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Fluctuations in nematics near Freedericksz and flexoelectric transitions
Colloids and Surfaces A: Physicochemical and Engineering Aspects 158 (1999) 355 – 361 www.elsevier.nl/locate/colsurfa
Fluctuations in nematics near Freedericksz and flexoelectric transitions V.P. Romanov *, G.K. Sklyarenko Faculty of Physics, St. Petersburg State Uni6ersity, St. Petersburg, 198904, Russia Received 26 January 1998; accepted 16 March 1999
Abstract The director fluctuations in nematic liquid crystals are considered in the presence of an electric field by taking into account the flexoelectric effect The orientation of the director on the surfaces is supposed to be fixed. For the planar orientation, the spatial correlation function is calculated. This function is analysed near the Freedericksz transition and near the threshold flexoelectric instability. The spectrum of director fluctuations varies significantly at fields close to threshold one. For electric fields close to threshold one, angular dependence of light scattering is studied. © 1999 Elsevier Science B.V. All rights reserved. Keywords: Electric field; Liquid crystal; Fluctuations; Freedericksz transition; Flexoelectric effect
1. Introduction Liquid crystals undergo various structural transformations in the presence of external electric and magnetic fields [1,2] owing to anisotropy of permittivity and magnetic susceptibility. Here, we bear in mind instabilities of uniformly oriented liquid crystals in fields above a threshold. Such a change in structure is equivalent to a phase transition of second order [1,4]. The threshold character of these phenomena is connected to spatial limitations of liquid crystal layers and a rigidity of boundary conditions. * Corresponding author. Tel.: +7-812-4287033; fax: +7812-2487240. E-mail address: [email protected] (V.P. Romanov)
The most investigated and well known effect of the orientational phase transition in nematic liquid crystals is the Freedericksz effect [1,2]. This transition takes place both in static fields and in a light wave field [4–6]. It was studied for various boundary conditions and orientations for periodic and aperiodic distortions [7–10]. In 1969, Meyer [11] showed that liquid crystals having a centre of symmetry possess the ability to show evidence of a piezoelectric effect which has received the name of flexoelectric one. This linear effect of modulated structure formation produced by the homogeneous electric field is due to the anisotropy of molecules with permanent dipole moments [1,2]. The present work is devoted to the study of thermal fluctuations in confined nematic liquid
0927-7757/99/$ - see front matter © 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 7 7 5 7 ( 9 9 ) 0 0 1 8 1 - 8
356
V.P. Romano6, G.K. Sklyarenko / Colloids and Surfaces A: Physicochem. Eng. Aspects 158 (1999) 355–361
crystals (NLC) in the presence of the flexoelectric effect. The standard method of the fluctuations’ research consists of an expansion in the spectrum of eigenfunctions [7 – 10,12 – 14]. The solution is written as an infinite series and each term of this series is obtained by the solution of a difficult transcendental equation. In Refs. [15,16], the method of obtaining the correlation function of director fluctuations in the closed form was suggested for arbitrary energy of interaction with a substrate.
Let a NLC possessing flexoelectric properties be confined between plane-parallel plates of thickness L with the rigid boundary conditions in the external electric field. We introduce the Cartesian coordinate frame with the origin in the cell centre and z axis normal to the plates. The variation of free energy DF consists of the elastic contribution DFfr, the contribution connected to the director orientation in the external field DFel and the contribution caused by occurrence of the flexoelectric polarization Dfflex [1,2]
&
1 DF =DFfr + DFel +DFflex = d 3r{K1(div n)2 2
"
− 2E[e1n div n +e3(rot n ×n)] ,
dn x, y, z= 9
L = 0. 2
(2)
