Calculations of the nuclear relaxation time T1 in a two-dimensional quasi-nematic system

Solid State Communications,

Vol. 10, PP. 571—574, 1972. Pergamon Press.

Printed in Great Britain

CALCULATIONS OF THE NUCLEAR RELAXATION TIME T1 IN A TWO-DIMENSIONAL QUASI-NEMATIC SYSTEM A. Caillé* Laboratoire de Physique des Solides, Université Paris-Sud, 91 Orsay, ~ (Received 23 December 1971 by PG. de Genries)

Orientational fluctuations of the quasi-nernatic order in a two-dimensional monomolecular layer contribute to the neclear relaxation time T1. The correlation functions are predicted to decay non-exponentially, an effect which could be measured from the dependence of T, on the nuclear Larmor frequency.

INTRODUCTION

located on one molecule. We consider a long spins F onof its axis.a,Wewith assume that the density molecule length two identical nuclear is low enough that we can neglect intermolecular contribution. We consider two cases with the magnetic field perpendicular to the layer (i) and parallel to it (ii). The first case is simpler since the magnetic field does not break the continuous rotational degeneracy in the plane of the layer. On the other hand, for a magnetic field in the plane of the layer, molecules having an anisotropic diamagnetic susceptibility tend to align themselves either parallel or perpendicular to the magnetic field. The calculations of the nuclear relaxation time could also serve to describe the electronic relaxation of a free radical on the chain in the framework of an E.P.R. experiment. The frequency range used in an E.P.R. measurement being higher, the model using a continuum would be pushed to the limit of its validity. Nevertheless, monomolecular or or bimolecular lipid phases are possible candidates if the layers could be treated as decoupled and if the chains are inclined with respect to the layer with a possible quasi-nematic ordering in the plane of the layer.

1 IN A RECENT report,for P. the G. de Gennes of a discusses the conditions existence two-dimensional quasi-nematic system. The most probable candidates are long molecules with an hvdrophilic group at each end lying on a waterair interface. This system doesn’t show long range orientational order in the absence of external agents. The orientational fluctuations are divergent in a two-dimensional system with continuous rotational degeneracy. Under suitable conditions of density, nevertheless there exists local orientational order of the long molecules which we will call a quasi-nematic. A1e calculate the contributions to the nuclear relaxation time T~ arising from the fluctuations in the quasi orientational order. The thermal orientational fluctuations give rise to interesting predictions for the inelastic light scattering1 in a two-dimensional quasinematic. The time dependent correlation does not have the simple exponential decay structure found in three-dimensional systems. These fluctuations may also cause nuclear relaxation through the modulation of the dipole—dipole interaction of, for example a pair of protons

2. CASE (i) *

Holder of a Post-doctoral Scholarship of the National Research Council of Canada. Laboratoire associé nu C.N.RS.

The part of the dipole—dipole interaction which contributes to T 1 when the magnetic field 571

572

A TWO-DIMENSIONAL QUASI-NEMATIC SYSTEM

is perpendicular to the layer (plane x-y) is: 2 a3 ~e~9 11* 12+ + H.c 1(1) = ~ (liy)

Vol. 10, No.6

is an independent ramdon noise variable and that any function of c, is calculated using a gaussian distribution of the form

where y is the nuclear gyromagnetic ratio. 0 is the angle that the molecule makes with the x-axis.

H.C stands for the hermitian conjugate. The nuclear relaxation time T 1 is given by the usual 2 expression: ~

1(1

~-~--

2

+

1) J(2) (2w).

a6

correlation function:

$

=

e_2t~~~ 0(i) di

~_~exp

(_5~/2~)

Using (7), GO) becomes: 1a2\ 0(t) = (.,-~.j)x/2exP[_..~-Ei(._i)7 exp

(7)

[_~ 7o]

(2)

J(2> (2w) is the Fourier transform of the time

(2w)

=

(3)

(8) where 7o is Euler constant (yo = 0.577...). For consistency with the continuum elastic theory, the integral over q is cut off as (1/a). Ej(x) is defined as:

where GO)

= . w is the Larmor frequency. There is a contribution to J (w)

coming from elevations with respect to the plane of the layer. This corresponds to a much shorter correlation time and is neglected in this study.

E 1(x)

=

—

j

e~ U

The parameter X is proportional to the ratio of the thermal energy to the elastic energy K(T)

X

=

2K~/~rK(T)

(9)

The spectrum of the local fluctuations was calculated by de Gennes) Assuming balancing between the friction torque on the molecules and

As expressen in reference 1, the validity of the simple model for the elastic energy is uncertain

the torque derived from an elastic free energy K, the statistical mean value of the time correlation of the angle 0 (z) is: 2(t) <[0(t) — 0(0)12> = 2~<(0q(0)~2> (1_e~2t).

for X > 2. Keeping this possible limitation in mind, the results are presented for all values of X. The correlation function G(t) starts from

c

(4)

The diffusion constant D includes two effects, The first part is (K/~)where y is the friction constant. This is the diffusion of orientation. There is also the contribution to translation diffusion of the molecules D’. When we treat these two motions uncoupled, we can write:

