Smectic a - smectic c transition : nearly tricritical mean-field or Ginzburg crossover ?

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Smectic a - smectic c transition : nearly tricritical mean-field or Ginzburg crossover ?

Y. Galerne

To cite this version:

Y. Galerne. Smectic a - smectic c transition : nearly tricritical mean-field or Ginzburg crossover ?.

Journal de Physique, 1985, 46 (5), pp.733-742. �10.1051/jphys:01985004605073300�. �jpa-00210015�

733

Smectic A - smectic C transition : nearly tricritical mean-field

or Ginzburg crossover ?

Y. Galerne

Laboratoire de Physique ^des Solides, ^Bât 510, Université Paris-Sud, ⁹¹⁴⁰⁵ Orsay, ^France (Reçu ^{le 3 octobre 1984,} accepté ^{le 3} janvier 1985)

Résumé. 2014 Une analyse détaillée, successivement classique ^et critique, ^est développée ^{sur toutes les mesures dis-} ponibles à la transition smectique A-smectique ^{C de} l’azoxy-di(03B1-méthylcinnamate ^de undécyl)-4,4’, c’est-à-dire ^sur

les variations en température ^du paramètre d’ordre, ^{de la} susceptibilité d’inclinaison, ^{et de la chaleur} spécifique. ^On

montre que la transition est critique dans l’intervalle de température ^{réduite 4 x 10-5 03C4 2 x} 10-3, ^{et devient}

de champ moyen après ^{le crossover de} Ginzburg, ^{évalué à} environ 1 K de la transition. Par conséquent, ^{le modèle}

de crossover entre les comportements ^de champ moyen tricritiques puis classiques ^{doit être écarté.}

Abstract. 2014 A detailed re-analysis, alternatively mean-field and critical, ^is presented ^{on all the} measurements available to-date at the smectic A-smectic C transition in azoxy-4, 4’-di-undecyl-03B1-methylcinnamate (AMC-11),

i.e. on the order parameter, ^tilt susceptibility, ^and specific-heat variations with temperature. ^It is shown that the transitional behaviour is critical over the reduced temperature range 4 ^x 10-5 ^03C4 2 x 10-3, ^{and becomes}

mean-field after the Ginzburg crossover, estimated in the _range of 1 K from the transition. Consequently, ^{the mean-}

field model of a crossover from a tricritical to a classical behaviour has to be discarded.

J. Physique ⁴⁶ (1985) ^{733-742 mAi 1985,}

Classification

Physics ^Abstracts

64.70M

During the last few _years, there has been considera- ble interest in studying the second-order phase ^tran- sitions, ^{and a} lot of results have been published ^{on the} subject. ^{But the} problem ^{is difficult} from both the theoretical and experimental points ^{of view,} resulting

in diverging conclusions about the nature of particu-

lar phase transitions, giving ^{rise to} controversy ^and shedding ^{some shadow on the} subject ^of phase ^tran-

sitions. The case of the smectic A-smectic C phase

transition in liquid crystals ^is typical : the AC-tran- sition is equally reported in literature as classical [1- 7], and critical [8-13]. ^We think that the situation

can be made clearer provided ^{that the} analyses ^{of the} experimental ^{data are driven more} rigorously. ^This

is what we try ^to do in this paper by presenting ^{a re-} analysis ^{of a set of} measurements performed ^{at the}

AC-transition of azoxy-4,4’-di-undecyl-a-methylcin-

namate (AMC-11). ^We have chosen AMC-11 because

complete ^{and accurate} measurements are available for this compound - ^order parameter, ^tilt suscep-

tibility, ^and specific heat have been measured with temperature ^{- which contrast with} particularly

controversial analyses [6]. One of these antagonistic analyses concludes that the critical behaviour is

experimentally accessible in the AC-transition, ^while

the other gives ^a fully mean-field interpretation ^{of the} data, using ^the possibility ^{of a crossover from a}

tricritical to a classical behaviour [14]. ^{In order to} get really ^unbiased conclusions, ^a complete re-analysis

of the experimental ^{data is} developed in this paper,

as symmetrically ^as possible between both models.