For small deviations of the director, the vectors n0 and dn are orthogonal, i.e. dn=(0, ny, nz ). Integrating Eq. (1) by parts in view of the boundary conditions in Eq. (2) up to accuracy of the second order terms over dn, Eq. (1) may be presented as follows DF =
2. Correlation matrix of a nematic in external field
+ K2(n · rot n)2 +K3(n × rot n)2 −
fluctuation. Because of the rigid boundary conditions we have
&
1 3 d rdn(r)Adnt(r), 2
(3)
where A is a differential operator, and index t means transposition. The correlation matrix of the director fluctuations G(r, r%) = B dn(r)dn+(r%) \ should satisfy the relation [3] (4)
AG(r, r%) = kbTId(r− r%),
where I is the unit matrix, T is the temperature, and kb is the Boltzmann constant. Fluctuation dn(r) is suitable to expand into the two-dimensional Fourier integral dn(r) =
&
1 d 2q exp[− i(q, rÞ)]dn(q, z), (2p)2
(5)
where q=(qÞ, 0) is the wave vector. After the Fourier-transformation, Eq. (4) has the form
oa (n · E)2 4p
AqG(q, z, z%)= kbTId(z −z%), (1)
where K1, K2, K3 are the Frank modules, n is the vector of director, E is the intensity of the electric field, oa =o −oÞ is the permittivity anisotropy, o , oÞ the permittivities along and normal to average orientation of nematic, e1, e3 are the flexoelectric coefficients. We consider the contribution to free energy depending upon the director fluctuations for the planar orientation. The homogeneous electric field is directed along the z axis: E = (0, 0, E). We present the vector of director in the form n= n0 + dn, where n0 =(1, 0, 0) is the equilibrium orientation of the director and dn is the director
(6)
where Aq is the Fourier-transform of operator A. Hence, the correlation matrix is obtained by the conversion of the Aq operator taking into account the boundary conditions
L G q, z = 9 , z% = 0. 2
(7)
In the geometry, we are interested in the variation of free energy (1) up to accuracy of the second order terms over dn has the form DF =
&
1 3 d r{K1((yny + (znz )2 + K2((ynz + (zny )2 2
+ K3[((xny )2 + ((xnz )2]−
oa 2 2 E n z − 2E[e1nz ((znz 4p
"
V.P. Romano6, G.K. Sklyarenko / Colloids and Surfaces A: Physicochem. Eng. Aspects 158 (1999) 355–361
+ (yny )+e3((xnz +ny(ynz +nz(znz )] .
(8)
357
If we introduce a new variable
i −1 D Cz 2
n
The matrix Aq, is equal to
G0 = exp
K 3q 2x +K1q 2y −K2( 2z Á à Aq = à −i(K 1 −K2)qy(z −i(e1 −e3)Eqy à Ä
and take into account the boundary conditions of Eq. (7), we have
Bq G0(q, z, z%)=kbT exp d(z− z%),
− i(K 1 −K2)qy(z +i(e1 −e3)Eqy K 3q 2x +K2q 2y −K1( 2z −
oa 2 E 4p
à à à Å
(9)
where the matrixes A0, C and D are equal to i(e 1 −e3)Eqy Á K 3q 2x +K1q 2y  à à A0 = à Ã, o −i(e 1 −e3)Eqy K 3q 2x +K2q 2y − a E 2 à 4p Ã Ä Å 0 − (K 1 −K2)qy
−K2 D= 0
−(K 1 −K2)qy , 0
i I(z + D − 1C 2
0 . −K 1
n
p
+H G =kbTD
D−1 (13)
d(z− z%),
−1
(10) where (11)
i −1 D Cz 2
(14)
n
i H exp − D − 1Cz 2
n
+ I( 2z
(15)
has the form
Á à − a sin2 a− b cos2 a+ f sin 2a + ( 2z Bq = à K 2 a− b à −i sin 2a +fcos 2a K1 2 Ä
'
'
 à à − a cos2 a− bsin2 a+ f sin2 a+ ( 2z à Ši
a=
2
1 H = (D − 1C)2 +D − 1A0. 4
Bq = exp
K 1 a− b sin 2a + fcos 2a K2 2
where
Multiplying both parts of Eq. (9) on matrix D − 1 and extracting a perfect square, we get
n
where the matrix
(A0 +iC(z +D( 2z )Gp =kbTId(z− z%),
i −1 D Cz 2
L G0 q, z = 9 , z% = 0, 2
For the solution of Eq. (6), it is convenient to present the matrix Aq in diagonal form. It is impossible to carry out this procedure by the transformation of similarity. Therefore, we transform Eq. (6) so that the left-hand side of this equation will contain an operator Bq admitting diagonalization. For this purpose, we present Eq. (6) as follows
C=
(12)
G
(16)
1 3 K2 1 K1 2 o K3 2 qx+ + − q y − a E 2, K1 2 4 K1 4 K2 4pK1
b= f=
1 3 K1 1 K2 2 K3 2 q + + − q , K2 x 2 4 K2 4 K1 y
(e1 − e3)
K1K2
Eqy,
a= −
1 (K1 − K2) q z. 2 K1K2 y
(17)
After diagonalization of the Bq matrix and the solution of obtained differential equations, we find the expressions for the elements of the correlation matrix
!
V.P. Romano6, G.K. Sklyarenko / Colloids and Surfaces A: Physicochem. Eng. Aspects 158 (1999) 355–361
358
G11 = ny (q, z)n+ y (q, z%) = −
1 1 (w cos 2a 2K2 p
!
