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On steady state solution of nonlinear Kerr electric birefringence and polarization for AC and NC superimposed fields
Physica A 176 ( 1991) 569-580 North-Holland
ON STEADY
STATE SOLUTION
BIREFRINGENCE SUPERIMPOSED
OF NONLINEAR
AND POLARIZATION
KERR ELECTRIC
FOR AC AND DC
FIELDS
N.M. HOUNKONNOU and A. RONVEAUX Laboratoire de Physique Matht?matique, Facrrltk Universitaires Notre-Dame de la Paix. Rue de Bruelies (il. B-SO00 Namtrr. Belgium
R.P. HAZOUME Laboratoire de Modklisation de S~&wes Physiques, lrtstitut de Mathhmatiqrces et de Sciences Physiques (IMSP). B. P. 61-Z.Porto-Now.
Bertirr
Received 22 November 1990 Revised manuscript received 20 March 1991
The steady state response of the nonlinear Kerr effect to a small ahernating field superimposed on a unidirectional field is anaiyzed using a second order perturbation theory. We draw inferences from the appearance of simple and double harmonics as the fundamental frequencies of the nonlinear Kerr effect. The study is based on the Smoluchowski equation of rotational Brownian motion.
1. Introduction The microscopic investigation of an initially isotropic molecular fluid. acted on by an external electric field, shows that there is molecular reorientation, i.e., birefringence or induced anisotropy in the system. This phenomenon is called the Kerr effect or optical Kerr effect when the field is optical. ;nt I r, order tn Lv rrlrelrnrot ylLI such physica! processes, it is necessary to describe the orientation motion of a molecule in a liquid. In a dense fluid, a molecule will experience many collisions as it rotates from position to position. This view of dynamics of molecular liquid suggests that the process is diffusion-like in nature. Although a molecule in a fluid rotates along a deterministic trajectory according to Eulez3 equations of motion, this motion is sufficiently complex so that it may be viewed as proceeding through a rather randomly chosen path. 037%-1371/91 /$!i3.50 @ 1991 - Elsevier Science Publishers B.V. (North-Holland)
570
N.M. Hounkonnou
et al. I Nonlinear Kerr electric birefringence
Since the orientational motion of a molecule of the liquid is affected by collisions and a similarly complex trajectory, it is possible to develop a diffusion model for orientational dynamics. Lang&n [I] proposed the first mathematical theory of Kerr electric birefringence (KIB), based on the orientation of anisotropic molecules due to the introduction of an orientation function of molecules possessing a symmetry axis, Born [2] generalized this theory by introducing the permanent dipole moment. All present theories involve these basic ideas. The Kerr electro-optic phenomenon has been studied intensively in a continuous regime, often in a pure alternating regime, and by Pauthenier [3], in an impulse regime. In the last two decades, the Laboratoire de Physique AppliquCe de l’Universit6 de Perpignan has contributed much to a better understanding of these two regimes [4-S]. Filippini [9] has measured experimentally the Kerr dispersion constant when an alternating field superimposed on a unidirectional field is applied to the liquid. Theoretical studies of Coffey and Paranjape [IO], Morita [ll], Morita and Watanabe [ 121 and those of Alexiewicz [13] on the dielectric relaxation and KEB in alternating and unidirectional fields have already appeared. Peterlin and Stuart [14] obtained solutions of the electric birefringence in a sinusoidal electric field E(t) = E,, cos( ot) for the cases of pure induced dipole and pure permanent dipole orientations. The solutions are limited to infinitely small fields. The genera1 case of induced and permanent dipoles coexisting on the particle was first solved by Ogawa and Oka [15] for a very low sinusoidal electric field E(t) = E,, sin( ccrt). Later the same problem was treated by Thurston and Bowling [ 161 for a sinusoidal electric field E(r) = E,, COS(U f). Most of these works have been made in the linear regime. In this paper, we extend the already existent theory of the linear Kerr effect, to treat the nonlinear regime in superimposed ac and dc fields for the general case of molecules with permanent and induced dipole moments. We adopt a perturbation approach to treat the genera1 nonlinear regime - within the classical framework of Smoluchowski theory - and propose steady-state analytical expressions for nonlinear electric polarization and birefringence in terms of KEB fundamental harmonics and characteristic eigenvectors obtained from a vector representation of the main svstem of equations. In section 2 we present a brief theoreticafintroductidn, with the main system of equations describing the nonlinear electric polarization and birefringence. Next, we give the perturbation approach to solve these equations in the vector representation and the essential analytical results (section 3). In the last section, we give genera1 conclusions and place our results in the context of an existent analytical treatment of these physical phenomena.