D

D’

=~-+

unity at = 0 and has a l~2 behavior for large time. This behavior is rapidly assumed because the exponential integral E, . (~-~) converges quickly to zero in the short tin~’et~ = ~ (a molecular time). In the generally accepted range of w << D we calculate the w dependence of 1/T 1 for different values of X. The results are summarised in the following table:

(5)

1

-~

The averages

obtained using the

X

static equipartition theorem for every degree of freedom: KBT <10q12> —

<~0~2>are

X > 2

(6)

=

w~e-~

F(w)

2)(2~_)

F(w) 2 Ix

-

1){i

+

g(X, ~) /2wa ~

D ____

X=2

ln~ 2 / D

we make that assumption that the variable =

0(t)

—

0(0)

(10)

X

=

1

X <<1

D

(~—2)

\~

)~‘]

Vol. 10, No.6

A TWO-DIMENSIONAL QUASI-NEMATIC SYSTEM

where g(X, 5)

with the above assumption of isotropic random

=

~~(4—s) r(i

~

2~...

—

with 0

<4

—

5

573

+

—

X)

motion, we have: 0 (14) We then find that nuclear relaxation time has the

—

< 2.

same behaviour as in case (i).’°

In the case of a very strong magnetic field 3. CASE ~

H, we keep only the term to first order in 0:

When the magnetic field is in the plane of the layer, we can in the two limits of very weaksolve and the veryproblem strong magnetic field. The magnetic field adds to the elastic energy K a term which comes from the diamagnetic arilsotropy: Fdj

XaH2SjO29

!

(11)

Hdd(t)

=

—

~ 2

~8(t)(11~12+12~11+) 3 a

=

We assume Xa > 0. Because of the term sin20, this becomes an unsolved statistical problem. For very weak field H, we can neglet Fd~ 2 dependence. For the very strong 3because field of the H limit, the system is aligned and one has a real nematic. We can then solve the problem since sin26 ~ 0~.The dipole—dipole interaction which contributes to the nuclear relaxation in this geometry becomes: Hdd

~2~2[~ =

~

L.

(15)

=

~ <10~> exp(—Dq2t)

where is given by the equipartition theorem:

(12)

-

H.C.

The time correlation function is obtained as in (4) with a diffusion model for the orientation and the translation: <0(0)00)>

0 is the angle the molecule makes with the direction of the magnetic field H. ~ is the difference between the parallel and perpendicular susceptibility

+



(16)

kBT

-

K(q2 * ~~2) (17) H is the magnetic coherence —

(k—)

‘21

length. The Fourier transform J ‘“(w) the time correlation function becomes in the limit of large H. knT 1 D D + log ~

2~K~~D22

(3cos 20— 1)11+12+

—

—

(18) 4. DISCUSSION

—

sin 20 (lIz

12+ 1~11+ ) +

H.C]

(13)

The magnetic field is along the z direction, In the limit of weak field H, we assume ~ tropic random motion for the orientation. Because of the assumption the three functions sin 20, cos 20 and 1 can be treated separately because of their orthogonality. For w different from zero, only the first two give a contribution. They give time correlation functions of the form ~ cos 20~ > and .These two expressions can be simply rewritten in terms of .This is possible because,

The non -exponential decay of the time autocorrelation function has already appeared in the theory of three-dimensional nematics.3 The twodimensional quasi-nematic adds to this behavior a dynamic exponent for the convergence of the correlation function. This exponent x is the ratio of the thermal energy to the elastic energy in a system where we have no long range order. For x large, the correlation time is small and the nuclear relaxation time is independent of frequency. For x ~ 2, we get a frequency dependence of (1/ 7 ) which goes from log w at x 2 to w at x very small.

574

A TWO-DIMENSIONAL QUASI-NEMATIC SYSTEM

Vol. 10, No. 6

In the presence of a strong magnetic field, we get a nematic alignment if the molecules have diamagnetic anisotropy. In this case, the frequency dependence of (1/T1) is independent of

a non-exponential decay of the auto-correlation function.

2 1 x and is given by (18). For w2>~’ D is pro~H 1 .portional to w This oriented phase also shows

Acknowledgements — We are grateful to Professor PG. de Gennes for suggesting this investigation and to the Orsay Liquid Crystal Group for helpful discussions.

~‘

.

~‘

REFERENCES 1. 2. 3.

GENNES P.G de, Disc. Faraday Soc. December (1971). ABRAGAM A., The Principles of Nuclear Magnetism, Chapter VIII. Oxford University Press, London (1961). PINCUS P., Solid State Commun. 7, 415 (1969).

Les fluctuations d’orientation de l’ordre quasi-nématique d’une couche monomoléculaire è deux dimensions contribuent au temps de relaxation nucléaire T 1. La fonction de correlation des La fonction de correlation des orientations ne décroit pas de facon exponentielle Ce phénoméne se traduit par une dépendance particulière de 7~ par rapport a la fréquence de Larmor.