Effective exponents of nematic liquid crystals

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Effective exponents of nematic liquid crystals

Lin Lei

To cite this version:

Lin Lei. Effective exponents of nematic liquid crystals. Journal de Physique, 1982, 43 (2), pp.251-253.

�10.1051/jphys:01982004302025100�. �jpa-00209392�

251

Effective exponents ^{of nematic} liquid crystals

Lin Lei

Institute of Physics, ^Chinese Academy ^of Sciences, Beijing, ^China (Rep ^le 23 juin 1980, ^révisé le 12 octobre 1981, accepté le 22 octobre 1981)

Résumé.

Les ^« exposants ^{» effectifs} 03B3’, ^{03BD’ et 0394’ sont calculés dans la} phase nématique ^{et des valeurs} différentes sont trouvées dans la phase isotrope. Nous proposons des ^mesures de ces « exposants ^».

Abstract

Effective ^« exponents ^» 03B3’, ^{v’ and 0394’} in the nematic phase ^are calculated and found to be different from their counterparts ^{in the} isotropic phase. Measurements of these exponents ^are proposed.

J. Physique ⁴³ (1982) ^251-253

^FTVRIER 1982,

Classification Physics ^Abstracts

61.30C - 64.70E

The nematic-isotropic liquid (N-I) phase ^transition

in liquid crystals ^{is first order in nature} [1, 2]. ^{In the}

I phase ^near transition point T,, the influence of the fluctuations of the order parameter ^{is more} apparent and the mean field approximation ^{has to be} improved [3, 4]. However, ^{in the N} phase ^near Tc, under certain circumstances the mean field approximation ^{is still}

useful. For example, ^this approximation predicts

that the critical exponents ^of the correlation length

are not equal ^to each other on the two sides of Tc [1, 5], which has been confirmed by the nuclear

magnetic ^resonance (NMR) experiments ^of Dong

and Tomchuk [6]. ^For PAA-d6, ^reference [5] predicted

that in the N phase ^of range 404 K-409.65 K ( = T)

the apparent exponent of correlation length, ⁷ (named v ^{in Ref.} [5]), ^should be 0.4 which was at odds with the experimental value of 0.66 available in reference [6] ^at that time. Subsequently, Dong [7]

reanalysed ^his experimental ^{data and obtained}

7

0.4, ⁱⁿ agreement ^{with our} theoretical value [5].

In view of these developments ^{and the} fundamental difficulties in the analysis ^{of NMR data of} v , ^we

here use the Landau-de Gennes (LdG) theory [8] ^to

calculate other exponents y’ and L1’ in the N phase

and _propose possible experiments ^{to check the}

results.

1. Theory. - The free _energy in the LdG theory

is given by

where S is the order parameter, A

a(T - T*),

h proportional ^{to the square} of the external field,

a, B, C, ^{T* are} positive ^material parameters, ^{T* the}

LE JOURNAL DE PHYSIQUE ^- T.

43,

2, FEVRIER 1982

supercooling temperature. ^{When h}

0, by OFIOS = 0

we obtain S

0 (I phase) and the order parameter in the N phase

where T+ = T* + B2/4 ^{aC. When} h # 0, ^the equa- tion of state S

S(T, h) ^is given by

Differentiating (3) ^with respect ^{to h we} get

giving ^{rise to} the well-known critical exponents

(T - T*) ^y

^{1 and d}

² [2-4, 9]. ^{In the N} phase,

S

S ⁱⁿ (4) ^and (5) ^and

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01982004302025100

252

The result ~4 ~/i(r) ^{above was first} obtained in reference [5]. Since ^the correlation length ^{in the N}

phase ^{A -’ ~2 ~} T v’,We ^obtain immediately y’

^{2 v’.} Similarly, y

2 v.

When T approaches ^{T+ from} below, fl N ^{T1 12}

and f2 ~ ^:r3/2. Therefore, y’

1/2, v’

1/4, ^{4 ’ = 1.}

Comparing with results in the I phase, ^{one obtains}

All these exponents ^{are mean} field values.

In real experimental situations (in ^{the N} phase),

since the N-I transition is first order, ^one generally

has T T, ^{T + . In other} words, ^the tempera-

ture T cannot be sufficiently ^{close to T~ for the}

above values of y’, v’ ^{and L1’ to} be measured. Yet,

in the experimental temperature range, through ^the slope of the linear fit to the log ^{S’ -} log s ^{curve one}

can still define ^{an apparent exponent q (and simi-}

larly ^{7 and} T). The value of these apparent expo- nents will depend ^{on the exact forms of} f ⁱ and f2

and _vary according ^{to the} temperature range under consideration and the materials involved. Fortuna-

tely, ^{if the} temperature is measured in terms of i ~

(rather ^{than E ---} (T+ - 7)/T’) ^then by (6) ^and (7),

both S’ and S" are functions of i. Consequently, ^for

different nematics, ^as long ^{as the} temperature range of 5 is the _same, one will obtain the same apparent exponents (7"71 A’). ^{In this sense,} apparent expo- nents of first order transitions in liquid crystals ^are

universal. However, ^this universality may break down when higher ^{order terms are} included in (1).

Alternatively, ^one may define effective exponents

at each point ^of temperature by

etc. Then, by (6) ^and (7),

where x ^--- :ri/2. Therefore, ^{when T} approaches ^T~

from below, T’ff changes ^{from 1 to} 1/2, veff ^from 1/2

to 1/4 ^and J’ff ^from 3/2 ^{to 1.}

2. Calculations. - In figure ^{la and} figure ^2a,

the calculated results of T’ff ^and d eff ^from (8) ^and (9)

are plotted, respectively. ^The point 5

1/8 ^{is the}

transition point Tc.

