CC BY
35
YAK 538.955
INFLUENCE OF THE SADDLE-SPLAY CONSTANT ON THE DIRECTOR FIELD DISTRIBUTION IN STRONG
MAGNETIC FIELDS
© 2014 A.A. Kudreyko, R.N. Migranova,1 A.R. Khafizov2
The role of anchoring effects in thin nematic films confined between two parallel plates was theoretically examined. The bulk and surface free energy densities were expanded up to O(e2) and the perturbated contributions were calculated. It is shown that the minimum of the free energy corresponds to the solution of the Euler-Lagrange equations and satisfies the Ericksen inequalities. The identified bifurcation points can estimate the influence of the saddle-splay constant k24 towards periodic perturbations of a director in the presence of strong magnetic field.
Key words: nematic liquid crystals, thin films, saddle-splay elasticity, Freedericksz transition, linear perturbation theory.
Introduction
Nematic liquid crystals (LCs) confined in restricted geometries are technologically important and have been the subject of extensive experimental and theoretical research for five decades. Freedericksz transitions caused by an external electric or magnetic field in nematic LCs have attracted attention since their discovery in 1933.
When the magnetic field is applied perpendicular to the uniformly aligned nematic, confined between two parallel plates, above a certain threshold, the director is affected on by deformations and tends to align along the field. The critical threshold of the magnetic field in one constant approximation is inversly proportional to the thickness of the nematic sample 2d, and is given by
where \a is the anisotropy of the diamagnetic susceptibility, which is assumed a positive number, and k is the elastic constant [1].
A surface term k24, which is often omitted represents elastic contribution to the surface free energy that, originally, has been indicated as a part of the elastic energy having the form of a divergence. This contribution - the so-called the saddle-splay
1Kudreyko Aleksey Al'fredovich (akudreyko@rusoil.net), Migranova Roksana Nailevna (ufangm@yahoo.co.uk), the Dept. of Physics, Ufa State Petroleum Technological University, Ufa, 450062, Russian Federation.
2Khafizov Ayrat Rimovich (hafizov57@mail.ru), the Dept. of Development and Exploitation of Gas and Condensate Deposits, Ufa State Petroleum Technological University, Ufa, 450062, Russian Federation.
term k24, which is important only for particular situations, in which a distortion has a two- or three-dimensional structures [2, 3].
One of the recent theoretical results [4] in the one-constant approximation shows that k24 causes instabilities if the ratio k24/k > 0.707. Another result from this study is the suggestion of how to measure k24.
The results in ref. [4] are essentially limited to the particular case in which the layer of nematic LC is infinite, and no proof that the distorted contributions of the distribution of polar and azimuthal angles yield the minimum of free energy. Hence, the study of possible mechanisms of interactions of the saddle-splay energy and strong magnetic fields (H ^ Hc) requires a quantitatively accurate description of the instability that goes beyond these limitations.
In this work we report the theory of the interaction of the saddle-splay elastic constant under the onset of periodic perturbations of a director in the presence of strong magnetic field. Our theoretical model is based on the Oseen-Zocher-Frank continuum theory of LCs. In the next Section we define the geometry of the problem and introduce free energy. In Section 2. we obtain the bulk and surface free energies in two-constant approximation. Then we determine bifurcation points by minimizing free energy. In Section 3. we analyze the influence of the saddle-splay term on the stability of the unperturbated state. In Section 3. we briefly outline the results of our research and the perspectives on open problems.
1. Geometry of the problem and free energy
Nematic LCs are described by director n, which is a unit vector, characterising the average orientation of molecules. The standart expression for the elastic free energy associated with n is given by
Fel = 1 /dV{kn(divn)2 + k22(n ■ curln)2 + k33[n x curln]2
y (1)
-(k22 + k24) div [n x curl n + (div n)n]} ,
where k11,k22,k33 and k24 are the splay, twist, bend and saddle-splay moduli respectively. In view of Gauss' theorem, the divergence terms only contribute to the surface free energy density. To guarantee a stable configuration of nematic LC in the absence of external fields, the saddle-splay constant must fulfill the Ericksen inequalities [2]:
kii > 0 , |k241 < k22 , k22 + k24 < 2kn , \kn — k22 — k241 < kn. (2)
We assume that kn = k33 = k and k24 = 0, then equation (1) takes the form Fel = 2 / dV{k(div n)2 + k22(n ■ curl n)2 + k[n x curl n]2
y (3)
— (k22 + k24) div [n x curl n + (div n)n]} .