1. Theoretical background.

The smectic A and smectic C liquid crystals ^are layered phases, in which the molecules are, statisti-

cally speaking, packed perpendicularly ^{to the} layers

and respectively ^tilted through ^an angle 0. ^This angle t/J, together with the azimuthal angle 0 (Fig. 1),

defines the tilt direction of the C-phase, leading ^{to the}

choice of 0 eif/> as the order-parameter of the AC- transition [15]. ^The AC-transition, controlled by ^a

two-dimensional order parameter, may therefore be

predicted ^to belong ^{to the same} universality ^class

as the 3D-X Y model of superfluid ^helium [16], pro- vided however that one can neglect the effects of the

anisotropies, and of the Landau-Peierls instability

of the layers [17]. ^A critical behaviour should thus be observed in a temperature range AT ^close enough

to the transition to agree with the Ginzburg ^crite-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01985004605073300

Fig. ^{1. -} (Right) : ýJ ^{is the} angle ^defined by ^{the normal to}

the layers ^{z and the director a The azimuthal} angle 0

measured from the zH plane specifies ^{the direction of the}

director.

(Left) : ^the laser-light, initially polarized (P) ^{to 450 with} respect ^to the incident plane, ^{comes out of the} sample ^with

an elliptical polarization. ^The ellipticity ^{of the} out-coming polarization ^{is measured and, from it, the tilt} angle ýJ ^is

deduced [12].

rion [18] :

where ACp is the mean-field heat capacity jump, Ço

is the bare correlation length, and ATG is the Ginz-

burg temperature. Farther from this limit, ^{the tran-}

sitional behaviour should become classical, ^and may then be described by ^a simple mean-field model. Let

us first recall the main features of these theories.

1.1 MEAN-FIELD THEORY. ^- In the mean-field theory [19], ^the fluctuations of the order parameter ^are neglected (i.e. ⁱⁿ particular : 0 ⁼ const.), and the free energy ^can be expanded ⁱⁿ power ^terms of the order parameter. Every ^choice of 0 being equivalent (for instance, 0 ^{= 0} and 0 = n), it results that the free- energy expansion ^contains only even-power ^terms

of if :

where a = a’ i, and a’ and b are constants, and

T - r

T = T c is the reduced temperature. ^Standard

T ^p

calculations yield the value of the order parameter :

of the heat capacity jump :

and of the bare correlation length (see ^{for instance}

Kadanoff et ale [18]) :

where K is the mean value of the Frank elastic cons- tants K - 5 ^x 10-’ dyne [15].

If the smectic phase is submitted to a local torque r, the free _energy expansion (2) ^{has to be} complemented

with the term-r.p. ^{In the} experiments (see below),

this torque ^{r is} currently produced by ^a magnetic

field applied ^{to 450 to} the smectic layers. The external torque ^is then, expressed ^{as r =}

4 Xa H2,

state function .p(T, H) ^is given by :

1.2 CRITICAL THEORY. ^- The critical theory ^results

in scaling laws which predict that 4/, ^{i and F} obey ^the

universal function [20] :

the function f depending ^{on the} dimensionalities of the order-parameter and of the physical space, and on the sign ^{of ’tao The exact} form of the function f being ^unknown, expansions ^{of it are used to make} equation (7) ^more tractable in the analysis ^{of the} experimental data. Two different expansions ^of f

are possible in the weak-field limit. One is obtained around 0, ^i.e. for t/J t/J 0 I i Ill, and this is the case

of a smectic A phase ^{submitted to a} weak external torque ^{r :}

where fl - ^{0.35 and y - 1.3 in} the 3D-X Y model.

Another possibility, ^which corresponds ^{to the}

smectic C, ^{is to} expand f ^{around 1. To first order,}

it becomes :

For r ⁼ 0, ^this expression ^{reduces to the usual power}

law :

I I ..17 "",

Note that both expansions (8) ^and (9) ^are valid in the weak-field limit only, and that in this approximation, they ^can be assembled in the same isothermal for- mula :

735

This expression is similar to the mean-field isothermal series (6), but with the coefficients a and b ^now being

power functions of T with the exponents y and y - ² fl, respectively. ^{This means that the} shape of the iso- therms i.e. their H-variations, ^{is the same} in the weak- field limit in both mean-field and critical theories,

and therefore that no separation ^can be made between the two theories by just analysing the H-variations of the isotherms around the AC-transition. In other

words, ^{if one wants to} determine the critical exponent ⁶ directly, strong ^field measurements are needed.

