Nonlinear dielectric relaxation from the Smoluchowski equation for rotation in three dimensions in alternating fields

Journal of Magnetism and Magnetic Materials 122 (1993) 187-192 North-Holland

AI I"

Nonlinear dielectric relaxation from the Smoluchowski equation for rotation in three dimensions in alternating fields Jean-Louis D6jardin Groupe de Physique Thdorique, Unit:ersitd de Perpignan, 52 At,enue de Villeneuve, 66 860 Perpignan Cdder, France

Analytic expressions for the nonlinear response of the dielectric relaxation are established. Three harmonic components are found arising from the application of a weak alternating field superimposed on a strong direct field. Dispersion plots and Cole-Cole diagrams illustrate the deviation from the classical Debye theory and give interesting information on molecular parameters.

1. Introduction T h e usual Debye theory of rotational diffusion of polar molecules in a pure alternating field leads to the well-known C o l e - C o l e diagrams which are perfect semicircles. This corresponds to the linear dielectric response for weak fields, which means that the results obtained for the electric polarization are linear in the field strength and that only one harmonic c o m p o n e n t exists, having angular frequency ~o. In this paper, we shall take into account the nonlinear behaviour of the dielectric response, by considering a double external electric stimulus, namely a unidirectional field E c on which is superimposed an alternating field E 0 cos(oJt). Moreover, the induced polarization of polar molecules is included, but any d i p o l e - d i p o l e coupling is ignored (dilute system). Also, the m e a n nonlinear polarizability is neglected since the present work is devoted to orientational molecular motions in a frequency scale up to 100 M H z only and not to electron motions. So, new original expressions describing the dielectric relaxation are obtained and three harmonic c o m p o n e n t s are found (in w, 2w, 3~o). Dispersion curves and C o l e - C o l e diagrams are presented for each of them, for a range of values of the p a r a m e t e r s P and 3'. These latter parameters measure the influence of induced dipolar m o m e n t s relative to that of p e r m a n e n t m o m e n t s and the electric field strengths, respectively: they are precisely defined in the following section. We must mention that this work may be used to find the solution of Brown's equation in the presence of an alternating magnetic field for the axially symmetric case of single domain ferromagnetic particles: this equation is then quite similar to the Smoluchowski equation which is applied to dielectric relaxation.

E(t) =E0cos(wt)

2. Theory W e assume that molecules of the liquid are small ellipsoids of rotation the geometric axes of which coincide with those of the electric polarizability tensor. The evolution equation of the orientational

Correspondence to." Jean-Louis DEjardin, Groupe de Physique ThEorique, UniversitE de Perpignan, 52 Avenue de Villeneuve,

66 860 Perpignan C~dex, France. TEl 68 66 20 62; Fax 68 66 20 19. I)304-8853/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved

188

J.-L. D~[ardin / Nonlinear dielectric relaxation in 3D

function f ( u , t) for such a molecule in the presence of the electric field E(t) equal to E c + E o cos(cot) is governed by the Smoluchowski equation, where inertial effects are ignored:

1 af D at

1

a

1,

sin 0 a0

(1)

where D is the rotational diffusion constant, 0 is an angle between the symmetry axis of the molecule and the direction of the external field, and k T is the thermal energy. Here, W is the orientational potential energy: W = - / . r E cos 0 - ~

AaE

(2)

2 COS 2 0 ,

where /, is the permanent dipole moment along the symmetry axis and k a = a l l - - o l ~ is the difference between the principal electric polarizabilities, parallel and perpendicular to the symmetry axis, respectively. As is well known, the electric polarization is proportional to the expected value of the first Legendre polynomial. For the steady-state, the solution of eq. (1) may be written in the form of convolution products, denoted by ® [1,2]. If we limit ourselves to terms up to the order of EB(t), we obtain {P'(u))NL(°¢) = D { e - x D ' ® [-~T(t)[ 1 - ( P x ( u ) ) L ( t ) ]

+ ~¢13(t)(Pl(U))k(t)] I,=='

(3)

where the subscripts NL and L stand for nonlinear and linear, respectively. (P2(u))L(t) is the mean value of the second Legendre polynomial, proportional to the electric birefringence and given by (P2(u))L(OC) = k[e st -'D' ® [ 3 y ( t ) { P , ( U ) ) L ( t ) + / J ( t ) ] } l , _

(4)

~,

where y(t) and /30) are the time-dependent parts of the orientational energy and the molecule in the field with respect to the thermal energy:

y ( t ) = Yo cos cot + Yc =

# Eo

kT

# E~

cos wt + - -

kT '

A~

Aa

Aa

/3(t) =/3 o cos2cot +/~c +/3Oc COS tot = k T E~ cos2wt + k T E~ + 2 k T E°Ec cos wt.

(5)

Eq. (3) may be written as follows:

3 {P,(U))NL( m ) = < P , ( u ) ) s T +

Y[ (X) COS j w t + X ; ' s i n j w t ) ,

j=l where (fl(lg))ST is time-independent component with time-dependent angular frequency such as ( P,( u) )ss

+ ~Pc + r ~ o -

1

90 1 +

~ cI

Y~"/c

~--~--t2][1 + 2 o j l{

{ (

5co2 i

7°)2

1 + 12D2I + 3 + ~

(7)

J.-L. D~jardin / Nonlinear dielectric relaxation in 3D

189

{ [ 72]

and Xj, X7 are the harmoniccomponents of the dielectric polarization, with P = (Aa/iz2)kT, defined as follows:

---

1+ (°) ]2\2D1

[

+P 1+

--

77--~

-O)

2" 1 +

1 ('~--~ 1 + ( ~ )