Consider one of the basic configurations of the director and the magnetic field for the Freedericksz transition (e.g. [3], p. 307). Let the distance in the bulk (see Fig. 1), which is needed to orient the director along the applied magnetic field H = Hex be expressed in the CGS system through the dimensionless coherence length
£e = -;
1 / k
d^ XaH2 '
where d is the half-thickness of the layer and eq C 1.
Glass plate
Glass plate
Fig. 1. Representation of the director n under the influence of the magnetic field H in a cell of the thickness 2d
The magnetic field contribution to the elastic free energy is represented by the functional
X.
F = F el - X f (n, H)2 dV, 2 JV
which yields the Euler equation [1, 2]
d2o . a a
-r- + sin 0 cos 0
dÇ 2
0,
(4)
where Z : 2/eg > Z > 0 is the scaled coordinate, which is related with the space coordinate -d<z<d as z = d — egdZ, 0 is the angle between the director and the z-axis (see Fig. 1).
When Z ^ 1/eg, we suppose 0 = n/2, (d0/dZ)z^i/Eg = 0 and p • (d0/dZ)z=o = = sin 0 cos 01z=0, where p = \JkH2\a/Wa is the dimensioless parameter characterising the relative strength of the magnetic field to the surface anchoring energy Wa. The solution of equation (4) yields the function
0(Z )
fAe2Z - 1 V Ae2( + 1
,
(5)
where A(p)
1+ P 1 - P '
It is widely known that the saddle-splay constant contributes to the equations for problems involving weak anchoring of the LC at the surface. Due to this reason, a mathematical condition 0 < p < 1 must hold.
2. Stability analysis
Supposing that the state (5) is linearly unstable, consider small perturbations of the director n induced by the magnetic field under the assumption that the configuration of the perturbations of the director in the yz plane is defined by
n = sin(9 + 0) cos <p ex + sin(9 + 0) sin <p ey + cos(9 + 0) ez where p(y,z), 0(y,z) are small 0(e) and periodic functions, i.e.
v(y, z) = g(z) cos(qy), 0(y, z) = f (z) sin(qy),
(6)
(7)
where q is the wavenumber.
The Taylor series expansion of the elastic and magnetic free energy up to 0(e2) gives
/f = 1 -2o(k( t) 2 + If) ( +
(k sin2^H) 2 + k22 cos20 (M) 2 + k22 (]) 2) +
k
(k22 - k) [sin2 0dydt + cos 20 (f) J , (k22 - k) sin 20If f - 20- 20 f -
(8)
(9)
(02 cos 29 - v2 sin2
and the surface energy obtained from the Rapini-Papoular functional [5] yields
f (2) = 02 W cos 29 - (k22 + k24) V sin2 9^ + ^240^ .
Under the assumption that the field-induced distortions (7) depend only on Z, the configuration of the director can be found by solving the Euler-Lagrange equations. The variational problem for f and g (7) written in terms of the scaled variable Z leads to the following system of differential equations:
Y( + 'I dz2 +
8Ae2
dg
A2e4Z -1 d(
)-(y - f - AAf
-g^2 = 0,
(10)
1 ( Ae2Z-1
2
Y [Ae2Z + 1^ + (Ae2Z + 1)2 1
4Ae2Z
df dz2
/
- / WÔ -
_ 8Ae2Z 2 (Ae2Z + 1)2
)
+ (Ae2Z + 1)2 -12Ae2Z ( (Ae2Î + 1)2
20Ae2ZqA2Z-!g + q(Ae2Z - 1)2 f
8A
- 1) f + 8Ae2Z(jee2-)2 /
(11)
+16^^ f
(1 - 1)
0,
k22
where y = ~r~ k
1 + q2e2ed2,
1
—+ q edd . Equations (10,11) written in
Y
terms of the one constant approximation will look identical with the corresponding variational problem given in ref. [4].