N. M. Hounkonnou
et al. I Nonlinear Kerr electric birefringence
571
2. General theory Upon the assumption
that the particle is axially symmetric and noninteract-
ing with other molecules, the orientational movement of the molecule may be described by the rotational Smoluchowski diffusion equation for the angular distribution function f = f(O, t):
at
af+
a
’
af -=--
D-I
sin 0 ae
de
’ aW
liTZi
f>l’
(1)
where D is the rotational diffusion constant about the transverse axis of the molecule, kT is the thermal energy, 0 is the angle between the symmetry axis and the field direction, t is the time, and W - the orientational energy of the molecule - is given by W= -pE(t)
cos 8 -
ga,-
(2)
a,)EZ(t)cOs'e,
where or. is the permanent dipole moment along the axis of symmetry of the molecule, Acu = cy, - cy2the difference of the molecular polarizability between parallel QI, and perpendicular cyZcomponents to the ,~~olecular axis. E(t) is the external electric field. (Y,and cy, may generally be functions of the electric field and can be expressed in a power series of E as follows [ 171: ai
=
a,. +
biE’ + ciE4 +
l
l
-
(i= 1,2).
(3)
In the following, however, we regard the polarizability as a constant by neglecting effects due to the hyperpolarizability. Eq. ( 1) may be solved by expanding the distribution function f = f(0. t) in Legendre polynomials [5-S, 11-13, 19-211
fM 0 = i r1
(4)
C,,(t) P,,(cos6) , =(I
where C,,(t) is a function of time and P,, is the Legendre r”z ,
polynomial
of degree
+1
C,,(r)= Substituting
2n + 2
f
1 I
f(e,t) P,,(cas
0) d(cos 0)
(5)
.
into eq. (1) and using the recursion
relations
between
the
N. M. Hounkortrrou et al. I Nortlinet~r Kerr electric birefrittgerzce
572
Legendre polynomials, [S-8, 11-13, 19-211:
D-’
one can simply obtain the recurrence
duw4)(t)
= -0
dt
relation as follows
+ w,,(u))o)
+Y
$;; :’~UuuMt)
+’
n(n + 1) 2n+l
- R+,WWl
212+ 1 (2n-1)(2n+3)
R(u))(t)
where
( P,lw m =1 U =
cos 8
$
Y=
’
p
(P,,(d)(t) =
=
P(t)
j’
f(u,
749= & E(t) = g
E’(t)
’
,
t) P,,@)du = y,,(t)
for the normalized distribution
function:
7
I
f(u, t) du = 1 ,
-1
in which angular brackets represent the ensemble average. The electric polarization and birefringence are directly connected to y,(t) and y?(t) [6, 12, 211. By taking up the terms of second power of E(t) and fourth power of E(t) for electric polarization and electric birefringence, respectively, we define the nonlinear Kerr effect. With respect to that, we obtain the following linear equations for JJ,(i) and j#): D-‘!,(t)
=
-2Y,(t)
D-$(r) = -b,(t) Y,(O) =
0
7
i=1,2.
+
$P(t)Y!(t) - b(t)
Y#)
+ b(t)
+ $P(t) y?(t) + ; y(t) y,(t) + i, P(t)
’
l
N. M. Hounkonnou
et al. I Nonlinear Kerr electric birefringence
573
Various papers have been devoted to approximate the analytical and numerical treatments of eq. (6) or its variants for the dielectric relaxation and Kerr electric birefringence in numerous regimes of external electric fields [ 1 l-14, 18, 19, 20-241. Hounkonnou [21,25] has recently used the Runge-Kutta method to solve this equation by including the nonlinear and inertial effects for a sudden application of an alternating field superimposed on a unidirectional field. He has given an exact analytical solution to the same equation, when one applies the unidirectional field [26]. Mathematically exact solutions of eqs. (7) for an alternating field superimposed on a unidirectional field are difficult to obtain. l[n this paper. we emphasize the usefulness of exact steady state solutions and use a second-order perturbation theory in order to solve eqs. (7) for E(r) = EC + E,, cos(wt) .
(8)
3. Perturbation approach Using the followmg reduced variables:
WI=;.
t1 =Dt, and writing
E(t,) = E,[l + E cos(o’t,)] P(t,) = &[l + &cos(oftJ2
y(t,) = y,[l + E cos(ortl)]
,
.
,
where
c =-
4 EC
pE y,= k’P
h!
c’
Pc==
Ef
.
l
the system (7) may be translated
into vector representation
as follows:
N. M. How~konnou et al. I Nonlirlenr Kerr electric birej’riqetm
574
where -2+
A(t,, E) =
gy
$p
dY(t, 4 7
dt,
’
with fi = P(t,), y = y(t,); E is a small parameter. The singular cases of ordinary linear Kerr electric birefringence (KEB) are also treated by Morita and Watanabe [ll, 12,221 and Dejardin and Debiais (201 by considering
The same situation is evaluated by Schwarz [23] for purely apolar molecules (y = 0). Okonski [24] has studied the purely polar molecules (p = 0). In the following, we consider the general case of molecules possessing permanent and induced dipole moments without any approximations to above mentioned molecular parameters y and p. By expanding the dependent variables Y(t,, E), A(t,, E), F(t,, E) as a power series in E, we generate a solution to any desired order in E. Hence, we have (10)
A(t,,&)=A~,+&A,+EZAZ+...,
where -2+ 0=
93,
f-l 5 Y,
-
3Yc
-6++p,
>’
N. M. Hourrkormou et ai. I hfortlirtear Kerr electric birefrirtgertce
575
where
4,=(p) P C
9
C
F, =cos(w’t,)
= cos(o’t,) f, ,
F, = cos$.ft,)
= cos$dt, ) fz .