1}

7"-~ 2 2[;+ (~-~)] 2T:Li 2-~1 2 + ~D7

yoy~

"-'~"~2

2 ;+(~

1 12D2

3D2

4511+(2~)2] [~+(2~)2][1+(~) ~] 2(1

('02

33+~D~

12D2 )

(°)2

+ 1 +P[1 +

60

2

1 + ('~--~)

,

[4 2 l

w/2D w/2D 1 "3 12"-D2 X['(w) (7o13) 1_1._(2__~) -('Y°3/90)+7"~ 2 2 1 ( - ~ - - ~ ) 1 + ( ~-o9) 2 1+ 1- + ( ~ - ~ =

-~ (°)2 +P[I+ 1+ 5-B

6]} (~)2 1+ -5-g

--21+[

-(Yo Tc2/4s) (.o/2D

[

+P 1 +

~~- + 1~2 ~2]

1+(25) l+(r~ 3

2

6(2+~) f.o 2

'

~)

(s)

J.-L. Ddjardin / Nonlinear dielectric relaxation 'in 3D

190

~°2 1-D~

v~,v,/9o

x~( ,o) -

1+ ~

1+

~-~

l- ~

12D2

l+

+

~

1+

2 + ~4D

2 I-

3D 2

+P[-3+

(

l+-dg 11

w2

6

6D 2

~o/D X~'(eo) = - (7~]7J90)

17 6 4-

1+(5)

5o)z

+ - -

24D 2

60

60

4

7

+P[-3 +

!

x3(,o) =

-

2

v~/18o

-{-

O)

2

O)

2

[

1 - 12D~

1

3w)

1+

14 3

w/2D X ; ( oa ) = - ( 7 3 / 1 8 0 )

1+

(9)

'

---

°'~

2D 2

~

(

3 11+

3°'-~ ] ~ ]

-~

~oe 2D 2

3w)2 ~

(lo) l+

Y6

l+

~-F

On setting P = 0 and 7~ = 0 in eqs. (7), (8) and (10), the results so obtained are in full agreement with those of Coffey et al. [3] (see also Rocard [4]), who have considered only polar molecules under the influence of a pure alternating field. Thus, it is possible to obtain the real Xj(oa) and the imaginary ~(w) parts of the electric polarization, normalized as

xj(,o) =x;(,o)/x;(o),

~(,o) =xT(,o)/x;(O),

11)

Z-L. Ddjardin / Nonlinear dielectric relaxation in 3D

191

where A'j(0) is the value of X](o~) when oJ ~ 0, namely A'~(0) = ( 1 / 3 ) Y 0 [ 1 - 3 ' 2 / 5 ( 1 / 4 + y z / y 2 ) (

_ 2 P + 1)],

(12)

A'~(0) = - (1/30)y02yc( - 2 P + 1),

(13)

x~(O) = - ( ' ) / 3 / 1 8 0 ) ( - 2 P

(14)

+ 1).

Given the values of P, Y0 and Yc, we may plot dispersion plots of Xj and Y: versus oJ (fig. 1). C o r r e s p o n d i n g C o l e - C o l e diagrams ~ ( X j ) are also drawn (fig. 2) and it is shown how they deviate from the classical linear Debye theory. Because each harmonic term may always be separated experimentally, we now have a triple set of data on the liquid u n d e r consideration. This increases the precision with which molecular parameters such as ~- (dielectric relaxation time), D and P may be determined. The m e t h o d s we have just outlined may also be applied (with appropriate modifications) to the study of the magnetization behaviour of ferromagnetic domains for small nonlinearities. We refer the reader to the work of I g n a t c h e n k o and G e k h t [5]. T h e y studied the dynamic hysteresis of a s u p e r p a r a m a g n e t by starting from a F o k k e r - P l a n c k equation similar to eq. (1) which describes the evolution of the density of the magnetic m o m e n t s on a unit sphere [6]. T h e variables are then separated in this equation to yield an

X(t~)

a

X(~)

c

4

b a

b

~ ~ '

5

a

4

loglo(~/2~)

S

6"-~-----~f-~,

Iogio ( ~ / 2 ~ )

a

X(m)

~

b

X(t~)

4 logto(~12~)

~

_~b/~

5

b

~

~'

logIo (0)121t)

Fig. 1. Normalized dispersion plots for P = -2, -0.7, 0 (Aa = 0), and 10 (~ -~ 0) corresponding to the three harmonic terms, a: to-component; b: 2w-component; c: 3o~-component.(Y0 = 0.1; 7¢ = 1.83; D has been chosen arbitrarily equal to 6× l06 s 1.)

J.-L. D6jardin / Nonlinear dielectric relaxation in 3D

192

Y(~)

Y((~)

/

.5

I

D

0

I

X(O.I)

0

1

x(o~)

I

X((~)

y((~)

Y(O))

c

b

.5

.5

I

0

1

X(O~)

I

0

Fig. 2. Plots of the imaginary part ~ vs. the real part Xj of the normalized dielectric relaxation functions for P = - 2, - 0.7, 0 and 10. a: w-component; b: 2w-component; c: 3w-component. (Y0 = 0.1; y~. = 1.83.)

infinite set of differential-difference equations which may be solved by perturbation theory in the manner we have outlined for the dielectric problem treated here. References [1] [2] [3] [4] [5] [6]

J.-L. Dfijardin and G. Debiais, Phys. Rev. A 40 (1989) 1560. J.-L. D~jardin and G. Debiais, Physica A 164 (1990) 182. W.T. Coffey and B.V. Paranjape, Proc. R. Ir. Acad. 78 (1978) 17. M.Y. Rocard, J. Phys. Radium 4 (1933) 247. V.A. lgnatchenko and R.S. Gekht, Sov. Phys. JETP 4(/(1974) 750. W.F. Brown, Phys. Rev. 130 (1963) 1677.