The obtained equations do not have an analytical solution, and we will treat (10,11) for the isotropic approximation kn = k22 = k33, assuming lower and upper boundaries. Then w1 = w2| ^ and consider a common value w.
The boundary conditions for weak anchoring are due to the minimization of the bulk and surface free energy terms. Hence, the influence of the surface on the
equilibrium director field of the LC is represented by the boundary conditions
f) dP,Z
+
df
(2)
dp
0, -
C=o
f) d^c
+
df
(2)
C=o
or
gc (0) = (1 + t f (0);
fz (0) = - (p - 2p) f (0) + ^V^2-! g(0):
(12)
(13)
where t = k24/k is the dimensionless saddle-splay modulus. To give a quantitative estimation, we carried out further calculations for 2d =40 • 10~4 cm, Xa = 10~7, H =1.5 • 104 G, k = 10~6 dyn and Wa = 9 • 10~3 erg/cm2, then ee — 0.1 and Z — 10. Likewise, the second pair of the boundary conditions is given by
f(2)
+
df
(2)
dpz dp
C=io
f) 3'ibc
+
df
(2)
(14)
C=io
or
(15)
gz (10) = -(1 + t f (10);
fz (10) = p f (10) + tV^2—! g(10).
With the aid of a computer algebra system, we obtain the distribution of perturbations for the polar and azimuthal angles
f (Z) = C
= C (^+1+Ae2Z (^-1))
Ae2< + 1
+ C2 "
z (^-1+Ae2Z (^ + 1)) Ae2Z + 1
g(Z) = C (u + 1+Ae2< (u-1)) + C (u-1 + Ae2< (^+1)) g(Z ) = °3-Ae.2C_ 1--r C4-Ae.2^ 1- '
(16)
where the integration constants Ci must satisfy the boundary conditions (13) and (15). The resulting equations represent the system of linear homogeneous equations
4
Y^ MijjCj = 0, which has a non-trivial solution if
i,j = 1
det M = 0. (17)
By computing the determinant of M, we get an implicit equation with respect to t and p. It is clear from (2) that (17) is regarded to the case t < 1. In order to test the model, it is necessary to clarify if solutions (16) of the Euler-Lagrange equations yield minimum energy with respect to the absence of perturbations (6). Therefore, if the sum of the bulk and surface free energy terms with the supposed perturbations (6) is less than the sum without perturbations, i.e. p(y,z) = ^(y,z) =0 (stable state), then the supposed perturbations exist. Thus, the condition
A d
f(2)dzdy +
f(2)dy
oo
< 0
(18)
must hold. If we suppose that Ci/Cj ~ 1, where i,j = 1...4, then the substitution of the solutions of the Euler-Lagrange equations (16) to (18) yields the expression
+ Fy = G(T,d,X,Wa,H,xa) < 0, where A is the wavelength of the periodic distortions of the director along the y axis. Using the parameters given above, we can plot the condition of the existence of perturbations (see Fig. 2).
For a boundary case when H ^ Hc, periodic distortions of the director are possible if \t\ ^ 1, which violates the Ericksen inequalities. It is also easy to challenge the model on the stability of with respect to Ci/Cj — 0.1... 10, and the corresponding results do fulfill conditions (2).
(2)
0
0
0
A
d
2
x
Fig. 2. Condition for the existence of perturbations in strong magnetic
(2) (2)
field. The area below the line corresponds to Fb + Fs < 0
3. Results and discussion
(2)
It is clear that for y>(y, z) = 0(y, z) = 0, then fs =0 . Therefore, the saddle-splay term does not contribute to the free energy for configurations in which the director is constant within a plane. The torsional strains considered here are two-dimensional, and so the contribution of k24 to is reasonable.
The state for u < 1 can not be achieved because it corresponds to complex wavenumbers q, i.e. u2 = 1 + q2Sgd2.