+ iu sin 2a) ×[J(P) −J(Q)] + [J(P) +J(Q)]}, 1 1 G22 =nz (q, z)n+ (w cos 2a z (q, z%) = 2K2 p −u sin 2a)×[J(P) −J(Q)][J(P) +J(Q)]}, G12 = nz (q, z)n+ z (q, z%) =
1
!
(18)
1 (w cos 2a 2 K1K2 p
−u sin 2a)× [J(P) −J(Q)], G21 =nz (q, z)n+ z (q, z%) = − G12. where u=f cos 2a − +
a−b sin 2a, 2
a−b cos 2a, 2
P=
'
p=
a+b +p, 2
J(r)=
w = f sin 2a
' '
Q=
a−b +f 2. 2
(19)
a+b −p, 2
(20)
kbT {cosh[r(z+ z%)] −cosh(rL) 2r sinh(rL)
cosh[r(z−z%)] +sinh(rL)sinh(r z− z% )}.
(21)
The obtained results for E = 0 coincide with the results of [12,15] for the rigid boundary conditions.
3. The threshold effects In the external electric field, the correlation function of the director fluctuations, Eq. (18), has poles of the first order. In one-constant approximation Ki =K(i= 1, 2, 3), these poles occur at Q= 9 i
p m, L
m= 1, 2, …
Note that at Q = 0, the expression for the correlation function remains finite. Under an increase of field E, the number of Q function poles is increasing at a constant q. The value E which corresponds to the first pole is equal to
'
K q2+
p2 L2
. (22) oa p2 2 (e1 − e ) q + K q + 2 4p L Eq. (22) shows that E0(qx, qy ) increases monotonously with an increase in qx. Therefore, the minimal field giving a pole of the correlation functions had to be calculated for qx = 0. This pole corresponds either to the Freedericksz transition or to the threshold flexoelectric instability arising in cells of finite thickness [1,2]. The last one consists in the occurrence of specific domain structure when the field exceeds the critical one Ec. The type of transition depends on liquid crystal parameters and the geometry of system. The minimum of function E(qy ) determines the threshold values Ec and qc [2] E0 =
D
2 2 3 y
oa K 4p , o (e1 − e3)2 − a K 4p p 2K(e1 − e3) Ec = . L oa 2 (e1 − e3) + K 4p
p qc = L
(e1 − e3)2 −
(23)
Eq. (23) shows that flexoelectric instability can arise only under the condition oa B
4p(e1 − e3)2 . K
(24)
Otherwise, E(qy ) is minimal at qy = 0, Ec = p
4pK/oa that corresponds to the Freedericksz L transition.
4. Intensity of light scattering Nematic liquid crystal by its nature is an optically uniaxial medium. The tensor of permittivity oab is connected with the director field by the relation oab (r) =oÞdab + oana (r)nb (r). Fluctuations of the director dn(r) cause fluctuations of the tensor oab (r)
V.P. Romano6, G.K. Sklyarenko / Colloids and Surfaces A: Physicochem. Eng. Aspects 158 (1999) 355–361
doab (r) =oa [nadnb (r) +nbdna (r)].
E%a (r)E% (r) + b
&
w4 = 4 d 3r%d 3r%Tag (r, r%)T+ bl(r, r¦)dogm (r%)doln (r¦) c × E 0mE 0n exp[iki (r% −r¦)],
(26)
where w is the angular frequency, c is the velocity of light, Tab (r%, r%%) is Green’s function of the Maxwell equations. For simplicity, we use the isotropic medium approximation, i.e. we will assume that at large distances, Green’s function has the form Tab (r) =
1 ikr e (dab −sasb ), 4pr
E%a (r)E%+ b (r) 1 w V (d −sasg )(dbl −sbsl ) L c 4(4p)2r 2 ag ×
&
dz%
− L/2
&
L/2
dz%e − iqsc, z (z − z%)dogm (qÞ sc, z%)
− L/2
× do (q , z%)E E , + ln
Þ sc
0 m
0 n
(28)
where V is the scattering volume, qsc =sk− ki is the vector of scattering, qÞ sc =(qsc,x, qsc, y, 0) is the transverse component of qsc. According to Eq. (25), the correlation function of permittivity tensor is connected with fluctuations of the director by the relation + Þ dogm (qÞ sc, z%)do lm(qsc, z%) Þ = oa [n 0g n 0ldnm (qÞ sc, z%)dnl (qsc, z¦) Þ +n 0mn 0n dng (qÞ sc, z%)dnl (qsc, z¦)
(29)
We consider a normal incidence ki = (0, 0, k). If the scattering occurs in the direction s= (sin u cos 8, sin u 8, cos u), the scattering vector is equal to qsc = k(sin u cos 8, sinu sin8, cos u −1), u qsc = 2k sin . 2
(30)
Let the light be linearly polarized along the yaxis, i.e. in Eqs. (28) and (29), it is necessary to put m = n= y. Then, in one-constant approximation the intensity of scattered light has the form I(u, E) = I0
1 w 4 Vo 2a (1−sin2 u cos2 8) L c 4 (4p)2r 2
×
&
L/2
dz%
&
− L/2
L/2
dz¦e − iqsc,z (z% − z¦)G11(qÞ sc, z% z¦).
