LATENT HEAT OF THE CHOLESTERIC TO SMECTIC A TRANSITION

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LATENT HEAT OF THE CHOLESTERIC TO SMECTIC A TRANSITION

T. Lubensky

To cite this version:

T. Lubensky. LATENT HEAT OF THE CHOLESTERIC TO SMECTIC A TRANSITION. Journal

de Physique Colloques, 1975, 36 (C1), pp.C1-151-C1-152. �10.1051/jphyscol:1975129�. �jpa-00215906�

Classification Physics Abstracts

7.130 — 7.480

LATENT HEAT OF THE CHOLESTERIC TO SMECTIC A TRANSITION (*)

T. C. LUBENSKY

Department of Physics and Laboratory for Research in the Structure of Matter, University of Pennsylvania, Philadelphia, Pa. 19174, U. S. A.

Abstract. — It is well known that the nematic to smectic A transition is analogous to the super- conducting transition and can be second order if director fluctuations are ignored. We argue here that the cholesteric to smectic A transition is analogous to the superconducting transition in an external magnetic field and is, therefore, always first order. We find the latent heat of this transition to be proportional to (#0 Z,) (1 -<*>' v where qo is the high temperature pitch wavenumber, L is a molecular length, a is the specific heat exponent, and v is the correlation length exponent.

In this paper, we will estimate the latent heat of the and y are the usual phenomenological coefficients, a cholesteric to smectic A transition assuming that the changes sign at the transition temperature r ^c :

nematic to smectic A transition is second order. The „

r

r T

latent heat is simply related to A7\, the amount by a = a! x where T = .

which the transition temperature, T

, to the smetic ^°

state is lowered by the presence of inter-molecular I n a n e x t e r n a l m a g n e t i c field H, the equilibrium state forces producing the cholesteric twist. We show that o f t h e s u p e r c 0 n d u c t o r minimizes the Gibbs free A7\ is of order 10 3 T NA . This number is small, but enerev

is possibly of the same order of magnitude as the

A7\ arising from director fluctuations [-1]. To calculate

^> T, H) = F - BH (2) the latent heat, we will exploit the analogy between ^w h ^{e r e} B = VxA. In the normal state, \j/ is zero and

a superconductor and a smectic A liquid crystal [2]

and argue that the twist term in,the cholesteric free G = G = - ^ ^ H

= — - uH

(3) energy plays the role of an external magnetic field in a n 2 dH

2

superconductor.

First we review the mean field theory of the super- w h e r e " = VWWT- V ^ m a g n e t i c fluctuations are conducting transition in a magnetic field [3]. The ^important, ^ = ^ In the superconducting state, Landau-Ginzburg fiee energy density for a super-

' v ~

'

> a n d

conductor in rationalized units is G = G

= — \ T

ACt

(4)

Fblf TA) = a\\l/\

+ -b\il/\

+ where AC is the specific heat jump. The system becomes 2 superconducting when G ^s becomes less than G

. This , > 2 occurs at a temperature AT

below T

given by

+ y V - i ^ A U + - i - ( V x A ) 2 (1)

\ he J 2„ 0 (ATi)2 = 7 ;/^L. ⁽⁵⁾

where \j/ is the order parameter, A the vector potential,

H

is the free space magnetic permeability, and a, b At T

{H) = T

+ AT

, ij/ jumps discontinuously from zero to a finite value, so that transition is first order.

(*) Supported in part by grants from the National Science

i i s a convenient measure of the Strength of the Foundation and the Office of Naval Research. first order transition [1].

JOURNAL DE PHYSIQUE Colloque Cl, supplément au n° 3, Tome 36, Mars 1975, page Cl-151

Résumé. — Il est bien connu que la transition nématique-smectique A est similaire à la transition supraconductrice et serait du 2 e ordre si on négligeait les fluctuations d'orientation. On montre ici que la transition cholestérique-smectique A est similaire à la transition supraconductrice sous champ magnétique et est, par conséquent, toujours du 1 ^er ordre. On trouve que la chaleur latente de cette transition est proportionnelle à (# ⁰ Ly !-«>/» où qo est le vecteur d'onde du pas de la phase à haute température, L une longueur moléculaire, a l'exposant critique de la chaleur spécifique et v l'expo- sant de la longueur de cohérence.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1975129

Cl-152 T. C. LUBENSKY Now consider the free energy for a smectic liquid

crystal with planar spacing d with no cholesteric twist term

where the subscripts // and I refer respectively to variations parallel and perpendicular to the director in the smectic state, n is the director, go = 2 z / d and K,, K, and K, are the Frank elastic constants. a changes sign at the smectic to nematic transition temperature TNs. F, is analogous to eq. (1) with the identification $

$, go

2 e l k , 6n * A, and (K,, K,)

p;'. The free energy of a smectic A with a cholesteric.phase contains an additional terh

G ⁼ F , -I- K: qo(n,.Vxn) (7) where K: is tlie high temperature twist elastic constant and go the high temperature twist wavenumber. The product K: go is determined by the molecular forces creating the cholesteric twist and is relatively tempe- rature independent [4]. Now, we note that G is analo- gous to the Gibbs potential eq. (2) with the identifica- tion go

- H. As in the superconducting case, the equilibrium state minimizes G. Hence if director fluctuations are unimportant within the mean field, we can say immediately that the transition to the smectic state will occur at a temperature AT, below Tc determined by

Since the nearly second order transitions to the smectic state exhibit critical fluctuations, the mean field does not adequately describe the transitions. It isd still true, however, that the equilibrium state mini- mizes the Gibbs free energy. In the scaling regime the free energy in the smectic state has the form

where sets the energy scale and a is the specific heat exponent. The free energy in the cholesteric phase is

where K, now includes all enhancements due to diamagnetic fluctuations. Thus, near Tc, KZ diverges with the correlation length exponent [2] v

The transition to the smectic state then occurs at the temperature AT, below T, when G, = G,,

where d is the dimensionality. In deriving eq. (1 2), we used the scaling relation dv = 2 - a.

We can make a crude numerical estimate of the magnitude of AT, by plugging in order of magnitude forms for the parameters in eq. (12)

where TNI is the isotropic to nematic transition tem- perature and L is a molecular length. Using eq. (13) and (12), we find

In three dimensions, v - ^{3, and go} L is of order 10-,.

This gives, using TNI -- TNA,

This is a fairly small number but may be of the same order of magnitude as the AT, arising from director fluctuations-in the nematic to smectic A transition [l].

Using eq: (9), one can estimate the latent heat per particle I of the transition

This corresponds to an entropy change per mole of order 10-3 R.

References

[l] HALPERIN,, B. I., LUBENSKY, T. C. and MA, Shang-keng, [3] For a standard treatment of this problem, see DE GEN- Phys. Rev. Lett. 32 (1974) 292 ;

P. G., Superconductivity of Metals and Alloys HALPERIN, B. I. and LUBENSKY, T; C., tofappear in Solid (Benjamin, New York) 1966, chap. 2.

State Commun.

121 DE GENNES, P. G., Sdid State Commun. 10 (1972) 753 ; [4] ALBEN, R., MoZ. Cryst. L i p . Cryst. 20 (1973)

*.

Mol. Cryst. Liqu. ,Cryst. 21 (1973) 49.