Y(t,, E) = Y[, + EY, + E’YZ +
l
-’
(12)
l
Substituting (lo), (11) and (12) into (9) and equating successive powers of E, we obtain
$0=4A + 4, 9, = A&
the coefficients
of
’
+ A,Y,, + F,
,
(13)
f2 = A,,u, + A,Y,, + Y,,(O,E) = 0
9
rz=0,1,2.
At any reduced time, the vector YJt,) can be written as the superposition of two solutions: the first describing the transient behavior Jr,) of the system. and the second its steady-state evolution Z,,( t, ):
W*) = w, ) + z,,(t*) *
(14)
We can obtain the characteristic time 7, at the end of which the transient regime vanishes, by solving the characteristic equation of
where A is an eigenvalue obtain
of
A’ + 2h(4 - g&)-2(-&&-
the identity matrix.
$yf+
g&-6)=0.
N. M. Hounkonnou
et al. I Nonlinear Kerr electric hirefringence
The discriminant A’ of (16) is
with R = y&3, ( yc and & G 1). We obtain approximately
A, = -2+ $$y;lR-
-6+#yflR+$yf=-l/7,,
A, = *1
=I
{yf = - l/r,,
z
and
rz -ACT,. =
Thus, after a reduced characteristic time r. 3 7,) the system evolves in a steady regime. We note that the mentioned characteristic time is in excellent agreement with the numerical results of Lee et al. [19] and with the analytical solution of Hounkonnou [26]. In the following, we take an interest in the steady-state JLtion. Thus, eq. (14) is reduced to
Therefore, it is straightforward to find (l’,, = 0):
Z&l 1= -A, ‘4,
(18)
l
F,, is well known. Hence the vector Z. is wholly defined. To first order of E, we have
e =- a
WV
A particular solution can be written (4 = 2, =
cos(dt,
+
6)
)
(194
where is represents a certain phase angle. Exploiting eqs, (19) and elementary trigonometric relations, we obtain
N. M. Hounkonnou
-@dcos6=
et al. I Nonlinear Kerr electric birefringence
-A,,b,sinS,
577
cw
Eq. (20a) leads to
of A,, corresponding
Eq. (20~) shows that 4 is an eigenvector A (A = A,, A = h,) as follows:
to the eigenvalue
A= w’cotg6.
The value 6 (6, and &) is thus determined, addition, the vector + is defined by (19b). Therefore the norms of the eigenvectors @=6,
(A+)
and
+=+,
since h is well known.
In
(A-A,)
are determined by cqs. (20b) and (20~) and the general wholly defined by the superposition of the particular solut
solution
This vector solution only depends on ac field harmonics and eigenvectors expressed in terms of molecular We now examine the solution to the second order of E. We have i,
=
+
(214
cos(w’t, + 6,)
We can also write as steady oscillating
2, =
cos(2dt,
+
A)
+
X
and constant
solutions, (21b) (214
578
N. M. Homkonnou
et al. I Nonlinear Kerr electric birefringence
According to the same way as in the case of the first-order cgkulation, exploiting eqs. (21) and elementary trigonometric transformations, we wholly define 2,” by the following relations: 422a)‘sinA=A,,acosA+ where cb;l= fi
- a,& ‘&,,
-In2o’cos
A = -A,$
d(~osS,a,~~+~os6,a,~~+~~),
sin A - i(sin 8, a,+, + sin 6, a,&),
Pa)
PO
Let us put &#= a, + 0, + J& (J2, solution of eqs. (22a) and (22b) when t#+= & = 0 and by analogy in a circular way for J2, and a,). We obtain cos A; sin Ai - sin A,,Ini = -20’ cos 6, i
sin A’. cos A; _ + sin COS 6i
i
WC)
and A,,@ = a3 2~’ cotg A;
(i= 1,2; j= 1,2).
Wd)
Eqs. (22~) and (22d) show again that pi (normalized by eqs. (22a) and (22b)) is an eigenvector of A(,, which corresponds to eigenvalue Ai:
hj = -20’
COS(A~ - Si)
sin(Ai + Si)
hi = 2~’ cotg A;
(j=
(j=1,2;i=1,2),
1,2).
We) Wf)
Thus, the vector 2, is wholly defined by the superposition of solutions: constant X7=.___I A /I_-4 n *>+\Ath~t-ofnr~ A dlIll 9Ci \C - i,2.3) fGT eXh ~ig~Gk%!*d~ h, WI r8(), u1.u L‘lk. YAV.V to SecGEd 1
.
order perturbation the steady solutron of the vector equation (9) is given by
W,) = z,,+ EZ&,) +
EZZ,(f,) .
Thus, the second order steady-state perturbation solutions of electric polarization and birefringence are expressed as functions of single and double a: field harmonics and molecular parameters.