− L/2
Here G11(qÞ sc, z%, z¦) =
! ' ' '
1 o E2 − a [J(P)−J(Q)]+ [J(P)+ J(Q)]}, 4K 4pg
g= E
4
L/2
Þ + n 0mn 0ldng (qÞ sc, z%)dnn (qsc, z¦)].
(31) (27)
here k =w/c o, where o is average permittivity, s= r/r is the direction to a point of observation. Substituting Eq. (27) into Eq. (26), we obtain
=
Þ + n 0g n 0n dnm (qÞ sc, z%)dnl (qsc, z¦)
(25)
These fluctuations of permittivity result in a scattering of light. The intensity of the scattered light I is proportional to E%a (r)E%+ b (r), where E% is the field of a scattered wave [17]. For a plane wave with amplitude E0 and wave vector ki, the value E%a (r)E%+ b (r) in single-scattering approximation is determined by an integral over scattering volume [17,18]
359
P=
Q=
oaE 4p
q Þ2 sc +
2
+ 4(e1 − e3)2q 2sc, y,
g o E2 − a , 2K 8pK
q Þ2 sc −
g o E2 − a . 2K 8pK
(32)
5. Discussion The angular dependence of the intensity of scattered light is shown in Fig. 1 for various values of the external electric field and typical values of nematic liquid crystal parameters [1,2]: (o1 − o3)= 0.96 · 10 − 11 K m − 1, K=0.7 · 10 − 11 H. The type of transition depends on the value of permittivity. The boundary value of oa is deter-
360
V.P. Romano6, G.K. Sklyarenko / Colloids and Surfaces A: Physicochem. Eng. Aspects 158 (1999) 355–361
Fig. 1. Angular dependence of light scattering intensity by a cell of nematic liquid crystal with the planar orientation for two values of permittivity and various values of electric field: (a)oa =0.1: (1) U =0; (2) U =5 V; (3) U =6 V; (4) U = 6.7 V; (5) U =6.8 V; (b) oa =0.7; (1) U =0; (2) U = 3.5 V; (3) U= 3.68 V; (4) U =3.71 V; (5) U =3.712 V.
V.P. Romano6, G.K. Sklyarenko / Colloids and Surfaces A: Physicochem. Eng. Aspects 158 (1999) 355–361
mined from the ratio in Eq. (24): oa =4p(e1 − e3)2/ K= 0.52. The thickness of the simple is fixed as L= 10 − 5 m, with the wave number k= 103 m − 1. Fig. 1(a) corresponds to oa =0.1 (the flexoelectric instability). In this case, the critical values of parameters are equal to Uc =EcL = 6.5 V, qÞ sc = qc = 2.6 · 105 m − 1. Fig. 1(b) shows the angular dependence of I(u) for oa = 0.7. In this case, qc =0, and at Uc = 3.341 V, the Freedericksz transition takes place. The figures show that the intensity of scattered light has a peak in the vicinity q Þ sc =qc. Its value increases at E Ec. The existence of the intensity peak gives a good possibility for experimental check of obtained results and measurement of a flexoelectric coefficient difference (e1 −e3).
References [1] P.G. de Gennes, The Physics of Liquid Crystals, Clarendon Press, Oxford, 1974. [2] S.A. Pikin, Structural Transformations in Liquid Crystals, Nauka, Moscow, 1981. [3] L.D. Landau, E.M. Lifshitz, Statistical Physics, part 1,
.