The heat-capacity variations due to the degree

of freedom of the order parameter, ACp(T), ^{may also}

be expressed ^{in the} scaling-laws approach. ^When neglecting ^the corrections-to-scaling, ACp(T) ^{is just}

written as :

where a ^{= -} 0.026 and

A -

model [7], ^{and C is a constant}

1. 3 MEAN-FIELD THEORY NEAR A TRICRITICAL POINT. -

The crossover between the classical and critical behaviours _may not be a good interpretation ^{of the} experimental ^data. Recently, Huang ^{and Viner} [14],

arguing ^{that the} Ginzburg ^{crossover could occur so}

close to the transition that the critical regime ^would

be invisible, suggested that the data could be described in a mean-field model developed ^{in the} vicinity ^{of a}

tricritical point, ^{i.e. as a crossover from a} tricritical mean-field to a classical behaviour. This case corres-

ponds ^{to an} abnormally small value of the coefficient b in the free _energy expansion (2), ^{which then has to be} complemented by ^a following ^{term in} 1/16 :

Minimizing ^F gives :

where to 3 b2 where to

4 a’c

In the limit I T I to, ^this algebraic form reduces to the classical expression (4/ = o I T B1/2) ^whereas

in the opposite limit r I >> to, 4/ ^behaves like r B1/4.

to is therefore the crossover temperature between the tricritical and classical, mean-field regimes.

From equation (13), ^one also derives the heat-

capacity variations :

In _summary, as soon as a pure regime, ^{classical or} critical, ^is excluded from satisfactorily explaining ^all

the experimental ^{data, two kinds of} crossovers can

be considered. One is the usual Ginzburg ^crossover

between the classical and the critical behaviours,

and another, completely mean-field, goes from a

tricritical to a classical regime.

2. *(T, H)-measurements.

2.1 MEANS ^OF MEASUREMENTS OF lji(T, H). ^{- A lot}

of experimental ^means have been used to measure the tilt angle 0 ^{as a} function of temperature ^{T and exter-} nal field H, ^{in a smectic} (A ^or C) liquid crystal :

electronic resonance [21], ^NMR [8], ^neutron [22]

and X-ray diffraction [23], ^and optical interferometry [12]. Probably, only ^{the last} two measurements are

accurate enough ^{to allow for a} discussion about the nature of the AC phase-transition. ^In principle, they

are not equivalent ^but complementary, ^since X-rays

are sensitive to the smectic layers ^whereas light ^is

sensitive to the _average molecular direction.

2.1.1 X-ray diffraction. ^- Though ^not yet ^available

in AMC-11 (or ^{not accurate} enough [24]), ^{let us}

first recall the X-ray experiments. X-rays ^{have been}

used in two manners to measure the tilt angle ^{at the}

AC-transition. The most usual method is rather indi- rect. It consists of measuring the thickness of the smectic layers ^{and then} calculating § ^{under the} assumption ^{that the} layer thickness varies _propor-

tionally ^to cos 0 [23, 4]. This method has the advan- tage ^{of not} needing ^a magnetic ^{field to} orient the molecules but, because the cos 0-law ^is quadratic

in the small values of t/J, ^the measurement is parti- cularly unaccurate close to the AC-transition. The other X-ray method does not suffer this drawback since the angle 0 ^is directly ^measured by ^the Bragg

reflections on the smectic planes, the molecules being supposedly perfectly oriented in a strong magnetic

field [4]. ^{But now} problems arise from the unperfect alignment ^{of the} molecules, ^{due to the} remanent

anchoring of the smectic layers ^to the solid surfaces

or to focal conics formed in the sample. ^{Another diffi-} culty ^{of this} measurement is that it requires ^a high magnetic ^field making the zero-field extrapolations

in the transitional _range hasardous.

2.1.2 Optical interferometry. ^{- An accurate} optical

method for measuring ^{the tilt} angle 0 ^{uses the inter-}

ference between two rays of orthogonal polarizations (ordinary ^and extraordinary rays) passing through

the sample ^{at an} angle ^{of 450 to} the normal to the

sample (Fig. 1). ^{In this manner, if the} optical ^indices

and the sample ^{thickness are} known, ^{it is} possible

to get ^a direct and accurate measurement of 0. ^This method, which has been used, ⁱⁿ particular, ^{in AMC-}

11 [12], ^needs transparent samples, ^i.e. very homo-

genously ^oriented samples. ^This difficulty, ^however,

is widely compensated by ^the possibility ^of making

measurements on a well defined system. ^{As shown} in figure ^{1, the} sample ^is homeotropically aligned

in smectic A, ^{with the} layers parallel ^{to the} plates ^of

the cell. In the C-phase, ^{a small} magnetic ^{field H is}

sufficient to suppress the director tilt degeneracy.