N. M. Hounkonnou
et al.
I Nonlinear
Kerr electric birefringence
579
4. Conclusions and remarks Analyzing the obtained results, we notice the appearance of a simple frequency for the first-order solution, whereas the second-order solution leads to a double frequency as the fundamental harmonics of KEB. This agrees well with the numerical calculations [21,25]. These general steady-state theoretical expressions provide the basis for discussions of the steady-state harmonic variations in the phenomena of electric polarization and birefringence. Using the experimental values of p and Aa, and choosing the ac and dc fields, the curves of electric polarization and birefringence can be drawn and compared to the steady experimental data. Conversely the molecular parameters p and Aa could be easily extracted from the steady solutions obtained here. But the inexistence of experimental data of the steady-state time evolution of nonlinear KEB in ac and dc superimposed fields does not permit such comparisons today. In the present state of the theoretical development of the Kerr relaxation processes in ac and dc field coupling, we have been unable to find in the literature exact results concerning the electric birefringence taking into account nonlinear effects. Indeed, the system of equations giving the electric polarization and birefringence is very difficult to solve analytically when both ac and dc perturbations are present. Some authors have proposed approximate solutions in the linear regimes [6-8,11,20]. However, to solve these complex systems account must be taken of the ncnlinear effects of species in a given physical medium. Numerical methods have been used in recent works [19,21,25] to solve this problem. Approximate analytical solutions to this problem have also been obtained by means of tedious Laplace transforms [20,27,28] but the resulting expressions are not simple to exploit. The main purpose of this paper has been to find another approach to solve this problem: the steady-state nonlinear electric polarization and birefringence in ac and dc field coupling are expressed explicitly in terms of molecular ayyxrbu parameters and ac fieid harmonics. To take this into account, -Wehave ~**l;cwl a second-order perturbation theory, which tu r~s opt to be sufficient to produce the fundamental harmonics of KEB when the fluid under investigation is perturWi by an alternating field [4-8, 11, 12, 21, 25,281. This perturbation theory is very general and its solution may be extended to any desired order in E. In particular, we have proved that the effects of higher harmonics in these fields (but less relevant experimentally [S]) are easily obtained by a simple recurrence procedure, even when the inertial effects are included [29
580
N. M. Hounkonnou
et al. / Nonlinecrr Kerr electric biwfringence
Although all these results are purely theoretical, they provide a basic formalism for the interpretation of experiments on the nonlinear dynamic molecular relaxation processes in ac and de coupled fields.
Acknowledgements We acknowledge the referee for his constructive criticism. NM. Hounkonnou thanks Professor A. Ronveaux and all members of his laboratory for their hospitality where this work has been accomplished.
References [I] P. Langevin, Radium 7 ( 1910) 249. [2] M. Born, Ann. Phys. 55 (1918) 77. [3] M.J. Pauthenier, J. Phys. Rad. 2 (1921) 253. [4] S. Brunet, thesis Montpellier, France (1971). [S] R. Marrony, thesis Montpellier, France (1971). [6] J.-L. Dejardin, thesis Montpellier, France (1971). [7] C. Delseny, thesis Perpignan, France (1980). (81 S. Benet, thesis Perpignan, France (1987). [9] J.C. Filippini, thesis Grenoble, France (1972). [lo] W.T. Coffey and B.V. Paranjape, Proc. R. Ir. Acad. 78A (1978) 17. [ 111 A. Morita, J. Phys. D 11 (1978) 1357. [ 121 H. Watanabe and A. Morita, Adv. Chem. Phys. 56 (1984) 255. [ 13) W. Alexiewicz, Mol. Phys. 59 (1986) 637. [14] A. Peterlin and H.A. Stuart, 2. Phys. 112 (1939) 129. [15] S. Ogawa and S. Oka, J. Phys. Sot. Jpn. 15 (1960) 658. [ 161 B. Thurston and D.L. Bowlng, J. Colloid Interface Sci. 30 (1969) 34. [17] A.D. Buckingham and J.A. Pople, Proc. Phys. Sot. A 68 (1955) 905. [18] H. Benoit, Ann. Phys. (Paris) 6 (1951) 561. [ 191 Y.H. Lee, D. Kim and S.H. Lee, J. Chem. Phys. 91 (1989) 5628. (20) J-L. Dejardin and G. Debiais, Physica A 161 (1990) 182. [21] N.M. Hounkonnou, thesis DEA Perpignan, France (1989). [22] H. Watanabe and A. Morita, J. Phys. D 13 (1980) L153. (231 G. Schwarz, Z. Phys. 145 (1956) 563. [24] C.T. O‘Korski and S. Krause, J. Phys. Chem. 74 (1970) 3243. (251 N.M. Hourkonnou, J. Chcm. Sot. Faraday Trans. 87 (lSS1) 297. [26) N.M. Hourikonnou, to be published. [27) A. Morita Phys. Rev. A 34 (1986) 1499. [28] A. Morita and H. Watanabe, Phys. Rev. A 35 (1987) 2690. [29] N.M. Holnkonnou and A. Ronveaux, to be published.