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Pergamon Press, Oxford, 1980. [4] A.S. Zolot’ko., V.F. Kitaeva., N. Kroo et al., Pis’ma Zh. Exp. Teor. Fiz. 32 (1980) 170 [JETP Lett. 32 (1980) 158]. [5] I.C. Khoo, Phys. Rev. A 25 (1982) 1040. [6] S.J. Elson, I. Solymar, D.J. Webb, Phys. Rev. E 48 (1993) 1172. [7] P. Galatola, M. Rajteri, Phys. Rev. E. 49 (1994) 623. [8] K. Eidner, M. Lewis, H.K.M. Vithana, et al., Phys. Rev. A 40 (1989) 6388. [9] M. San Miguel, F. Sagues, Phys. Rev. A 39 (1989) 6567. [10] P. Galatola, C. Oldano, M. Rajteri, Phys. Rev. E 49 (1994) 1458. [11] R.B. Meyer, Phys. Rev. Lett. 22 (1969) 918. [12] B.Ya. Zel’dovich, N.V. Tabiryan, Zh. Eksp. Teor. Fiz. 81 (1981) 1738 [Sov. Phys. JETP 54 (1981) 922]. [13] T.Ya. Marusii, Yu. A. Reznikov, V. Yu. Reshetoyak et al., Zh. Eksp. Teor. Fiz. 91 (1986) 851 [Sov. Phys. JETP 64 (1986) 502]. [14] F. Lonberg, R.B. Meyer, Phys. Rev. Lett. 55 (1985) 718. [15] V.P. Romanov, A.N. Shalaginov, Zh. Eksp. Teor. Fiz. 102 (1992) 884 [Sov. Phys. JETP 76 (1992) 483]. [16] A. Yu. Val’kov, V.P. Romanov, A.N. Shalaginov, Usp. Fiz. Nauk 164 (1994) 149 [Physics-Uspekhi 37 (1994) 139]. [17] L.D. Landau, E.M. Lifshitz, Electrodynamics of continuous media, Pergamon Press, Oxford, 1984. [18] V.L. Kuz’min, V.P. Romanov, L.A. Zubkov, Phys. Rep. 248 (1994) 71.
Fluctuations in nematics near Freedericksz and flexoelectric transitions V.P. Romanov *, G.K. Sklyarenko Faculty of Physics, St. Petersburg State Uni6ersity, St. Petersburg, 198904, Russia Received 26 January 1998; accepted 16 March 1999
Abstract The director fluctuations in nematic liquid crystals are considered in the presence of an electric field by taking into account the flexoelectric effect The orientation of the director on the surfaces is supposed to be fixed. For the planar orientation, the spatial correlation function is calculated. This function is analysed near the Freedericksz transition and near the threshold flexoelectric instability. The spectrum of director fluctuations varies significantly at fields close to threshold one. For electric fields close to threshold one, angular dependence of light scattering is studied. © 1999 Elsevier Science B.V. All rights reserved. Keywords: Electric field; Liquid crystal; Fluctuations; Freedericksz transition; Flexoelectric effect
1. Introduction Liquid crystals undergo various structural transformations in the presence of external electric and magnetic fields [1,2] owing to anisotropy of permittivity and magnetic susceptibility. Here, we bear in mind instabilities of uniformly oriented liquid crystals in fields above a threshold. Such a change in structure is equivalent to a phase transition of second order [1,4]. The threshold character of these phenomena is connected to spatial limitations of liquid crystal layers and a rigidity of boundary conditions. * Corresponding author. Tel.: +7-812-4287033; fax: +7812-2487240. E-mail address: [email protected] (V.P. Romanov)
The most investigated and well known effect of the orientational phase transition in nematic liquid crystals is the Freedericksz effect [1,2]. This transition takes place both in static fields and in a light wave field [4–6]. It was studied for various boundary conditions and orientations for periodic and aperiodic distortions [7–10]. In 1969, Meyer [11] showed that liquid crystals having a centre of symmetry possess the ability to show evidence of a piezoelectric effect which has received the name of flexoelectric one. This linear effect of modulated structure formation produced by the homogeneous electric field is due to the anisotropy of molecules with permanent dipole moments [1,2]. The present work is devoted to the study of thermal fluctuations in confined nematic liquid
0927-7757/99/$ - see front matter © 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 7 7 5 7 ( 9 9 ) 0 0 1 8 1 - 8
356
V.P. Romano6, G.K. Sklyarenko / Colloids and Surfaces A: Physicochem. Eng. Aspects 158 (1999) 355–361
crystals (NLC) in the presence of the flexoelectric effect. The standard method of the fluctuations’ research consists of an expansion in the spectrum of eigenfunctions [7 – 10,12 – 14]. The solution is written as an infinite series and each term of this series is obtained by the solution of a difficult transcendental equation. In Refs. [15,16], the method of obtaining the correlation function of director fluctuations in the closed form was suggested for arbitrary energy of interaction with a substrate.