The director n lies then in the plane ^{H, z} (z ^{normal to}

the layers), ^{its azimuthal} angle 0 being equal ^to

zero. As discussed elsewhere [25], this oriented confi-

guration ^{is not} completely free of defects. Because the sample thickness is quantized ^{in terms of number}

of layers, the defects produce ^a constraint field in the

sample, ^which may have large ^{effects at} the transition.

However, the thicker the sample, ^the smaller, ^this

undesirable field becomes and _may therefore be

neglected in well-oriented samples thicker than

- 300 _ym. Naturally, ⁱⁿ disoriented samples ^where

the effective distance is the typical size of focal conics,

much smaller than the sample thickness, ^{the cons-}

traint and its related effects _may become important.

2.2 METHOD OF ANALYSIS FOR THE O(T, H) ^MEASURE-

MENTS.

2. 2 .1 Isotherms. -The 1/1 ( T, H) measurements which constitute a 3-dimensional set of data, ^{are easier to} analyse ^if organized in families of 1-variable functions

as isotherms (Fig. ^2).

2.2.2 Weak-field approximation. ^- Except ^{in the} milliKelvin scales around the transition, figure ^{2 shows}

that the isotherms, I/1(H2) ^{are linear.} This indicates that in this experiment (H ¹⁶ kG) ^the magnetic

field remains small enough ^{for the} weak-field approxi-

mation to be valid. Consequently, ^the expansion (11)

is justified ^{in both} models, mean-field and critical,

and may be used for fitting ^{the data.} The variations of the parameters a(T) ^and b(T) ^{are then deter-}

mined by separate ^{fits on each} isotherm, and the

following zero-field extrapolations ^{are deduced :} 1/1 ( T) ^{the tilt} angle in absence of external influence in smectic C, and a(T) the inverse of the tilt suscepti- bility in smectic A. Each of these two zero-field mea-

surements, .p(T) ^and a(T), ^are able, through ^their

temperature variations, ^to determine the behaviour at the AC-transition. Reciprocally, they ^{contain about}

all the information which can be extracted from the

I/1(T, H) measurements (Fig. 2). ^It has been noticed

in § ^1.2 that, in the weak-field limit, ^the shape ^of

the isotherms is unable to differentiate the classical

Fig. ^{2. - Network of isotherms} VI(T, Io measured in a

sample ^of 340 J.1m thickness. The experimental ^{data are}

taken when increasing ^or decreasing ^the magnetic field,

as shown by ^the arrows, at ^a constant temperature ^{T indi-} cated in the upper scales. The full lines correspond ^{to the} least-squares fit of the data to the expansion (11) ^{which is}

valid everywhere except ^{close to} the transition (hachured part) ^where the mean-field approximation may be ques- tioned

from the critical behaviour. Neither were the varia- tions of the shape ^{of the} isotherms, ^{which are describ-}

ed by the variations of b(T), ^{able to make a} separation

between both behaviours, because the temperature

range in which they ^are measurable in the experiment

is too narrow (- 10 mK).

2.2. 3 Ginzburg ^{crossover and} analysis process of O(T)

and a(T). ^{- As recalled} above, ^the transitional beha- viour is theoretically critical from transition _up to the Ginzburg limit, and then becomes mean-field.

Such a critical to mean-field crossover is therefore

implicitly included in the critical model itself, ^and

in other words, ^{a critical} analysis which discards the possibility ^{of a} Ginzburg ^{crossover should not be} really ^correct. Several methods of analysis ^{can cor-} rectly ^{account for this} difficulty. The method which is used in § 2. 3 and 2.4 consists in analysing ^{the data}

over reduced _ranges of temperature, and then collect-

ing ^and comparing ^{the fitted} parameters ^into stability

tests. Such a method is able to show in which manner

the theoretical law agrees with the experimental data;

for example, ^it allows ^{a crossover} temperature ^to

occur inside the _range of analysis ^{or, as is} needed,

the fitted law to be correct only asymptotically. ^These possibilities make the method appropriate ^{to the} problem. Another of its advantages is also that it can

be used symmetrically for different theoretical laws.

It should therefore be possible ^{to reach unbiased}

conclusions concerning ^{the two} competing ^models

at the AC-transition : critical with a Ginzburg ^cross-

over or mean-field near a tricritical point.