ON STEADY
STATE SOLUTION
BIREFRINGENCE SUPERIMPOSED
OF NONLINEAR
AND POLARIZATION
KERR ELECTRIC
FOR AC AND DC
FIELDS
N.M. HOUNKONNOU and A. RONVEAUX Laboratoire de Physique Matht?matique, Facrrltk Universitaires Notre-Dame de la Paix. Rue de Bruelies (il. B-SO00 Namtrr. Belgium
R.P. HAZOUME Laboratoire de Modklisation de S~&wes Physiques, lrtstitut de Mathhmatiqrces et de Sciences Physiques (IMSP). B. P. 61-Z.Porto-Now.
Bertirr
Received 22 November 1990 Revised manuscript received 20 March 1991
The steady state response of the nonlinear Kerr effect to a small ahernating field superimposed on a unidirectional field is anaiyzed using a second order perturbation theory. We draw inferences from the appearance of simple and double harmonics as the fundamental frequencies of the nonlinear Kerr effect. The study is based on the Smoluchowski equation of rotational Brownian motion.
1. Introduction The microscopic investigation of an initially isotropic molecular fluid. acted on by an external electric field, shows that there is molecular reorientation, i.e., birefringence or induced anisotropy in the system. This phenomenon is called the Kerr effect or optical Kerr effect when the field is optical. ;nt I r, order tn Lv rrlrelrnrot ylLI such physica! processes, it is necessary to describe the orientation motion of a molecule in a liquid. In a dense fluid, a molecule will experience many collisions as it rotates from position to position. This view of dynamics of molecular liquid suggests that the process is diffusion-like in nature. Although a molecule in a fluid rotates along a deterministic trajectory according to Eulez3 equations of motion, this motion is sufficiently complex so that it may be viewed as proceeding through a rather randomly chosen path. 037%-1371/91 /$!i3.50 @ 1991 - Elsevier Science Publishers B.V. (North-Holland)
570
N.M. Hounkonnou
et al. I Nonlinear Kerr electric birefringence
Since the orientational motion of a molecule of the liquid is affected by collisions and a similarly complex trajectory, it is possible to develop a diffusion model for orientational dynamics. Lang&n [I] proposed the first mathematical theory of Kerr electric birefringence (KIB), based on the orientation of anisotropic molecules due to the introduction of an orientation function of molecules possessing a symmetry axis, Born [2] generalized this theory by introducing the permanent dipole moment. All present theories involve these basic ideas. The Kerr electro-optic phenomenon has been studied intensively in a continuous regime, often in a pure alternating regime, and by Pauthenier [3], in an impulse regime. In the last two decades, the Laboratoire de Physique AppliquCe de l’Universit6 de Perpignan has contributed much to a better understanding of these two regimes [4-S]. Filippini [9] has measured experimentally the Kerr dispersion constant when an alternating field superimposed on a unidirectional field is applied to the liquid. Theoretical studies of Coffey and Paranjape [IO], Morita [ll], Morita and Watanabe [ 121 and those of Alexiewicz [13] on the dielectric relaxation and KEB in alternating and unidirectional fields have already appeared. Peterlin and Stuart [14] obtained solutions of the electric birefringence in a sinusoidal electric field E(t) = E,, cos( ot) for the cases of pure induced dipole and pure permanent dipole orientations. The solutions are limited to infinitely small fields. The genera1 case of induced and permanent dipoles coexisting on the particle was first solved by Ogawa and Oka [15] for a very low sinusoidal electric field E(t) = E,, sin( ccrt). Later the same problem was treated by Thurston and Bowling [ 161 for a sinusoidal electric field E(r) = E,, COS(U f). Most of these works have been made in the linear regime. In this paper, we extend the already existent theory of the linear Kerr effect, to treat the nonlinear regime in superimposed ac and dc fields for the general case of molecules with permanent and induced dipole moments. We adopt a perturbation approach to treat the genera1 nonlinear regime - within the classical framework of Smoluchowski theory - and propose steady-state analytical expressions for nonlinear electric polarization and birefringence in terms of KEB fundamental harmonics and characteristic eigenvectors obtained from a vector representation of the main svstem of equations. In section 2 we present a brief theoreticafintroductidn, with the main system of equations describing the nonlinear electric polarization and birefringence. Next, we give the perturbation approach to solve these equations in the vector representation and the essential analytical results (section 3). In the last section, we give genera1 conclusions and place our results in the context of an existent analytical treatment of these physical phenomena.