Each curve in Figs. 3 divides the parametric space p — u into two regions and the instability threshold is determined by the matrix M. In view of Fig. 3a, the minimum of each curve represents the critical values of the dimensionless parameter pc. The instability threshold between the stable state det M > 0 (below each curve) and periodically distorted state det M < 0 occurs in the vicinity of the glass surface for 0.53 < pc < 0.578.
The role of the saddle-splay term within the present problem statement can be seen by letting k24 ^ k, (t ^ 1), then the stability diagram (17) grows faster, and means that wavelength of perturbations A increases (Fig. 3b).
The obtained results show that for the determination of the saddle-splay constant one need to measure the distance 2d between two plates, elastic constants and by changing the magnetic field, observe periodic perturbations, and measure the wavelength at the plate surface. The determination of u and t can give a value of the saddle-splay constant with a certain error because the presented theoretical results are based on the one constant approximation.
Conclusions
In this paper we have studied the saddle-splay term, which is represented in the Frank free energy (1) and influences on the surface free energy. Periodic perturbations of the director yield deviation from the base state (5) when the saddle-splay energy
a)
--x=0.65-T=0.64 T=0.67
0,680,660,640,620,600, 0,560,54-
b)
x=0.85--tM).8-x=0.75
0,7-
2 3 4 5 6
CD
1,5 2 2,5 3 3,5
(0
Fig. 3. Stability diagrams (det M = 0) for various values of the saddle-splay ratio t = k24/k
becomes important. This issue becomes important only for high magnetic fields, i.e. H > 104 G, which are difficult to achieve in many laboratories. The similarity of the effects induced by magnetic fields and electric fields shows that this effect can also be viewed in the electric field with ratio E = y,4nxa/£aH, where ea = 0.1 is the dielectric anisotropy. Thus, the behavior of LCs in magnetic field 104 G is equivalent to electric field with strength E = 35.4 V/cm.
Likewise in [4], this paper contributes to the theory of the surface terms on the periodic Freedericksz transition. However, the analysis performed in the present paper has shown that there is no low threshold for t when the bifurcation occurs in the layer of nematic LC with a finite thickness. Moreover, the critical values of parameter pc satisfy the range criterion 0.53 ^ pc ^ 0.578.
The differential equations (10,11) written in terms of the two-constant approximation can give more accurate results at the expense of algorithmic complexity. In particular, the solution of the variational problem will cause two wave vectors and more complex boundary conditions.
Finally, we remark that the experimental description of the boundary orientational relaxation of the director caused by k24 modulus requires the study in optical transmittance of aligned nematic liquid crystals [6]. The authors will continue to treat the problem in this direction.
Acknowledgements
The authors are grateful to Dr. Oksana Manyuhina (Nordic Institute for Theoretical Physics, Sweden) and Dr. Maxim Khazimullin (Institute of Molecule and Crystal Physics, Ufa) for helpful insight on theoretical computations.
The contribution of Dr. A. Kudreyko is supported by the Russian Foundation for Basic Research under project no. 14-02-97026.
References
[1] Gennes P.G. de. The Physics of Liquid Crystals //J. Prost. Second Edition. Oxford University Press, 1993. 596 p.
[2] Stewart I.W. The Static and Dynamic Continuum Theory of Liquid Crystals. A Mathematical Introduction. Taylor & Francis, 2004.
[3] Blinov L.M. Structure and Properties of Liquid Crystals. Springer, Netherlands, 2011.
[4] Manyuhina O.V. Saddle-splay elasticity and field induced soliton in nematics // Journal of Physics: Condensed Matter 24. 195102. 2012.
[5] Rapini A., Papoular M. Distorsion d'une Lamelle Nematique Sous Champ Magnetique Conditions d'ancrage aux Parois // J. Phys. Colloques 30. 1969. C4-54-C4-56.
[6] Optical Properties of Aligned Nematic Liquid Crystals in Electric Field / S. Yilmaz [et al.] // Journal of Modern Physics. 2011. 2. P. 248-255.
Paper received 3/X///2013. Paper accepted 2/IV/2014.