Let a NLC possessing flexoelectric properties be confined between plane-parallel plates of thickness L with the rigid boundary conditions in the external electric field. We introduce the Cartesian coordinate frame with the origin in the cell centre and z axis normal to the plates. The variation of free energy DF consists of the elastic contribution DFfr, the contribution connected to the director orientation in the external field DFel and the contribution caused by occurrence of the flexoelectric polarization Dfflex [1,2]
&
1 DF =DFfr + DFel +DFflex = d 3r{K1(div n)2 2
"
− 2E[e1n div n +e3(rot n ×n)] ,
dn x, y, z= 9
L = 0. 2
(2)
For small deviations of the director, the vectors n0 and dn are orthogonal, i.e. dn=(0, ny, nz ). Integrating Eq. (1) by parts in view of the boundary conditions in Eq. (2) up to accuracy of the second order terms over dn, Eq. (1) may be presented as follows DF =
2. Correlation matrix of a nematic in external field
+ K2(n · rot n)2 +K3(n × rot n)2 −
fluctuation. Because of the rigid boundary conditions we have
&
1 3 d rdn(r)Adnt(r), 2
(3)
where A is a differential operator, and index t means transposition. The correlation matrix of the director fluctuations G(r, r%) = B dn(r)dn+(r%) \ should satisfy the relation [3] (4)
AG(r, r%) = kbTId(r− r%),
where I is the unit matrix, T is the temperature, and kb is the Boltzmann constant. Fluctuation dn(r) is suitable to expand into the two-dimensional Fourier integral dn(r) =
&
1 d 2q exp[− i(q, rÞ)]dn(q, z), (2p)2
(5)
where q=(qÞ, 0) is the wave vector. After the Fourier-transformation, Eq. (4) has the form
oa (n · E)2 4p
AqG(q, z, z%)= kbTId(z −z%), (1)
where K1, K2, K3 are the Frank modules, n is the vector of director, E is the intensity of the electric field, oa =o −oÞ is the permittivity anisotropy, o , oÞ the permittivities along and normal to average orientation of nematic, e1, e3 are the flexoelectric coefficients. We consider the contribution to free energy depending upon the director fluctuations for the planar orientation. The homogeneous electric field is directed along the z axis: E = (0, 0, E). We present the vector of director in the form n= n0 + dn, where n0 =(1, 0, 0) is the equilibrium orientation of the director and dn is the director
(6)
where Aq is the Fourier-transform of operator A. Hence, the correlation matrix is obtained by the conversion of the Aq operator taking into account the boundary conditions
L G q, z = 9 , z% = 0. 2
(7)
In the geometry, we are interested in the variation of free energy (1) up to accuracy of the second order terms over dn has the form DF =
&
1 3 d r{K1((yny + (znz )2 + K2((ynz + (zny )2 2
+ K3[((xny )2 + ((xnz )2]−
oa 2 2 E n z − 2E[e1nz ((znz 4p
"
V.P. Romano6, G.K. Sklyarenko / Colloids and Surfaces A: Physicochem. Eng. Aspects 158 (1999) 355–361
+ (yny )+e3((xnz +ny(ynz +nz(znz )] .
(8)
357
If we introduce a new variable
i −1 D Cz 2
n
The matrix Aq, is equal to
G0 = exp
K 3q 2x +K1q 2y −K2( 2z Á à Aq = à −i(K 1 −K2)qy(z −i(e1 −e3)Eqy à Ä
and take into account the boundary conditions of Eq. (7), we have
Bq G0(q, z, z%)=kbT exp d(z− z%),
− i(K 1 −K2)qy(z +i(e1 −e3)Eqy K 3q 2x +K2q 2y −K1( 2z −
oa 2 E 4p
à à à Å
(9)
where the matrixes A0, C and D are equal to i(e 1 −e3)Eqy Á K 3q 2x +K1q 2y  à à A0 = à Ã, o −i(e 1 −e3)Eqy K 3q 2x +K2q 2y − a E 2 à 4p Ã Ä Å 0 − (K 1 −K2)qy
−K2 D= 0
−(K 1 −K2)qy , 0
i I(z + D − 1C 2
0 . −K 1
n
p
+H G =kbTD
D−1 (13)
d(z− z%),
−1
(10) where (11)
i −1 D Cz 2
(14)
n
i H exp − D − 1Cz 2
n
+ I( 2z
(15)
has the form
Á à − a sin2 a− b cos2 a+ f sin 2a + ( 2z Bq = à K 2 a− b à −i sin 2a +fcos 2a K1 2 Ä
'
'
 à à − a cos2 a− bsin2 a+ f sin2 a+ ( 2z à Ši
a=
2
1 H = (D − 1C)2 +D − 1A0. 4
Bq = exp
K 1 a− b sin 2a + fcos 2a K2 2
where
Multiplying both parts of Eq. (9) on matrix D − 1 and extracting a perfect square, we get
n
where the matrix
(A0 +iC(z +D( 2z )Gp =kbTId(z− z%),
i −1 D Cz 2
L G0 q, z = 9 , z% = 0, 2
For the solution of Eq. (6), it is convenient to present the matrix Aq in diagonal form. It is impossible to carry out this procedure by the transformation of similarity. Therefore, we transform Eq. (6) so that the left-hand side of this equation will contain an operator Bq admitting diagonalization. For this purpose, we present Eq. (6) as follows
C=
(12)
G
(16)
1 3 K2 1 K1 2 o K3 2 qx+ + − q y − a E 2, K1 2 4 K1 4 K2 4pK1
b= f=
1 3 K1 1 K2 2 K3 2 q + + − q , K2 x 2 4 K2 4 K1 y
(e1 − e3)
K1K2
Eqy,
a= −
1 (K1 − K2) q z. 2 K1K2 y
(17)
After diagonalization of the Bq matrix and the solution of obtained differential equations, we find the expressions for the elements of the correlation matrix
!