N. M. Hounkonnou
et al. I Nonlinear Kerr electric birefringence
571
2. General theory Upon the assumption
that the particle is axially symmetric and noninteract-
ing with other molecules, the orientational movement of the molecule may be described by the rotational Smoluchowski diffusion equation for the angular distribution function f = f(O, t):
at
af+
a
’
af -=--
D-I
sin 0 ae
de
’ aW
liTZi
f>l’
(1)
where D is the rotational diffusion constant about the transverse axis of the molecule, kT is the thermal energy, 0 is the angle between the symmetry axis and the field direction, t is the time, and W - the orientational energy of the molecule - is given by W= -pE(t)
cos 8 -
ga,-
(2)
a,)EZ(t)cOs'e,
where or. is the permanent dipole moment along the axis of symmetry of the molecule, Acu = cy, - cy2the difference of the molecular polarizability between parallel QI, and perpendicular cyZcomponents to the ,~~olecular axis. E(t) is the external electric field. (Y,and cy, may generally be functions of the electric field and can be expressed in a power series of E as follows [ 171: ai
=
a,. +
biE’ + ciE4 +
l
l
-
(i= 1,2).
(3)
In the following, however, we regard the polarizability as a constant by neglecting effects due to the hyperpolarizability. Eq. ( 1) may be solved by expanding the distribution function f = f(0. t) in Legendre polynomials [5-S, 11-13, 19-211
fM 0 = i r1
(4)
C,,(t) P,,(cos6) , =(I
where C,,(t) is a function of time and P,, is the Legendre r”z ,
polynomial
of degree
+1
C,,(r)= Substituting
2n + 2
f
1 I
f(e,t) P,,(cas
0) d(cos 0)
(5)
.
into eq. (1) and using the recursion
relations
between
the
N. M. Hounkortrrou et al. I Nortlinet~r Kerr electric birefrittgerzce
572
Legendre polynomials, [S-8, 11-13, 19-211:
D-’
one can simply obtain the recurrence
duw4)(t)
= -0
dt
relation as follows
+ w,,(u))o)
+Y
$;; :’~UuuMt)
+’
n(n + 1) 2n+l
- R+,WWl
212+ 1 (2n-1)(2n+3)
R(u))(t)
where
( P,lw m =1 U =
cos 8
$
Y=
’
p
(P,,(d)(t) =
=
P(t)
j’
f(u,
749= & E(t) = g
E’(t)
’
,
t) P,,@)du = y,,(t)
for the normalized distribution
function:
7
I
f(u, t) du = 1 ,
-1
in which angular brackets represent the ensemble average. The electric polarization and birefringence are directly connected to y,(t) and y?(t) [6, 12, 211. By taking up the terms of second power of E(t) and fourth power of E(t) for electric polarization and electric birefringence, respectively, we define the nonlinear Kerr effect. With respect to that, we obtain the following linear equations for JJ,(i) and j#): D-‘!,(t)
=
-2Y,(t)
D-$(r) = -b,(t) Y,(O) =
0
7
i=1,2.
+
$P(t)Y!(t) - b(t)
Y#)
+ b(t)
+ $P(t) y?(t) + ; y(t) y,(t) + i, P(t)
’
l
N. M. Hounkonnou
et al. I Nonlinear Kerr electric birefringence
573
Various papers have been devoted to approximate the analytical and numerical treatments of eq. (6) or its variants for the dielectric relaxation and Kerr electric birefringence in numerous regimes of external electric fields [ 1 l-14, 18, 19, 20-241. Hounkonnou [21,25] has recently used the Runge-Kutta method to solve this equation by including the nonlinear and inertial effects for a sudden application of an alternating field superimposed on a unidirectional field. He has given an exact analytical solution to the same equation, when one applies the unidirectional field [26]. Mathematically exact solutions of eqs. (7) for an alternating field superimposed on a unidirectional field are difficult to obtain. l[n this paper. we emphasize the usefulness of exact steady state solutions and use a second-order perturbation theory in order to solve eqs. (7) for E(r) = EC + E,, cos(wt) .
(8)
3. Perturbation approach Using the followmg reduced variables:
WI=;.
t1 =Dt, and writing
E(t,) = E,[l + E cos(o’t,)] P(t,) = &[l + &cos(oftJ2
y(t,) = y,[l + E cos(ortl)]
,
.
,
where
c =-
4 EC
pE y,= k’P
h!
c’
Pc==
Ef
.
l
the system (7) may be translated
into vector representation
as follows:
N. M. How~konnou et al. I Nonlirlenr Kerr electric birej’riqetm
574
where -2+
A(t,, E) =
gy
$p
dY(t, 4 7
dt,
’
with fi = P(t,), y = y(t,); E is a small parameter. The singular cases of ordinary linear Kerr electric birefringence (KEB) are also treated by Morita and Watanabe [ll, 12,221 and Dejardin and Debiais (201 by considering
The same situation is evaluated by Schwarz [23] for purely apolar molecules (y = 0). Okonski [24] has studied the purely polar molecules (p = 0). In the following, we consider the general case of molecules possessing permanent and induced dipole moments without any approximations to above mentioned molecular parameters y and p. By expanding the dependent variables Y(t,, E), A(t,, E), F(t,, E) as a power series in E, we generate a solution to any desired order in E. Hence, we have (10)
A(t,,&)=A~,+&A,+EZAZ+...,
where -2+ 0=
93,
f-l 5 Y,
-
3Yc
-6++p,
>’
N. M. Hourrkormou et ai. I hfortlirtear Kerr electric birefrirtgertce
575
where
4,=(p) P C
9
C
F, =cos(w’t,)
= cos(o’t,) f, ,
F, = cos$.ft,)
= cos$dt, ) fz .