737

2.3 t/J(T) ^{ANALYSIS. - The} zero-field measurements

O(T) ^are analysed following ^{the same} procedure ^and using similar criteria in both models.

2. 3 .1 Critical analysis. ^- A series of interferometric measurements of the tilt angle t/J(T) is shown and

analysed ⁱⁿ figure ^3. This series of measurements has

already been discussed in reference [12], ^{and we} pre- sent it again [26], because it was obtained in our best

sample, but also because its analysis ^{has been} ques- tioned by Birgeneau ^{et ale} [6]. ^In figure ^{3a the} experi-

mental data are shown with their least-squares ^fit

to the _power law [10] :

Fig. ^{3. -} a) ^{Series of} interferometric measurements of the tilt angle t/J ^versus temperature (in Kelvin), ^{fitted to} the power law [10]. The data such that T T, (crosses) ^{are not taken} into account in the least squares fit. The ^{arrows on} the dots

display ^{the errors from the} fit, multiplied by ^100.

b) ^Fitted p-values and their standard deviations as a func- tion of Tc - T,. ^{The arrow from} figure ^3a shows the result of the corresponding ^{fit As} Ft ^{decreases towards} T c’ P ^sta-

bilizes to its asymptotic value fl ^{= 0.356} ± ^0.0015 (see

Ref. [12] ^{for more} details).

c) ^{Same test of} stability applied ^{to the fitted} parameter T c.

performed by adjusting ^{the three} independent para- meters P, t/J 0 ^and T C’ ^{In order to} give ^a better view of the experimental ^data, logarithmic ^{scales are} used, ^{and an arrow} is drawn on each dot, equal ^to

100 times its deviation from the fit In this _way, ^one

can see that the ^error in 0 is less than 2 x 10-4 rad.

One can also see that the points ^are randomly spread apart each side of the best fit, ^{which is a mark of} quality ^{of the} fitting. ^This quality has however to be

confirmed, ^for example, by testing ^the stability ^{of the}

fit. The test of stability ^{which is} performed ^{here was} originally [12] ^intended to extract the asymptotic

behaviour of the AC-transition from the _necessary saturation of the tilt angle t/J(T). It consists (§ 2.2.3)

of a series of least-squares ^{fits to the} power law [10]

applied ^{to a} reduced number of experimental ^data.

Figures ^3b and 3c show the values of the parameters P

and Tc ^{found as functions of the} cutoff-temperature T,

below which the data are not introduced into the fit. In this _manner, we verify ^{that the} parameters ^of the fit stabilize near the transition. More exactly,

we find that, in the _range of temperature Tc - T,

0.5 K, T c becomes stable within the uncertainty ^of

determination ðTc f’-I ^{3 x 10-4 K} (or ^{10-’ in the}

reduced temperature). ^{The same kind of test is also} applied satisfactorily ^when suppressing ^the points

from the other end, ^{close to} the transition. All these tests demonstrate the validity ^{of the} power law in the range of the reduced temperatures : ^{4 x} 10- 5 ’t

(= T/Tc - 1) ^{2 x 10-3.} The value of the critical exponent which is found in this analysis (fl ⁼ 0.356)

appears ^to be similar, though slightly ^different [27],

to that measured at the transition of superfluid ^helium (13 f’-I 0.35) ^{as indicated} by ^{de Gennes} [16].

2. 3. 2 Mean-field analysis ^{around a} tricritical point. ^-

In the nearly tricritical mean-field model, ^{the order-} parameter variations [28] ^with temperature ^are given by equation (14) :

This algebraic function contains an equal ^number (three) ^of independent parameters ^{as the} power law [10]

in the critical analysis. An unbiased comparison

between the two questioned behaviours is therefore

possible, provided ^{that the same} procedure ^{as in the}

critical analysis is followed. A similar display ^is

used (Fig. 4). ^{As one can} readily ^{see in the} logarithmic

scales of figure ^4a, the result of the least _squares fit of the experimental ^{data to} equation (14) ^is quite unsatisfactory [29]. ^The points ^{are coiled in an « S-} shape » around the best fit, and exhibit a standard deviation 5 times larger than in the _power law ana-

lysis (Fig. 3a). ^{The test of} stability ^of T c’ ^{which is} presented ⁱⁿ figure ^4b, completes this first impression.

It shows that the Tc-value found in the classical

analysis systematically ^drifts by ¹⁰ mK per decade.