V.P. Romano6, G.K. Sklyarenko / Colloids and Surfaces A: Physicochem. Eng. Aspects 158 (1999) 355–361
358
G11 = ny (q, z)n+ y (q, z%) = −
1 1 (w cos 2a 2K2 p
!
+ iu sin 2a) ×[J(P) −J(Q)] + [J(P) +J(Q)]}, 1 1 G22 =nz (q, z)n+ (w cos 2a z (q, z%) = 2K2 p −u sin 2a)×[J(P) −J(Q)][J(P) +J(Q)]}, G12 = nz (q, z)n+ z (q, z%) =
1
!
(18)
1 (w cos 2a 2 K1K2 p
−u sin 2a)× [J(P) −J(Q)], G21 =nz (q, z)n+ z (q, z%) = − G12. where u=f cos 2a − +
a−b sin 2a, 2
a−b cos 2a, 2
P=
'
p=
a+b +p, 2
J(r)=
w = f sin 2a
' '
Q=
a−b +f 2. 2
(19)
a+b −p, 2
(20)
kbT {cosh[r(z+ z%)] −cosh(rL) 2r sinh(rL)
cosh[r(z−z%)] +sinh(rL)sinh(r z− z% )}.
(21)
The obtained results for E = 0 coincide with the results of [12,15] for the rigid boundary conditions.
3. The threshold effects In the external electric field, the correlation function of the director fluctuations, Eq. (18), has poles of the first order. In one-constant approximation Ki =K(i= 1, 2, 3), these poles occur at Q= 9 i
p m, L
m= 1, 2, …
Note that at Q = 0, the expression for the correlation function remains finite. Under an increase of field E, the number of Q function poles is increasing at a constant q. The value E which corresponds to the first pole is equal to
'
K q2+
p2 L2
. (22) oa p2 2 (e1 − e ) q + K q + 2 4p L Eq. (22) shows that E0(qx, qy ) increases monotonously with an increase in qx. Therefore, the minimal field giving a pole of the correlation functions had to be calculated for qx = 0. This pole corresponds either to the Freedericksz transition or to the threshold flexoelectric instability arising in cells of finite thickness [1,2]. The last one consists in the occurrence of specific domain structure when the field exceeds the critical one Ec. The type of transition depends on liquid crystal parameters and the geometry of system. The minimum of function E(qy ) determines the threshold values Ec and qc [2] E0 =
D
2 2 3 y
oa K 4p , o (e1 − e3)2 − a K 4p p 2K(e1 − e3) Ec = . L oa 2 (e1 − e3) + K 4p
p qc = L
(e1 − e3)2 −
(23)
Eq. (23) shows that flexoelectric instability can arise only under the condition oa B
4p(e1 − e3)2 . K
(24)
Otherwise, E(qy ) is minimal at qy = 0, Ec = p
4pK/oa that corresponds to the Freedericksz L transition.
4. Intensity of light scattering Nematic liquid crystal by its nature is an optically uniaxial medium. The tensor of permittivity oab is connected with the director field by the relation oab (r) =oÞdab + oana (r)nb (r). Fluctuations of the director dn(r) cause fluctuations of the tensor oab (r)
V.P. Romano6, G.K. Sklyarenko / Colloids and Surfaces A: Physicochem. Eng. Aspects 158 (1999) 355–361
doab (r) =oa [nadnb (r) +nbdna (r)].