Y(t,, E) = Y[, + EY, + E’YZ +
l
-’
(12)
l
Substituting (lo), (11) and (12) into (9) and equating successive powers of E, we obtain
$0=4A + 4, 9, = A&
the coefficients
of
’
+ A,Y,, + F,
,
(13)
f2 = A,,u, + A,Y,, + Y,,(O,E) = 0
9
rz=0,1,2.
At any reduced time, the vector YJt,) can be written as the superposition of two solutions: the first describing the transient behavior Jr,) of the system. and the second its steady-state evolution Z,,( t, ):
W*) = w, ) + z,,(t*) *
(14)
We can obtain the characteristic time 7, at the end of which the transient regime vanishes, by solving the characteristic equation of
where A is an eigenvalue obtain
of
A’ + 2h(4 - g&)-2(-&&-
the identity matrix.
$yf+
g&-6)=0.
N. M. Hounkonnou
et al. I Nonlinear Kerr electric hirefringence
The discriminant A’ of (16) is
with R = y&3, ( yc and & G 1). We obtain approximately
A, = -2+ $$y;lR-
-6+#yflR+$yf=-l/7,,
A, = *1
=I
{yf = - l/r,,
z
and
rz -ACT,. =
Thus, after a reduced characteristic time r. 3 7,) the system evolves in a steady regime. We note that the mentioned characteristic time is in excellent agreement with the numerical results of Lee et al. [19] and with the analytical solution of Hounkonnou [26]. In the following, we take an interest in the steady-state JLtion. Thus, eq. (14) is reduced to
Therefore, it is straightforward to find (l’,, = 0):
Z&l 1= -A, ‘4,
(18)
l
F,, is well known. Hence the vector Z. is wholly defined. To first order of E, we have
e =- a
WV
A particular solution can be written (4 = 2, =
cos(dt,
+
6)
)
(194
where is represents a certain phase angle. Exploiting eqs, (19) and elementary trigonometric relations, we obtain
N. M. Hounkonnou
-@dcos6=
et al. I Nonlinear Kerr electric birefringence
-A,,b,sinS,
577
cw
Eq. (20a) leads to
of A,, corresponding
Eq. (20~) shows that 4 is an eigenvector A (A = A,, A = h,) as follows:
to the eigenvalue
A= w’cotg6.
The value 6 (6, and &) is thus determined, addition, the vector + is defined by (19b). Therefore the norms of the eigenvectors @=6,
(A+)
and
+=+,
since h is well known.
In
(A-A,)
are determined by cqs. (20b) and (20~) and the general wholly defined by the superposition of the particular solut
solution
This vector solution only depends on ac field harmonics and eigenvectors expressed in terms of molecular We now examine the solution to the second order of E. We have i,
=
+
(214
cos(w’t, + 6,)
We can also write as steady oscillating
2, =
cos(2dt,
+
A)
+
X
and constant
solutions, (21b) (214
578
N. M. Homkonnou
et al. I Nonlinear Kerr electric birefringence
According to the same way as in the case of the first-order cgkulation, exploiting eqs. (21) and elementary trigonometric transformations, we wholly define 2,” by the following relations: 422a)‘sinA=A,,acosA+ where cb;l= fi
- a,& ‘&,,
-In2o’cos
A = -A,$
d(~osS,a,~~+~os6,a,~~+~~),
sin A - i(sin 8, a,+, + sin 6, a,&),
Pa)
PO
Let us put &#= a, + 0, + J& (J2, solution of eqs. (22a) and (22b) when t#+= & = 0 and by analogy in a circular way for J2, and a,). We obtain cos A; sin Ai - sin A,,Ini = -20’ cos 6, i
sin A’. cos A; _ + sin COS 6i
i
WC)
and A,,@ = a3 2~’ cotg A;
(i= 1,2; j= 1,2).
Wd)
Eqs. (22~) and (22d) show again that pi (normalized by eqs. (22a) and (22b)) is an eigenvector of A(,, which corresponds to eigenvalue Ai:
hj = -20’
COS(A~ - Si)
sin(Ai + Si)
hi = 2~’ cotg A;
(j=
(j=1,2;i=1,2),
1,2).
We) Wf)
Thus, the vector 2, is wholly defined by the superposition of solutions: constant X7=.___I A /I_-4 n *>+\Ath~t-ofnr~ A dlIll 9Ci \C - i,2.3) fGT eXh ~ig~Gk%!*d~ h, WI r8(), u1.u L‘lk. YAV.V to SecGEd 1
.
order perturbation the steady solutron of the vector equation (9) is given by
W,) = z,,+ EZ&,) +
EZZ,(f,) .
Thus, the second order steady-state perturbation solutions of electric polarization and birefringence are expressed as functions of single and double a: field harmonics and molecular parameters.
N. M. Hounkonnou
et al.