In treating ^the experimental data, ^and knowing Tc

and T*, ^one may ^use T+ = Tc ⁺ (T, - T*)/8 ^to

determine T+. ^The determination of yeff f requires

more accurate data (since differentiation with respect to 5 is involved, ^see definition of yeff above). ^There- fore, many experimentalists ^are still interested in the

apparent exponents (ÿ’, etc.) ^{in a certain} temperature

Fig. ^1.

a) Variation of yeff with reduced temperature ^T.

b) log 10/1

vs. log 1 o z. ^{The dots are} calculated results. The

slope of the linear fit gives ^the apparent exponent y’ ^for

a given temperature range.

Fig. ^2.

a) Variation of d eff with reduced temperature ^i.

b) log 1 o fi ^vs. log 1 o i. ^{The dots are} calculated results.

The slope of the linear fit gives ^the apparent exponent d~’

for a given temperature range.

253

range. To this end, ^we plot respectively ⁱⁿ figures ^{1 b}

and 2b, loglo f 1(i) ^and log 10 f2(-T) ^{as functions of} 10glo T. For the sake of clarity, only selected calcu- lated points (and ^not the whole curve) represented by ^{dots are} presented ⁱⁿ figures I b and 2b. The straight

line represents ^the best linear fit to the theoretical

values, ^the slopes ^of which give respectively ^the apparent exponents T’ ^and T. With MBBA in mind,

we take T,

^{47 °C and} T, - ^T*

1 °C, ^{the tem-} perature range 37 °C T 47 °C corresponds ^to

0.125 5 10.125. From figures _1 b ^{and 2b we}

obtain approximately y

^{0.75 and} d’ - 1.2. Conse-

quently, ^V

0.38. These theoretical results can be checked experimentally.

Comparing figures ^{1 a and} 1 b it is obvious that yeff definitely gives ^{much more} information than y ^as

obtained from figure 1 b. From figure 1 b, ^{one sees}

that it is not always possible ^{to obtain a} good ^linear

fit if the temperature range is too wide. In addition a

perfectly straight ^{line in} log f i ^vs. log ^{T will corres-} pond ^{to a constant} yeff ^{for the same} temperature

range. In figure ^{la where} yeff ^{does not} approach ^a

constant in the _range of ? shown, ^one may still obtain

an approximate constant Y ^{for the same} range Of T in figure ^{1 b.} Therefore, ^{one must be} very careful in

discussing ^the meaning ^and accuracy of apparent exponents deduced in the _way of figure ^{1 b. The same}

is true for figure ^2. These observations are also rele- vant and should be kept ⁱⁿ mind in the discussion of

experimental ^data [2, 10] (y, d, etc.) ^{in the I} phase.

3. Discussion.

In the indirect measurement of V by ^NMR experiments [6, 7], ^the analysis ^{of the}

relaxation time data is known to be complicated ^and

difficult. The main reason is that there is usually

more than one relaxation mechanism in liquid crystals ^and the theoretical description ^{is too} compli-

cated to be of sufficient accuracy. In this regard,

measurements of y ^and T through ^the quantities ^S’

and S" _may be more direct and much easier in _compa- rison. Experimentally, ^one only ^{has to} determine the order parameter S in the nematic phase ^{as functions}

of T and h (similar ^{to the case in the I} phase [10]).

And there are quite ^a number of different _ways that S

can be measured [11].

The kind of asymmetry ^{of the} exponents ^{on the}

two sides of Tc presented ^{here are} characteristic of the first order nature of the N-I transition and are not expected ^{from the} tricritical hypothesis [9, 12].

Therefore, experimental determination of these _expo- nents y and d’ in the nematic phase ^{will not} only provide ^a check of the LdG model but will also give

further tests on the nontricritical nature of the N-I transition.

Acknowledgments.

This work ^was carried out

during the author’s visit at the Laboratoire de Phy- sique ^des Solides, Universite Paris-Sud, Orsay, ^France Feb.-May, ^{1980. We thank} Jacques Friedel and

Georges Durand for useful discussions, ^{and Roland}

Ribotta for hospitality.

References

[1] ^LIN LEI, Kexue Tongbao ²³ (1978) ^715.

[2] ^LIN LEI, Phys. Rev. Lett. 43 (1979) ^1604.

[3] ^LIN LEI, ^WANG XINYI, ^Acta Physica ^{Sinica 29} (1980) 1427 ;

LIN LEI, ^WANG XINYI, in Proc. Ann. Conf. ^Condensed

Matter Phys., European Phys. Soc., Antwerpen, April ^1980.

[4] ^LIN LEI, in Liquid Crystals, S. Chandrasekhar (Heyden, London) ^1980.

[5] ^LIN LEI, ^CAI JUNDAO, Scientia Sinica (Forgn. Lang. ed.)

22 (1979) ^1258.

[6] DONG, ^R. Y., TOMCHUK, E., Phys. ^{Rev. A 17} (1978)

2062.

[7] DONG, ^R. Y., Phys. ^{Rev. A 21} (1980) ^1064.

[8] DEGENNES, ^P. G., ^Mol. Cryst. Liq. Cryst. 12 (1971) ^193.

[9] KEYES, ^P. H., Phys. ^{Lett. 67A} (1978) ^132.

[10] KEYES, ^P. H., SHANE, ^J. R., Phys. Rev. Lett. 42 (1979)

722.

[11] DEGENNES, ^P. G., Physics of Liquid Crystals (Claren- don, Oxford) ^{1974 ;} STEPHEN, ^M. J., STRALEY, ^J. P.,

Rev. Mod. Phys. ⁴⁶ (1974) ^617.

[12] ^LIN LEI, ^Kexue Tongbao (Forgn. Lang. ed.) ²⁵ (1980)

798.