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PHASE TRANSITIONSRELAXATION TIME OF THE
CYBOTACTIC GROUPS AT THE SA-N PHASE
TRANSITION
F. Brochard
To cite this version:
RELAXATION TIME OF THE CYBOTACTIC GROUPS
AT THE S,-N PHASE TRANSITION
F. BROCHARD
Laboratoire de Physique des Solides
UniversitC de Paris-Sud, B2t. 510, 91405 Orsay, France
RBsum6. - Dans deux etudes antMeures [ I , 21 nous avons Btudid les proprietes dynamiques de la transition SA-N et prevu pour le temps de relaxation du paramittre d'ordre zgl(T) = Sz/& oh S2
est la vitesse du second son et 5 la longueur de coherence. En ordre de grandeur,
En fait les modes de second son sont amortis pour q< = 1 et I'on a
ou B est la constante de rigidite associee a la compression des couches smectiques et Gune viscositk renormalisee. Loin de Tc, z;'(T) = Bolqo
-
108 s-1. On attend doncPour l'exposant critique x, TDLT et les lois d'echelle dynamiques donnent x = 1. Cette relaxation
trks lente du paramktre d'ordre a kt6 observb par ultrasons et RMN. Elle conduit a des viscositQ
critiques cent fois superieures k nos previsions anterieures [2]. Les resultats expbrimentaux exigent
un facteur de cette importance.
Abstract. - In previous work [ I , 21 the relaxation time of the smectic order parameter tynear the
SA-N phase transition was estimated as zgl(T) = Sz/T, where S2 is the second sound velocity and 5
is the coherence length. This leads to
q ( T ) 1011
T
S - 1 .T
In fact the second sound modes are overdamped for q5 = 1 and we find then l/z$ = ~ l y w e r e B
is the rigidity coefficient for the compression of the smectic layers and the renormalized viscosity Far from Tc,
We expect then
1 AT s-1.
- (T) =
)
108,
(
z*
For the critical exponent x, TDLT and dynamical scaling lead both to x = 1. This slow relaxation of the order parameter agrees with ultrasonic and NMR data. It leads to an increase of the critical viscosities by a factor of one hundred from our earlier predictions 121. Experimental data require a factor of this magnitude.
1; Introduction. - In reference [I] we studied the as for the lambda transition of helium. S2 is the second critical dynamical behaviour of a second order S,-N sound velocity and
5
the coherence length. Eq. (1) leadstransition by assuming dynamical scaling laws for the to
fluctuations of the order parameter $. We postulated AT
that the relaxation time z*(T) of I) is given by Z ; ' ( T ) = 101l-s T
.
7;' = S 2 / 5
,
(1) I n reference [2] we calculated the anomalies for theC3-86 F. BROCHARD
viscosities using Kpbo formulas and a decoupling approximation introduced by Ferrel for binary mix- tures. We found
where B is the rigidity coefficient for the compression of the layers. Using the value ( 1 ) for z@(T) we found
AT - 0 . 3 3
6 = 1 0
( )
poises,
i. e. a very small anomaly.We have today many experimental data for the critical viscosities. The critical exponent 0.33 is not incompatible with most of them ; but the observed coefficient is in all cases a hundred time bigger than our prediction [4]. The data require
Recently e,,,(T) has been measured by ultrasonic absorption and NMR : z*(T) is found to be extremly long (7; ' ( T )
-
1 M H z at ATIT-
The obser- vation that 6;(-
z$(T) after eq. (2)) and z@(T) require the same factor to fit the experimental data lead us to reconsider our estimation ( 1 ) for z*(T).We explain here why eq. ( 1 ) for z@(T) is not valid. We show first that the second sound overdamping leads to much longer relaxation times, as observed by ultrasonic measurements. We show then that the technique used in reference (2) is no more successful in calculating {. It leads to
f
= y" ! Last we estimate by assuming dynamical scaling laws and we eva- luate z*(T).2. Overdamping of second sound. - 2.1 SMECTIC
EIGENMODES IN THE HYDRODYNAMIC LIMIT (q5
<
1).-
In the limit qt
<
1 , the theory of MPP [7] holds and the dispersion relation for the eigenmode frequencies can be written [ 8 ] aswhere is the total viscosity which involves the five coefficients of MPP and 0; = Bq2/p.
The nature of the modes depends of the value of q5 as compared to a dimensionless coefficient p giben by :
Far below T,,
' 0 pd' (= LO-'
-
10-6).p=-
-
2 'lo
Above Tc, BC2 = l? [7] is the renormalized Frank elastic constant and p is the dimensionless parameter
p =
zpG2
introduced for nematic [7] ( p is the ratiobetween slow orientational and fast shear viscous eigenmode frequencies).
a) For qt < ,u"~, we have the second sound
b) ,For q t
>
p1'2, we have two overdamped modes :- a fast viscous shear mode : o = iijq2/P ;
- a slow viscoelastic mode : w = iB/{. (6)
2.2 q[ = 1 : RELAXATION TIME Z J I ' ( T ) OF $.