I Nonlinear
Kerr electric birefringence
579
4. Conclusions and remarks Analyzing the obtained results, we notice the appearance of a simple frequency for the first-order solution, whereas the second-order solution leads to a double frequency as the fundamental harmonics of KEB. This agrees well with the numerical calculations [21,25]. These general steady-state theoretical expressions provide the basis for discussions of the steady-state harmonic variations in the phenomena of electric polarization and birefringence. Using the experimental values of p and Aa, and choosing the ac and dc fields, the curves of electric polarization and birefringence can be drawn and compared to the steady experimental data. Conversely the molecular parameters p and Aa could be easily extracted from the steady solutions obtained here. But the inexistence of experimental data of the steady-state time evolution of nonlinear KEB in ac and dc superimposed fields does not permit such comparisons today. In the present state of the theoretical development of the Kerr relaxation processes in ac and dc field coupling, we have been unable to find in the literature exact results concerning the electric birefringence taking into account nonlinear effects. Indeed, the system of equations giving the electric polarization and birefringence is very difficult to solve analytically when both ac and dc perturbations are present. Some authors have proposed approximate solutions in the linear regimes [6-8,11,20]. However, to solve these complex systems account must be taken of the ncnlinear effects of species in a given physical medium. Numerical methods have been used in recent works [19,21,25] to solve this problem. Approximate analytical solutions to this problem have also been obtained by means of tedious Laplace transforms [20,27,28] but the resulting expressions are not simple to exploit. The main purpose of this paper has been to find another approach to solve this problem: the steady-state nonlinear electric polarization and birefringence in ac and dc field coupling are expressed explicitly in terms of molecular ayyxrbu parameters and ac fieid harmonics. To take this into account, -Wehave ~**l;cwl a second-order perturbation theory, which tu r~s opt to be sufficient to produce the fundamental harmonics of KEB when the fluid under investigation is perturWi by an alternating field [4-8, 11, 12, 21, 25,281. This perturbation theory is very general and its solution may be extended to any desired order in E. In particular, we have proved that the effects of higher harmonics in these fields (but less relevant experimentally [S]) are easily obtained by a simple recurrence procedure, even when the inertial effects are included [29
580
N. M. Hounkonnou
et al. / Nonlinecrr Kerr electric biwfringence
Although all these results are purely theoretical, they provide a basic formalism for the interpretation of experiments on the nonlinear dynamic molecular relaxation processes in ac and de coupled fields.
Acknowledgements We acknowledge the referee for his constructive criticism. NM. Hounkonnou thanks Professor A. Ronveaux and all members of his laboratory for their hospitality where this work has been accomplished.
References [I] P. Langevin, Radium 7 ( 1910) 249. [2] M. Born, Ann. Phys. 55 (1918) 77. [3] M.J. Pauthenier, J. Phys. Rad. 2 (1921) 253. [4] S. Brunet, thesis Montpellier, France (1971). [S] R. Marrony, thesis Montpellier, France (1971). [6] J.-L. Dejardin, thesis Montpellier, France (1971). [7] C. Delseny, thesis Perpignan, France (1980). (81 S. Benet, thesis Perpignan, France (1987). [9] J.C. Filippini, thesis Grenoble, France (1972). [lo] W.T. Coffey and B.V. Paranjape, Proc. R. Ir. Acad. 78A (1978) 17. [ 111 A. Morita, J. Phys. D 11 (1978) 1357. [ 121 H. Watanabe and A. Morita, Adv. Chem. Phys. 56 (1984) 255. [ 13) W. Alexiewicz, Mol. Phys. 59 (1986) 637. [14] A. Peterlin and H.A. Stuart, 2. Phys. 112 (1939) 129. [15] S. Ogawa and S. Oka, J. Phys. Sot. Jpn. 15 (1960) 658. [ 161 B. Thurston and D.L. Bowlng, J. Colloid Interface Sci. 30 (1969) 34. [17] A.D. Buckingham and J.A. Pople, Proc. Phys. Sot. A 68 (1955) 905. [18] H. Benoit, Ann. Phys. (Paris) 6 (1951) 561. [ 191 Y.H. Lee, D. Kim and S.H. Lee, J. Chem. Phys. 91 (1989) 5628. (20) J-L. Dejardin and G. Debiais, Physica A 161 (1990) 182. [21] N.M. Hounkonnou, thesis DEA Perpignan, France (1989). [22] H. Watanabe and A. Morita, J. Phys. D 13 (1980) L153. (231 G. Schwarz, Z. Phys. 145 (1956) 563. [24] C.T. O‘Korski and S. Krause, J. Phys. Chem. 74 (1970) 3243. (251 N.M. Hourkonnou, J. Chcm. Sot. Faraday Trans. 87 (lSS1) 297. [26) N.M. Hourikonnou, to be published. [27) A. Morita Phys. Rev. A 34 (1986) 1499. [28] A. Morita and H. Watanabe, Phys. Rev. A 35 (1987) 2690. [29] N.M. Holnkonnou and A. Ronveaux, to be published.