E%a (r)E% (r) + b
&
w4 = 4 d 3r%d 3r%Tag (r, r%)T+ bl(r, r¦)dogm (r%)doln (r¦) c × E 0mE 0n exp[iki (r% −r¦)],
(26)
where w is the angular frequency, c is the velocity of light, Tab (r%, r%%) is Green’s function of the Maxwell equations. For simplicity, we use the isotropic medium approximation, i.e. we will assume that at large distances, Green’s function has the form Tab (r) =
1 ikr e (dab −sasb ), 4pr
E%a (r)E%+ b (r) 1 w V (d −sasg )(dbl −sbsl ) L c 4(4p)2r 2 ag ×
&
dz%
− L/2
&
L/2
dz%e − iqsc, z (z − z%)dogm (qÞ sc, z%)
− L/2
× do (q , z%)E E , + ln
Þ sc
0 m
0 n
(28)
where V is the scattering volume, qsc =sk− ki is the vector of scattering, qÞ sc =(qsc,x, qsc, y, 0) is the transverse component of qsc. According to Eq. (25), the correlation function of permittivity tensor is connected with fluctuations of the director by the relation + Þ dogm (qÞ sc, z%)do lm(qsc, z%) Þ = oa [n 0g n 0ldnm (qÞ sc, z%)dnl (qsc, z¦) Þ +n 0mn 0n dng (qÞ sc, z%)dnl (qsc, z¦)
(29)
We consider a normal incidence ki = (0, 0, k). If the scattering occurs in the direction s= (sin u cos 8, sin u 8, cos u), the scattering vector is equal to qsc = k(sin u cos 8, sinu sin8, cos u −1), u qsc = 2k sin . 2
(30)
Let the light be linearly polarized along the yaxis, i.e. in Eqs. (28) and (29), it is necessary to put m = n= y. Then, in one-constant approximation the intensity of scattered light has the form I(u, E) = I0
1 w 4 Vo 2a (1−sin2 u cos2 8) L c 4 (4p)2r 2
×
&
L/2
dz%
&
− L/2
L/2
dz¦e − iqsc,z (z% − z¦)G11(qÞ sc, z% z¦).
− L/2
Here G11(qÞ sc, z%, z¦) =
! ' ' '
1 o E2 − a [J(P)−J(Q)]+ [J(P)+ J(Q)]}, 4K 4pg
g= E
4
L/2
Þ + n 0mn 0ldng (qÞ sc, z%)dnn (qsc, z¦)].
(31) (27)
here k =w/c o, where o is average permittivity, s= r/r is the direction to a point of observation. Substituting Eq. (27) into Eq. (26), we obtain
=
Þ + n 0g n 0n dnm (qÞ sc, z%)dnl (qsc, z¦)
(25)
These fluctuations of permittivity result in a scattering of light. The intensity of the scattered light I is proportional to E%a (r)E%+ b (r), where E% is the field of a scattered wave [17]. For a plane wave with amplitude E0 and wave vector ki, the value E%a (r)E%+ b (r) in single-scattering approximation is determined by an integral over scattering volume [17,18]
359
P=
Q=
oaE 4p
q Þ2 sc +
2
+ 4(e1 − e3)2q 2sc, y,
g o E2 − a , 2K 8pK
q Þ2 sc −
g o E2 − a . 2K 8pK
(32)
5. Discussion The angular dependence of the intensity of scattered light is shown in Fig. 1 for various values of the external electric field and typical values of nematic liquid crystal parameters [1,2]: (o1 − o3)= 0.96 · 10 − 11 K m − 1, K=0.7 · 10 − 11 H. The type of transition depends on the value of permittivity. The boundary value of oa is deter-
360
V.P. Romano6, G.K. Sklyarenko / Colloids and Surfaces A: Physicochem. Eng. Aspects 158 (1999) 355–361
Fig. 1. Angular dependence of light scattering intensity by a cell of nematic liquid crystal with the planar orientation for two values of permittivity and various values of electric field: (a)oa =0.1: (1) U =0; (2) U =5 V; (3) U =6 V; (4) U = 6.7 V; (5) U =6.8 V; (b) oa =0.7; (1) U =0; (2) U = 3.5 V; (3) U= 3.68 V; (4) U =3.71 V; (5) U =3.712 V.
V.P. Romano6, G.K. Sklyarenko / Colloids and Surfaces A: Physicochem. Eng. Aspects 158 (1999) 355–361
mined from the ratio in Eq. (24): oa =4p(e1 − e3)2/ K= 0.52. The thickness of the simple is fixed as L= 10 − 5 m, with the wave number k= 103 m − 1. Fig. 1(a) corresponds to oa =0.1 (the flexoelectric instability). In this case, the critical values of parameters are equal to Uc =EcL = 6.5 V, qÞ sc = qc = 2.6 · 105 m − 1. Fig. 1(b) shows the angular dependence of I(u) for oa = 0.7. In this case, qc =0, and at Uc = 3.341 V, the Freedericksz transition takes place. The figures show that the intensity of scattered light has a peak in the vicinity q Þ sc =qc. Its value increases at E Ec. The existence of the intensity peak gives a good possibility for experimental check of obtained results and measurement of a flexoelectric coefficient difference (e1 −e3).
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