-
The fluctuations of $ relax in a microscopic time z*(T). Above Tc, z*(T) is the life time of the cybotactic groups. The equation for ' ( T ) is the dispersion relation ( 3 ) for qt = 1, i. e.
We can have the two following cases :
Case 1 : p ( T ) > 1 . z* is given by eq. (5) : Far from Tc : AT %10"s-' and z i l ( ~ ) = l O " - s
.
d - T Case 2 : p(T)<
1.Z+ is given by the slow mode of eq. (6) :
Far from Tc : Bo
- N 10' and 2;'
=
10'-
Yo
.
(8 bis)We show now by an argument of self consistency that case 1 is excluded. We suppose p
>
1. This case is just the case studied in reference [2] and we have"r
B(ST
l .Including this result in p, we find that p isjdways smal- ler than one ( p N 1 0 - 4 - 1 ~ - 6 far from Tc and tends
to one as T + T,).
Conclusion.
-
At all temperatcre, ' ( T ) =BIG.
Whatever the value of fi, this relaxation time is much slower than S2/5.We discuss now ;(T) to know the critical exponent x
defined in eq. (8 bis).
3. Critical behavior of the viscosities.
-
3 . 1 KUBOFORMULAS. -The Kubo formulas relate the transport
Above To : Using s2
--
< - l f 2 ( I ) , we can writeBelow Tc :
= ( q t ) - l J 2 q3/2
+
i(yUq)lJ2 q3I2.
From this expression, we deduce1
>
z = q .
(10)- I
a4 = Zi6 = 0 We expect then the following behaviour :
1 71 - for q l % 1, o*
--
q3I2 ;- -
r - F
([ is the permeation coefficient).
- for q t = 1, w*(T)--
tp3/'.
We verify that It is sufficient to investigate one coefficient to get the o + ( T ) =,
-
tpzi2
by using eq. (11).
others. The Kubo formula for the twist nematic visco- 'Isity leads [2] to We estimate yo by using the dimensionless parameter p,
F,
= Bz+(T).
defined by eq. (4). Using--
tl/',
p is regular. We assume that p remains constant though the transition In reference [2], we took z+(T) = S2/5-
and we findy",
= BS;'4:
(eq. 2). We have seen that in fact T;'(T) = B/? and the Kubo formula give only the result6
=6
! However we can predict by assuming dynamical scaling laws for the fluctuations of+.
3.2 DYNAMICAL SCALING RESULTS. - The technique
of scaling laws is illustrated by figure 1.
If dynamical scaling holds for the S,-N phase tran- sition, the spectral width of the dynamic correlation function
<
I)$,,,
>
of the order parameter has to be an generalized homogeneous function :Below Tc, and for q< -+ 0, coJ, is given by (5) :
Y 2
w * = s 2 q
+
i - q.
P
As B t 2 = Bo dc, this assumption leads to yo =
6,
a bare smectic viscosity. This assumption fit rather well the experimental data [4].Conclusion.
-
If dynamical scaling holds :By taking for yo a bare viscosity of the smectic, as required by experimental data, we find
C3-88 F. BROCHARD 4. Discussion.
-
The second sound overdampingat q( = 1 explains the very large value of the relaxation time measured by NMR and ultrasonic measurements. Unfortunatly our earlier mode mode coupling calcula- tion is no more successful in calculating the anomalies of the viscosities. Assuming thattime scaling laws are maintained we expect r] = r], J(/d, where yo is a normal
viscosity. This conjecture agrees with L. Leger's [4] measurements. Of course it should always be recalled that the whole helium type picture of the A-N transi- tion is still doubtful : in particular the critical exponent
found for the elastic constant B is B
-
( A T / T ) ~ . ~ ~ instead of ( A T / T ) ' . ~ ~ (except by ultrasonic measure- ments where Bacri finds 0.66). If the static exponent 0.33 is confirmed the present description of the S,-N phase transition is completly invalid and the present considerations are probably meaningless.Acknowledgments. - The authors thanks P. Marti- noty for prior communications of its experimental results and L. Lkger and M. Delaye for stimulating discussions.
References [I] BROCHARD, F., J. Physique 34 (1973) 41 1-422.
[2] JAHNIG, F., BROCHARD, F., J. Physique 35 (1974) 106-118. [3] HOHENBERG, P. C., Varenna Summer School on critical
phenomena (1970).
[4] LEGER, L., MARTINET, A., to be published in J. Physique
Colloq. 37 (1976) 63-89.
HUANG, C . C. and al., Phys. Rev. Lett. 33 (1974) 405. DFLAYE, M., J. Physique Colloq. 37 (1976) C 3-99. [5] MARTINOTY, P., J. Physique Colloq. 37 (1976) C 3-113.
[6] BLINC, R. and al., J. Physique Colloq. 37 (1976) C 3-73.
[7] MARTIN, P., PERSHAN, P. S., PARODI, O., Phys. Rev. A 6
(1972) 240.
[8] DE GENNES, P. G., Thephysics of Liquid Crystals (Clarendon Press, Oxford) 1974.
191 RIBOTTA, R., C . R. Hebd SPan. Acad. Sci. 279B (1974) 295.
BIRECKI, H. and al. to be published.
