a result the

anisotropic hyperscaling

relation 2 _a

= vii + 2 v~ no

longer

holds for the

X-ray

results and is

replaced by

2 _a

~ vi + 2 v~.

Support

for this

theory

comes from

Dasgupta [28],

who

performed

Monte Carlo simulations _{on a} discretised version of the de Gennes model of the

NS~ phase

^{transition in the} type ^II

superconducting

limit. In these

simulations,

the

specific

heat shows _a weak

singularity

consistent with non-inverted 3D XY behaviour. The correlation function shows indeed that _vii

~ v~. Furthermore, the _structure factor shows _a faster than Lorentzian

decay, corresponding

to the

quartic

term in the direction

perpendicular

to the director. These computer simulations

give

the

possibility

to compare these results with the correlation function _as calculated in the

superconductor

_gauge. In the

superconductor

_gauge there _was indeed _no

anisotropy

for the

growth

of the correlation

lengths

and _no

quartic

term in the transversal linewidth in _{contrast to} the situation in the

liquid crystal

_gauge.

Recently,

Patton and Andereck

[29]

have obtained similar

predictions

^from calculations based _on the

de Gennes model without

mapping

the model onto the

superconducting analog.

In these

calculations the correlation

lengths

are real

thermodynamic quantities

in _contrast with the results of both the gauge

theory

and the Monte Carlo simulations. Therefore,

according

to this

theory

^the

hyperscaling

relation should still hold.

The critical exponents _y and _vii for the

NS~~ phase

transition of BOPCBOB agree

nicely

with

the 3D XY values _as discussed

before, independent

on whether _a

mosaicity

correction is

included

(see

Tab.

IV).

The value for v~ is smaller than the 3D XY value _as should be

expected

from the gauge

theory [14].

The value for _v~ is

larger

than y/4, which agrees with the

predictions taking

into _account the

increasing importance

of the

coupling

between the smectic order parameter ^{and the director} fluctuations

[15]. Together

^with the 3D XY value for the critical exponent ^{for the heat}

capacity [12],

this has been the first

recognized

case with full

3D XY behaviour.

Recently,

_more systems ^{have been found with} a similar behaviour for the

critical exponents as mentioned above

[19].

In _most of these systems ^the

hyperscaling

relation is _not fulfilled.

The critical exponents ^{for the}

NS~~ phase

^{transition of 7.6CB show a different}

picture (see

Tab.

IV), despite

the 3DXY value for the critical exponent ^of the heat

capacity [10].

For 7.6CB the value for _y is on the

high

side but could still agree within the _error limits with the 3D XY value the values for v~ and v~ agree with the

predictions

^{for the}

liquid crystal

gauge.

However, the value for _vii is much

larger

than the 3D XY value. This is similar _to the situation for the

NS~ phase

transitions in the

weakly polar compounds

40.7

[21]

and

iS5 _[22].

_{In these}

cases also values for _y

m

1.5)

^and _{vii m}

0.8)

are found that _are much

larger

than the 3D XY value. This cannot be

explained by

the gauge

theory.

It _{seems as} if there _{are two} classes of

NS~ phase

transitions _a class of three

ring compounds (that

are

highly polar)

with _a smectic

Aj phase

that exhibit 3DXY

behaviour,

and _a class of _two

ring compounds

with either _a smectic

A~

or an

A~ phase

that has

higher

values for the critical exponents. ^{If this division is}

true in

general,

it _seems that not

only

the symmetry ^{of the}

phase

is

important

for the critical

behaviour but also

microscopic

_aspects, which _are not included in the present

theory.

A

difference in critical behaviour of the

splay

constant could be

responsible

for this deviant behaviour of the

NS~~ phase

^transition

^[30].

A somewhat different _treatment of the

experimental

data has been advocated

by

Garland and co-workers

[20].

From their

point

of

view,

it is essential that correction _to

scaling

terms are

included in the

analysis

of the data. Critical exponents are defined in the limit of very small reduced temperatures, ^{while for}

large

reduced temperatures, mean field values _are obtained. In

general, depending

on the reduced temperature range of the measurements, effective exponents ^{with values in between} XY and _mean field values may be obtained. The

preasymptotic theory

of

Bagnuls

and Berviller

[31] provides

a way to include correction _to

scaling

terms in the

analysis

and _to calculate this _{cross over} for

isotropic

systems.

Unfortunately,

at present no

equivalent theory

is available for

anisotropic

_systems. As _a consequence, this _means for the

NS~ phase

transition that

only

the

specific

heat, smectic

susceptibility

and the correlation volume

iii f )

_can be fitted and _not the separate correlation

lengths.

In the

preasymptotic theory [31]

the correction _to

scaling

terms for these three

quantities

are all interrelated

by

a non-universal temperature

scaling

_{parameter Ho.} Its value _can be determined from the calorimetric measurements while

keeping

the critical exponents ^fixed

at the 3D XY values. Garland and co-workers have shown that this

interpretation gives

_a

good description

for the data for BOPCBOB and _some other systems ^with

NS~~ phase

transitions

120j.

We have

attempted

to

interpret

the data for 7.6CB

according

to the

preasymptotic

model.

Unfortunately,

_no calorimetric data _are available in _a reduced temperature range far

enough

away from the

phase

transition temperature to determine

independently

the value of Ho. Therefore this

quantity

_must be treated _{as an} extra

fitting

parameter. ^The

quality

of the

preasymptotic

fit of the

susceptibility

_« is _worse than the power law fit. This could be

expected

as the effective critical exponent ^is

already larger

than its XY value. The

preasymptotic

fit for the correlation volume

ii f )

is of the _same

quality

as the

asymptotic fit,

_as could be

expected

since the effective exponent was

already nearly

at the XY value. From the fits for wand

iii

f)

values of Ho were obtained of 0 and

0.01, respectively.

This _means that correction _to

scaling

terms

probably play

no

significant

role for 7.6CB.

It should be realized that the structure factor

given

in

equation (3)

has been derived in _a mean

field

approach,

whilst the critical exponents are

compared

with renormalization group

calculations.

According

to renormalisation group

theory,

the correlation function should

obey

some

general scaling

laws which _are not

obeyed by equation ^(3).

^An alternative form could be

[32]

_:

S(q)

= ~

"

~ ~ ~

(6)

' +

iii(

(Ql Qo)1 ^~~ +

(ii

_Qi ^~

which is similar _to the form used first

by

^Mcmillan

[33].

Several

experiments

have been

performed using

similar forms for the structure

factor,

which did not

change drastically

the

/